diff --git a/.github/workflows/axial-density-review.yml b/.github/workflows/axial-density-review.yml
index 9c55eaff9..90a82a6d3 100644
--- a/.github/workflows/axial-density-review.yml
+++ b/.github/workflows/axial-density-review.yml
@@ -12,8 +12,8 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@v7
-      - uses: actions/setup-python@v7
+      - uses: actions/checkout@v7.0.1
+      - uses: actions/setup-python@v7.0.0
         with:
           python-version: '3.11'
       - name: Exact algebra in both modes with semantic mutations
@@ -31,7 +31,7 @@ jobs:
             done
           done
           git rev-parse HEAD > "$RUNNER_TEMP/axial-evidence/checkout_commit.txt"
-      - uses: actions/upload-artifact@v7
+      - uses: actions/upload-artifact@v7.0.1
         if: always()
         with:
           name: axial-density-replay
diff --git a/.github/workflows/bf-six-pin-scalar.yml b/.github/workflows/bf-six-pin-scalar.yml
index 9fef6d254..ccba07efa 100644
--- a/.github/workflows/bf-six-pin-scalar.yml
+++ b/.github/workflows/bf-six-pin-scalar.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-24.04
     timeout-minutes: 5
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11'
       - name: Normal and optimized refusal tests
diff --git a/.github/workflows/c2-exact.yml b/.github/workflows/c2-exact.yml
index b8574323d..9267257a7 100644
--- a/.github/workflows/c2-exact.yml
+++ b/.github/workflows/c2-exact.yml
@@ -21,10 +21,10 @@ jobs:
       matrix:
         flags: ['-B -S', '-B -O -S']
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify manifest and main-resident pins
diff --git a/.github/workflows/c2-finite-jet-transfer.yml b/.github/workflows/c2-finite-jet-transfer.yml
index f6d6f18c9..8a070d983 100644
--- a/.github/workflows/c2-finite-jet-transfer.yml
+++ b/.github/workflows/c2-finite-jet-transfer.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Authenticate the historical sources and run exact controls
diff --git a/.github/workflows/c2-torus-transfer.yml b/.github/workflows/c2-torus-transfer.yml
new file mode 100644
index 000000000..f19cc875e
--- /dev/null
+++ b/.github/workflows/c2-torus-transfer.yml
@@ -0,0 +1,88 @@
+name: c2 torus transfer (replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c2_torus_transfer_20261001/**'
+      - '.github/workflows/c2-torus-transfer.yml'
+      - 'frontiers/c2_exact_20261001/**'
+      - 'coefficients/side24_v1/PROOF.md'
+      - 'frontiers/candidate_third_order_20261001/PROOF.md'
+      - 'frontiers/cusp_coefficient_certified_20261001/RESULTS.json'
+      - 'frontiers/elder_third_order_20261001/PROOF.md'
+      - 'frontiers/cusp_torus_transfer_20261001/PROOF.md'
+      - 'frontiers/c2_finite_jet_transfer_20261001/PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  transfer-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 45
+    strategy:
+      fail-fast: false
+      matrix:
+        flags: ['-B -S', '-B -O -S']
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest, main-resident pins and byte-identical copies
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, pathlib
+          root = pathlib.Path('frontiers/c2_torus_transfer_20261001').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          def ident(data):
+              return len(data), hashlib.sha256(data).hexdigest(), hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              if ident(path.read_bytes()) != (entry['bytes'], entry['sha256'], entry['git_blob']):
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if not entry['current_required']:
+                  continue
+              if ident(pathlib.Path(entry['path']).read_bytes()) != (entry['bytes'], entry['sha256'], entry['git_blob']):
+                  raise ValueError('consumed source identity mismatch on this tree: ' + entry['path'])
+          for copy, orig in pins['copies'].items():
+              if (root / copy).read_bytes() != pathlib.Path(orig).read_bytes():
+                  raise ValueError('copy differs from its pinned original: ' + copy)
+          print('manifest, pins and copies verified')
+          PY
+      - name: Ball-arithmetic replay against RESULTS.json, mutants, controls
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+          FLAGS: ${{ matrix.flags }}
+        run: |
+          set -euo pipefail
+          cd frontiers/c2_torus_transfer_20261001
+          pids=()
+          for m in no-image no-theta drop-ratio eta-half fp-const; do
+            ( set +e; python $FLAGS transfer.py --mutant "$m" > "/tmp/mutant_$m.log" 2>&1; echo $? > "/tmp/mutant_$m.rc" ) &
+            pids+=($!)
+          done
+          python $FLAGS transfer.py --check
+          python $FLAGS controls.py
+          wait "${pids[@]}"
+          for m in no-image no-theta drop-ratio eta-half fp-const; do
+            rc=$(cat "/tmp/mutant_$m.rc")
+            if [ "$rc" -ne 1 ]; then echo "mutant $m not rejected (rc=$rc)"; cat "/tmp/mutant_$m.log"; exit 1; fi
+            echo "mutant $m rejected"
+          done
+          cd "$GITHUB_WORKSPACE"
+          git diff --exit-code
+          test -z "$(git status --porcelain)"
diff --git a/.github/workflows/c6-cluster-coefficients-numerics.yml b/.github/workflows/c6-cluster-coefficients-numerics.yml
index 0331e4e39..268ca3fb8 100644
--- a/.github/workflows/c6-cluster-coefficients-numerics.yml
+++ b/.github/workflows/c6-cluster-coefficients-numerics.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify manifest and main-resident pins, replay exact controls and GH40, reject mutants
diff --git a/.github/workflows/c6-cluster-law.yml b/.github/workflows/c6-cluster-law.yml
index 9b053545d..85bbd08ea 100644
--- a/.github/workflows/c6-cluster-law.yml
+++ b/.github/workflows/c6-cluster-law.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/c6-count-cap-boundary.yml b/.github/workflows/c6-count-cap-boundary.yml
index c5a297d62..c3888e7d6 100644
--- a/.github/workflows/c6-count-cap-boundary.yml
+++ b/.github/workflows/c6-count-cap-boundary.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources and both-mode semantic controls
diff --git a/.github/workflows/c6-factorial-moment.yml b/.github/workflows/c6-factorial-moment.yml
index 24031c563..6d3a14155 100644
--- a/.github/workflows/c6-factorial-moment.yml
+++ b/.github/workflows/c6-factorial-moment.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/c6-fourier-cutoff.yml b/.github/workflows/c6-fourier-cutoff.yml
index 5a6ccdd93..4f10f5432 100644
--- a/.github/workflows/c6-fourier-cutoff.yml
+++ b/.github/workflows/c6-fourier-cutoff.yml
@@ -26,10 +26,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact packet, both Python modes, and six upstream identities
diff --git a/.github/workflows/c6-marked-fourier.yml b/.github/workflows/c6-marked-fourier.yml
index 183d345b5..33ad7f266 100644
--- a/.github/workflows/c6-marked-fourier.yml
+++ b/.github/workflows/c6-marked-fourier.yml
@@ -32,11 +32,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           fetch-depth: 0
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Both-mode controls and exact upstream custody
diff --git a/.github/workflows/c6-palm-route.yml b/.github/workflows/c6-palm-route.yml
index ac21f1fcb..9dba863a2 100644
--- a/.github/workflows/c6-palm-route.yml
+++ b/.github/workflows/c6-palm-route.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/c6-rare-cluster-laws.yml b/.github/workflows/c6-rare-cluster-laws.yml
index 2c7b8b261..9c4dc65c6 100644
--- a/.github/workflows/c6-rare-cluster-laws.yml
+++ b/.github/workflows/c6-rare-cluster-laws.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch immutable public premise object
diff --git a/.github/workflows/c6-remote-coefficient-numerics.yml b/.github/workflows/c6-remote-coefficient-numerics.yml
index fdb6f61ce..0f622023e 100644
--- a/.github/workflows/c6-remote-coefficient-numerics.yml
+++ b/.github/workflows/c6-remote-coefficient-numerics.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify manifest and main-resident pins, replay exact controls and sample kernels, reject mutants
diff --git a/.github/workflows/c6-residual-closure.yml b/.github/workflows/c6-residual-closure.yml
index 1bde9c3b3..c99afc4f1 100644
--- a/.github/workflows/c6-residual-closure.yml
+++ b/.github/workflows/c6-residual-closure.yml
@@ -28,10 +28,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes (baseline or installed register), reject mutants
diff --git a/.github/workflows/c6-sharpened.yml b/.github/workflows/c6-sharpened.yml
index 363d116be..1e5fd374a 100644
--- a/.github/workflows/c6-sharpened.yml
+++ b/.github/workflows/c6-sharpened.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/c6-tail-constant-certified.yml b/.github/workflows/c6-tail-constant-certified.yml
index 3ac90f6b0..53b09857d 100644
--- a/.github/workflows/c6-tail-constant-certified.yml
+++ b/.github/workflows/c6-tail-constant-certified.yml
@@ -18,10 +18,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 30
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources and pins, replay both modes, reject mutants
diff --git a/.github/workflows/c6-witness-collision-reconciliation.yml b/.github/workflows/c6-witness-collision-reconciliation.yml
index 1f74fb0ca..a176b69a2 100644
--- a/.github/workflows/c6-witness-collision-reconciliation.yml
+++ b/.github/workflows/c6-witness-collision-reconciliation.yml
@@ -30,10 +30,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes (baseline or installed register), reject mutants
diff --git a/.github/workflows/c7-nonvanishing-review.yml b/.github/workflows/c7-nonvanishing-review.yml
index f981a3e09..9043f1d68 100644
--- a/.github/workflows/c7-nonvanishing-review.yml
+++ b/.github/workflows/c7-nonvanishing-review.yml
@@ -18,10 +18,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources and both-mode intended refusals
diff --git a/.github/workflows/c7-total-bounded.yml b/.github/workflows/c7-total-bounded.yml
index a4a139067..3d025b1ae 100644
--- a/.github/workflows/c7-total-bounded.yml
+++ b/.github/workflows/c7-total-bounded.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/c7-zero-gap.yml b/.github/workflows/c7-zero-gap.yml
index 00e01f63f..5282f9174 100644
--- a/.github/workflows/c7-zero-gap.yml
+++ b/.github/workflows/c7-zero-gap.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources, both Python modes, semantic and path negatives
diff --git a/.github/workflows/c8-cap-failure-coefficient-dims.yml b/.github/workflows/c8-cap-failure-coefficient-dims.yml
index 571abb55d..b74f83b24 100644
--- a/.github/workflows/c8-cap-failure-coefficient-dims.yml
+++ b/.github/workflows/c8-cap-failure-coefficient-dims.yml
@@ -18,10 +18,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify manifest and main-resident pins, replay the certified values in both interpreter modes, reject mutants
diff --git a/.github/workflows/c8-cap-failure-coefficient-planar.yml b/.github/workflows/c8-cap-failure-coefficient-planar.yml
new file mode 100644
index 000000000..e072c919b
--- /dev/null
+++ b/.github/workflows/c8-cap-failure-coefficient-planar.yml
@@ -0,0 +1,72 @@
+name: C8 planar cap-failure coefficient (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c8_cap_failure_coefficient_planar_20260930/**'
+      - '.github/workflows/c8-cap-failure-coefficient-planar.yml'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md'
+      - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json'
+      - 'frontiers/local_elder_geometry_20260930/PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 75
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest and main-resident pins, recompute every certified enclosure in both interpreter modes, reject mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import concurrent.futures, hashlib, json, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/c8_cap_failure_coefficient_planar_20260930').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']:
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if not entry['current_required']:
+                  continue
+              data = pathlib.Path(entry['path']).read_bytes()
+              blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']:
+                  raise ValueError('consumed source identity mismatch on this tree: ' + entry['path'])
+          script = str(root / 'cap_coefficient.py')
+          for flags in (['-B', '-S'], ['-B', '-O', '-S']):
+              out = subprocess.run([sys.executable, *flags, script, '--check', '--procs', '2'], cwd=root, capture_output=True, timeout=2400)
+              if out.returncode != 0:
+                  raise RuntimeError('check failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:])
+              print(flags, out.stdout.decode().strip()[-300:])
+          # each mutant fails on the first k value (about four minutes); run them two at a time
+          def mutant(args):
+              flags, m = args
+              return m, flags, subprocess.run([sys.executable, *flags, script, '--check', '--procs', '1', '--mutant', m], cwd=root, capture_output=True, timeout=2400).returncode
+          jobs = [(flags, m) for flags in (['-B', '-S'], ['-B', '-O', '-S']) for m in ['no-third-jet', 'sign-lo', 'variance-yyy']]
+          with concurrent.futures.ThreadPoolExecutor(max_workers=2) as ex:
+              for m, flags, rc in ex.map(mutant, jobs):
+                  if rc != 1:
+                      raise RuntimeError('mutant ' + m + ' not rejected under ' + ' '.join(flags))
+          print(json.dumps({'scientific_effect': 'NONE', 'certified_enclosures_replayed': True, 'mutants_rejected': 6}))
+          PY
+          git diff --exit-code
diff --git a/.github/workflows/c8-elder-failure-coefficient.yml b/.github/workflows/c8-elder-failure-coefficient.yml
new file mode 100644
index 000000000..a7f06cfe4
--- /dev/null
+++ b/.github/workflows/c8-elder-failure-coefficient.yml
@@ -0,0 +1,83 @@
+name: C8 elder-failure coefficient (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c8_elder_failure_coefficient_20261001/**'
+      - '.github/workflows/c8-elder-failure-coefficient.yml'
+      - 'frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md'
+      - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json'
+      - 'frontiers/local_elder_geometry_20260930/PROOF.md'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'frontiers/c6_cluster_law_20260929/PROOF.md'
+      - 'frontiers/planar_cubic_cluster_20260929/PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 150
+    strategy:
+      fail-fast: false
+      matrix:
+        flags: ['-B -S', '-B -O -S']
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest and main-resident pins
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, pathlib
+          root = pathlib.Path('frontiers/c8_elder_failure_coefficient_20261001').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              data = path.read_bytes()
+              blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']:
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if not entry['current_required']:
+                  continue
+              data = pathlib.Path(entry['path']).read_bytes()
+              blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']:
+                  raise ValueError('consumed source identity mismatch on this tree: ' + entry['path'])
+          print('manifest and pins verified')
+          PY
+      - name: Self-test, full replay against RESULTS.json, mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+          FLAGS: ${{ matrix.flags }}
+        run: |
+          set -euo pipefail
+          cd frontiers/c8_elder_failure_coefficient_20261001
+          python $FLAGS -m unittest test_replay_metadata -v
+          python $FLAGS certificate.py --check --procs "$(nproc)"
+          for m in vw-sign ww-drop corner-swap mid-branch k-weight; do
+            set +e
+            python $FLAGS certificate.py --check --mutant "$m" --procs "$(nproc)" > /dev/null
+            rc=$?
+            set -e
+            if [ "$rc" -ne 1 ]; then echo "mutant $m not rejected (rc=$rc)"; exit 1; fi
+            echo "mutant $m rejected"
+          done
+          cd "$GITHUB_WORKSPACE"
+          git diff --exit-code
+          test -z "$(git status --porcelain)"
diff --git a/.github/workflows/c8-lifetime-difference-constant.yml b/.github/workflows/c8-lifetime-difference-constant.yml
new file mode 100644
index 000000000..6f20ce0b7
--- /dev/null
+++ b/.github/workflows/c8-lifetime-difference-constant.yml
@@ -0,0 +1,75 @@
+name: C8 lifetime difference constant and second-order ledger (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c8_lifetime_difference_constant_20261001/**'
+      - '.github/workflows/c8-lifetime-difference-constant.yml'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md'
+      - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md'
+      - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md'
+      - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md'
+      - 'frontiers/c8_cap_failure_coefficient_dims_20261001/RESULTS.json'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 240
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest, main-resident pins and engine copies; replay the certificate in both interpreter modes; reject mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, os, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/c8_lifetime_difference_constant_20261001').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']:
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          def blob(data):
+              return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if entry.get('current_required'):
+                  data = pathlib.Path(entry['path']).read_bytes()
+              elif entry.get('packet_copy'):
+                  data = (root / entry['packet_copy']).read_bytes()
+              else:
+                  continue
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob(data) != entry['git_blob']:
+                  raise ValueError('pinned source identity mismatch: ' + entry['path'])
+          script = str(root / 'theorem_b.py')
+          procs = str(max(2, os.cpu_count() or 2))
+          # every cap band in one mode, the sampled replay in the other
+          for flags, mode in ((['-B', '-S'], '--check-full'), (['-B', '-O', '-S'], '--check')):
+              out = subprocess.run([sys.executable, *flags, script, mode, '--procs', procs], cwd=root, capture_output=True, timeout=10800)
+              if out.returncode != 0:
+                  raise RuntimeError(mode + ' failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:])
+              print(flags, mode, out.stdout.decode().strip()[-600:])
+          for m in ('no-om-term', 'no-bad-set', 'no-det', 'no-torus', 'cap-half'):
+              rc = subprocess.run([sys.executable, '-B', '-S', script, '--check', '--procs', procs, '--mutant', m], cwd=root, capture_output=True, timeout=3600).returncode
+              if rc != 1:
+                  raise RuntimeError('mutant ' + m + ' not rejected')
+          print(json.dumps({'scientific_effect': 'NONE', 'cap_bands_replayed': 274, 'mutants_rejected': 5}))
+          PY
+          git diff --exit-code
diff --git a/.github/workflows/c8-theorem-a-constants-d3.yml b/.github/workflows/c8-theorem-a-constants-d3.yml
new file mode 100644
index 000000000..559e2a397
--- /dev/null
+++ b/.github/workflows/c8-theorem-a-constants-d3.yml
@@ -0,0 +1,68 @@
+name: C8 Theorem A constants in d = 3 (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c8_theorem_a_constants_d3_20261001/**'
+      - '.github/workflows/c8-theorem-a-constants-d3.yml'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md'
+      - 'frontiers/c8_cap_failure_coefficient_dims_20261001/RESULTS.json'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 120
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest and main-resident pins, replay every band and the C table in both interpreter modes, reject mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, os, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/c8_theorem_a_constants_d3_20261001').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']:
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if not entry['current_required']:
+                  continue
+              data = pathlib.Path(entry['path']).read_bytes()
+              blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']:
+                  raise ValueError('consumed source identity mismatch on this tree: ' + entry['path'])
+          script = str(root / 'theorem_a3.py')
+          procs = str(max(2, os.cpu_count() or 2))
+          # every one of the 121 bands in one mode, the sampled replay in the other
+          for flags, mode in ((['-B', '-S'], '--check-full'), (['-B', '-O', '-S'], '--check')):
+              out = subprocess.run([sys.executable, *flags, script, mode, '--procs', procs], cwd=root, capture_output=True, timeout=5400)
+              if out.returncode != 0:
+                  raise RuntimeError(mode + ' failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:])
+              print(flags, mode, out.stdout.decode().strip()[-600:])
+          for m in ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term', 'no-lambda2'):
+              rc = subprocess.run([sys.executable, '-B', '-S', script, '--check', '--procs', procs, '--mutant', m], cwd=root, capture_output=True, timeout=2400).returncode
+              if rc != 1:
+                  raise RuntimeError('mutant ' + m + ' not rejected')
+          print(json.dumps({'scientific_effect': 'NONE', 'bands_replayed': 121, 'mutants_rejected': 5}))
+          PY
+          git diff --exit-code
diff --git a/.github/workflows/c8-theorem-a-constants-planar.yml b/.github/workflows/c8-theorem-a-constants-planar.yml
new file mode 100644
index 000000000..104e5e501
--- /dev/null
+++ b/.github/workflows/c8-theorem-a-constants-planar.yml
@@ -0,0 +1,66 @@
+name: C8 planar Theorem A constants (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c8_theorem_a_constants_planar_20260930/**'
+      - '.github/workflows/c8-theorem-a-constants-planar.yml'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 120
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest and main-resident pins, replay every band and the C table in both interpreter modes, reject mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/c8_theorem_a_constants_planar_20260930').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']:
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if not entry['current_required']:
+                  continue
+              data = pathlib.Path(entry['path']).read_bytes()
+              blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob != entry['git_blob']:
+                  raise ValueError('consumed source identity mismatch on this tree: ' + entry['path'])
+          script = str(root / 'theorem_a.py')
+          # every one of the 153 bands in one mode, the sampled replay in the other
+          for flags, mode in ((['-B', '-S'], '--check-full'), (['-B', '-O', '-S'], '--check')):
+              out = subprocess.run([sys.executable, *flags, script, mode, '--procs', '2'], cwd=root, capture_output=True, timeout=3600)
+              if out.returncode != 0:
+                  raise RuntimeError(mode + ' failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:])
+              print(flags, mode, out.stdout.decode().strip()[-400:])
+          for m in ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term'):
+              rc = subprocess.run([sys.executable, '-B', '-S', script, '--check', '--procs', '2', '--mutant', m], cwd=root, capture_output=True, timeout=1800).returncode
+              if rc != 1:
+                  raise RuntimeError('mutant ' + m + ' not rejected')
+          print(json.dumps({'scientific_effect': 'NONE', 'bands_replayed': 153, 'mutants_rejected': 4}))
+          PY
+          git diff --exit-code
diff --git a/.github/workflows/c8-torus-vanishing-remainder.yml b/.github/workflows/c8-torus-vanishing-remainder.yml
new file mode 100644
index 000000000..f01604880
--- /dev/null
+++ b/.github/workflows/c8-torus-vanishing-remainder.yml
@@ -0,0 +1,80 @@
+name: C8 torus vanishing remainder (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c8_torus_vanishing_remainder_20261001/**'
+      - '.github/workflows/c8-torus-vanishing-remainder.yml'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md'
+      - 'imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md'
+      - 'reviews/d1_section9_borel_repair_20260925/REPAIR.md'
+      - 'reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 120
+    strategy:
+      fail-fast: false
+      matrix:
+        flags: ['-B -S', '-B -O -S']
+    name: certified-replay (${{ matrix.flags }})
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify manifest, main-resident pin and packet copies; replay the certificate; reject the mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+          FLAGS: ${{ matrix.flags }}
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, os, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/c8_torus_vanishing_remainder_20261001').resolve()
+          manifest = json.loads((root / 'SOURCE_FILES.json').read_text())['files']
+          present = sorted(p.name for p in root.iterdir())
+          if present != sorted([e['path'] for e in manifest] + ['SOURCE_FILES.json']):
+              raise ValueError('packet tree differs from manifest: ' + repr(present))
+          def blob(data):
+              return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+          for entry in manifest:
+              path = root / entry['path']
+              if pathlib.PurePosixPath(entry['path']).name != entry['path'] or path.is_symlink():
+                  raise ValueError('flat regular source required: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob(data) != entry['git_blob']:
+                  raise ValueError('source identity mismatch: ' + entry['path'])
+          pins = json.loads((root / 'SOURCE_MAP.json').read_text())
+          for entry in pins['sources']:
+              if entry.get('current_required'):
+                  data = pathlib.Path(entry['path']).read_bytes()
+              elif entry.get('packet_copy'):
+                  data = (root / entry['packet_copy']).read_bytes()
+              else:
+                  continue
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or blob(data) != entry['git_blob']:
+                  raise ValueError('pinned source identity mismatch: ' + entry['path'])
+          flags = os.environ['FLAGS'].split()
+          script = str(root / 'theorem_v.py')
+          out = subprocess.run([sys.executable, *flags, script, '--check'], cwd=root, capture_output=True, timeout=3600)
+          if out.returncode != 0:
+              raise RuntimeError('--check failed: ' + out.stdout.decode()[-400:] + out.stderr.decode()[-400:])
+          print(flags, out.stdout.decode().strip())
+          muts = ('torus-const-only', 'exp-radius', 'no-lin', 'no-om-term', 'coupling-half')
+          procs = [subprocess.Popen([sys.executable, *flags, script, '--mutant', m], cwd=root,
+                                    stdout=subprocess.DEVNULL, stderr=subprocess.DEVNULL) for m in muts]
+          for m, p in zip(muts, procs):
+              if p.wait(timeout=3600) != 1:
+                  raise RuntimeError('mutant ' + m + ' not rejected')
+          print(json.dumps({'scientific_effect': 'NONE', 'mutants_rejected': len(muts)}))
+          PY
+          git diff --exit-code
diff --git a/.github/workflows/c8-window-coefficient-d3-torus.yml b/.github/workflows/c8-window-coefficient-d3-torus.yml
index dc49ec669..0194bba0b 100644
--- a/.github/workflows/c8-window-coefficient-d3-torus.yml
+++ b/.github/workflows/c8-window-coefficient-d3-torus.yml
@@ -18,10 +18,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 30
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources and pins, replay both modes, reject mutants
diff --git a/.github/workflows/c8-window-coefficient-planar.yml b/.github/workflows/c8-window-coefficient-planar.yml
index d3102c319..b5061549c 100644
--- a/.github/workflows/c8-window-coefficient-planar.yml
+++ b/.github/workflows/c8-window-coefficient-planar.yml
@@ -18,10 +18,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 30
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources and pins, replay both modes, reject mutants
diff --git a/.github/workflows/candidate-third-order.yml b/.github/workflows/candidate-third-order.yml
index 3bd2c8cd2..624a74faa 100644
--- a/.github/workflows/candidate-third-order.yml
+++ b/.github/workflows/candidate-third-order.yml
@@ -26,10 +26,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/cap-first-exit-lean.yml b/.github/workflows/cap-first-exit-lean.yml
index 9d96705e3..d2bf2186d 100644
--- a/.github/workflows/cap-first-exit-lean.yml
+++ b/.github/workflows/cap-first-exit-lean.yml
@@ -20,10 +20,10 @@ jobs:
       LEAN_NUM_THREADS: 2
       PYTHONDONTWRITEBYTECODE: '1'
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Source gate and unit tests in both Python modes (no Lean)
@@ -34,7 +34,7 @@ jobs:
           python -B -O -S -m unittest discover -s "$P" -p 'test_*.py' -v
           python -B -S "$P/gate.py"
       - name: Install pinned Lean and retrieve the Mathlib cache
-        uses: leanprover/lean-action@50fcf42d2e460296f1a34b402e990d1b24f8b596
+        uses: leanprover/lean-action@f061402b660e0c34644504b324e830f2991d4865
         with:
           lake-package-directory: frontiers/cap_first_exit_lean_20261002
           auto-config: 'false'
@@ -49,7 +49,7 @@ jobs:
         run: git diff --exit-code
       - name: Preserve logs and receipt
         if: always()
-        uses: actions/upload-artifact@ea165f8d65b6e75b540449e92b4886f43607fa02
+        uses: actions/upload-artifact@043fb46d1a93c77aae656e7c1c64a875d1fc6a0a
         with:
           name: cap-first-exit-lean-${{ github.run_id }}-${{ github.run_attempt }}
           path: frontiers/cap_first_exit_lean_20261002/.lake/evidence/
diff --git a/.github/workflows/cluster-exponential-weight.yml b/.github/workflows/cluster-exponential-weight.yml
index b01274852..c25a5a287 100644
--- a/.github/workflows/cluster-exponential-weight.yml
+++ b/.github/workflows/cluster-exponential-weight.yml
@@ -17,11 +17,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           fetch-depth: 0
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Finite controls and immutable source custody (not proof acceptance)
diff --git a/.github/workflows/contact-angular.yml b/.github/workflows/contact-angular.yml
index 3cfe9673c..df21255fa 100644
--- a/.github/workflows/contact-angular.yml
+++ b/.github/workflows/contact-angular.yml
@@ -10,10 +10,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 5
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify exact sources and run both modes
@@ -43,7 +43,7 @@ jobs:
           git diff --exit-code
       - name: Retain evidence including failures
         if: always()
-        uses: actions/upload-artifact@ea165f8d65b6e75b540449e92b4886f43607fa02
+        uses: actions/upload-artifact@043fb46d1a93c77aae656e7c1c64a875d1fc6a0a
         with:
           name: contact-angular-${{ github.run_id }}
           path: ${{ runner.temp }}/contact-angular-results
diff --git a/.github/workflows/cusp-coefficient-certified.yml b/.github/workflows/cusp-coefficient-certified.yml
index 26db78d28..24412ef8f 100644
--- a/.github/workflows/cusp-coefficient-certified.yml
+++ b/.github/workflows/cusp-coefficient-certified.yml
@@ -23,10 +23,10 @@ jobs:
       matrix:
         flags: ['-B -S', '-B -O -S']
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify manifest and main-resident pins
diff --git a/.github/workflows/cusp-second-order.yml b/.github/workflows/cusp-second-order.yml
index 11d088378..7060635b1 100644
--- a/.github/workflows/cusp-second-order.yml
+++ b/.github/workflows/cusp-second-order.yml
@@ -26,10 +26,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/cusp-torus-transfer.yml b/.github/workflows/cusp-torus-transfer.yml
index b00d26777..d6cb94e6e 100644
--- a/.github/workflows/cusp-torus-transfer.yml
+++ b/.github/workflows/cusp-torus-transfer.yml
@@ -18,11 +18,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch exact historical source commits
diff --git a/.github/workflows/d1-cap-elder-partner-second-pass.yml b/.github/workflows/d1-cap-elder-partner-second-pass.yml
index 1d0baf1ca..4b688ce16 100644
--- a/.github/workflows/d1-cap-elder-partner-second-pass.yml
+++ b/.github/workflows/d1-cap-elder-partner-second-pass.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources and pins, replay both modes, reject mutants
diff --git a/.github/workflows/d1-cde-fulldepth-review.yml b/.github/workflows/d1-cde-fulldepth-review.yml
index fa03e30ab..089df7714 100644
--- a/.github/workflows/d1-cde-fulldepth-review.yml
+++ b/.github/workflows/d1-cde-fulldepth-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d1-chain-reconciliation.yml b/.github/workflows/d1-chain-reconciliation.yml
index 167cf9206..b32264cbb 100644
--- a/.github/workflows/d1-chain-reconciliation.yml
+++ b/.github/workflows/d1-chain-reconciliation.yml
@@ -19,10 +19,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d1-external-cap-audit.yml b/.github/workflows/d1-external-cap-audit.yml
index 531472589..2c3a834fd 100644
--- a/.github/workflows/d1-external-cap-audit.yml
+++ b/.github/workflows/d1-external-cap-audit.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Authenticate the frozen source set
diff --git a/.github/workflows/d1-theorem-a.yml b/.github/workflows/d1-theorem-a.yml
index 45f71d766..55dad2971 100644
--- a/.github/workflows/d1-theorem-a.yml
+++ b/.github/workflows/d1-theorem-a.yml
@@ -16,10 +16,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d2-remainder-fulldepth-review.yml b/.github/workflows/d2-remainder-fulldepth-review.yml
index ac73540c3..dbbba3570 100644
--- a/.github/workflows/d2-remainder-fulldepth-review.yml
+++ b/.github/workflows/d2-remainder-fulldepth-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify record tree, reviewed source identities, replay both modes, reject mutants
diff --git a/.github/workflows/d5-dimension-lift.yml b/.github/workflows/d5-dimension-lift.yml
index e81691867..06a9c445b 100644
--- a/.github/workflows/d5-dimension-lift.yml
+++ b/.github/workflows/d5-dimension-lift.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d5-intermediate-window-review.yml b/.github/workflows/d5-intermediate-window-review.yml
index 9c8c1fcd1..ecf5c711e 100644
--- a/.github/workflows/d5-intermediate-window-review.yml
+++ b/.github/workflows/d5-intermediate-window-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d5-pin-disk.yml b/.github/workflows/d5-pin-disk.yml
index 01b2bd183..70697b1d1 100644
--- a/.github/workflows/d5-pin-disk.yml
+++ b/.github/workflows/d5-pin-disk.yml
@@ -16,10 +16,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d5-punctured-pin-continuum-review.yml b/.github/workflows/d5-punctured-pin-continuum-review.yml
index 7e7885d04..3500d189c 100644
--- a/.github/workflows/d5-punctured-pin-continuum-review.yml
+++ b/.github/workflows/d5-punctured-pin-continuum-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/d5-reconciliation.yml b/.github/workflows/d5-reconciliation.yml
index 4ae0bb8b4..2474267a9 100644
--- a/.github/workflows/d5-reconciliation.yml
+++ b/.github/workflows/d5-reconciliation.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/downstream-gate.yml b/.github/workflows/downstream-gate.yml
index 13ba3b5c2..7ed7500fe 100644
--- a/.github/workflows/downstream-gate.yml
+++ b/.github/workflows/downstream-gate.yml
@@ -120,7 +120,7 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 5
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
       - name: Require both checks and source-bound current-run formal success
diff --git a/.github/workflows/elder-cusp-parity.yml b/.github/workflows/elder-cusp-parity.yml
index 16b1c8be4..29be167a6 100644
--- a/.github/workflows/elder-cusp-parity.yml
+++ b/.github/workflows/elder-cusp-parity.yml
@@ -30,10 +30,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/elder-dimension-lift-review.yml b/.github/workflows/elder-dimension-lift-review.yml
index 6a708cba4..801df8dd5 100644
--- a/.github/workflows/elder-dimension-lift-review.yml
+++ b/.github/workflows/elder-dimension-lift-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/elder-dimension-lift.yml b/.github/workflows/elder-dimension-lift.yml
index fcb467a25..b4301cba8 100644
--- a/.github/workflows/elder-dimension-lift.yml
+++ b/.github/workflows/elder-dimension-lift.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources, both modes and intended semantic failures
diff --git a/.github/workflows/elder-lower-all-d.yml b/.github/workflows/elder-lower-all-d.yml
index 31734edcf..e036ccb64 100644
--- a/.github/workflows/elder-lower-all-d.yml
+++ b/.github/workflows/elder-lower-all-d.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/elder-lower-review.yml b/.github/workflows/elder-lower-review.yml
index dbbddd04c..82cffd9aa 100644
--- a/.github/workflows/elder-lower-review.yml
+++ b/.github/workflows/elder-lower-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/elder-partner-extremes.yml b/.github/workflows/elder-partner-extremes.yml
index 77741c122..b3c91d6f1 100644
--- a/.github/workflows/elder-partner-extremes.yml
+++ b/.github/workflows/elder-partner-extremes.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch exact public source commits
diff --git a/.github/workflows/elder-third-order.yml b/.github/workflows/elder-third-order.yml
index 98928ee2f..f24740733 100644
--- a/.github/workflows/elder-third-order.yml
+++ b/.github/workflows/elder-third-order.yml
@@ -29,10 +29,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/extreme-cluster-sampling.yml b/.github/workflows/extreme-cluster-sampling.yml
index d47c6fcba..e86438898 100644
--- a/.github/workflows/extreme-cluster-sampling.yml
+++ b/.github/workflows/extreme-cluster-sampling.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch exact public source objects
diff --git a/.github/workflows/far-elder-rate.yml b/.github/workflows/far-elder-rate.yml
index efe17c813..7be1fdeea 100644
--- a/.github/workflows/far-elder-rate.yml
+++ b/.github/workflows/far-elder-rate.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/fixed-r-inverse-lifetime.yml b/.github/workflows/fixed-r-inverse-lifetime.yml
index 9216a6e52..92de6cb23 100644
--- a/.github/workflows/fixed-r-inverse-lifetime.yml
+++ b/.github/workflows/fixed-r-inverse-lifetime.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch exact public parent model
diff --git a/.github/workflows/formal-lean.yml b/.github/workflows/formal-lean.yml
index 97d78b819..bf4f2c0e9 100644
--- a/.github/workflows/formal-lean.yml
+++ b/.github/workflows/formal-lean.yml
@@ -33,7 +33,7 @@ jobs:
     env:
       LEAN_NUM_THREADS: 2
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
       - name: Gate tests and exact source identity
@@ -42,7 +42,7 @@ jobs:
           python3 -O -m unittest discover -s formal/tests -v
           python3 formal/gate.py
       - name: Install pinned Lean and retrieve dependency cache
-        uses: leanprover/lean-action@50fcf42d2e460296f1a34b402e990d1b24f8b596
+        uses: leanprover/lean-action@f061402b660e0c34644504b324e830f2991d4865
         with:
           lake-package-directory: formal
           auto-config: 'false'
@@ -58,7 +58,7 @@ jobs:
         run: python3 -B -S tools/required_formal_check.py receipt
       - name: Preserve logs and exact-run receipt
         if: always()
-        uses: actions/upload-artifact@ea165f8d65b6e75b540449e92b4886f43607fa02
+        uses: actions/upload-artifact@043fb46d1a93c77aae656e7c1c64a875d1fc6a0a
         with:
           name: formal-evidence-${{ github.run_id }}-${{ github.run_attempt }}
           path: formal/.lake/formal-evidence/
diff --git a/.github/workflows/gpdata114-inert-custody.yml b/.github/workflows/gpdata114-inert-custody.yml
index c0b665b88..c3b1d2c82 100644
--- a/.github/workflows/gpdata114-inert-custody.yml
+++ b/.github/workflows/gpdata114-inert-custody.yml
@@ -17,10 +17,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify inert bytes and decoder controls only
diff --git a/.github/workflows/intermediate-window.yml b/.github/workflows/intermediate-window.yml
index 2c5b78820..6b27723b0 100644
--- a/.github/workflows/intermediate-window.yml
+++ b/.github/workflows/intermediate-window.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources, two modes, assertion-rejected mutations
diff --git a/.github/workflows/lifetime-moment-boundary.yml b/.github/workflows/lifetime-moment-boundary.yml
index db56ae55c..6c9fa1d8c 100644
--- a/.github/workflows/lifetime-moment-boundary.yml
+++ b/.github/workflows/lifetime-moment-boundary.yml
@@ -19,10 +19,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 5
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - run: python -B -S frontiers/lifetime_moment_boundary_20260930/verify.py
diff --git a/.github/workflows/local-cluster-tightness.yml b/.github/workflows/local-cluster-tightness.yml
index 365bad977..072a5a4de 100644
--- a/.github/workflows/local-cluster-tightness.yml
+++ b/.github/workflows/local-cluster-tightness.yml
@@ -26,11 +26,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           fetch-depth: 0
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Both modes, semantic controls and exact current/historical sources
diff --git a/.github/workflows/moving-remote-collision.yml b/.github/workflows/moving-remote-collision.yml
index b955379bd..8ab75c1a5 100644
--- a/.github/workflows/moving-remote-collision.yml
+++ b/.github/workflows/moving-remote-collision.yml
@@ -41,11 +41,11 @@ jobs:
       matrix:
         mode: [normal, optimized]
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           fetch-depth: 0
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources and run bounded exact controls
diff --git a/.github/workflows/p15-bipartite-overlap.yml b/.github/workflows/p15-bipartite-overlap.yml
index 57266f07a..5ef13e961 100644
--- a/.github/workflows/p15-bipartite-overlap.yml
+++ b/.github/workflows/p15-bipartite-overlap.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify exact sources and both-mode semantic controls
diff --git a/.github/workflows/p15-overlap-tail-review.yml b/.github/workflows/p15-overlap-tail-review.yml
index c7d35ac14..788120770 100644
--- a/.github/workflows/p15-overlap-tail-review.yml
+++ b/.github/workflows/p15-overlap-tail-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/p15-tail-load.yml b/.github/workflows/p15-tail-load.yml
index e55e456da..da2d606c8 100644
--- a/.github/workflows/p15-tail-load.yml
+++ b/.github/workflows/p15-tail-load.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources, both Python modes and seven semantic mutants
diff --git a/.github/workflows/pin-micro-covariance.yml b/.github/workflows/pin-micro-covariance.yml
index 683b8da20..4579bb5de 100644
--- a/.github/workflows/pin-micro-covariance.yml
+++ b/.github/workflows/pin-micro-covariance.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 5
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Run exact rational check in both modes
diff --git a/.github/workflows/planar-cubic-cluster.yml b/.github/workflows/planar-cubic-cluster.yml
index b1b7d84d3..6021e3485 100644
--- a/.github/workflows/planar-cubic-cluster.yml
+++ b/.github/workflows/planar-cubic-cluster.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify packet, both sources and both Python modes
diff --git a/.github/workflows/planar-rejected-endpoint-margin.yml b/.github/workflows/planar-rejected-endpoint-margin.yml
index b35e3e7ad..ed768c694 100644
--- a/.github/workflows/planar-rejected-endpoint-margin.yml
+++ b/.github/workflows/planar-rejected-endpoint-margin.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify stored identities and replay the published author controls, both modes
diff --git a/.github/workflows/planar-soft-layer-chain.yml b/.github/workflows/planar-soft-layer-chain.yml
index 23d0e7cfb..933f771fa 100644
--- a/.github/workflows/planar-soft-layer-chain.yml
+++ b/.github/workflows/planar-soft-layer-chain.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify stored identities and replay every published checker, both modes
diff --git a/.github/workflows/radial-rate.yml b/.github/workflows/radial-rate.yml
index fa1a9abe5..51a4bb364 100644
--- a/.github/workflows/radial-rate.yml
+++ b/.github/workflows/radial-rate.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch exact public source objects
diff --git a/.github/workflows/register-alignment.yml b/.github/workflows/register-alignment.yml
index ac806471c..74bbb0b2a 100644
--- a/.github/workflows/register-alignment.yml
+++ b/.github/workflows/register-alignment.yml
@@ -31,10 +31,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes (baseline or installed register), reject mutants
diff --git a/.github/workflows/register-execution-readiness.yml b/.github/workflows/register-execution-readiness.yml
index c6262ab5e..844532f64 100644
--- a/.github/workflows/register-execution-readiness.yml
+++ b/.github/workflows/register-execution-readiness.yml
@@ -21,10 +21,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Compose the three proposals on a scratch copy; run the three checkers and their mutants in the installed state
diff --git a/.github/workflows/remainder-rate.yml b/.github/workflows/remainder-rate.yml
index 1aacac1c5..d32aa63cb 100644
--- a/.github/workflows/remainder-rate.yml
+++ b/.github/workflows/remainder-rate.yml
@@ -25,10 +25,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remainder-vanishing.yml b/.github/workflows/remainder-vanishing.yml
index e046d962d..e238b8a3e 100644
--- a/.github/workflows/remainder-vanishing.yml
+++ b/.github/workflows/remainder-vanishing.yml
@@ -24,10 +24,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-collision.yml b/.github/workflows/remote-collision.yml
index dcf4566bc..ebd51494b 100644
--- a/.github/workflows/remote-collision.yml
+++ b/.github/workflows/remote-collision.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-distance-moments-review.yml b/.github/workflows/remote-distance-moments-review.yml
index 62456d897..7ccebfad6 100644
--- a/.github/workflows/remote-distance-moments-review.yml
+++ b/.github/workflows/remote-distance-moments-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-height-decoupling.yml b/.github/workflows/remote-height-decoupling.yml
index 52ab5b906..c291a13b5 100644
--- a/.github/workflows/remote-height-decoupling.yml
+++ b/.github/workflows/remote-height-decoupling.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify exact sources and both-mode semantic controls
diff --git a/.github/workflows/remote-height-marks-review.yml b/.github/workflows/remote-height-marks-review.yml
index f6bea0160..a019deed2 100644
--- a/.github/workflows/remote-height-marks-review.yml
+++ b/.github/workflows/remote-height-marks-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-height-review.yml b/.github/workflows/remote-height-review.yml
index d527d5efa..c947ba577 100644
--- a/.github/workflows/remote-height-review.yml
+++ b/.github/workflows/remote-height-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-inverse-separation-review.yml b/.github/workflows/remote-inverse-separation-review.yml
index 39efea01d..54a3f6916 100644
--- a/.github/workflows/remote-inverse-separation-review.yml
+++ b/.github/workflows/remote-inverse-separation-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-inverse-separation.yml b/.github/workflows/remote-inverse-separation.yml
index b6131b954..273e3336d 100644
--- a/.github/workflows/remote-inverse-separation.yml
+++ b/.github/workflows/remote-inverse-separation.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify exact packet and both-mode intended negatives
diff --git a/.github/workflows/remote-mixed-inverse-review.yml b/.github/workflows/remote-mixed-inverse-review.yml
index b5f1b7fe8..1939dd177 100644
--- a/.github/workflows/remote-mixed-inverse-review.yml
+++ b/.github/workflows/remote-mixed-inverse-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-mixed-inverse.yml b/.github/workflows/remote-mixed-inverse.yml
index 8551c08e4..ddcb90dce 100644
--- a/.github/workflows/remote-mixed-inverse.yml
+++ b/.github/workflows/remote-mixed-inverse.yml
@@ -24,10 +24,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact packet, both modes and upstream source bytes
diff --git a/.github/workflows/remote-pair-law-review.yml b/.github/workflows/remote-pair-law-review.yml
index 9a4c5c2d9..0229ba5e6 100644
--- a/.github/workflows/remote-pair-law-review.yml
+++ b/.github/workflows/remote-pair-law-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/remote-singleton.yml b/.github/workflows/remote-singleton.yml
index 3cdda862d..9e2cfda7b 100644
--- a/.github/workflows/remote-singleton.yml
+++ b/.github/workflows/remote-singleton.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources, two Python modes and semantic mutations
diff --git a/.github/workflows/sard-g-a2-fixed-frame-review.yml b/.github/workflows/sard-g-a2-fixed-frame-review.yml
index 3b194f1ce..5942d27d2 100644
--- a/.github/workflows/sard-g-a2-fixed-frame-review.yml
+++ b/.github/workflows/sard-g-a2-fixed-frame-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/sard-g-a2-fixed-frame.yml b/.github/workflows/sard-g-a2-fixed-frame.yml
index 3f207ebf5..d76490ca3 100644
--- a/.github/workflows/sard-g-a2-fixed-frame.yml
+++ b/.github/workflows/sard-g-a2-fixed-frame.yml
@@ -17,10 +17,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact successor sources and intended semantic failures in both modes
diff --git a/.github/workflows/sard-g-a2-parameter.yml b/.github/workflows/sard-g-a2-parameter.yml
index 05d079a86..12a1ba778 100644
--- a/.github/workflows/sard-g-a2-parameter.yml
+++ b/.github/workflows/sard-g-a2-parameter.yml
@@ -12,10 +12,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - run: |
diff --git a/.github/workflows/sard-g-a2-regularity.yml b/.github/workflows/sard-g-a2-regularity.yml
index bb5fe34d8..5791830f7 100644
--- a/.github/workflows/sard-g-a2-regularity.yml
+++ b/.github/workflows/sard-g-a2-regularity.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/sard-g-pinned-transfer.yml b/.github/workflows/sard-g-pinned-transfer.yml
index deb283f56..542bbd134 100644
--- a/.github/workflows/sard-g-pinned-transfer.yml
+++ b/.github/workflows/sard-g-pinned-transfer.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/sard-g-r1r2-review.yml b/.github/workflows/sard-g-r1r2-review.yml
index 9d9329fdc..475888b2f 100644
--- a/.github/workflows/sard-g-r1r2-review.yml
+++ b/.github/workflows/sard-g-r1r2-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/sard-g-r3r4-review.yml b/.github/workflows/sard-g-r3r4-review.yml
index 77c81e099..8ea00c717 100644
--- a/.github/workflows/sard-g-r3r4-review.yml
+++ b/.github/workflows/sard-g-r3r4-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/sard-robust-charts.yml b/.github/workflows/sard-robust-charts.yml
index 2ec278e32..22f1a838e 100644
--- a/.github/workflows/sard-robust-charts.yml
+++ b/.github/workflows/sard-robust-charts.yml
@@ -18,10 +18,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact packet and both-mode geometric negatives
diff --git a/.github/workflows/separated-fold-multiplicity.yml b/.github/workflows/separated-fold-multiplicity.yml
index 91ae08f43..bedf4b2d2 100644
--- a/.github/workflows/separated-fold-multiplicity.yml
+++ b/.github/workflows/separated-fold-multiplicity.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Authenticate historical interfaces and replay exact controls
diff --git a/.github/workflows/side24-periodization-firstvar.yml b/.github/workflows/side24-periodization-firstvar.yml
index d3e96a4ba..15d650c56 100644
--- a/.github/workflows/side24-periodization-firstvar.yml
+++ b/.github/workflows/side24-periodization-firstvar.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/side24-periodization-remainder.yml b/.github/workflows/side24-periodization-remainder.yml
index 51219e70a..59227d630 100644
--- a/.github/workflows/side24-periodization-remainder.yml
+++ b/.github/workflows/side24-periodization-remainder.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/side24-remainder.yml b/.github/workflows/side24-remainder.yml
index bd8b8a3f3..e788de7f4 100644
--- a/.github/workflows/side24-remainder.yml
+++ b/.github/workflows/side24-remainder.yml
@@ -20,10 +20,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Exact sources, both Python modes, semantic negative controls
diff --git a/.github/workflows/side24-v1-coefficient-review.yml b/.github/workflows/side24-v1-coefficient-review.yml
index d0f7d52ac..54fe93873 100644
--- a/.github/workflows/side24-v1-coefficient-review.yml
+++ b/.github/workflows/side24-v1-coefficient-review.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants, replay the reviewed package
diff --git a/.github/workflows/soft-closed-form.yml b/.github/workflows/soft-closed-form.yml
index da14cfb6e..efb3b211d 100644
--- a/.github/workflows/soft-closed-form.yml
+++ b/.github/workflows/soft-closed-form.yml
@@ -22,10 +22,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/soft-fold-limit.yml b/.github/workflows/soft-fold-limit.yml
index e9cc7f822..839b99a47 100644
--- a/.github/workflows/soft-fold-limit.yml
+++ b/.github/workflows/soft-fold-limit.yml
@@ -31,10 +31,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/soft-rejected-pairs.yml b/.github/workflows/soft-rejected-pairs.yml
index 9359bc11a..a930f5334 100644
--- a/.github/workflows/soft-rejected-pairs.yml
+++ b/.github/workflows/soft-rejected-pairs.yml
@@ -31,10 +31,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/spectral-cluster-closure.yml b/.github/workflows/spectral-cluster-closure.yml
index 9fed2e14d..efa7e789c 100644
--- a/.github/workflows/spectral-cluster-closure.yml
+++ b/.github/workflows/spectral-cluster-closure.yml
@@ -15,11 +15,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch exact public non-main source objects
diff --git a/.github/workflows/third-order-rate.yml b/.github/workflows/third-order-rate.yml
index 666bdbc6b..85859c3ca 100644
--- a/.github/workflows/third-order-rate.yml
+++ b/.github/workflows/third-order-rate.yml
@@ -31,10 +31,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/.github/workflows/two-scale-cluster-geometry.yml b/.github/workflows/two-scale-cluster-geometry.yml
index 5dae60249..44911831e 100644
--- a/.github/workflows/two-scale-cluster-geometry.yml
+++ b/.github/workflows/two-scale-cluster-geometry.yml
@@ -20,11 +20,11 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
           fetch-depth: 0
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Fetch immutable non-base source objects
diff --git a/.github/workflows/unrestricted-selection-difference.yml b/.github/workflows/unrestricted-selection-difference.yml
index 88038843d..b30b9f80c 100644
--- a/.github/workflows/unrestricted-selection-difference.yml
+++ b/.github/workflows/unrestricted-selection-difference.yml
@@ -15,10 +15,10 @@ jobs:
     runs-on: ubuntu-latest
     timeout-minutes: 10
     steps:
-      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
         with:
           persist-credentials: false
-      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
         with:
           python-version: '3.11.16'
       - name: Verify sources, replay both modes, reject mutants
diff --git a/frontiers/c2_torus_transfer_20261001/NOTE.md b/frontiers/c2_torus_transfer_20261001/NOTE.md
new file mode 100644
index 000000000..5ce5bad2a
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/NOTE.md
@@ -0,0 +1,312 @@
+# The third-order coefficient `c2` on the finite torus: transfer for every `L >= 10`, `d = 1, 2, 3`
+
+**Object:** CL-C2-TORUS-TRANSFER-20261001-v1.
+**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 1 October 2026. Author of the
+reference packet `frontiers/c2_exact_20261001` ([C2X]).
+**Scope claim:** Math-#223 comment 5934236720. The task was posted in Math-#223 comment 5933574379 (with the finite-part
+starting inequality used in Lemma F) and listed as unsupplied in main#229 (5933485698).
+**Disposition:** author-side. An explicit bound for every real `L >= L0`, every orthonormal frame and `d = 1, 2, 3` on
+the difference between the coefficient `c2` of the normalized periodic Gaussian kernel and its closed-form reference value.
+It is certified in exact rational ball arithmetic with the standard library. There is also one structural lemma
+(Lemma E) valid for every stationary kernel.
+**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX`, `GRAPH`, claim, catalog, prize or source-body change. Same GitHub
+account as every lane; zero organizational-independence credit. Nothing in `frontiers/c2_exact_20261001/` or in
+Math-#216, #218, #220 or #224 is touched.
+**Delivered under:** Dylan Roy's explicit instructions in this session, given after the 2026-09-27 owner stop (quoted in
+`SOURCE_MAP.json`, `delivered_under`). `OWNER_STOP.md` addresses Cursor agents and automations and is untouched.
+
+## 0. Statement
+
+**The torus.** This is the field of [SIDE24] (1)–(2) and Math-#224 (T1), for any real side `L` and any `d`:
+
+    K_L(z) = sum_{n in Z^d} phi(z + L n) / sum_{n in Z^d} phi(L n),        phi(z) = exp(-|z|^2/2).
+
+Fix an orthonormal frame with axial vector `u`. The pins sit at `M = -r u/2` and `S = r u/2`. Let `A~_r^{K,u}(b, k)` be the
+fixed-cone surrogate of the two-point kernel of [R] §§2–4 for the covariance `K` in this frame, exactly as in [C2X] §1.
+Its fold expansion is `A~_r = A_0 + r^2 A_2 + O(r^3)` (§3). Put
+
+    F^u_K(k) = integral_R A_2^{K,u}(b, k) db,
+    T(F) = f.p. integral_0^oo F(k) k^(-4/3) dk = integral_0^oo [F(k) - F(0)] k^(-4/3) dk,
+    c2[K, u] = (1/3) |S^(d-1)| T(F^u_K),       c2[K] = (1/3) integral_(S^(d-1)) T(F^u_K) d sigma(u).
+
+`c2[K]` is the expression of Math-#218 (0.1). For `K = phi` it does not depend on `u` and equals the closed form of [C2X]
+§0. #218's Theorem T, which identifies this expression as a coefficient of the candidate density, is a candidate
+theorem for `d >= 2`. It was merged at `fb6ee97`, which is not a status promotion. In `d = 1` the theorem below is a transfer of the explicitly defined fixed-cone
+expression, not an extension of #218's density theorem (review 5383099127, item 3).
+
+**Theorem C2T.** For `d in {1, 2, 3}`, every `L0` in the table, every real `L >= L0` and every orthonormal frame,
+
+    |c2[K_L, u] - c2[phi]| <= eps_d(L0),   hence   |c2[K_L] - c2[phi]| <= eps_d(L0),
+
+with these values (rounded up):
+
+| `d` | `L >= 10` | `L >= 12` | `L >= 16` | `L >= 24` |
+|---|---|---|---|---|
+| `1` | `1.41035e-13` | `1.42699e-22` | `5.66439e-46` | `4.09547e-114` |
+| `2` | `2.09404e-12` | `2.02307e-21` | `7.57841e-45` | `5.20139e-113` |
+| `3` | `1.00039e-10` | `9.20576e-20` | `3.23748e-43` | `2.09418e-111` |
+
+**Relative to `c2[phi]`** (using the lower endpoints of [C2X]'s enclosures), the error is below:
+- `6.2e-13`, `9.5e-12` and `6.3e-10` for `L >= 10`;
+- `1.8e-113`, `2.4e-112` and `1.3e-110` for `L >= 24`.
+
+**Corollary (SIDE24).** [C2X] widened its published enclosures by `1e-14` relative beyond the certified enclosure (§5
+there). That margin is much larger than `eps_d(24) <= 2.1e-111`, so the published intervals contain the torus values as
+they stand:
+- `c2[K_24] in [0.2215244106266632, 0.2215244106267110]` in `d = 2`;
+- `c2[K_24] in [0.1612340491269447, 0.1612340491269810]` in `d = 3`.
+
+With [SIDE24]'s enclosures of `c_{d,24}`, the interval quotients give
+`c2[K_24]/c_{2,24} in [3.017759261945395, 3.017759261946047]` and
+`c2[K_24]/c_{3,24} in [3.859496174547459, 3.859496174548329]`.
+
+**Three-term law, with its sources.** The law is `nu/(c ell^(-1/3)) - 1 = (c1/c) ell^(7/12) + (c2/c) ell^(2/3) + ...`.
+On the SIDE24 torus in `d = 3` its coefficients are `-5.071062...` and `3.859496174548...`, to the stated digits. The sources
+of each piece are:
+- **`c1/c`** comes from [CU] (Math-#219, certified arithmetic, merged at `5a836a8`). It is consumed, not accepted here.
+- **Math-#224** plays two separate roles:
+  - its (T1) defines the torus kernel used throughout;
+  - its `c1` transfer is merged at `d3a0b6c` (`PROOF.md` blob `de3d85fd`, byte-identical to the consumed head `a5e69ee`);
+    it is cited as the analogue, and nothing here inherits status from it.
+- **The identification as density coefficients** is conditional on the chain in §8.
+
+**Lemma E (no `r^1` term for every stationary kernel).** Let `K` be any smooth stationary covariance (no isotropy or
+evenness needed) whose pin and transverse covariances are nondegenerate at `r = 0`. Then `A~_r(b, k)` is smooth in `r` up
+to `r = 0`, and at fixed `v_1 = b - k r^3/2` its Taylor expansion in `r` contains only even powers: for every `N`,
+`A~_r(b, k) = sum_{j<N} r^(2j) a_j(b - k r^3/2, k) + O(r^(2N))`. If `K` is moreover real-analytic, as the Gaussian `phi`
+and the torus kernels `K_L` are, then `A~_r(b, k) = Atilde(r^2, b - k r^3/2, k)` exactly near `r = 0`, with `Atilde`
+real-analytic. In either case
+
+    A~_r = A_0 + r^2 A_2 - (k/2) r^3 d_b A_0 + O(r^4):
+
+there is no `r^1` term, and the `r^3` coefficient is `-(k/2) d_b A_0`. (v1.3: smoothness alone does not give analyticity.
+A smooth positive-spectrum covariance can have a Taylor series of radius zero, for example spectral density
+`exp(-sqrt|xi|)`; 5938497821.)
+
+On the torus this decides, for the surrogate, the question left open in Math-#227 ("whether the torus `A_r` has an `r^1` term
+at all"): it has none. With the consumed comparison `A_r = A~_r + O(r^3)`, the typed `A_r` has none either. This does not
+remove #227's certified `beta r` allowance, accept #227's premises, or close C8 ([L'], cited only).
+
+**Relation to Math-#232** ([FJ], merged at `7fe06b0`). #232 gives the general eight-jet local stability interface (a
+birth-marginalized finite-jet formula with an analytic budget, `< 1e-60` for `L >= 24`). This packet gives sharper
+torus-specific constants for every `L >= 10`, together with Lemma E. The two are compatible, and neither supersedes the other.
+
+## 1. What is consumed, and how the proof runs
+
+- **[C2X] §§1–2 (main, `f9ebba1`).** The pin rows, the target `v_r`, the fixed-cone surrogate and its exact series derivation.
+  `exact.py` and `pseries.py` are carried as byte-identical copies, and this packet's pipeline is a re-implementation over balls
+  that `reference_checks` compares with them identically (§3).
+- **[SIDE24] (1)–(2) (main).** The normalized periodic kernel.
+- **[T] (Math-#218, merged at `fb6ee97`) (0.1) and Lemma F.** The definition of `c2`, and its identification as the `ell^(1/3)`
+  coefficient of the candidate density for `d >= 2` (an author-side candidate). Consumed, not reviewed.
+- **[CU] (Math-#219, merged at `5a836a8`).** The certified `c1/c` enclosure quoted in §0. Consumed, not accepted.
+- **[D] §2.2 (Math-#216, unmerged).** `A_r = A~_r + O(r^3)`; consumed, as in [C2X].
+
+**The route.**
+1. The jet covariances of `K_L` lie in explicit balls around those of `phi` (Lemma J).
+2. The exact derivation of [C2X], run in ball arithmetic, encloses the coefficients of the `A_2`-integrand
+   `kappa e^(-E(b, k, x)) H(b, k, x)` of `K_L` (Lemma P).
+3. The finite part is controlled additively by `Delta(0)`, a weighted integral of `Delta'` and an integral of `|Delta|`
+   (Lemma F).
+4. Each of these is bounded by explicit absolute Gaussian moments (Lemma G).
+
+There is no multiplicative sandwich: `A_2` is signed, and `c2` is a finite part.
+
+## 2. Lemma J: the jets of `K_L`
+
+`Cov(d^a f(0), d^b f(0)) = (-1)^|b| d^(a+b) K(0)` in the frame coordinates. `K_L` is even, so `d^g K_L(0) = 0` exactly when
+`|g|` is odd. For even `|g|`, write `delta = sum_(n != 0) phi(. + p_n)` with `p_n = L R n` (`R` the frame rotation) and
+`theta_1 = sum_(n != 0) phi(L n)`. Then
+
+    d^g K_L(0) - d^g phi(0) = [d^g delta(0) - theta_1 d^g phi(0)] / (1 + theta_1),
+    |d^g delta(0)| <= sum_(n != 0) prod_i |He_(g_i)(p_(n,i))| phi(p_n) <= sum_(n != 0) prod_i He*_(g_i)(L |n|) e^(-L^2 |n|^2 / 2).
+
+Here `d^m e^(-t^2/2) = (-1)^m He_m(t) e^(-t^2/2)`, `He*_m(s) = sum_j m!/(j!(m-2j)! 2^j) s^(m-2j)` majorizes `|He_m|` on
+`|t| <= s` and is nondecreasing, and `|p_(n,i)| <= |p_n| = L|n|` in every frame.
+
+- **Shells `|n|^2 <= 4`** are enumerated exactly; every such `n` has `|n_i| <= 2`.
+- **The tail `|n|^2 = s >= 5`** uses `#{n : |n|^2 = s} <= (3 sqrt s)^d` and `He*_m(t) <= (t + m)^m <= (2t)^m` for
+  `m <= t`, with `|g| <= L sqrt5`. The `s`-th term is at most `3^d (2L)^|g| s^a e^(-L^2 s/2)`, where `a = (d + |g|)/2`.
+  Consecutive ratios are at most `(6/5)^a e^(-L^2/2) <= 1/2`, so the tail is at most `2 * 3^d (2L)^|g| 5^a e^(-5L^2/2)`.
+
+**Monotonicity in `L`.** Each shell term has logarithmic `L`-derivative at most `|g|/L - L s <= 0` when `|g| <= L^2`,
+because `t He*'(t) <= m He*(t)`. The tail bound and `theta_1` also decrease. So the value at `L0` bounds every `L >= L0`.
+
+The radius used for `d^g K_L(0)` is the image bound plus `theta_1 |d^g phi(0)|`. The certificate queries jets up to order `20`
+and requires `|g|^2 <= 5 L0^2` and `|g| <= L0^2`. At `L0 = 10`, `theta_1 <= 1.16e-21` in `d = 3`.
+
+## 3. Lemma P: the ball pipeline
+
+A ball `B(c, rho)` is the rational interval `[c - rho, c + rho]`.
+- Sums, products and reciprocals (when `|c| > rho`) are the standard outward rules.
+- The radii are rounded up by an outward dyadic ceiling rule (`rup`), with at least 64-bit relative precision. This
+  affects precision only, not outwardness.
+- The operands are integers, Fractions or balls throughout. Float scalars are not certified: `B(1) + 0.1` would round the
+  center without a radius. The certificate never uses one. Reuse elsewhere should coerce or reject floats (review C49).
+- The centers are exact. With all radii zero the arithmetic is exactly [C2X]'s, so the centers of every output are the
+  reference values.
+
+`transfer.kernel(d, jetcov)` repeats [C2X] §2 with `pseries.R = 8` (series exact through `r^5`) and polynomials truncated
+at `r^4`:
+1. the pin covariance `S_r`, its inverse and determinant, by Gauss–Jordan over series of balls;
+2. conditioning of the Hessian entries on the pins and then on the **full** transverse block. In `d = 3` that block is
+   `T = [[x1, x3], [x3, x2]]`, with no rotation reduction, because the torus is not isotropic;
+3. `F = E[det H_M det H_S | T, U_r = v_r]` by Isserlis;
+4. `(det ratio)^(-1/2) exp(-(dq + dQ)/2) F`. The `r^0` parts of the determinant ratio and of `q`, `Q` are removed
+   structurally rather than by subtracting equal balls, which would leave spurious radii.
+
+The output is the integrand `G` (balls), `E = (q_0 + Q_0)/2` (a homogeneous quadratic form in `w = (b, k, x)`), `det S_0` and
+`det T_0`. For the surrogate,
+
+    A~_r^{K,u}(b, k) = kappa_K integral_(T < 0) e^(-E) G dx / r^2,   kappa_K = -12 (2 pi)^(-(p + n_T)/2) (det S_0 det T_0)^(-1/2),
+
+and the `A_2`-integrand is `kappa_K e^(-E) H` with `H = [r^4] G`. As in [C2X] §2.3, `A~_r` is real-analytic near
+`r = 0`, because `K_L` and `phi` are real-analytic and the `r = 0` covariances are nondegenerate. Every pivot is certified by `|c| > rho`.
+
+**Reference check (exact).** With zero radii, `kernel` reproduces [C2X]'s `kernel_data` identically:
+- the coefficients of `G` through `r^4`, `E` and `det S_0` in `d = 1, 2`;
+- in `d = 3`, the same on the slice `x3 = 0`, which is the diagonal slice [C2X] evaluates.
+
+## 4. Lemma E: evenness in `r` at fixed `v_1`
+
+Relabel the two points: `r -> -r` sends `M = -r u/2` to `+r u/2` and `S` to `-r u/2`. Every pin row of [R] is invariant under
+this relabelling as a linear form in the jets at `0`, because each is symmetric in `(M, S)` together with the sign of `r`.
+The rows are `(f(M) + f(S))/2`, `(f(S) - f(M))/r`, `(f_u(S) - f_u(M))/r`,
+`6 r^(-2)(f_u(M) + f_u(S) - 2(f(S) - f(M))/r)`, and the transverse pairs.
+
+The relabelling swaps `H_M` and `H_S`, which leaves `det H_M det H_S` unchanged, and it fixes `T = D_Theta^2 f(0)`. The
+target is `v_r = (b - k r^3/2, -k r^2, 0, 12k, 0, ...)`, whose entries other than `v_1 = b - k r^3/2` depend on `r` only
+through `r^2`. Hence the integrand `e^(-(q + Q)/2) (det ratio)^(-1/2) F`, as a function of `(r, v_1, k, x)`, is even in `r` as a
+formal power series. The coefficients are finite sums of jet covariances, so no property of `K` beyond stationarity and
+smoothness is used.
+- **Smooth `K`.** The rows are averages of derivatives against smooth kernels, so the covariances, the regression and the
+  cone integral are smooth in `r` up to `0`. Integrating the even formal series over the fixed cone gives the even
+  finite-order expansion of Lemma E, to every order.
+- **Real-analytic `K`.** The series converge near `r = 0`, which gives the exact form
+  `A~_r = Atilde(r^2, b - k r^3/2, k)` with `Atilde` real-analytic.
+
+**Exact checks (controls.py).** For anisotropic product kernels `prod_i e^(-s_i z_i^2/2)` (rational jets, not isotropic,
+`d = 1, 2, 3`, two `s` each) the exact integrand has `G_0 = G_1 = G_3 = 0` and
+`G_5 = -(k/2)(d_b G_2 - (d_b E) G_2)`, as Lemma E predicts.
+
+## 5. Lemmas F and G: the finite part and the Gaussian bounds
+
+**Lemma F.** For `C^1` functions `F_1, F_2` on `[0, oo)` with integrable decay, put `Delta = F_1 - F_2`. Then
+
+    |T(F_1) - T(F_2)| <= 3 |Delta(0)| + 3 integral_0^1 |Delta'(s)| s^(-1/3) ds + integral_1^oo |Delta(k)| k^(-4/3) dk.
+
+*Proof.* `T(F) = integral_0^1 [F - F(0)] k^(-4/3) + integral_1^oo F k^(-4/3) - 3F(0)`. Next,
+`|Delta(k) - Delta(0)| <= integral_0^k |Delta'|`. Exchanging the order of integration gives
+`integral_s^1 k^(-4/3) dk = 3(s^(-1/3) - 1) <= 3 s^(-1/3)`.
+
+This is the inequality of 5933574379 with its second-derivative term replaced by a first-derivative one, so no evenness in
+`k` is needed.
+
+**Lemma G.** Write `g_K = kappa_K e^(-E_K) H_K` and, for the derivative, `g'_K = kappa_K e^(-E_K) H'_K`, where
+`H' = d_k H - (d_k E) H` is formed in ball arithmetic. Pointwise,
+
+    |g_L - g_ref| <= |kappa_ref| e^(-E_ref) [ (|e_kappa| + |dE|) e^(|dE|) |H_L| + |H_L - H_ref| ],
+
+where:
+- `|kappa_ref| = 12 (2 pi)^(-(p + n_T)/2) (det S_0 det T_0)^(-1/2)` is the magnitude of the negative constant
+  `kappa_ref`; `bound_core` bounds it above by `kap` (review C49-F01, 5382406982);
+- `kappa_L = kappa_ref (1 + e_kappa)`, with `|e_kappa| <= rho/(1 - rho)` and `rho` the relative radius of
+  `det S_0 det T_0`;
+- `|dE| = |E_L - E_ref| <= sum_i eta_i w_i^2`, from the radii of `E`'s coefficients via
+  `|w_i w_j| <= (w_i^2 + w_j^2)/2`;
+- `|H_L| <= sum (|c| + rho)|w^e|` and `|H_L - H_ref| <= sum rho |w^e|` over `H`'s monomials.
+
+**Gaussian floor.** A rational `t` with `M - t diag(M)` positive semidefinite, where `E_ref = w^T M w`, is certified by
+exact `LDL^T` and bisection. For diagonal `M` it is `t = 1` by an exact branch, with zero residual, as in `d = 1`. It gives
+`e^(-E_ref + |dE|) <= prod_i e^(-a_i w_i^2)` with `a_i = t M_ii - eta_i > 0`. Strictness comes from `a_i > 0`, not from
+the residual matrix (5384221624).
+
+The bounds then enlarge the cone and the `b`-line to all of `R^(1 + n_T)`:
+- `|Delta(0)|` uses the monomials with no `k`;
+- `integral_0^1 |Delta'| s^(-1/3)` uses `s^(j - 1/3) <= s^(j-1)` on `(0, 1]` for `j >= 1`, and `integral_0^1 s^(-1/3) = 3/2`
+  for `j = 0`;
+- `integral_1^oo |Delta| k^(-4/3)` is bounded by `integral_0^oo |Delta|`.
+
+Each is a finite sum of products `integral |t|^n e^(-a t^2) dt = Gamma((n+1)/2) a^(-(n+1)/2)`, bounded above in
+rationals: `sqrt(pi) < sqrt(355/113)`, `pi > 333/106`, and `isqrt` bounds. Finally
+`eps = (1/3)|S^(d-1)| * [3 Delta_0 + 3 Delta'_w + Delta_int]`, with `|S^1| <= 2 * 355/113` and `|S^2| <= 4 * 355/113`. It
+bounds every frame, and therefore the average over `u`.
+
+## 6. Numbers (`RESULTS.json`)
+
+| `d` | `L0` | `t` | `min a_i` | `max eta_i` | `3 Delta_0` | `3 Delta'_w` | `Delta_int` | `eps` |
+|---|---|---|---|---|---|---|---|---|
+| `1` | `10` | `1` | `0.75` | `9.4e-16` | `1.3e-14` | `2.0e-13` | `1.7e-15` | `1.41e-13` |
+| `2` | `10` | `1/2` | `0.125` | `1.9e-15` | `3.4e-14` | `9.6e-13` | `8.9e-15` | `2.09e-12` |
+| `3` | `10` | `0.3675` | `0.0919` | `2.8e-15` | `5.3e-13` | `2.3e-11` | `2.2e-13` | `1.00e-10` |
+
+At `L0 = 24` every radius and bound is smaller by a factor of about `1e-101`. `H` has 9, 50 and 1006 monomials in `d = 1, 2, 3`.
+
+**Slack.** The bound is conservative. Controls with explicit perturbations (§7) show it exceeds the true change of `c2` by
+factors of `1e4` (`d = 1`) to `1e8` (`d = 3`). The causes are ball radii that grow through elimination and Isserlis without
+cancellation, the absolute moments, and the cone enlarged to the whole space. The true torus change is of order `e^(-L^2/2)` times polynomial factors, so none of this matters for the conclusions.
+
+## 7. Verification and controls
+
+- `python3 -B -S transfer.py --check` recomputes `reference_checks` and all 12 bounds and compares them with `RESULTS.json`
+  exactly. It takes about two minutes, mostly the `d = 3` ball Isserlis step, and passes identically under `-B -O -S`.
+- **Five seeded defects must each exit 1:**
+  - `no-image`: the image sums are dropped;
+  - `no-theta`: the normalization term is dropped;
+  - `drop-ratio`: the transverse determinant ratio is dropped;
+  - `eta-half`: the exponent radius is halved;
+  - `fp-const`: the finite-part constant `1` replaces `3`.
+- **`controls.py`** (exact and floating; not part of the certificate). It takes about 13 seconds:
+  - **Lemma E**, exact: six anisotropic kernels (§4).
+  - **Reference:** a floating integrator applied to [C2X]'s exact data reproduces `c2` inside [C2X]'s certified intervals
+    in `d = 1, 2, 3`.
+  - **Falsification of Lemmas F, G and P at a measurable scale.** For the even isotropic mixtures
+    `K = (1 - eps) phi + eps phi(sqrt s .)`, with `s = 2` or `1/2` and `eps = 1e-3` or `1e-5`:
+    - `c2[K]` is computed directly, by the exact derivation with `K`'s jets and then closed-form floating integration;
+    - the bound of `bound_core`, fed with the balls `(phi-jet, |K-jet - phi-jet|)`, must exceed `|c2[K] - c2[phi]|`;
+    - all 12 cases pass. The true changes are `1e-7` to `1e-3`.
+
+## 8. What this does not do
+
+- **The definition and identification are consumed.**
+  - **Candidate density.** That `c2[K_L]` is its `ell^(1/3)` coefficient is [T]'s claim: Math-#218, a candidate theorem
+    for `d >= 2`, merged at `fb6ee97`.
+  - **Elder density.** The identification is [E3]'s (Math-#220), a theorem merged at `0d79778` (`PROOF.md` blob
+    `c8767dde`, byte-identical to the consumed head `70dcf31`). It consumes #207 (merged at `566b1a1`, `PROOF.md` blob
+    `f6df5a73`), #218 (merged at `fb6ee97`, `PROOF.md` blob `70ca57ef`) and the merged #187. The complete chain
+    #191/#198 → #207 → #218 → #220 is now on `main`.
+  - Neither identification is reviewed here, and the transfer of the fixed-cone expression does not depend on either
+    (review 5383099127, item 2).
+  - So is the typed replacement `A_r = A~_r + O(r^3)` ([D] §2.2) on the torus. Lemma E and Theorem C2T are statements
+    about the surrogate.
+- **`d = 1, 2, 3` only;** the pipeline runs unchanged in `d = 4` given time. Lattices other than `L Z^d` are not treated,
+  although Lemma J only needs `|p_n| >= L|n|`-type counts.
+- **Conservative constants** (§6). They are not estimates of the true torus deviation.
+- **No review** of [C2X], [T], [D], [SIDE24], [CU], [E3], [L'], [FJ] or Math-#224. Same GitHub account as every lane; zero
+  organizational-independence credit. **I will not merge.**
+
+## 9. Provenance
+
+- **Sources** (`SOURCE_MAP.json`). The workflow verifies these on the checked-out tree:
+  - [C2X] `frontiers/c2_exact_20261001/{exact.py, pseries.py, NOTE.md, RESULTS.json}` on `main` (since `f9ebba1`; unchanged at `7fe06b0`). `exact.py` and
+    `pseries.py` are also carried byte-identically in this directory.
+  - [SIDE24] `coefficients/side24_v1/PROOF.md`.
+  - [T] Math-#218 `frontiers/candidate_third_order_20261001/PROOF.md` (blob `70ca57ef`), on `main` since `fb6ee97` and
+    byte-identical to the consumed head `0cf048d`.
+  - [CU] Math-#219 `frontiers/cusp_coefficient_certified_20261001/RESULTS.json` (blob `8bbd0b0c`), on `main` since
+    `5a836a8` and byte-identical to the consumed head `7a04873`. The `c1/c` interval is read from this file.
+  - [E3] Math-#220 `frontiers/elder_third_order_20261001/PROOF.md` (blob `c8767dde`), on `main` since `0d79778` and
+    byte-identical to the consumed head `70dcf31`. Cited for the elder identification only.
+  - [CT] Math-#224 `frontiers/cusp_torus_transfer_20261001/PROOF.md` (blob `de3d85fd`), on `main` since `d3a0b6c` and
+    byte-identical to the consumed head `a5e69ee`. Consumed for (T1); cited as the `c1` analogue.
+- **Recorded and not checked on this tree:**
+  - [CU] Math-#219 `NOTE.md` at `7a04873` (blob `7dccfa97`). On `main` it has since changed by wording only (v1.4–v1.5);
+    the certified values are in the unchanged `RESULTS.json`.
+  - [D] Math-#216 `NOTE.md` (blob `aa078a9c`);
+  - [L'] Math-#227 at `2d1ec7c`: `NOTE.md` (blob `138521e8`), cited only.
+- **On `main` since `7fe06b0`, cited only:** [FJ] Math-#232
+  `frontiers/c2_finite_jet_transfer_20261001/PROOF.md` (blob `54cc4a1a`).
+- **Files:**
+  - the certificate: `transfer.py`, `ball.py`, `exact.py`, `pseries.py`;
+  - `controls.py`;
+  - `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`.
diff --git a/frontiers/c2_torus_transfer_20261001/README.md b/frontiers/c2_torus_transfer_20261001/README.md
new file mode 100644
index 000000000..68eec096f
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/README.md
@@ -0,0 +1,19 @@
+# `c2` on the finite torus: transfer for every `L >= 10` (Math-#218 (0.1), against `frontiers/c2_exact_20261001`)
+
+`CL-C2-TORUS-TRANSFER-20261001-v1`. Author-side; scientific effect NONE. See `NOTE.md`.
+
+For the normalized periodic Gaussian kernel `K_L` (SIDE24 is `L = 24`), every orthonormal frame and every real `L >= L0`,
+`|c2[K_L] - c2[phi]| <= eps_d(L0)`:
+
+| `d` | `L >= 10` | `L >= 24` |
+|---|---|---|
+| `1` | `1.41035e-13` | `4.09547e-114` |
+| `2` | `2.09404e-12` | `5.20139e-113` |
+| `3` | `1.00039e-10` | `2.09418e-111` |
+
+Lemma E: for every smooth stationary kernel the fixed-cone surrogate has an even finite-order expansion in `r` at fixed
+`b - k r^3/2`, so it has no `r^1` term and its `r^3` coefficient is `-(k/2) d_b A_0`. For real-analytic kernels, which
+include `phi` and `K_L`, it is exactly `A~(r^2, b - k r^3/2, k)`.
+
+Replay: `python3 -B -S transfer.py --check` (exact rational ball arithmetic, standard library only; about two minutes).
+Controls: `python3 -B -S controls.py`.
diff --git a/frontiers/c2_torus_transfer_20261001/RESULTS.json b/frontiers/c2_torus_transfer_20261001/RESULTS.json
new file mode 100644
index 000000000..919a567b2
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/RESULTS.json
@@ -0,0 +1,226 @@
+{
+ "bounds": {
+  "d1_L10": {
+   "Delta0": "4.44080e-15",
+   "Delta_int": "1.68784e-15",
+   "L0": 10,
+   "a_min": "7.50000e-1",
+   "d": 1,
+   "dDelta_w": "6.55138e-14",
+   "e_kappa": "8.05025e-17",
+   "eps": "1.41035e-13",
+   "eta_max": "9.40524e-16",
+   "kappa_up": "8.77514e-2",
+   "max_jet_order": 20,
+   "max_radius_H": "3.13354e-13",
+   "n_monomials_H": 9,
+   "rel": "6.13074e-13",
+   "t_diag": "1.00000e+0",
+   "theta1": "3.85750e-22"
+  },
+  "d1_L12": {
+   "Delta0": "4.54251e-24",
+   "Delta_int": "1.71392e-24",
+   "L0": 12,
+   "a_min": "7.50000e-1",
+   "d": 1,
+   "dDelta_w": "6.62356e-23",
+   "e_kappa": "6.27881e-26",
+   "eps": "1.42699e-22",
+   "eta_max": "7.39127e-25",
+   "kappa_up": "8.77514e-2",
+   "max_jet_order": 20,
+   "max_radius_H": "3.04757e-22",
+   "n_monomials_H": 9,
+   "rel": "6.20309e-22",
+   "t_diag": "1.00000e+0",
+   "theta1": "1.07604e-31"
+  },
+  "d1_L16": {
+   "Delta0": "1.82806e-47",
+   "Delta_int": "6.83459e-48",
+   "L0": 16,
+   "a_min": "7.50000e-1",
+   "d": 1,
+   "dDelta_w": "2.62661e-46",
+   "e_kappa": "1.57599e-49",
+   "eps": "5.66439e-46",
+   "eta_max": "1.87017e-48",
+   "kappa_up": "8.77514e-2",
+   "max_jet_order": 20,
+   "max_radius_H": "1.14804e-45",
+   "n_monomials_H": 9,
+   "rel": "2.46230e-45",
+   "t_diag": "1.00000e+0",
+   "theta1": "5.14442e-56"
+  },
+  "d1_L24": {
+   "Delta0": "1.33798e-115",
+   "Delta_int": "4.96197e-116",
+   "L0": 24,
+   "a_min": "7.50000e-1",
+   "d": 1,
+   "dDelta_w": "1.89740e-114",
+   "e_kappa": "5.56180e-118",
+   "eps": "4.09547e-114",
+   "eta_max": "6.64022e-117",
+   "kappa_up": "8.77514e-2",
+   "max_jet_order": 20,
+   "max_radius_H": "7.90074e-114",
+   "n_monomials_H": 9,
+   "rel": "1.78030e-113",
+   "t_diag": "1.00000e+0",
+   "theta1": "1.67579e-125"
+  },
+  "d2_L10": {
+   "Delta0": "1.13884e-14",
+   "Delta_int": "8.94035e-15",
+   "L0": 10,
+   "a_min": "1.25000e-1",
+   "d": 2,
+   "dDelta_w": "3.18908e-13",
+   "e_kappa": "1.81351e-16",
+   "eps": "2.09404e-12",
+   "eta_max": "1.88105e-15",
+   "kappa_up": "3.93992e-3",
+   "max_jet_order": 20,
+   "max_radius_H": "6.32063e-13",
+   "n_monomials_H": 50,
+   "rel": "9.45285e-12",
+   "t_diag": "5.00000e-1",
+   "theta1": "7.71500e-22"
+  },
+  "d2_L12": {
+   "Delta0": "1.12587e-23",
+   "Delta_int": "8.66171e-24",
+   "L0": 12,
+   "a_min": "1.25000e-1",
+   "d": 2,
+   "dDelta_w": "3.07836e-22",
+   "e_kappa": "1.37156e-25",
+   "eps": "2.02307e-21",
+   "eta_max": "1.47826e-24",
+   "kappa_up": "3.93992e-3",
+   "max_jet_order": 20,
+   "max_radius_H": "6.12547e-22",
+   "n_monomials_H": 50,
+   "rel": "9.13248e-21",
+   "t_diag": "5.00000e-1",
+   "theta1": "2.15208e-31"
+  },
+  "d2_L16": {
+   "Delta0": "4.34211e-47",
+   "Delta_int": "3.25673e-47",
+   "L0": 16,
+   "a_min": "1.25000e-1",
+   "d": 2,
+   "dDelta_w": "1.15187e-45",
+   "e_kappa": "3.32413e-49",
+   "eps": "7.57841e-45",
+   "eta_max": "3.74033e-48",
+   "kappa_up": "3.93992e-3",
+   "max_jet_order": 20,
+   "max_radius_H": "2.30056e-45",
+   "n_monomials_H": 50,
+   "rel": "3.42103e-44",
+   "t_diag": "5.00000e-1",
+   "theta1": "1.02889e-55"
+  },
+  "d2_L24": {
+   "Delta0": "3.06010e-115",
+   "Delta_int": "2.24309e-115",
+   "L0": 24,
+   "a_min": "1.25000e-1",
+   "d": 2,
+   "dDelta_w": "7.89749e-114",
+   "e_kappa": "1.14043e-117",
+   "eps": "5.20139e-113",
+   "eta_max": "1.32805e-116",
+   "kappa_up": "3.93992e-3",
+   "max_jet_order": 20,
+   "max_radius_H": "1.58088e-113",
+   "n_monomials_H": 50,
+   "rel": "2.34800e-112",
+   "t_diag": "5.00000e-1",
+   "theta1": "3.35158e-125"
+  },
+  "d3_L10": {
+   "Delta0": "1.78082e-13",
+   "Delta_int": "2.24526e-13",
+   "L0": 10,
+   "a_min": "9.18862e-2",
+   "d": 3,
+   "dDelta_w": "7.70788e-12",
+   "e_kappa": "3.14352e-16",
+   "eps": "1.00039e-10",
+   "eta_max": "2.82157e-15",
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+   "max_radius_H": "1.90639e-12",
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+   "rel": "6.20455e-10",
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+   "theta1": "1.15725e-21"
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+   "Delta_int": "2.07220e-22",
+   "L0": 12,
+   "a_min": "9.18862e-2",
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+   "Delta_int": "7.32175e-46",
+   "L0": 16,
+   "a_min": "9.18862e-2",
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+   "dDelta_w": "2.48771e-44",
+   "e_kappa": "5.34636e-49",
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+   "eta_max": "5.61049e-48",
+   "kappa_up": "7.05724e-5",
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+   "rel": "2.00794e-42",
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+   "Delta_int": "4.76136e-114",
+   "L0": 24,
+   "a_min": "9.18862e-2",
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+   "dDelta_w": "1.60666e-112",
+   "e_kappa": "1.76946e-117",
+   "eps": "2.09418e-111",
+   "eta_max": "1.99207e-116",
+   "kappa_up": "7.05724e-5",
+   "max_jet_order": 20,
+   "max_radius_H": "4.74408e-113",
+   "n_monomials_H": 1006,
+   "rel": "1.29885e-110",
+   "t_diag": "3.67545e-1",
+   "theta1": "5.02737e-125"
+  }
+ },
+ "object": "CL-C2-TORUS-TRANSFER-20261001-v1",
+ "reference_checks": {
+  "1": true,
+  "2": true,
+  "3": true
+ }
+}
diff --git a/frontiers/c2_torus_transfer_20261001/SOURCE_FILES.json b/frontiers/c2_torus_transfer_20261001/SOURCE_FILES.json
new file mode 100644
index 000000000..b7506522d
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/SOURCE_FILES.json
@@ -0,0 +1,59 @@
+{
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+  },
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diff --git a/frontiers/c2_torus_transfer_20261001/SOURCE_MAP.json b/frontiers/c2_torus_transfer_20261001/SOURCE_MAP.json
new file mode 100644
index 000000000..5365bc2d8
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/SOURCE_MAP.json
@@ -0,0 +1,142 @@
+{
+ "object": "CL-C2-TORUS-TRANSFER-20261001-v1",
+ "delivered_under": "explicit owner instructions of Dylan Roy in Claude Code session session_015wNj8LPTKXsaT68G3DgPPh, given after OWNER_STOP (2026-09-27): 2026-09-28 \"you have my full permisions and approval to make the proper changes and uploads\"; 2026-09-30 \"Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know. Ultra tokens, ultra depth @GitHub @Superpowers\" (repeated 2026-09-30 as \"Continue to Work on closing ...\"). Pull requests only, never merged by their author; no register, GRAPH, STATUS, lemma_closed, prize or premise edits. OWNER_STOP.md addresses Cursor agents and automations and is untouched.",
+ "task": "Math-#223 comment 5933574379 (task and the finite-part starting inequality), main#229 comment 5933485698 (listed as unsupplied); scope claim Math-#223 comment 5934236720",
+ "sources": [
+  {
+   "label": "[C2X-code]",
+   "path": "frontiers/c2_exact_20261001/exact.py",
+   "ref": "main f9ebba1",
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+  },
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+   "ref": "main f9ebba1",
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+  },
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+   "ref": "main f9ebba1",
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+  },
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+   "path": "frontiers/c2_exact_20261001/RESULTS.json",
+   "ref": "main f9ebba1",
+   "consumed": "the certified enclosures of c2 (lower endpoints used for the relative bounds; C2_REF_LO in transfer.py)",
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+  },
+  {
+   "label": "[SIDE24]",
+   "path": "coefficients/side24_v1/PROOF.md",
+   "ref": "main f9ebba1",
+   "consumed": "the normalized periodic covariance K_24 and the enclosures of c_(d,24) (corollary quotients)",
+   "current_required": true,
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+  },
+  {
+   "label": "[T]",
+   "path": "frontiers/candidate_third_order_20261001/PROOF.md",
+   "ref": "Math-#218, merged at fb6ee97 (byte-identical to the consumed head 0cf048d)",
+   "consumed": "(0.1): the expression c2; Lemma F and Theorem T's identification of it as the candidate-density ell^(1/3) coefficient for d >= 2 (candidate theorem; merged at fb6ee97, which is not a status promotion; cited, not reviewed)",
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+  },
+  {
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+   "path": "frontiers/third_order_coefficient_20261001/NOTE.md",
+   "ref": "Math-#216 head 04c08e1 (unmerged)",
+   "consumed": "section 2.2: the typed indicator replaced by 1{T < 0} at O(r^3) (consumed premise, as in [C2X])",
+   "current_required": false,
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+  },
+  {
+   "label": "[CT]",
+   "path": "frontiers/cusp_torus_transfer_20261001/PROOF.md",
+   "ref": "Math-#224, merged at d3a0b6c (blob byte-identical to the consumed head a5e69ee)",
+   "consumed": "(T1): the normalized periodic kernel for every L; the analogous c1 transfer (cited)",
+   "current_required": true,
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+  },
+  {
+   "label": "[CU]",
+   "path": "frontiers/cusp_coefficient_certified_20261001/NOTE.md",
+   "ref": "Math-#219 head 7a04873 (merged at 5a836a8; NOTE.md changed there by wording only, so this NOTE pin is historical)",
+   "consumed": "the certified c1/c enclosure quoted in NOTE section 0 (consumed, not accepted)",
+   "current_required": false,
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+  },
+  {
+   "label": "[CU-results]",
+   "path": "frontiers/cusp_coefficient_certified_20261001/RESULTS.json",
+   "ref": "Math-#219, merged at 5a836a8 (RESULTS.json byte-identical to the consumed head 7a04873)",
+   "consumed": "the stored c1/c interval (consumed, not accepted)",
+   "current_required": true,
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+  },
+  {
+   "label": "[E3]",
+   "path": "frontiers/elder_third_order_20261001/PROOF.md",
+   "ref": "Math-#220, merged at 0d79778 (blob byte-identical to the consumed head 70dcf31); consumes #207 (merged at 566b1a1), #218 (merged at fb6ee97) and merged #187",
+   "consumed": "the identification of c2 as the elder-density ell^(1/3) coefficient (cited, not reviewed)",
+   "current_required": true,
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+   "git_blob": "c8767ddecf1041d234f685e95af582073a7be393"
+  },
+  {
+   "label": "[L']",
+   "path": "frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md",
+   "ref": "Math-#227 head 2d1ec7c (unmerged)",
+   "consumed": "cited only: the open r^1 question that Lemma E answers for the surrogate; #227's beta r allowance is unaffected",
+   "current_required": false,
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+   "git_blob": "138521e82d6c4088a61b97a89020164dbbae6c40"
+  },
+  {
+   "label": "[FJ]",
+   "path": "frontiers/c2_finite_jet_transfer_20261001/PROOF.md",
+   "ref": "main 7fe06b0 (merged Math-#232)",
+   "consumed": "cited only: the general eight-jet stability interface; compatible, neither supersedes the other",
+   "current_required": true,
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+  }
+ ],
+ "copies": {
+  "exact.py": "frontiers/c2_exact_20261001/exact.py",
+  "pseries.py": "frontiers/c2_exact_20261001/pseries.py"
+ },
+ "note": "current_required entries are verified byte-for-byte on the checked-out tree by the workflow, and the copies against them; the unmerged pins record the exact bytes read and are not checked against the tree."
+}
diff --git a/frontiers/c2_torus_transfer_20261001/ball.py b/frontiers/c2_torus_transfer_20261001/ball.py
new file mode 100644
index 000000000..72205f8ad
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/ball.py
@@ -0,0 +1,158 @@
+"""Rational ball arithmetic and rational bounds for CL-C2-TORUS-TRANSFER-20261001 (NOTE.md section 2).  Standard library.
+
+A ball B(c, r) is the set [c - r, c + r] with c, r rational, r >= 0.  Centers are carried EXACTLY: with every radius
+zero the arithmetic is that of Fractions, so a ball pipeline reproduces the exact reference derivation in its centers.
+Radii are rounded UP to dyadic rationals with RB significant bits after each operation, which only enlarges the ball.
+Every operation returns a ball containing every result of the operation applied to members of the operand balls."""
+from fractions import Fraction as Fr
+import math
+
+RB = 64
+
+
+def rup(x):
+    """smallest dyadic m / 2^e >= x with m < 2^RB (x >= 0 rational)"""
+    x = Fr(x)
+    if x <= 0:
+        if x < 0:
+            raise ValueError('negative radius')
+        return Fr(0)
+    e = RB - (x.numerator.bit_length() - x.denominator.bit_length())
+    if e >= 0:
+        m = -((-x.numerator << e) // x.denominator)
+        return Fr(m, 1 << e)
+    m = -((-x.numerator) // (x.denominator << -e))
+    return Fr(m << -e)
+
+
+class B:
+    __slots__ = ('c', 'r')
+
+    def __init__(self, c, r=0):
+        self.c = Fr(c)
+        self.r = rup(r)
+
+    def __bool__(self):
+        return bool(self.c) or bool(self.r)
+
+    def __add__(self, o):
+        if isinstance(o, B):
+            return B(self.c + o.c, self.r + o.r)
+        return B(self.c + o, self.r)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return B(-self.c, self.r)
+
+    def __sub__(self, o):
+        return self + (-o)
+
+    def __rsub__(self, o):
+        return (-self) + o
+
+    def __mul__(self, o):
+        if isinstance(o, B):
+            return B(self.c * o.c, abs(self.c) * o.r + abs(o.c) * self.r + self.r * o.r)
+        o = Fr(o)
+        return B(self.c * o, abs(o) * self.r)
+    __rmul__ = __mul__
+
+    def inv(self):
+        a = abs(self.c)
+        if not a > self.r:
+            raise ZeroDivisionError('ball contains 0')
+        return B(1 / self.c, self.r / (a * (a - self.r)))
+
+    def __truediv__(self, o):
+        if isinstance(o, B):
+            return self * o.inv()
+        return self * (1 / Fr(o))
+
+    def __rtruediv__(self, o):
+        return self.inv() * o
+
+    def mag(self):
+        """upper bound of |x| on the ball"""
+        return abs(self.c) + self.r
+
+    def __repr__(self):
+        return 'B(%s, %s)' % (float(self.c), float(self.r))
+
+
+def center(x):
+    return x.c if isinstance(x, B) else Fr(x)
+
+
+def radius(x):
+    return x.r if isinstance(x, B) else Fr(0)
+
+
+def mag(x):
+    return x.mag() if isinstance(x, B) else abs(Fr(x))
+
+
+def nonzero_center(x):
+    return center(x) != 0
+
+
+# ---------------------------------------------------------------------------------------------- rational bounds
+PI_UP = Fr(355, 113)            # pi < 355/113
+PI_DN = Fr(333, 106)            # 333/106 < pi
+
+
+def exp_up(x):
+    """upper bound of e^x for rational x (exact partial sums: e^x >= sum_{j<=J} x^j/j! for x >= 0)"""
+    x = Fr(x)
+    if x <= 0:
+        # e^-y = (e^(-y/n))^n, n = 2^m with y/n <= 1;  e^(-t) <= 1/sum_{j<=30} t^j/j!;  each squaring rounds up
+        y = -x
+        n, m = 1, 0
+        while y / n > 1:
+            n, m = 2 * n, m + 1
+        t = y / n
+        s, term = Fr(0), Fr(1)
+        for j in range(31):
+            s += term
+            term = term * t / (j + 1)
+        v = rup(1 / s)
+        for _ in range(m):
+            v = rup(v * v)
+        return v
+    # e^x = 1 / e^-x  <=  1 / (lower bound of e^-x);  e^-x >= alternating partial sum ending on a negative term ... use
+    # e^x = (e^(x/n))^n with x/n <= 1/2 and e^t <= 1/(1 - t - t^2/2 ... ) avoided: e^t <= sum_{j<=J} t^j/j! + tail
+    n = 1
+    while x / n > Fr(1, 2):
+        n *= 2
+    t = x / n
+    J = 30
+    s, term = Fr(0), Fr(1)
+    for j in range(J + 1):
+        s += term
+        term = term * t / (j + 1)
+    s += 2 * term                  # tail sum_{j>J} t^j/j! <= 2 t^(J+1)/(J+1)! for t <= 1/2
+    return rup(s ** n)
+
+
+def exp_dn(x):
+    """lower bound of e^x"""
+    return 1 / exp_up(-Fr(x))
+
+
+def isqrt_frac_up(x, bits=80):
+    """upper bound of sqrt(x), x >= 0 rational"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    sc = 1 << (2 * bits)
+    n = -((-x.numerator * sc) // x.denominator)
+    s = math.isqrt(n)
+    if s * s < n:
+        s += 1
+    return Fr(s, 1 << bits)
+
+
+def isqrt_frac_dn(x, bits=80):
+    x = Fr(x)
+    sc = 1 << (2 * bits)
+    n = (x.numerator * sc) // x.denominator
+    return Fr(math.isqrt(n), 1 << bits)
diff --git a/frontiers/c2_torus_transfer_20261001/controls.py b/frontiers/c2_torus_transfer_20261001/controls.py
new file mode 100644
index 000000000..863023c25
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/controls.py
@@ -0,0 +1,199 @@
+"""Floating controls for CL-C2-TORUS-TRANSFER-20261001 (NOTE.md section 6; not part of the certificate).
+
+Control 1 (reference).  The floating integrator below, applied to the exact Gaussian-kernel data of c2_exact, returns c2
+inside the certified intervals of frontiers/c2_exact_20261001/RESULTS.json.
+
+Control 0 (Lemma E, exact).  For anisotropic product kernels prod_i exp(-s_i z_i^2/2) (rational jets; not isotropic),
+the exact integrand of the fixed-cone surrogate has G_0 = G_1 = G_3 = 0 and G_5 = -(k/2)(d_b G_2 - (d_b E) G_2): the
+identities that A~_r = A~(r^2, b - k r^3/2, k) implies.  These are exact identities of rational polynomials.
+
+Control 2 (falsification of Lemmas F, G, P at a measurable scale).  For the isotropic, even, normalized mixture kernels
+    K_eps(z) = (1 - eps) exp(-|z|^2/2) + eps exp(-s |z|^2/2),     s = 2 or 1/2,
+c2[K_eps] is computed directly: c2_exact's exact derivation with the jets of K_eps (rational), then the floating
+integration in closed form (k: Gamma finite parts; b: completing the square; cone: half-line / quadrant moments).  The
+certified bound of transfer.bound_core, fed with balls (phi-jet, |K_eps-jet - phi-jet|), must exceed |c2[K_eps] - c2[phi]|.
+Both the bound and the difference are linear in eps for small eps; their ratio is the slack of the bound.
+
+    python3 -B -S controls.py"""
+from fractions import Fraction as Fr
+import math
+import os
+import sys
+
+sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
+import pseries as P          # noqa: E402
+import exact as X            # noqa: E402
+import transfer as T         # noqa: E402
+from ball import B           # noqa: E402
+
+GAUSS_RHO = P.rho_der
+C2_INTERVALS = {1: ('0.2300445802661503', '0.2300445802661998'), 2: ('0.2215244106266632', '0.2215244106267110'),
+                3: ('0.1612340491269447', '0.1612340491269810')}
+
+
+def mix_rho(eps, s):
+    def rho(g):
+        v = GAUSS_RHO(g)
+        return v * ((1 - eps) + eps * s ** (sum(g) // 2)) if v else 0
+    return rho
+
+
+def half_moment_f(j, c):
+    """int_0^oo t^j e^(-c t^2) dt"""
+    return math.gamma((j + 1) / 2) * c ** (-(j + 1) / 2) / 2
+
+
+def quadrant_moments_f(al, be, ga, nmax):
+    D = 4 * al * ga - be * be
+    M = {(0, 0): (math.pi / 2 - math.atan(be / math.sqrt(D))) / math.sqrt(D)}
+    for n in range(nmax):
+        for i in range(n + 1):
+            j = n - i
+            R1 = M[(i - 1, j)] * i if i else half_moment_f(j, ga)
+            R2 = M[(i, j - 1)] * j if j else half_moment_f(i, al)
+            M.setdefault((i + 1, j), (R1 * 2 * ga - R2 * be) / D)
+            M.setdefault((i, j + 1), (R2 * 2 * al - R1 * be) / D)
+    return M
+
+
+def c2_float(d, data):
+    """c2 = (|S^(d-1)|/3) const int db f.p. int dk k^(-4/3) int_cone exp(-E) H w  for isotropic-kernel data of
+    exact.kernel_data (rotation-reduced in d = 3), with general exponent coefficients."""
+    G = data['G']
+    H = G.rcoef(4)
+    E = (data['q0'] + data['Q0']) * Fr(1, 2)
+    ak = float(E.coef_of('k', 2).scalar())
+    if not (E.coef_of("k", 1).is_zero() and E.degree("k") == 2):
+        raise ArithmeticError("unexpected k-dependence of the exponent")
+    # k: f.p. int_0^oo k^(j - 4/3) e^(-ak k^2) dk = Gamma((j - 1/3)/2) ak^(-(j - 1/3)/2) / 2
+    Hk = {}
+    for e, c in H.t.items():
+        j = e[2]
+        a = (j - Fr(1, 3)) / 2
+        val = float(c) * math.gamma(float(a)) * ak ** (-float(a)) / 2
+        key = (e[1],) + e[3:]
+        Hk[key] = Hk.get(key, 0.0) + val
+    Ek = E.coef_of('k', 0)
+    ab = float(Ek.coef_of('b', 2).scalar())
+    Lx, Rx = Ek.coef_of('b', 1), Ek.coef_of('b', 0)
+    # complete the square: b -> b - Lx/(2 ab); expand (b - l)^n with l a linear form in x (floats)
+    lin = {e[3:]: float(c) / (2 * ab) for e, c in Lx.t.items()}         # l(x) = sum lin[e] x^e
+    Hb = {}
+    for key, val in Hk.items():
+        n, xe = key[0], key[1:]
+        # (b - l)^n = sum_m C(n, m) b^m (-l)^(n-m); keep even m
+        lpow = {(0, 0): 1.0}
+        powers = [dict(lpow)]
+        for _ in range(n):
+            nxt = {}
+            for e1, c1 in powers[-1].items():
+                for e2, c2 in lin.items():
+                    e = (e1[0] + e2[0], e1[1] + e2[1])
+                    nxt[e] = nxt.get(e, 0.0) + c1 * c2
+            powers.append(nxt)
+        for m in range(0, n + 1, 2):
+            bm = 2 * half_moment_f(m, ab)
+            for e, c in powers[n - m].items():
+                kk = (xe[0] + e[0], xe[1] + e[1])
+                Hb[kk] = Hb.get(kk, 0.0) + val * math.comb(n, m) * bm * c * (-1) ** (n - m)
+    Ex = Rx - Lx * Lx * (Fr(1) / (4 * Ek.coef_of('b', 2).scalar()))
+    total = 0.0
+    if d == 1:
+        total = sum(Hb.values())
+    elif d == 2:
+        g = float(Ex.coef_of('x1', 2).scalar())
+        for key, val in Hb.items():
+            total += val * half_moment_f(key[0], g) * (-1) ** key[0]
+    else:
+        al = float(Ex.coef_of('x1', 2).scalar())
+        ga = float(Ex.coef_of('x2', 2).scalar())
+        bx = float(Ex.coef_of('x1', 1).coef_of('x2', 1).scalar())
+        # x1 = -s, x2 = -s - t:  al s^2 + bx s (s + t) + ga (s + t)^2 ; weight x1 - x2 = t
+        A_ = al + bx + ga
+        Bt = bx + 2 * ga
+        C_ = ga
+        nmax = max(k[0] + k[1] for k in Hb) + 3
+        M = quadrant_moments_f(A_, Bt, C_, nmax)
+        for key, val in Hb.items():
+            i, j = key
+            # x1^i x2^j = (-s)^i (-s - t)^j
+            for m in range(j + 1):
+                coef = math.comb(j, m) * (-1) ** (i + j)
+                total += val * coef * M[(i + m, j - m + 1)]
+    nn = data['p'] + (0 if d == 1 else (1 if d == 2 else 3))
+    const = -12 * (2 * math.pi) ** (-nn / 2) / math.sqrt(float(data['detS0'] * data['detT0']))
+    if d == 3:
+        const *= math.pi
+    sphere = {1: 2, 2: 2 * math.pi, 3: 4 * math.pi}[d]
+    return sphere / 3 * const * total
+
+
+def aniso_rho(sv):
+    def rho(g):
+        v = GAUSS_RHO(g)
+        if not v:
+            return Fr(0)
+        out = Fr(v)
+        for gi, si in zip(g, sv):
+            out *= si ** (gi // 2)
+        return out
+    return rho
+
+
+def evenness_checks():
+    fails = 0
+    nr0 = T.NR
+    T.NR = 5                                       # series exact through r^5 at pseries.R = 8
+    try:
+        for d in (1, 2, 3):
+            for sv in ([Fr(1), Fr(3, 2), Fr(5, 4)], [Fr(2), Fr(1, 3), Fr(7, 5)]):
+                D = T.kernel(d, aniso_rho(sv[:d]))
+                G, E = D['G'], D['E']
+                G2 = G.rcoef(2)
+                rhs5 = (G2.deriv('b') - E.deriv('b') * G2) * T.Poly.var('k') * Fr(-1, 2)
+                ok = G.rcoef(0).is_zero() and G.rcoef(1).is_zero() and G.rcoef(3).is_zero() and \
+                    (G.rcoef(5) - rhs5).is_zero() and not G2.is_zero()
+                fails += not ok
+                print('Lemma E, d=%d, s=%s: G0=G1=G3=0 and G5 = -(k/2)(d_b - E_b)G2: %s'
+                      % (d, [str(x) for x in sv[:d]], ok))
+    finally:
+        T.NR = nr0
+    return fails
+
+
+def run():
+    fails = evenness_checks()
+    out = {}
+    for d in (1, 2, 3):
+        P.rho_der = GAUSS_RHO
+        ref = c2_float(d, X.kernel_data(d))
+        lo, hi = (float(x) for x in C2_INTERVALS[d])
+        ok = lo - 1e-12 <= ref <= hi + 1e-12
+        fails += not ok
+        print('d=%d reference c2 (floating) %.15f  in certified interval: %s' % (d, ref, ok))
+        for s in (Fr(2), Fr(1, 2)):
+            for eps in (Fr(1, 1000), Fr(1, 100000)):
+                P.rho_der = mix_rho(eps, s)
+                c2m = c2_float(d, X.kernel_data(d))
+                P.rho_der = GAUSS_RHO
+                mix = mix_rho(eps, s)
+
+                def jc(g, mix=mix):
+                    if sum(g) % 2:
+                        return B(0)
+                    v = GAUSS_RHO(g)
+                    return B(v, abs(Fr(mix(g)) - v))
+                bd = T.bound_core(d, T.kernel(d, jc))
+                diff = abs(c2m - ref)
+                ok = float(bd['eps']) >= diff
+                fails += not ok
+                out[(d, s, eps)] = (diff, float(bd['eps']))
+                print('  s=%s eps=%s: |c2[K]-c2[phi]| = %.3e   bound = %.3e   slack %.1f   %s'
+                      % (s, eps, diff, float(bd['eps']), float(bd['eps']) / diff if diff else float('inf'),
+                         'ok' if ok else 'FAIL'))
+    print('controls: %d failures' % fails)
+    return fails
+
+
+if __name__ == '__main__':
+    sys.exit(1 if run() else 0)
diff --git a/frontiers/c2_torus_transfer_20261001/exact.py b/frontiers/c2_torus_transfer_20261001/exact.py
new file mode 100644
index 000000000..b06819802
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/exact.py
@@ -0,0 +1,432 @@
+"""Exact derivation of c and c2 for the Gaussian kernel, d = 1, 2, 3 (NOTE.md sections 2-3).  Standard library only.
+
+Polynomials (Poly) in (r, b, k, x1, x2) over Q, truncated at r^NR; closed-form numbers (Num)
+    sum q * pi^(h/2) * sqrt(m) * theta^e,   q rational, h integer, m squarefree, e in {0, 1},
+where theta = M(0, 0) is the one cone atom of d = 3 (section 3.3).  Each coefficient is returned as a Num times one
+Gamma atom:  c = Num * Gamma(1/6) 12^(-1/6),  c2 = Num * Gamma(5/6) 12^(-5/6)."""
+from fractions import Fraction as Fr
+import itertools
+import math
+
+import pseries as P
+
+NR = 6
+VARS = ('r', 'b', 'k', 'x1', 'x2')
+NV = len(VARS)
+ONE = (0,) * NV
+MUTANT = None
+
+
+# ------------------------------------------------------------------------------------------- multivariate polynomials
+class Poly:
+    __slots__ = ('t',)
+
+    def __init__(self, t=None):
+        self.t = {e: c for e, c in (t or {}).items() if c}
+
+    @staticmethod
+    def var(name, power=1):
+        e = [0] * NV
+        e[VARS.index(name)] = power
+        return Poly({tuple(e): Fr(1)})
+
+    @staticmethod
+    def const(c):
+        return Poly({ONE: Fr(c)})
+
+    def __add__(self, o):
+        if not isinstance(o, Poly):
+            o = Poly.const(o)
+        t = dict(self.t)
+        for e, c in o.t.items():
+            t[e] = t.get(e, 0) + c
+        return Poly(t)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return Poly({e: -c for e, c in self.t.items()})
+
+    def __sub__(self, o):
+        return self + (-o if isinstance(o, Poly) else Poly.const(-o))
+
+    def __mul__(self, o):
+        if not isinstance(o, Poly):
+            o = Fr(o)
+            return Poly({e: c * o for e, c in self.t.items()})
+        t = {}
+        for e1, c1 in self.t.items():
+            for e2, c2 in o.t.items():
+                if e1[0] + e2[0] > NR:
+                    continue
+                e = tuple(a + b for a, b in zip(e1, e2))
+                t[e] = t.get(e, 0) + c1 * c2
+        return Poly(t)
+    __rmul__ = __mul__
+
+    def rcoef(self, n):
+        return Poly({(0,) + e[1:]: c for e, c in self.t.items() if e[0] == n})
+
+    def degree(self, name):
+        i = VARS.index(name)
+        return max((e[i] for e in self.t), default=0)
+
+    def coef_of(self, name, p):
+        i = VARS.index(name)
+        return Poly({e[:i] + (0,) + e[i + 1:]: c for e, c in self.t.items() if e[i] == p})
+
+    def scalar(self):
+        if set(self.t) - {ONE}:
+            raise ValueError('not a constant: %r' % (self.t,))
+        return self.t.get(ONE, Fr(0))
+
+    def subs(self, name, poly):
+        i = VARS.index(name)
+        out, powers = Poly(), [Poly.const(1)]
+        for e, c in self.t.items():
+            while len(powers) <= e[i]:
+                powers.append(powers[-1] * poly)
+            out = out + Poly({e[:i] + (0,) + e[i + 1:]: c}) * powers[e[i]]
+        return out
+
+    def is_zero(self):
+        return not self.t
+
+
+def from_series(s):
+    return Poly({(i,) + (0,) * (NV - 1): c for i, c in enumerate(s.c[:NR + 1]) if c})
+
+
+# ------------------------------------------------------------------------------------------- closed-form numbers
+def squarefree(n):
+    q, m, p = 1, 1, 2
+    while p * p <= n:
+        while n % (p * p) == 0:
+            n //= p * p
+            q *= p
+        if n % p == 0:
+            n //= p
+            m *= p
+        p += 1
+    return q, m * n
+
+
+def sqrt_frac(x):
+    x = Fr(x)
+    if x <= 0:
+        raise ValueError('sqrt of a non-positive rational')
+    q, m = squarefree(x.numerator * x.denominator)
+    return Fr(q, x.denominator), m
+
+
+class Num:
+    __slots__ = ('t',)
+
+    def __init__(self, t=None):
+        self.t = {k: v for k, v in (t or {}).items() if v}
+
+    @staticmethod
+    def rat(q):
+        return Num({(0, 1, 0): Fr(q)})
+
+    @staticmethod
+    def sqrt(x):
+        q, m = sqrt_frac(x)
+        return Num({(0, m, 0): q})
+
+    @staticmethod
+    def pi_half(h):
+        return Num({(h, 1, 0): Fr(1)})
+
+    def __add__(self, o):
+        t = dict(self.t)
+        for k, v in o.t.items():
+            t[k] = t.get(k, 0) + v
+        return Num(t)
+
+    def __neg__(self):
+        return Num({k: -v for k, v in self.t.items()})
+
+    def __sub__(self, o):
+        return self + (-o)
+
+    def __eq__(self, o):
+        return isinstance(o, Num) and not (self - o).t
+
+    def __mul__(self, o):
+        if not isinstance(o, Num):
+            return Num({k: v * Fr(o) for k, v in self.t.items()})
+        t = {}
+        for (h1, m1, e1), v1 in self.t.items():
+            for (h2, m2, e2), v2 in o.t.items():
+                g = math.gcd(m1, m2)
+                key = (h1 + h2, (m1 // g) * (m2 // g), e1 + e2)
+                t[key] = t.get(key, 0) + v1 * v2 * g
+        return Num(t)
+    __rmul__ = __mul__
+
+    def text(self):
+        parts = []
+        for (h, m, e), v in sorted(self.t.items()):
+            s = str(v)
+            if m != 1:
+                s += ' sqrt(%d)' % m
+            if h:
+                s += ' pi^(%s)' % Fr(h, 2)
+            if e:
+                s += ' theta' + ('' if e == 1 else '^%d' % e)
+            parts.append(s)
+        return ' + '.join(parts) if parts else '0'
+
+
+# ------------------------------------------------------------------------------------------- the kernel A_r
+def forms_for(d):
+    forms = {}
+    for nm, sg in (('M', -1), ('S', 1)):
+        for i in range(d):
+            for j in range(i, d):
+                be = [0] * d
+                be[i] += 1
+                be[j] += 1
+                forms[(nm, i, j)] = P.deriv_at(tuple(be), sg, d)
+    trans = []
+    for i in range(1, d):
+        for j in range(i, d):
+            be = [0] * d
+            be[i] += 1
+            be[j] += 1
+            forms[('T', i, j)] = {tuple(be): P.S.const(1)}
+            trans.append(('T', i, j))
+    return forms, trans
+
+
+def matchings(lst):
+    if not lst:
+        yield []
+        return
+    f, rest = lst[0], lst[1:]
+    for mm in matchings(rest):
+        yield mm
+    for i, y in enumerate(rest):
+        for mm in matchings(rest[:i] + rest[i + 1:]):
+            yield [(f, y)] + mm
+
+
+def perm_sign(p):
+    return -1 if sum(1 for i in range(len(p)) for j in range(i + 1, len(p)) if p[i] > p[j]) % 2 else 1
+
+
+def series_pow(x, alpha):
+    """(1 + x)^alpha for a Poly x without r^0 part"""
+    out, term, coef = Poly.const(1), Poly.const(1), Fr(1)
+    for n in range(1, NR + 1):
+        coef = coef * (Fr(alpha) - (n - 1)) / n
+        term = term * x
+        out = out + term * coef
+    return out
+
+
+def series_exp(x):
+    out, term = Poly.const(1), Poly.const(1)
+    for n in range(1, NR + 1):
+        term = term * x * Fr(1, n)
+        out = out + term
+    return out
+
+
+def kernel_data(d):
+    """A~_r(b, k) = const * [cone integral of exp(-(q0 + Q0)/2) G w] / r^2  (NOTE.md 2.3; the fixed-cone surrogate), with
+       w = 1 (d = 1), da over a < 0 (d = 2), (x1 - x2) dx over x2 < x1 < 0 (d = 3; the factor pi is in const)."""
+    forms, trans = forms_for(d)
+    pin = P.Pinned(d, forms)
+    p, nt = pin.p, len(trans)
+    r, b, k = Poly.var('r'), Poly.var('b'), Poly.var('k')
+    tgt = 11 if MUTANT == 'target' else 12
+    v = [b - k * r * r * r * Fr(1, 2), -(k * r * r), Poly(), k * tgt] + [Poly()] * (p - 4)
+    mean = {u: sum((from_series(pin.G[u][j]) * v[j] for j in range(p)), Poly()) for u in pin.names}
+    hn = [u for u in pin.names if u not in trans]
+    if nt:
+        CTT = [[pin.C[(x, y)] for y in trans] for x in trans]
+        CTTinv = P.mat_inv(CTT)
+        detT = P.mat_det(CTT)
+        B = {u: [sum((pin.C[(u, trans[j])] * CTTinv[j][i] for j in range(nt)), P.S()) for i in range(nt)] for u in hn}
+        cp = {(u, w): from_series(pin.C[(u, w)] - sum((B[u][i] * pin.C[(trans[i], w)] for i in range(nt)), P.S()))
+              for u in hn for w in hn}
+        tval = {('T', 1, 1): Poly.var('x1'), ('T', 1, 2): Poly(), ('T', 2, 2): Poly.var('x2')}
+        zT = [tval[nm] - mean[nm] for nm in trans]
+        aff = {u: mean[u] + sum((from_series(B[u][i]) * zT[i] for i in range(nt)), Poly()) for u in hn}
+    else:
+        CTTinv, detT, zT = [], P.S.const(1), []
+        cp = {(u, w): from_series(pin.C[(u, w)]) for u in hn for w in hn}
+        aff = dict((u, mean[u]) for u in hn)
+    F = Poly()          # E[det H_M det H_S | T, U_r = v_r] by Isserlis
+    for pp in itertools.permutations(range(d)):
+        for qq in itertools.permutations(range(d)):
+            fac = [('M', min(i, pp[i]), max(i, pp[i])) for i in range(d)] + \
+                  [('S', min(i, qq[i]), max(i, qq[i])) for i in range(d)]
+            sg = perm_sign(pp) * perm_sign(qq)
+            for mt in matchings(list(range(2 * d))):
+                used = set(i for pr in mt for i in pr)
+                c = Poly.const(sg)
+                for (i, j) in mt:
+                    c = c * cp[(fac[i], fac[j])]
+                for i in range(2 * d):
+                    if i not in used:
+                        c = c * aff[fac[i]]
+                F = F + c
+    q = sum((v[i] * from_series(pin.Sinv[i][j]) * v[j] for i in range(p) for j in range(p)), Poly())
+    Q = sum((zT[i] * from_series(CTTinv[i][j]) * zT[j] for i in range(nt) for j in range(nt)), Poly())
+    q0, Q0 = q.rcoef(0), Q.rcoef(0)
+    detS0, detT0 = pin.detS.c[0], detT.c[0]
+    ratio = from_series(pin.detS) * (1 / detS0) * from_series(detT) * (1 / detT0) - Poly.const(1)
+    G = series_pow(ratio, Fr(-1, 2)) * series_exp(((q - q0) + (Q - Q0)) * Fr(-1, 2)) * F
+    nn = p + nt
+    const = Num.rat(-12) * Num.sqrt(Fr(1, 2 ** nn)) * Num.pi_half(-nn) * Num.sqrt(Fr(1) / (detS0 * detT0))
+    if d == 3:
+        const = const * Num.pi_half(2)
+    return {'G': G, 'q0': q0, 'Q0': Q0, 'const': const, 'p': p, 'detS0': detS0, 'detT0': detT0, 'F': F}
+
+
+# ------------------------------------------------------------------------------------------- the integrations
+SPHERE = {1: Num.rat(2), 2: Num.pi_half(2) * 2, 3: Num.pi_half(2) * 4}
+
+
+def half_moment(j, c):
+    """int_0^oo t^j e^(-c t^2) dt = Gamma((j+1)/2) c^(-(j+1)/2)/2 (c > 0 rational)"""
+    if j % 2 == 0:
+        g = Fr(1)
+        for i in range(j // 2):
+            g *= Fr(2 * i + 1, 2)
+        return Num.pi_half(1) * Num.sqrt(Fr(1) / c) * (g / c ** (j // 2) / 2)
+    m = (j + 1) // 2
+    return Num.rat(Fr(math.factorial(m - 1)) / c ** m / 2)
+
+
+def quadrant_moments(al, be, ga, nmax):
+    """M(i, j) = int int_(s, t > 0) s^i t^j exp(-(al s^2 + be s t + ga t^2)) ds dt, i + j <= nmax, from
+    (i)  2 al M(i+1, j) + be M(i, j+1) = i M(i-1, j) + [i = 0] H_j(ga),
+    (ii) be M(i+1, j) + 2 ga M(i, j+1) = j M(i, j-1) + [j = 0] H_i(al)     (H the half-line moments),
+    integration by parts in s and in t.  M(0, 0) = theta = (pi/2 - arctan(be/sqrt(D)))/sqrt(D), D = 4 al ga - be^2.
+    Every M(i+1, j) with j >= 1 is reached twice (from (i, j) and from (i+1, j-1)); the two values must agree."""
+    D = 4 * al * ga - be * be
+    if not (D > 0 and al > 0 and ga > 0):
+        raise ValueError('cone exponent not positive definite')
+    M = {(0, 0): Num({(0, 1, 1): Fr(1)})}
+    for n in range(nmax):
+        for i in range(n + 1):
+            j = n - i
+            R1 = (M[(i - 1, j)] * i if i else half_moment(j, ga))
+            R2 = (M[(i, j - 1)] * j if j else half_moment(i, al))
+            vals = (((i + 1, j), (R1 * (2 * ga) - R2 * be) * (Fr(1) / D)),
+                    ((i, j + 1), (R2 * (2 * al) - R1 * be) * (Fr(1) / D)))
+            for key, val in vals:
+                if key in M:
+                    if not M[key] == val:
+                        raise ArithmeticError('quadrant recurrence inconsistent at %r' % (key,))
+                else:
+                    M[key] = val
+    return M, D
+
+
+def gamma_ratio(a, a0):
+    """Gamma(a)/Gamma(a0) for a - a0 an integer (exact)"""
+    out, x = Fr(1), Fr(a0)
+    while x < a:
+        out *= x
+        x += 1
+    while x > a:
+        x -= 1
+        out /= x
+    return out
+
+
+def coefficient(d, which, data=None):
+    """which = 0: c = (|S^(d-1)|/3) int db int_0^oo A_0 k^(-2/3) dk;
+       which = 2: c2 = (|S^(d-1)|/3) int db f.p. int_0^oo A_2 k^(-4/3) dk  (the finite part = the subtraction of A_2(b, 0)).
+    Returns (Num, Gamma atom, info)."""
+    D_ = data or kernel_data(d)
+    G = D_['G']
+    for n in (0, 1):
+        if not G.rcoef(n).is_zero():
+            raise ArithmeticError('integrand does not vanish to order r^2')
+    if not G.rcoef(3).is_zero():
+        raise ArithmeticError('A_r has an r^1 term')
+    H = G.rcoef(which + 2)
+    E = (D_['q0'] + D_['Q0']) * Fr(1, 2)
+    # --- k: E = ak k^2 + (k-free part)
+    ak = E.coef_of('k', 2).scalar()
+    if not (E.coef_of('k', 1).is_zero() and E.degree('k') == 2 and ak == 12):
+        raise ArithmeticError('unexpected k-dependence of the exponent')
+    a0 = Fr(1, 6) if which == 0 else Fr(5, 6)
+    expo = Fr(-2, 3) if which == 0 else Fr(-4, 3)
+    Hk = Poly()
+    for j in range(H.degree('k') + 1):
+        cj = H.coef_of('k', j)
+        if cj.is_zero():
+            continue
+        if j % 2:
+            raise ArithmeticError('odd power of k')
+        # int_0^oo k^(j + expo) e^(-12 k^2) dk (finite part) = Gamma(a) 12^(-a)/2,  a = (j + expo + 1)/2
+        a = (j + expo + 1) / 2
+        if MUTANT == 'finite-part' and j == 0 and which == 2:
+            a = a0                       # drops the subtraction of A_2(b, 0): Gamma(-1/6) replaced by Gamma(5/6)
+        shift = a - a0
+        Hk = Hk + cj * (gamma_ratio(a, a0) * Fr(12) ** (-int(shift)) / 2)
+    katom = 'Gamma(1/6) 12^(-1/6)' if which == 0 else 'Gamma(5/6) 12^(-5/6)'
+    # --- b: k-free exponent = ab b^2 + b L(x) + R(x); complete the square
+    Ek = E.coef_of('k', 0)
+    ab = Ek.coef_of('b', 2).scalar()
+    if Ek.degree('b') != 2:
+        raise ArithmeticError('exponent not quadratic in b')
+    Lx, Rx = Ek.coef_of('b', 1), Ek.coef_of('b', 0)
+    shifted = Hk.subs('b', Poly.var('b') - Lx * (Fr(1) / (2 * ab)))
+    Hb = {}
+    for e, c in shifted.t.items():
+        if e[1] % 2:
+            continue
+        key = e[3:]
+        Hb[key] = Hb.get(key, Num()) + half_moment(e[1], ab) * (2 * c)
+    if not Hb:
+        raise ArithmeticError('empty integrand: the r-truncation is too low')
+    Ex = Rx - Lx * Lx * (Fr(1) / (4 * ab))
+    total, info = Num(), {}
+    if d == 1:
+        if not Ex.is_zero():
+            raise ArithmeticError('d = 1: residual exponent')
+        for val in Hb.values():
+            total = total + val
+    elif d == 2:
+        g = Ex.coef_of('x1', 2).scalar()
+        if not (Ex.degree('x1') == 2 and Ex.coef_of('x1', 1).is_zero() and Ex.coef_of('x1', 0).is_zero()):
+            raise ArithmeticError('d = 2: cone exponent')
+        for key, val in Hb.items():
+            total = total + val * half_moment(key[0], g) * (-1) ** key[0]       # int over a < 0
+        info['gamma_a'] = g
+    else:
+        X1, X2 = -Poly.var('x1'), -Poly.var('x1') - Poly.var('x2')    # x1 = -s, x2 = -s - t  (s -> x1, t -> x2 slots)
+
+        def subst(poly):
+            out = Poly()
+            for e, c in poly.t.items():
+                term = Poly({e[:3] + (0, 0): c})
+                for _ in range(e[3]):
+                    term = term * X1
+                for _ in range(e[4]):
+                    term = term * X2
+                out = out + term
+            return out
+        Est = subst(Ex)
+        al = Est.coef_of('x1', 2).coef_of('x2', 0).scalar()
+        ga = Est.coef_of('x2', 2).coef_of('x1', 0).scalar()
+        be = Est.coef_of('x1', 1).coef_of('x2', 1).scalar()
+        chk = Poly.var('x1', 2) * al + Poly.var('x1') * Poly.var('x2') * be + Poly.var('x2', 2) * ga
+        if not (Est - chk).is_zero():
+            raise ArithmeticError('d = 3: cone exponent not a quadratic form')
+        nmax = max(key[0] + key[1] for key in Hb) + 2
+        M, Dc = quadrant_moments(al, -be if MUTANT == 'cone-sign' else be, ga, nmax)
+        for key, val in Hb.items():
+            mono = subst(Poly({(0, 0, 0) + key: Fr(1)})) * Poly.var('x2')           # weight x1 - x2 = t
+            for e, c in mono.t.items():
+                total = total + val * M[(e[3], e[4])] * c
+        info.update({'alpha': al, 'beta': be, 'gamma': ga, 'D': Dc})
+    value = SPHERE[d] * Fr(1, 3) * D_['const'] * total
+    return value, katom, info
diff --git a/frontiers/c2_torus_transfer_20261001/pseries.py b/frontiers/c2_torus_transfer_20261001/pseries.py
new file mode 100644
index 000000000..fac3c562b
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/pseries.py
@@ -0,0 +1,203 @@
+"""Exact truncated power series in the separation r, and the pinned two-point structure (NOTE.md section 2).
+
+Field: centred Gaussian on R^d with covariance K(z) = exp(-|z|^2/2), so Cov(d^a f(0), d^b f(0)) = (-1)^|b| d^(a+b) K(0),
+d^g K(0) = prod_i (-1)^(g_i/2) (g_i - 1)!! for even g and 0 otherwise.  Points M = -r e1/2, S = +r e1/2.
+
+Every quantity is a linear form in the jets at 0 whose coefficients are power series in r:
+    d^beta f(x e1) = sum_j x^j/j! d^(beta + j e1) f(0),   x = -+ r/2.
+A Series keeps the coefficients of r^0..r^R exactly (Fractions).  The pin rows divide by r, r and r^3; each division is
+exact (the low coefficients vanish identically, which `shift` asserts), and costs at most three orders, so every derived
+series is exact through r^(R - 3) = r^7.  The certificate needs r^6 (exact.NR)."""
+from fractions import Fraction as Fr
+
+R = 10
+
+
+class S:
+    """a power series c_0 + c_1 r + ... + c_R r^R over Q (higher orders dropped)"""
+    __slots__ = ('c',)
+
+    def __init__(self, c=None):
+        c = list(c or [])
+        self.c = (c + [Fr(0)] * (R + 1))[:R + 1]
+
+    @staticmethod
+    def const(x):
+        return S([Fr(x)])
+
+    def __add__(self, o):
+        if not isinstance(o, S):
+            o = S.const(o)
+        return S([a + b for a, b in zip(self.c, o.c)])
+    __radd__ = __add__
+
+    def __neg__(self):
+        return S([-a for a in self.c])
+
+    def __sub__(self, o):
+        return self + (-o if isinstance(o, S) else S.const(-o))
+
+    def __rsub__(self, o):
+        return (-self) + o
+
+    def __mul__(self, o):
+        if not isinstance(o, S):
+            return S([a * o for a in self.c])
+        out = [Fr(0)] * (R + 1)
+        for i, a in enumerate(self.c):
+            if a:
+                for j in range(R + 1 - i):
+                    if o.c[j]:
+                        out[i + j] += a * o.c[j]
+        return S(out)
+    __rmul__ = __mul__
+
+    def inv(self):
+        a0 = self.c[0]
+        if a0 == 0:
+            raise ZeroDivisionError('series with zero constant term')
+        out = [Fr(0)] * (R + 1)
+        out[0] = 1 / a0
+        for n in range(1, R + 1):
+            out[n] = -sum(self.c[k] * out[n - k] for k in range(1, n + 1)) / a0
+        return S(out)
+
+    def is_zero(self):
+        return not any(self.c)
+
+
+def dfact(n):
+    out = 1
+    while n > 1:
+        out *= n
+        n -= 2
+    return out
+
+
+def rho_der(g):
+    """d^g K(0) for K = exp(-|z|^2/2)"""
+    if any(x % 2 for x in g):
+        return 0
+    out = 1
+    for x in g:
+        out *= (-1) ** (x // 2) * dfact(x - 1)
+    return out
+
+
+def cov(u, w):
+    """Cov of two linear forms (dict jet -> S)"""
+    tot = S()
+    for a, sa in u.items():
+        for b, sb in w.items():
+            v = rho_der(tuple(x + y for x, y in zip(a, b)))
+            if v:
+                tot = tot + sa * sb * ((-1) ** sum(b) * v)
+    return tot
+
+
+def deriv_at(beta, sign, d):
+    """d^beta f at x = sign r/2 along e1, as jets: sum_(j <= R) (sign/2)^j r^j/j! d^(beta + j e1) f(0)"""
+    out = {}
+    fact = 1
+    for j in range(R + 1):
+        if j:
+            fact *= j
+        c = [Fr(0)] * (R + 1)
+        c[j] = Fr(sign, 2) ** j / fact
+        out[(beta[0] + j,) + tuple(beta[1:])] = S(c)
+    return out
+
+
+def lin(*terms):
+    """sum of coef * form"""
+    out = {}
+    for coef, form in terms:
+        for g, s in form.items():
+            out[g] = out.get(g, S()) + s * coef
+    return {g: s for g, s in out.items() if not s.is_zero()}
+
+
+def shift(form, p):
+    """divide by r^p; the coefficients of r^0..r^(p-1) must vanish identically"""
+    out = {}
+    for g, s in form.items():
+        if any(x != 0 for x in s.c[:p]):
+            raise ArithmeticError('inexact division by r^%d at jet %r' % (p, g))
+        out[g] = S(s.c[p:])
+    return out
+
+
+def pin_rows(d):
+    """The symmetric pin vector U_r of [R] section 2 (Math-#216 c2_check.py Pinned.pin_rows):
+        (f(M) + f(S))/2,  (f(S) - f(M))/r,  (f_x(S) - f_x(M))/r,  6/r^2 (f_x(M) + f_x(S) - 2 (f(S) - f(M))/r),
+        and for each transverse direction y: (f_y(M) + f_y(S))/2, (f_y(S) - f_y(M))/r.
+    Target v_r = (b - k r^3/2, -k r^2, 0, 12 k, 0, ..., 0)."""
+    e0 = (0,) * d
+    ex = (1,) + (0,) * (d - 1)
+    fa, fc = deriv_at(e0, -1, d), deriv_at(e0, 1, d)
+    fxa, fxc = deriv_at(ex, -1, d), deriv_at(ex, 1, d)
+    rows = [lin((Fr(1, 2), fa), (Fr(1, 2), fc)),
+            shift(lin((1, fc), (-1, fa)), 1),
+            shift(lin((1, fxc), (-1, fxa)), 1)]
+    num = lin((1, fxa), (1, fxc), (-2, shift(lin((1, fc), (-1, fa)), 1)))
+    rows.append({g: s * 6 for g, s in shift(num, 2).items()})
+    for t in range(1, d):
+        ey = tuple(1 if i == t else 0 for i in range(d))
+        ya, yc = deriv_at(ey, -1, d), deriv_at(ey, 1, d)
+        rows += [lin((Fr(1, 2), ya), (Fr(1, 2), yc)), shift(lin((1, yc), (-1, ya)), 1)]
+    return rows
+
+
+def mat_inv(M):
+    """inverse of a square matrix of series whose r^0 part is invertible (Gauss-Jordan over Q[[r]])"""
+    n = len(M)
+    A = [row[:] + [S.const(int(i == j)) for j in range(n)] for i, row in enumerate(M)]
+    for c in range(n):
+        p = next(i for i in range(c, n) if A[i][c].c[0] != 0)
+        A[c], A[p] = A[p], A[c]
+        iv = A[c][c].inv()
+        A[c] = [x * iv for x in A[c]]
+        for i in range(n):
+            if i != c and not A[i][c].is_zero():
+                f = A[i][c]
+                A[i] = [x - f * y for x, y in zip(A[i], A[c])]
+    return [row[n:] for row in A]
+
+
+def mat_det(M):
+    n = len(M)
+    A = [row[:] for row in M]
+    det = S.const(1)
+    for c in range(n):
+        p = next(i for i in range(c, n) if A[i][c].c[0] != 0)
+        if p != c:
+            A[c], A[p] = A[p], A[c]
+            det = -det
+        det = det * A[c][c]
+        iv = A[c][c].inv()
+        for i in range(c + 1, n):
+            if not A[i][c].is_zero():
+                f = A[i][c] * iv
+                A[i] = [x - f * y for x, y in zip(A[i], A[c])]
+    return det
+
+
+class Pinned:
+    """conditional law of linear forms given U_r: mean G v (G[u][j] series), covariance C[(u, w)] (series)"""
+
+    def __init__(self, d, forms):
+        self.d = d
+        self.rows = pin_rows(d)
+        p = len(self.rows)
+        self.p = p
+        Sm = [[cov(a, b) for b in self.rows] for a in self.rows]
+        self.Sinv = mat_inv(Sm)
+        self.detS = mat_det(Sm)
+        self.names = list(forms)
+        CF = {u: [cov(forms[u], row) for row in self.rows] for u in self.names}
+        self.G = {u: [sum((CF[u][q] * self.Sinv[q][j] for q in range(p)), S()) for j in range(p)] for u in self.names}
+        self.C = {}
+        for i, u in enumerate(self.names):
+            for w in self.names[i:]:
+                val = cov(forms[u], forms[w]) - sum((self.G[u][q] * CF[w][q] for q in range(p)), S())
+                self.C[(u, w)] = self.C[(w, u)] = val
diff --git a/frontiers/c2_torus_transfer_20261001/transfer.py b/frontiers/c2_torus_transfer_20261001/transfer.py
new file mode 100644
index 000000000..05dc3fba1
--- /dev/null
+++ b/frontiers/c2_torus_transfer_20261001/transfer.py
@@ -0,0 +1,707 @@
+"""CL-C2-TORUS-TRANSFER-20261001: the third-order coefficient c2 on the finite torus, d = 1, 2, 3, every L >= L0 and
+every orthonormal frame, against the closed form of frontiers/c2_exact_20261001 (NOTE.md).  Standard library only.
+
+    python3 -B -S transfer.py                 print the bounds
+    python3 -B -S transfer.py --write         write RESULTS.json
+    python3 -B -S transfer.py --check         recompute everything and compare with RESULTS.json (exit 1 on mismatch)
+    python3 -B -S transfer.py --mutant NAME   run --check with a seeded defect (must exit 1)
+
+Pipeline (NOTE.md sections 1-4):
+  1. jet covariances of the normalized periodic kernel K_L as balls around the Gaussian ones (Lemma J);
+  2. the exact derivation of c2_exact (pin rows, conditioning, Isserlis, exp/det expansion) re-run in ball arithmetic,
+     with the full transverse block T = (T11, T22, T12) in d = 3 (no rotation reduction): the r^4 coefficient
+     H(b, k, x) of the integrand, the exponent E = (q0 + Q0)/2 and the determinants, as balls (Lemma P);
+  3. the finite-part lemma and explicit Gaussian moment bounds for the difference of the two integrands (Lemmas F, G)."""
+from fractions import Fraction as Fr
+import itertools
+import json
+import math
+import os
+import sys
+
+sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
+from ball import (B, rup, center, radius, mag, exp_up, isqrt_frac_dn, PI_UP, PI_DN)    # noqa: E402
+import pseries as P                                                                      # noqa: E402
+
+P.R = 8                       # series exact through r^(R - 3) = r^5; the certificate needs r^4
+NR = 4                        # polynomial truncation in r: the r^4 coefficient is the A_2 integrand
+VARS = ('r', 'b', 'k', 'x1', 'x2', 'x3')
+NV = len(VARS)
+ONE = (0,) * NV
+MUTANT = None
+MUTANTS = ('no-image', 'no-theta', 'drop-ratio', 'eta-half', 'fp-const')
+L0S = (10, 12, 16, 24)
+HERE = os.path.dirname(os.path.abspath(__file__))
+
+
+def require(cond, msg):
+    if not cond:
+        raise AssertionError(msg)
+
+
+# ============================================================================== 1. jet covariances (Lemma J)
+def rho_der(g):
+    """d^g phi(0) for phi(z) = exp(-|z|^2/2) (exact integer)"""
+    return P.rho_der(g)
+
+
+def he_major(m):
+    """coefficients of the majorant He*_m(s) = sum_j m!/(j! (m-2j)! 2^j) s^(m-2j) of the Hermite polynomial He_m:
+    |He_m(t)| <= He*_m(|t|), He* nondecreasing on [0, oo)"""
+    return {m - 2 * j: Fr(math.factorial(m), math.factorial(j) * math.factorial(m - 2 * j) * 2 ** j)
+            for j in range(m // 2 + 1)}
+
+
+def he_major_at(m, s):
+    return sum((c * s ** p for p, c in he_major(m).items()), Fr(0))
+
+
+def shells(d, S=4):
+    """{s: #{n in Z^d : |n|^2 = s}} for 1 <= s <= S (every such n has |n_i| <= 2)"""
+    cnt = {}
+    for n in itertools.product(range(-2, 3), repeat=d):
+        s = sum(x * x for x in n)
+        if 1 <= s <= S:
+            cnt[s] = cnt.get(s, 0) + 1
+    return cnt
+
+
+class Images:
+    """Image sums of the periodization, bounded for every real L >= L0 and every orthonormal frame (Lemma J)."""
+
+    def __init__(self, d, L0):
+        self.d, self.L0 = d, Fr(L0)
+        self.sh = shells(d)
+        self.cache = {}
+        self.theta1 = self.image(0)
+
+    def image(self, m_tot, g=None):
+        """upper bound of sum_{n != 0} prod_i |He_{g_i}(p_i)| phi(p),  p = L R n  (|p| = L |n|, any rotation R)"""
+        d, L = self.d, self.L0
+        g = g if g is not None else (0,) * d
+        key = tuple(sorted(g))
+        if key in self.cache:
+            return self.cache[key]
+        n = sum(g)
+        require(n * n <= 5 * L * L and n <= L * L, 'jet order outside the range of Lemma J')
+        tot = Fr(0)
+        for s, c in sorted(self.sh.items()):
+            sq = L * L * s
+            t = Fr(c) * exp_up(-sq / 2)
+            from ball import isqrt_frac_up
+            hs = isqrt_frac_up(sq)
+            for gi in g:
+                t *= he_major_at(gi, hs)
+            tot += t
+        a2 = d + n                            # 2a, a = (d + |g|)/2 <= 20
+        require(a2 <= 40, 'tail exponent')
+        tail = Fr(3) ** d * (2 * L) ** n * 2 * Fr(5) ** ((a2 + 1) // 2) * exp_up(-5 * L * L / 2)
+        val = rup(tot + tail)
+        self.cache[key] = val
+        return val
+
+    def jetcov(self, g):
+        """d^g K_L(0) as a ball around d^g phi(0)"""
+        ref = rho_der(g)
+        if sum(g) % 2:
+            return B(0)                      # K_L is even: every odd total order vanishes exactly
+        if MUTANT == 'no-image':
+            return B(ref, self.theta1 * abs(ref))
+        if MUTANT == 'no-theta':
+            return B(ref, self.image(sum(g), g))
+        return B(ref, self.image(sum(g), g) + self.theta1 * abs(ref))
+
+
+# ============================================================================== 2. series and polynomials over balls
+class S:
+    """power series c_0 + ... + c_R r^R with Fraction or ball coefficients"""
+    __slots__ = ('c',)
+
+    def __init__(self, c=None):
+        c = list(c or [])
+        self.c = (c + [Fr(0)] * (P.R + 1))[:P.R + 1]
+
+    @staticmethod
+    def const(x):
+        return S([x])
+
+    def __add__(self, o):
+        if not isinstance(o, S):
+            o = S.const(o)
+        return S([a + b for a, b in zip(self.c, o.c)])
+    __radd__ = __add__
+
+    def __neg__(self):
+        return S([-a for a in self.c])
+
+    def __sub__(self, o):
+        return self + (-o)
+
+    def __mul__(self, o):
+        if not isinstance(o, S):
+            return S([a * o for a in self.c])
+        out = [Fr(0)] * (P.R + 1)
+        for i, a in enumerate(self.c):
+            if a:
+                for j in range(P.R + 1 - i):
+                    if o.c[j]:
+                        out[i + j] = out[i + j] + a * o.c[j]
+        return S(out)
+    __rmul__ = __mul__
+
+    def inv(self):
+        a0 = self.c[0]
+        ia = a0.inv() if isinstance(a0, B) else 1 / Fr(a0)
+        out = [Fr(0)] * (P.R + 1)
+        out[0] = ia
+        for n in range(1, P.R + 1):
+            acc = Fr(0)
+            for k in range(1, n + 1):
+                if self.c[k] and out[n - k]:
+                    acc = acc + self.c[k] * out[n - k]
+            out[n] = -(acc * ia)
+        return S(out)
+
+    def is_zero(self):
+        return not any(self.c)
+
+
+def from_pseries(s):
+    return S(list(s.c))
+
+
+def cov(u, w, jetcov):
+    tot = S()
+    for a, sa in u.items():
+        for b, sb in w.items():
+            g = tuple(x + y for x, y in zip(a, b))
+            v = jetcov(g)
+            if v:
+                tot = tot + from_pseries(sa * sb) * (v * (-1) ** sum(b))
+    return tot
+
+
+def mat_inv(M):
+    n = len(M)
+    A = [row[:] + [S.const(Fr(int(i == j))) for j in range(n)] for i, row in enumerate(M)]
+    for c in range(n):
+        p = next(i for i in range(c, n) if center(A[i][c].c[0]) != 0)
+        A[c], A[p] = A[p], A[c]
+        iv = A[c][c].inv()
+        A[c] = [x * iv for x in A[c]]
+        for i in range(n):
+            if i != c and not A[i][c].is_zero():
+                f = A[i][c]
+                A[i] = [x - f * y for x, y in zip(A[i], A[c])]
+    return [row[n:] for row in A]
+
+
+def mat_det(M):
+    n = len(M)
+    A = [row[:] for row in M]
+    det = S.const(Fr(1))
+    for c in range(n):
+        p = next(i for i in range(c, n) if center(A[i][c].c[0]) != 0)
+        if p != c:
+            A[c], A[p] = A[p], A[c]
+            det = -det
+        det = det * A[c][c]
+        iv = A[c][c].inv()
+        for i in range(c + 1, n):
+            if not A[i][c].is_zero():
+                f = A[i][c] * iv
+                A[i] = [x - f * y for x, y in zip(A[i], A[c])]
+    return det
+
+
+class Poly:
+    """polynomial in (r, b, k, x1, x2, x3), truncated at r^NR, Fraction or ball coefficients"""
+    __slots__ = ('t',)
+
+    def __init__(self, t=None):
+        self.t = {e: c for e, c in (t or {}).items() if c}
+
+    @staticmethod
+    def var(name, power=1):
+        e = [0] * NV
+        e[VARS.index(name)] = power
+        return Poly({tuple(e): Fr(1)})
+
+    @staticmethod
+    def const(c):
+        return Poly({ONE: c})
+
+    def __add__(self, o):
+        if not isinstance(o, Poly):
+            o = Poly.const(o)
+        t = dict(self.t)
+        for e, c in o.t.items():
+            t[e] = t[e] + c if e in t else c
+        return Poly(t)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return Poly({e: -c for e, c in self.t.items()})
+
+    def __sub__(self, o):
+        return self + (-o if isinstance(o, Poly) else Poly.const(-o))
+
+    def __mul__(self, o):
+        if not isinstance(o, Poly):
+            return Poly({e: c * o for e, c in self.t.items()})
+        t = {}
+        for e1, c1 in self.t.items():
+            for e2, c2 in o.t.items():
+                if e1[0] + e2[0] > NR:
+                    continue
+                e = tuple(a + b for a, b in zip(e1, e2))
+                v = c1 * c2
+                t[e] = t[e] + v if e in t else v
+        return Poly(t)
+    __rmul__ = __mul__
+
+    def rcoef(self, n):
+        return Poly({(0,) + e[1:]: c for e, c in self.t.items() if e[0] == n})
+
+    def r_positive(self):
+        """the part with r-degree >= 1 (removes the r^0 terms; no subtraction of equal balls)"""
+        return Poly({e: c for e, c in self.t.items() if e[0] >= 1})
+
+    def deriv(self, name):
+        i = VARS.index(name)
+        return Poly({e[:i] + (e[i] - 1,) + e[i + 1:]: c * e[i] for e, c in self.t.items() if e[i]})
+
+    def subs_zero(self, name):
+        i = VARS.index(name)
+        return Poly({e: c for e, c in self.t.items() if e[i] == 0})
+
+    def centers(self):
+        return Poly({e: center(c) for e, c in self.t.items()})
+
+    def is_zero(self):
+        return not self.t
+
+
+def from_series(s):
+    return Poly({(i,) + (0,) * (NV - 1): c for i, c in enumerate(s.c[:NR + 1]) if c})
+
+
+def series_pow(x, alpha):
+    """(1 + x)^alpha for a Poly x with no r^0 part (exact through r^NR)"""
+    require(all(e[0] >= 1 for e in x.t), 'series_pow: r^0 part')
+    out, term, coef = Poly.const(Fr(1)), Poly.const(Fr(1)), Fr(1)
+    for n in range(1, NR + 1):
+        coef = coef * (Fr(alpha) - (n - 1)) / n
+        term = term * x
+        out = out + term * coef
+    return out
+
+
+def series_exp(x):
+    require(all(e[0] >= 1 for e in x.t), 'series_exp: r^0 part')
+    out, term = Poly.const(Fr(1)), Poly.const(Fr(1))
+    for n in range(1, NR + 1):
+        term = term * x * Fr(1, n)
+        out = out + term
+    return out
+
+
+# ============================================================================== 3. the kernel (Lemma P)
+def forms_for(d):
+    forms = {}
+    for nm, sg in (('M', -1), ('S', 1)):
+        for i in range(d):
+            for j in range(i, d):
+                be = [0] * d
+                be[i] += 1
+                be[j] += 1
+                forms[(nm, i, j)] = P.deriv_at(tuple(be), sg, d)
+    trans = []
+    for i in range(1, d):
+        for j in range(i, d):
+            be = [0] * d
+            be[i] += 1
+            be[j] += 1
+            forms[('T', i, j)] = {tuple(be): P.S.const(1)}
+            trans.append(('T', i, j))
+    return forms, trans
+
+
+def matchings(lst):
+    if not lst:
+        yield []
+        return
+    f, rest = lst[0], lst[1:]
+    for mm in matchings(rest):
+        yield mm
+    for i, y in enumerate(rest):
+        for mm in matchings(rest[:i] + rest[i + 1:]):
+            yield [(f, y)] + mm
+
+
+def perm_sign(p):
+    return -1 if sum(1 for i in range(len(p)) for j in range(i + 1, len(p)) if p[i] > p[j]) % 2 else 1
+
+
+TVAL = {('T', 1, 1): 'x1', ('T', 2, 2): 'x2', ('T', 1, 2): 'x3'}
+
+
+def kernel(d, jetcov):
+    """A~_r(b, k) = kappa * integral over the cone {T < 0} (x-coordinates, Lebesgue) of exp(-E) G / r^2, with
+    E = (q0 + Q0)/2 and kappa = -12 (2 pi)^(-(p + nT)/2) (det S0 det T0)^(-1/2) (NOTE.md 1.3; c2_exact NOTE 2.3)."""
+    forms, trans = forms_for(d)
+    rows = P.pin_rows(d)
+    p, nt = len(rows), len(trans)
+    Sm = [[cov(a, b_, jetcov) for b_ in rows] for a in rows]
+    Sinv, detS = mat_inv(Sm), mat_det(Sm)
+    names = list(forms)
+    CF = {u: [cov(forms[u], row, jetcov) for row in rows] for u in names}
+    G = {u: [sum((CF[u][q] * Sinv[q][j] for q in range(p)), S()) for j in range(p)] for u in names}
+    C = {}
+    for i, u in enumerate(names):
+        for w in names[i:]:
+            val = cov(forms[u], forms[w], jetcov) - sum((G[u][q] * CF[w][q] for q in range(p)), S())
+            C[(u, w)] = C[(w, u)] = val
+    r, b, k = Poly.var('r'), Poly.var('b'), Poly.var('k')
+    v = [b - k * r * r * r * Fr(1, 2), -(k * r * r), Poly(), k * 12] + [Poly()] * (p - 4)
+    mean = {u: sum((from_series(G[u][j]) * v[j] for j in range(p)), Poly()) for u in names}
+    hn = [u for u in names if u not in trans]
+    if nt:
+        CTT = [[C[(x, y)] for y in trans] for x in trans]
+        CTTinv, detT = mat_inv(CTT), mat_det(CTT)
+        Bc = {u: [sum((C[(u, trans[j])] * CTTinv[j][i] for j in range(nt)), S()) for i in range(nt)] for u in hn}
+        cp = {(u, w): from_series(C[(u, w)] - sum((Bc[u][i] * C[(trans[i], w)] for i in range(nt)), S()))
+              for u in hn for w in hn}
+        zT = [Poly.var(TVAL[nm]) - mean[nm] for nm in trans]
+        aff = {u: mean[u] + sum((from_series(Bc[u][i]) * zT[i] for i in range(nt)), Poly()) for u in hn}
+    else:
+        CTTinv, detT, zT = [], S.const(Fr(1)), []
+        cp = {(u, w): from_series(C[(u, w)]) for u in hn for w in hn}
+        aff = dict((u, mean[u]) for u in hn)
+    F = Poly()
+    for pp in itertools.permutations(range(d)):
+        for qq in itertools.permutations(range(d)):
+            fac = [('M', min(i, pp[i]), max(i, pp[i])) for i in range(d)] + \
+                  [('S', min(i, qq[i]), max(i, qq[i])) for i in range(d)]
+            sg = perm_sign(pp) * perm_sign(qq)
+            for mt in matchings(list(range(2 * d))):
+                used = set(i for pr in mt for i in pr)
+                c = Poly.const(Fr(sg))
+                for (i, j) in mt:
+                    c = c * cp[(fac[i], fac[j])]
+                for i in range(2 * d):
+                    if i not in used:
+                        c = c * aff[fac[i]]
+                F = F + c
+    q = sum((v[i] * from_series(Sinv[i][j]) * v[j] for i in range(p) for j in range(p)), Poly())
+    Q = sum((zT[i] * from_series(CTTinv[i][j]) * zT[j] for i in range(nt) for j in range(nt)), Poly())
+    q0, Q0 = q.rcoef(0), Q.rcoef(0)
+    detS0, detT0 = detS.c[0], detT.c[0]
+    # (det S_r / det S_0)(det T_r / det T_0) - 1 from the r >= 1 coefficients only (its r^0 part is exactly 0)
+    rS = Poly({(i,) + (0,) * (NV - 1): detS.c[i] * (1 / detS0 if not isinstance(detS0, B) else detS0.inv())
+               for i in range(1, NR + 1) if detS.c[i]})
+    i0 = (1 / detT0) if not isinstance(detT0, B) else detT0.inv()
+    rT = Poly({(i,) + (0,) * (NV - 1): detT.c[i] * i0 for i in range(1, NR + 1) if detT.c[i]})
+    ratio = rS + rT + (rS * rT)
+    if MUTANT == 'drop-ratio':
+        ratio = rS
+    dqQ = (q.r_positive() + Q.r_positive()) * Fr(-1, 2)
+    Gint = series_pow(ratio, Fr(-1, 2)) * series_exp(dqQ) * F
+    return {'G': Gint, 'E': (q0 + Q0) * Fr(1, 2), 'p': p, 'nt': nt, 'detS0': detS0, 'detT0': detT0}
+
+
+# ============================================================================== 4. bounds (Lemmas F, G)
+WV = ('b', 'k', 'x1', 'x2', 'x3')
+
+
+def strip_r(poly):
+    out = {}
+    for e, c in poly.t.items():
+        require(e[0] == 0, 'strip_r: r-dependent term')
+        out[e[1:]] = c
+    return out
+
+
+def quad_matrix(Ec, nv):
+    """symmetric matrix M with E(w) = w^T M w for the homogeneous quadratic E (centers)"""
+    M = [[Fr(0)] * nv for _ in range(nv)]
+    for e, c in Ec.items():
+        require(sum(e) == 2, 'exponent not homogeneous quadratic')
+        idx = [i for i in range(nv) for _ in range(e[i])]
+        i, j = idx
+        if i == j:
+            M[i][i] += c
+        else:
+            M[i][j] += c / 2
+            M[j][i] += c / 2
+    return M
+
+
+def is_pd(M):
+    n = len(M)
+    A = [row[:] for row in M]
+    for c in range(n):
+        if A[c][c] <= 0:
+            return False
+        for i in range(c + 1, n):
+            f = A[i][c] / A[c][c]
+            for j in range(c, n):
+                A[i][j] -= f * A[c][j]
+    return True
+
+
+def diag_scale_lower(M):
+    """a rational t in (0, 1] with M - t diag(M) positive semidefinite (exact LDL checks, bisection to 2^-30), so that
+    w^T M w >= t sum_i M_ii w_i^2"""
+    n = len(M)
+    require(is_pd(M), 'reference exponent not positive definite')
+
+    def ok(t):
+        return t == 0 or is_pd([[M[i][j] * (1 - t if i == j else 1) for j in range(n)] for i in range(n)]) or \
+            (t == 1 and all(M[i][j] == 0 for i in range(n) for j in range(n) if i != j))
+    if ok(Fr(1)):
+        return Fr(1)
+    lo, hi = Fr(0), Fr(1)
+    for _ in range(30):
+        mid = (lo + hi) / 2
+        if ok(mid):
+            lo = mid
+        else:
+            hi = mid
+    require(lo > 0, 'diagonal scaling')
+    return lo
+
+
+def gamma_half_up(n):
+    """upper bound of Gamma(n/2), n >= 1"""
+    if n % 2 == 0:
+        return Fr(math.factorial(n // 2 - 1))
+    m = (n - 1) // 2                                  # Gamma(m + 1/2) = (2m-1)!! sqrt(pi) / 2^m
+    df = 1
+    for i in range(1, 2 * m, 2):
+        df *= i
+    from ball import isqrt_frac_up
+    return Fr(df, 2 ** m) * isqrt_frac_up(PI_UP)
+
+
+def pow_neg_half_up(a, n):
+    """upper bound of a^(-n/2), a > 0 rational"""
+    if n % 2 == 0:
+        return 1 / a ** (n // 2)
+    return 1 / (a ** (n // 2) * isqrt_frac_dn(a))
+
+
+def moment_up(n, a, half=False):
+    """upper bound of int_R |t|^n e^(-a t^2) dt = Gamma((n+1)/2) a^(-(n+1)/2)  (half: over [0, oo), one half)"""
+    v = gamma_half_up(n + 1) * pow_neg_half_up(a, n + 1)
+    return v / 2 if half else v
+
+
+def maj_poly(H, which):
+    """nonnegative majorant coefficients: which = 'mag' -> |c| + rad, 'rad' -> rad"""
+    out = {}
+    for e, c in H.items():
+        v = mag(c) if which == 'mag' else radius(c)
+        if v:
+            out[e] = out.get(e, Fr(0)) + v
+    return out
+
+
+def mono_mul(P1, P2):
+    out = {}
+    for e1, c1 in P1.items():
+        for e2, c2 in P2.items():
+            e = tuple(a + b for a, b in zip(e1, e2))
+            out[e] = out.get(e, Fr(0)) + c1 * c2
+    return out
+
+
+def gauss_bound(Pm, a, mode, nx):
+    """upper bound of the integral of sum_e Pm[e] |w^e| e^(-sum_i a_i w_i^2) over b in R, x in R^nx, with k:
+    'k0' k = 0;  'kw' integral over k in [0, 1] against k^(-1/3) (k^(j - 1/3) <= k^(j-1) for j >= 1, extended to
+    [0, oo);  for j = 0, e^(-a k^2) <= 1 and int_0^1 k^(-1/3) dk = 3/2);  'kint' integral over k in [0, oo)."""
+    tot = Fr(0)
+    for e, c in Pm.items():
+        eb, ek, ex = e[0], e[1], e[2:2 + nx]
+        require(all(v == 0 for v in e[2 + nx:]), 'unused transverse variable')
+        if mode == 'k0' and ek:
+            continue
+        t = c * moment_up(eb, a[0])
+        for i, v in enumerate(ex):
+            t *= moment_up(v, a[2 + i])
+        if mode == 'kint':
+            t *= moment_up(ek, a[1], half=True)
+        elif mode == 'kw':
+            t *= moment_up(ek - 1, a[1], half=True) if ek else Fr(3, 2)
+        tot += t
+    return tot
+
+
+def eta_of(dE, nv):
+    """eta_i with |dE(w)| <= sum_i eta_i w_i^2 from the coefficient radii of the quadratic dE
+    (|w_i w_j| <= (w_i^2 + w_j^2)/2)"""
+    acc = [Fr(0)] * nv
+    for e, c in dE.items():
+        rad = radius(c)
+        if not rad:
+            continue
+        require(sum(e) == 2, 'exponent radius on a non-quadratic monomial')
+        idx = [i for i in range(nv) for _ in range(e[i])]
+        i, j = idx
+        if i == j:
+            acc[i] += rad
+        else:
+            acc[i] += rad / 2
+            acc[j] += rad / 2
+    return [x / 2 for x in acc] if MUTANT == 'eta-half' else acc
+
+
+SPHERE_UP = {1: Fr(2), 2: 2 * PI_UP, 3: 4 * PI_UP}
+C2_REF_LO = {1: Fr('0.2300445802661503'), 2: Fr('0.2215244106266632'), 3: Fr('0.1612340491269447')}   # c2_exact
+
+
+def bound_core(d, D):
+    """Lemmas F and G: an upper bound of sup_u |c2[K, u] - c2[phi]| from the ball kernel data D of K (NOTE.md 4)."""
+    nx = D['nt']
+    nv = 2 + nx
+    H = strip_r(D['G'].rcoef(4))
+    E = strip_r(D['E'])
+    M = quad_matrix({e: center(c) for e, c in E.items()}, nv)
+    t = diag_scale_lower(M)                  # E_ref(w) >= t sum_i M_ii w_i^2
+    eta = eta_of(E, nv)                      # |E_K(w) - E_ref(w)| <= sum_i eta_i w_i^2
+    a = [t * M[i][i] - eta[i] for i in range(nv)]
+    require(all(x > 0 for x in a), 'exponent perturbation too large')
+    # kappa_K / kappa_ref = (D_K / D_ref)^(-1/2),  D = det S0 det T0
+    DK = D['detS0'] * D['detT0']
+    Dref = center(DK)
+    require(Dref > 0, 'reference determinant')
+    rho = radius(DK) / Dref
+    require(rho < Fr(1, 2), 'determinant radius')
+    e_kappa = rho / (1 - rho)                # |kappa_K/kappa_ref - 1| <= (1 - rho)^(-1/2) - 1 <= rho/(1 - rho)
+    nn = D['p'] + nx
+    kap = 12 / isqrt_frac_dn(Dref)           # kappa_ref <= kap
+    two_pi = 2 * PI_DN
+    kap = kap / two_pi ** (nn // 2)
+    if nn % 2:
+        kap = kap / isqrt_frac_dn(two_pi)
+    # H' = dH/dk - (dE/dk) H:  d/dk [exp(-E) H] = exp(-E) H'
+    Hp = Poly({(0,) + e: c for e, c in H.items()})
+    Ep = Poly({(0,) + e: c for e, c in E.items()})
+    Hd = strip_r(Hp.deriv('k') - Ep.deriv('k') * Hp)
+    fac = {(0,) * 5: e_kappa}                # |e_kappa| + sum_i eta_i w_i^2
+    for i in range(nv):
+        e = [0] * 5
+        e[i] = 2
+        fac[tuple(e)] = eta[i]
+
+    def bound(Hx, mode):
+        return kap * gauss_bound(dict_add(mono_mul(fac, maj_poly(Hx, 'mag')), maj_poly(Hx, 'rad')), a, mode, nx)
+    fp_const = 3 if MUTANT != 'fp-const' else 1
+    b0, b1, b2 = bound(H, 'k0'), bound(Hd, 'kw'), bound(H, 'kint')
+    T_bound = fp_const * b0 + 3 * b1 + b2
+    eps = rup(SPHERE_UP[d] / 3 * T_bound)
+    return {'d': d, 't_diag': t, 'a_min': min(a), 'eta_max': max(eta), 'e_kappa': e_kappa, 'kappa_up': kap, 'Delta0': b0, 'dDelta_w': b1,
+            'Delta_int': b2, 'eps': eps, 'max_radius_H': max((radius(c) for c in H.values()), default=Fr(0)),
+            'n_monomials_H': len(H)}
+
+
+def transfer_bound(d, L0):
+    img = Images(d, L0)
+    out = bound_core(d, kernel(d, img.jetcov))
+    out.update({'L0': L0, 'theta1': img.theta1, 'rel': rup(out['eps'] / C2_REF_LO[d]),
+                'max_jet_order': max((sum(g) for g in img.cache), default=0)})
+    return out
+
+
+def dict_add(A, Bd):
+    out = dict(A)
+    for e, c in Bd.items():
+        out[e] = out.get(e, Fr(0)) + c
+    return out
+
+
+# ============================================================================== 5. reference checks
+def reference_checks():
+    """With every radius zero the pipeline is exact; it must reproduce c2_exact's kernel_data (d = 1, 2 identically;
+    d = 3 on the slice x3 = T12 = 0, which is the diagonal slice c2_exact evaluates)."""
+    import exact as X
+    out = {}
+    for d in (1, 2, 3):
+        ref = X.kernel_data(d)
+        mine = kernel(d, lambda g: Fr(rho_der(g)))
+        Gm = mine['G'].subs_zero('x3') if d == 3 else mine['G']
+        Em = mine['E'].subs_zero('x3') if d == 3 else mine['E']
+        ok = True
+        for n in range(NR + 1):
+            a = {e[:5]: c for e, c in Gm.rcoef(n).t.items()}
+            bb = {e: c for e, c in ref['G'].rcoef(n).t.items()}
+            ok = ok and a == bb
+        Eref = (ref['q0'] + ref['Q0']) * Fr(1, 2)
+        ok = ok and {e[:5]: c for e, c in Em.t.items()} == dict(Eref.t)
+        ok = ok and mine['detS0'] == ref['detS0']
+        if d < 3:
+            ok = ok and mine['detT0'] == ref['detT0']
+        out[d] = ok
+        require(ok, 'reference pipeline mismatch in d = %d' % d)
+    return out
+
+
+# ============================================================================== 6. driver
+def dec_up(x, digits=6):
+    """decimal string >= x (x >= 0 rational) with `digits` significant digits"""
+    x = Fr(x)
+    if x == 0:
+        return '0'
+    e = 0
+    while x >= 10 ** (e + 1):
+        e += 1
+    while x < 10 ** e:
+        e -= 1
+    scale = Fr(10) ** (digits - 1 - e)
+    m = -((-x.numerator * scale.numerator) // (x.denominator * scale.denominator))
+    if m >= 10 ** digits:
+        m, e = m // 10 + (m % 10 > 0), e + 1
+    s = str(m)
+    return '%s.%se%+d' % (s[0], s[1:], e)
+
+
+def fmt(x):
+    return dec_up(abs(x)) if x >= 0 else '-' + dec_up(-x)
+
+
+def compute():
+    res = {'object': 'CL-C2-TORUS-TRANSFER-20261001-v1', 'reference_checks': {}, 'bounds': {}}
+    res['reference_checks'] = {str(k): v for k, v in reference_checks().items()}
+    for d in (1, 2, 3):
+        for L0 in L0S:
+            t = transfer_bound(d, L0)
+            res['bounds']['d%d_L%d' % (d, L0)] = {k: (fmt(v) if isinstance(v, Fr) else v) for k, v in t.items()}
+    return res
+
+
+def main():
+    global MUTANT
+    args = sys.argv[1:]
+    if '--mutant' in args:
+        MUTANT = args[args.index('--mutant') + 1]
+        require(MUTANT in MUTANTS, 'unknown mutant')
+        args = ['--check']
+    res = compute()
+    path = os.path.join(HERE, 'RESULTS.json')
+    if '--write' in args:
+        with open(path, 'w') as f:
+            json.dump(res, f, indent=1, sort_keys=True)
+            f.write('\n')
+    elif '--check' in args:
+        with open(path) as f:
+            stored = json.load(f)
+        if stored != res:
+            print('MISMATCH against RESULTS.json')
+            sys.exit(1)
+        print('check passed')
+    else:
+        print(json.dumps(res, indent=1, sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_cap_failure_coefficient_planar_20260930/NOTE.md b/frontiers/c8_cap_failure_coefficient_planar_20260930/NOTE.md
new file mode 100644
index 000000000..2838b76b2
--- /dev/null
+++ b/frontiers/c8_cap_failure_coefficient_planar_20260930/NOTE.md
@@ -0,0 +1,336 @@
+# The cap-failure probability in the proof of Theorem A has an exact leading coefficient in the plane
+
+**Object:** `CL-C8-CAP-FAILURE-COEFFICIENT-PLANAR-20260930-v1`. **Author lane:** Anthropic / Claude.
+**Kind:** author-side limit theorem (Theorem G) with certified enclosures of its coefficient (exact rational interval
+arithmetic; standard library only). **Scientific effect:** NONE. Catalog entry C8 (`reviews/candidates_pending_20260928/CANDIDATES.md`:
+"explicit finite `C`, `r_*`, `z_*`, `c_{B,K}`, `c_{d,L}` on a declared band; OPEN") stays OPEN: this packet supplies neither
+`C` nor `r_*`; it determines the exact size of the cap-failure probability `Q^W(G_r^c)` that the proof of Theorem A bounds
+([LP] (7.8)), hence a lower bound for every cap-route constant `C_cap`; it gives no lower bound for the constant of Theorem A
+itself, which bounds the smaller probability `1 - p_r` (v1.2; section 5, item 1).
+No STATUS, PROOF_INDEX, GRAPH or catalog edit. Same GitHub account as every lane; zero organizational-independence credit.
+The author will not merge.
+
+Sources (pinned in `SOURCE_MAP.json`): [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`
+(sections 1, 3, 4, 5, 7: the law `Q`, the weight `W_r`, the normalizer `Z_r`, the contact frame, the coupling (4.2), the
+limit (5.4), the good event `G_r` and the bounds (7.5)–(7.8)); [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md`
+(the cylinder `D`, the partial-block norms `M_3`, `M_4`, condition (1), and the recorded constants (2)–(3)); [NUM]
+`frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json` (the exact planar contact structure and the values of
+`alpha_1`, `alpha_2`); [S] `frontiers/local_elder_geometry_20260930/PROOF.md` (Theorem S, (S2): `r^-3 (1 - p_r) -> alpha_1 + alpha_2`);
+[Z] `frontiers/c8_normalizer_floor_planar_20260930/NOTE.md` at Math-#195 (the midpoint-jet coordinates; read, not consumed).
+
+## 0. Statement
+
+Setting of [LP] section 1 in `d = 2`: the torus `X = (R / L Z)^2`, the kernel `K_L` (periodized Gaussian, `K_L(0) = 1`),
+an orthonormal frame `R` with axial direction `u`, the pins `M = -(r/2) u`, `S = (r/2) u`, `f(M) = b`, `f(S) = b - k r^3`,
+`grad f(M) = grad f(S) = 0`, the Gaussian regression law `Q = Q_{r,b,k,R}`, the typed weight
+`W_r = |det H_M det H_S| 1{H_M < 0, det H_S < 0}`, `Z_r = E_Q W_r`, `dQ^W = (W_r / Z_r) dQ`. In the frame's coordinates
+`(x, y)` let `D = [-2r, 2r]^2` and, for `j = 3, 4`, `M_j = max_{a + c = j} sup_D |partial_x^a partial_y^c f|` ([CAP] section 1;
+in `d = 2` the partial-block norm is the maximum of the coordinate partials). The good event of [LP] section 7 / [CAP] (1) is
+
+    G_r = { -f_yy(M) > (4/(3k)) r M_3^2,  r M_4 <= 3k/10 }
+
+(on the typed support `-f_yy(M) = lambda_min(-A_M)`). Theorem A of [LP] is `0 <= 1 - p_r <= C r^3` for `0 < r <= r_*`
+([LP] (1.1); `p_r` is the `Q^W` probability that the global ordinary elder death partner of `M` is `S`). Its proof establishes
+the stronger cap bound [LP] (7.8), `Q^W(G_r^c) <= C_3 r^3 + C_4 r^4 <= C r^3`, and deduces (1.1) from `1 - p_r <= Q^W(G_r^c)`
+([LP] section 8: on `G_r` the global elder partner is `S`). A **cap-route constant** is any `C_cap` with
+`Q^W(G_r^c) <= C_cap r^3` for `0 < r <= r_*`; every cap-route constant is a constant for [LP] (1.1), not conversely.
+
+**Theorem G.** Fix `b in R`, `k > 0`, `L`, `R`. Then
+
+    Q^W(G_r^c) = c_G(b, k) r^3 + o(r^3),        r -> 0,
+
+    c_G(b, k) = R(b) J(k),
+    R(b) = p_w(0) / E[w^2 1{w < 0}],
+    J(k) = (216 k^3)^(-1) E[ integral_{max(a, c)}^{8 M^2} (s - a)(s - c) ds ],
+    a = v^2,   c = 6k om - v^2,   M = max(12k, 2|v|, |om|, |Y|),
+
+where `w` has the law of `f_yy(0)` given `(f, f_xx, f_xy)(0) = (b, 0, 0)`, `p_w` is its density, and `(v, om, Y)` has the law of
+`(-f_xxy/2, f_xyy, f_yyy)(0)` given `(f_x, f_y, f_xxx)(0) = (0, 0, 12k)`, all under the unconditioned field with kernel
+`K_L(R .)`; `w` is independent of `(v, om, Y)`. For the reference kernel `exp(-|z|^2/2)` (equivalently `L -> infinity`),
+
+    w ~ N(-b, 2),   v ~ N(0, 1/2),   om ~ N(0, 2),   Y ~ N(0, 6),   independent,
+
+    R(b) = [phi(b/sqrt2) / sqrt2] / [(2 + b^2) Phi(b/sqrt2) + sqrt2 b phi(b/sqrt2)],
+
+and `J(k) = 2359296 k^3 (1 + delta(k))`, where on the certified grid `delta(1/6) = 52.7`, `delta(1/2) = 0.0232`,
+`delta(3/4) = 1.3e-4`, `delta(1) = -1.3e-6`, `delta(3/2) = -3.0e-7`, `delta(2) = -8.6e-8` (both signs occur; `delta(k) -> 0` as
+`k -> infinity` by Remark (b), without a uniform rate proved here).
+
+**Certified values (reference kernel only).** `J(k)` at `k in {1/6, 1/2, 3/4, 1, 3/2, 2}`, `R(b)` at `b in {0, 1/4, 1/2, 3/4, 1, 6/5}`
+and the products, all as rational enclosures (section 4). Two headline numbers:
+
+    c_G(0, 2)     in  [5324360.4426, 5324360.4427]      (the maximum over the Math-#195 band B = [0, 1], K = [1/2, 2] among the grid points),
+    c_G(6/5, 1/6) in  [35736.7352, 35736.8785]    (SIDE24's fixed-axis parameters, the only pair for which an explicit C is on record;
+                                                  reference-kernel value only: no bound on SIDE24's finite-L jet-law deviation is claimed or enclosed here).
+
+**Corollary (what this says about `C_cap` and `r_*`).** Every cap-route constant `C_cap` at `(b, k)` (for a given kernel and
+frame) satisfies `C_cap >= c_G(b, k)` for that kernel and frame. This is not a lower bound for every constant `C` of [LP] (1.1):
+the only relation between the two events is `1 - p_r <= Q^W(G_r^c)`, and the smaller probability has the smaller coefficient
+`alpha_1 + alpha_2` of [S] (S2) (Remark (d); section 5, items 1 and 4). The certified numbers are the reference-kernel
+values `c_G^{ref}`; for the torus kernel `K_L` in a frame `R` the coefficient is `c_G^{L,R}`, the same formula on the exact
+torus jet law, whose deviation from `c_G^{ref}` is **not enclosed here** (section 6). So, as a
+reference-kernel statement (equivalently `L -> infinity`): every band containing `(0, 2)` needs `C_cap > 5.3 x 10^6`, and the
+cap route cannot give a nontrivial bound (`C_cap r^3 < 1`) above `r = c_G^{-1/3}`, which is `5.7 x 10^-3` at `(0, 2)`,
+`3.4 x 10^-2` at `(1, 1/2)`, and `3.0 x 10^-2` at `(6/5, 1/6)`; for a finite torus only the symbolic statements hold, with
+`c_G^{L,R}` in place of `c_G^{ref}`; no numerical value is claimed there. The recorded cap-route constant of [CAP] (2)–(3),
+`C3_new + C4_new / 20 ≈ 2.36 x 10^23` (`< 2.4 x 10^23`) at `(6/5, 1/6)`, exceeds the reference-kernel sharp cap coefficient
+`c_G^{ref}(6/5, 1/6)` by a factor above `6 x 10^18` (the comparison with SIDE24's own finite-`L` coefficient is not enclosed). Against the true failure coefficient `alpha_1 + alpha_2` of [S] (S2) (the
+actual elder-pairing failure, values from [NUM]) the cap criterion overstates the failure rate by the `b`-free factor
+`36 k^2 J(k) / (J_1(k) + J_2(k))`: `7.4 x 10^4` at `k = 1/2`, `4.9 x 10^5` at `k = 1`, `2.4 x 10^6` at `k = 2` (section 5).
+
+## 1. Notation and the exact identities
+
+Write `lambda = -f_yy(M)`, and the six midpoint-jet coordinates of [Z] section 2 (exact for every `C^2` function with the
+pins; only their definitions are used here):
+
+    w = f_yy(M),                     om = (f_yy(S) - f_yy(M)) / r,
+    T = (f_xx(M) + 6 k r) / r^2,     tau = (f_xx(S) - f_xx(M) - 12 k r) / r^3,
+    v = f_xy(M) / r,                 nu = (f_xy(S) + f_xy(M)) / r^2.
+
+They recover `f_xx(M) = r^2 T - 6kr`, `f_xx(S) = r^2 (T + r tau) + 6kr`, `f_xy(M) = r v`, `f_xy(S) = r (r nu - v)`,
+`f_yy(S) = w + r om`. Each of `T`, `tau`, `nu`, `v`, `om` is a linear functional of `f` annihilating the polynomials of
+degree `<= 3, 4, 3, 2, 2` respectively along the pin segment (the pins supply `f_x(M) = f_x(S) = f_y(M) = f_y(S) = 0`
+and `f(S) - f(M) = -k r^3`), so by Taylor's formula with integral remainder
+
+    |T| <= C ||f||_{C^4(D)},  |tau| <= C ||f||_{C^5(D)},  |nu| <= C ||f||_{C^4(D)},  |v| <= C ||f||_{C^3(D)},  |om| <= C ||f||_{C^3(D)}   (1.1)
+
+with absolute constants, for `r <= 1`, and as `r -> 0` (pathwise, for a `C^5` function) `T -> f_xxxx(0)/12`,
+`v -> -f_xxy(0)/2`, `om -> f_xyy(0)`, `nu -> f_xxxy(0)/... ` (a fixed combination of the 4-jet), `tau` to a fixed combination of the 5-jet.
+The determinants are exactly
+
+    det H_M = r^2 [ (6k - r T) u_r - v^2 ],                                      u_r := lambda / r = -w / r,
+    det H_S = r^2 [ (6k + r T + r^2 tau)(om - u_r) - (r nu - v)^2 ],                                        (1.2)
+
+so on `{u_r = u}`:  `W_r = r^4 |(6k - rT) u - v^2| |(6k + rT + r^2 tau)(om - u) - (r nu - v)^2| 1{type}`, with `type` =
+`{f_xx(M) < 0, det H_M > 0, det H_S < 0}`.
+
+## 2. Proof of Theorem G
+
+**2.1 Coupling.** Let `G` be the unconditioned field (kernel `K_L(R .)`) and `U_r(G)` the six pin functionals in the
+preconditioned form [LP] (3.1), with target `v_r` and covariance `Sigma_r`. The Gaussian regression coupling
+
+    f_r = G + kappa_r(z) . (v_r - U_r(G)),        kappa_r(z) = Cov(G(z), U_r) Sigma_r^{-1},                   (2.1)
+
+realizes `Q_r` for every `r` on one probability space ([LP] (4.2)). As `r -> 0`, `U_r(G) -> U_0(G) = (f, f_x, f_xx, f_xxx, f_y, f_xy)(0)`
+(Taylor), `v_r -> v_0 = (b, 0, 0, 12k, 0, 0)`, `Sigma_r -> Sigma_0` (positive definite, [LP] section 3), and
+`Cov(G(z), U_{r,i}) -> Cov(G(z), U_{0,i})` in every `C^q(X)` (difference quotients of derivatives of `K_L`). Hence
+`f_r -> f_0 := G + kappa_0 . (v_0 - U_0(G))` in `C^q(X)` almost surely and in every `L^p`, for every `q`; `f_0` has the
+contact law `Q_0` (the field given the contact jet `U_0 = v_0`). All `Q_r`-moments of `||f_r||_{C^q(X)}` are bounded
+uniformly in `r <= 1` ([LP] (4.1)).
+
+**2.2 Regression on `w`.** Under `Q_r`, `w = f_r,yy(M)` is `N(mu_w(r), sigma_w(r)^2)` with `(mu_w, sigma_w) -> (mu_w(0), sigma_w(0))`,
+the parameters of `f_yy(0)` given the contact pins (`(-b, sqrt2)` for the reference kernel). Put `g_r(z) = Cov_Q(f_r(z), w) / sigma_w(r)^2`
+and `f_r^perp = f_r - (w - mu_w(r)) g_r`, independent of `w`, with `g_r -> g_0` in every `C^q` and `f_r^perp -> f_0^perp` as in 2.1.
+For any functional `Psi`,
+
+    E_Q[Psi(f_r)] = integral p_{w,r}(omega) E[Psi(f_r^perp + (omega - mu_w(r)) g_r)] d omega.                  (2.2)
+
+The law of `f_r^perp + (0 - mu_w(0)) g_0` in the limit is the contact law given `f_yy(0) = 0`: by the parity of `K_L`
+(`K_L(-z) = K_L(z)`, so all odd-order derivatives of `K_L` vanish at 0), the odd jets `(f_x, f_y, f_xxx, f_xxy, f_xyy, f_yyy)(0)`
+are independent of the even jets `(f, f_xx, f_xy, f_yy)(0)`, and the conditioning on `f_yy(0) = 0` leaves the law of the odd
+3-jet given `(f_x, f_y, f_xxx) = (0, 0, 12k)` unchanged. That law is `N(mu_odd, Sigma_odd)`; for the reference kernel the odd
+covariances are `E f_x^2 = E f_y^2 = 1`, `E f_x f_xxx = -3`, `E f_x f_xyy = E f_y f_xxy = -1`, `E f_y f_yyy = -3`,
+`E f_xxx^2 = E f_yyy^2 = 15`, `E f_xxy^2 = E f_xyy^2 = 3`, `E f_xxx f_xyy = E f_xxy f_yyy = 3`, the rest zero, so the two groups
+`(f_x, f_xxx, f_xyy)` and `(f_y, f_xxy, f_yyy)` are independent, and the regression gives `f_xyy | (f_x = 0, f_xxx = 12k) ~ N(0, 2)`,
+`(f_xxy, f_yyy) | f_y = 0 ~ N(0, diag(2, 6))`; likewise `f_yy | (f, f_xx, f_xy) = (b, 0, 0) ~ N(-b, 2)` ([NUM] `exact_structure`).
+Hence `v = -f_xxy/2 ~ N(0, 1/2)`, `om = f_xyy ~ N(0, 2)`, `Y = f_yyy ~ N(0, 6)`, independent, as stated.
+
+**2.3 The three negligible parts.** (i) `Q^W(r M_4 > 3k/10) <= (10 r / 3k)^p E_Q[(W_r/r^2) M_4^p] / (Z_r / r^2) = O(r^p)` for every
+`p`, by the uniform moments and the floor [LP] (5.5) (this is [LP] (7.7) with `p = 4`). (ii) On the depth failure
+`{lambda <= (4/(3k)) r M_3^2}`, the part `{lambda > sqrt r}` is contained in `{M_3 >= (3k/4)^(1/2) r^(-1/4)}`, so its `Q^W`-mass is
+at most `E_Q[(W_r/r^2) M_3^p] (4/(3k))^(p/2) r^(p/4) / z_* = O(r^(p/4))`, which is `o(r^3)` for `p > 12`. (iii) `Z_r / r^2 -> z_0(b, k) = 36 k^2 E[w^2 1{w < 0}]`
+by [LP] (5.4). It remains to show
+
+    r^(-5) E_Q[ W_r 1{lambda <= (4/(3k)) r M_3^2} 1{lambda <= sqrt r} ]  ->  p_w(0) E[ I(v, om, Y) ],
+    I = integral_{max(a, c)}^{8 M^2} (s - a)(s - c) ds / (6k).                                                (2.3)
+
+**2.4 The main term.** Apply (2.2) with `omega = -r u`, `u in (0, r^(-1/2)]`:
+
+    r^(-5) E_Q[...] = integral_0^{r^(-1/2)} p_{w,r}(-r u) E[ r^(-4) W_r(u) 1{u <= (4/(3k)) M_3(u)^2} ] du,
+
+where `W_r(u)`, `M_3(u)` are evaluated on the field `f_r(u) := f_r^perp + (-r u - mu_w(r)) g_r`. Since `r u <= sqrt r`, the
+field `f_r(u)` differs from `f_r^perp - mu_w(r) g_r` by at most `sqrt r ||g_r||_{C^5}` in `C^5(X)`, uniformly in `u`.
+
+*Pointwise limit.* Fix `u > 0`. By 2.1, 2.2 and (1.1), the coordinates of `f_r(u)` converge almost surely:
+`v -> v`, `om -> om`, `T`, `tau`, `nu` to finite limits, `M_3(u) -> max(|f_xxx|, |f_xxy|, |f_xyy|, |f_yyy|)(0) = max(12k, 2|v|, |om|, |Y|) = M`
+(the supremum of a continuous function over the shrinking square `D_r`, the third partials of `f_r(u)` converging uniformly to
+those of the limit field, whose `f_xxx(0)` is pinned at `12k`). By (1.2), `r^(-4) W_r(u) -> (6ku - v^2)(6k(u - om) + v^2)`
+on `{6ku > v^2, 6k(om - u) - v^2 < 0}` and to `0` on the complement, except on the null set `{6ku = v^2} cup {6k(u - om) = -v^2}`
+(`(v, om)` has a density); `f_xx(M) = r^2 T - 6kr < 0` eventually. With `s = 6ku`, `a = v^2`, `c = 6k om - v^2`, the limit is
+`(s - a)(s - c) 1{s > max(a, c)}`. The indicator `1{u <= (4/(3k)) M_3(u)^2}` converges to `1{u <= (4/(3k)) M^2} = 1{s <= 8 M^2}`
+for every `u` with `P(u = (4/(3k)) M^2) = 0`, i.e. for all `u` except `u = 192 k` (the atom of `M` at `12k`); a single `u` is
+Lebesgue-null. And `p_{w,r}(-ru) -> p_w(0)`.
+
+*Domination.* Let `N_r = 1 + ||f_r^perp||_{C^5(X)} + ||g_r||_{C^5(X)}`. For `r <= 1` and `u <= r^(-1/2)`, all of `|T|, |tau|, |nu|, |v|, |om|`
+on `f_r(u)` are bounded by `C N_r` by (1.1), so `r^(-4) W_r(u) <= C (1 + u)^2 N_r^4`, and `M_3(u) <= C N_r`, so
+`1{u <= (4/(3k)) M_3(u)^2} <= 1{u <= C N_r^2}`. Therefore the `u`-integrand is bounded by `sup_r sup p_{w,r} . C (1 + u)^2 N_r^4 1{u <= C N_r^2}`,
+whose `du`-integral is at most `C N_r^{10}`, with `sup_r E N_r^{10} < infinity` by 2.1. Explicitly, taking the expectation
+first: at fixed `u` the expected integrand is at most `C (1 + u)^2 min(E N_r^4, (C/u)^4 E N_r^{12})` (Markov on
+`{N_r^2 >= u/C}`), which is bounded uniformly in `r <= 1` by the integrable function `C' (1 + u)^2 min(1, u^-4)`
+(`sup_r E N_r^{12} < infinity` by 2.1); and at fixed `u` the integrand converges almost surely under the coupling and is
+uniformly integrable (`sup_r E N_r^8 < infinity`). Vitali's theorem at fixed `u` and dominated convergence in `u` give (2.3),
+where the limit field is the contact field given `f_yy(0) = 0` and `E` is over the law of 2.2.
+
+**2.5 Conclusion.** `Q^W(G_r^c) = [E_Q(W_r 1_{depth, lambda <= sqrt r}) + o(r^5)] / Z_r = r^3 [p_w(0) E I + o(1)] / [z_0 + o(1)]`, and
+`p_w(0) E[I] / z_0 = p_w(0) / (36 k^2 E[w^2 1{w<0}]) . E[I] = R(b) J(k)`, with `J(k) = E[I] / (36 k^2) = (216 k^3)^(-1) E[integral (s-a)(s-c) ds]`.
+The lower limit `max(a, c) <= 8 M^2` always: `M >= 2|v|` gives `8 M^2 >= 32 v^2 >= a`, and `M >= max(12k, |om|)` gives
+`8 M^2 >= 96 k |om| >= 6k om >= c`. QED.
+
+**Remarks.** (a) `R(b)` for the reference kernel: with `beta = b / sqrt2`, `p_w(0) = phi(beta)/sqrt2` and
+`E[w^2 1{w<0}] = (2 + b^2) Phi(beta) + sqrt2 b phi(beta)` (integrate `(sqrt2 t - b)^2 phi(t)` over `t < beta`); `R` is strictly
+decreasing on `b >= 0` (numerator decreasing; the denominator's derivative is `2 b Phi(beta) + 2 sqrt2 phi(beta) > 0`).
+(b) The leading size of `J`: since `M >= 12k` is forced by the pinned `f_xxx(0)`, `E[F(8M^2)] >= (512/3)(12k)^6 - 8 E[v^4 M^2]`,
+and `J(k) = (512/3)(12k)^6 / (216 k^3) (1 + delta(k)) = 2359296 k^3 (1 + delta(k))` with `delta -> 0` as `k` grows (the
+free third derivatives `2|v|, |om|, |Y|` rarely exceed `12k`); `delta(k) > 0` for `k <= 3/4` and `|delta(k)| < 2e-6` for
+`k >= 1` on the grid (both signs; the term `-8 E[v^4 M^2]` is negative). (c) For the torus kernel in a general frame the
+same formula holds with the exact Gaussian laws `w ~ N(mu_w, sigma_w^2)`, `(v, om, Y) ~ N(mu_odd, Sigma_odd)` read off the
+derivatives of `K_L(R .)` at 0; these differ from the reference values by `O(L^6 e^{-L^2/2})` (image sums), which is not
+enclosed here (section 6). (d) Theorem G is a statement about the sufficient criterion `G_r`, not about the elder pairing:
+the actual failure `1 - p_r` is `(alpha_1 + alpha_2) r^3 + o(r^3)` by [S] (S2), and `G_r^c` contains the failure event
+([LP] section 8), so `c_G >= alpha_1 + alpha_2` necessarily; the size of the gap is in section 5. Hence `c_G` bounds
+cap-route constants only: a constant for [LP] (1.1) itself is bounded below by `alpha_1 + alpha_2` where (S2) holds, not by `c_G`.
+
+## 3. Reduction of `J(k)` to three one-dimensional integrals
+
+With `F(s) = s^3/3 - (a + c) s^2/2 + a c s`, `integral_{max(a,c)}^{8M^2} (s-a)(s-c) ds = F(8M^2) - F(max(a, c))`. Since `M` depends on
+`om` only through `|om|`, and `a + c = 6k om`, `a c = v^2 (6k om - v^2)` are odd in `om` up to the `-v^4` term,
+`E[(a + c) M^4] = 0` and `E[a c M^2] = -E[v^4 M^2]`. Hence
+
+    216 k^3 J(k) = (512/3) I1 - 8 I2 - I3,    I1 = E[M^6],   I2 = E[v^4 M^2],   I3 = E[F(max(a, c))].
+
+Write `m = 12k`, `F_1(t) = P(|N(0,2)| <= t) = 2 Phi(t/sqrt2) - 1` (the law of `2|v|` and of `|om|`), `F_3(t) = 2 Phi(t/sqrt6) - 1`
+(the law of `|Y|`), `sigma_v^2 = 1/2`.
+
+- `I1 = m^6 + integral_m^infinity 6 t^5 [1 - F_1(t)^2 F_3(t)] dt`  (`P(M > t) = 1 - F_1^2 F_3` for `t >= m`).
+- `I2 = H E[v^4 1{|v| <= 6k}] + 4 E[v^6 1{|v| > 6k}] + integral_m^infinity 2t [1 - F_1(t) F_3(t)] E[v^4 1{6k < |v| < t/2}] dt`, where
+  `H = m^2 + integral_m^infinity 2t [1 - F_1 F_3] dt = E[max(m, |om|, |Y|)^2]`: condition on `v`; `E[max(m_v, |om|, |Y|)^2] = m_v^2 + integral_{m_v}^infinity 2t (1 - F_1 F_3)`
+  with `m_v = max(m, 2|v|)`, and exchange the order of integration on `{2|v| > m}`.
+- `I3 = 2 integral_0^infinity G(v) rho_v(v) dv`, where for fixed `v` the `om`-expectation is closed: with `theta = v^2/(3k)`
+  (the point `c = a`), `F(max(a,c)) = -(2/3) v^6 + 3k v^4 om` on `{om <= theta}` and `= [3 v^2 c^2 - c^3]/6` on `{om > theta}`, `c = 6k om - v^2`;
+  the truncated moments `E[om^j 1{om <= theta}]`, `E[om^j 1{om > theta}]`, `j <= 3`, are `Phi`, `phi` at `theta/sqrt2` times polynomials.
+
+Every remaining integral is one-dimensional with an integrand built from polynomials, `Phi(t/sigma)` and `phi(t/sigma)`.
+The truncated Gaussian moments use `M_j(x) = integral_x^infinity u^j phi(u) du`, `M_0 = 1 - Phi`, `M_1 = phi`, `M_j = x^(j-1) phi(x) + (j-1) M_(j-2)`.
+
+## 4. Certification and values
+
+`cap_coefficient.py` (standard library; exact rational interval arithmetic with outward rounding to `2^-160`; `exp(-t)` by the
+alternating series in fixed point with an integer rounding budget, `pi` by Machin, `Phi` by the fixed-point series of [Z]).
+Each one-dimensional integral over `[m, m + 30]` (in `t`) or `[0, 8]` (in `v`) is enclosed by cells of width `1/512` with the
+mean-value bound
+
+    integral_cell f  in  h f(mid) +- (h^2/8) width(f'(cell)),
+
+`f'` obtained by forward-mode interval differentiation (a pair (value, derivative) propagated through `+`, `x`, `Phi`, `phi`, `M_j`);
+the tails beyond the ranges are bounded in closed form (`1 - F_1^2 F_3 <= 2(1 - F_1) + (1 - F_3)`, `integral_T^infinity t^5 (1 - Phi(t/sigma)) dt <= (sigma^6/6) M_6(T/sigma)`, and
+polynomial majorants of `|F(max(a,c))|` against the incomplete moments of `v`). The enclosures of `I1`, `I2`, `I3` are combined exactly;
+`J`, `R` and `c_G` are printed as decimal brackets (rounded outward).
+
+| `k` | `J(k)` | width | `J / (2359296 k^3)` |
+|---|---|---|---|
+| `1/6` | `[586678.1211613003, 586680.4730695802]` | `2.35190827979` | `[53.71198619, 53.71220152]` |
+| `1/2` | `[301744.4364470241, 301744.4616663935]` | `0.025219369345` | `[1.02316771, 1.02316780]` |
+| `3/4` | `[995454.6123240485, 995454.6129283207]` | `0.000604272143` | `[1.00012720, 1.00012721]` |
+| `1` | `[2359292.9611053703, 2359292.9611122569]` | `0.000006886435` | `[0.99999871, 0.99999872]` |
+| `3/2` | `[7962621.6645357431, 7962621.6645378716]` | `0.000002128399` | `[0.99999970, 0.99999971]` |
+| `2` | `[18874366.3403946311, 18874366.3403964820]` | `0.000001850785` | `[0.99999991, 0.99999992]` |
+
+| `b` | `R(b)` |
+|---|---|
+| `0` | `[0.282094791773, 0.282094791774]` |
+| `1/4` | `[0.211225157119, 0.211225157120]` |
+| `1/2` | `[0.155804629374, 0.155804629375]` |
+| `3/4` | `[0.113118131654, 0.113118131655]` |
+| `1` | `[0.080766267688, 0.080766267689]` |
+| `6/5` | `[0.060913700223, 0.060913700224]` |
+
+`c_G(b, k) = R(b) J(k)` (four decimals, rounded outward):
+
+| `b` \ `k` | `1/6` | `1/2` | `3/4` | `1` | `3/2` | `2` |
+|---|---|---|---|---|---|---|
+| `0` | `[165498.8424, 165499.5059]` | `[85120.5339, 85120.5411]` | `[280812.5615, 280812.5618]` | `[665544.2565, 665544.2566]` | `[2246214.1004, 2246214.1005]` | `[5324360.4426, 5324360.4427]` |
+| `1/4` | `[123921.1783, 123921.6752]` | `[63736.0159, 63736.0214]` | `[210265.0568, 210265.0571]` | `[498342.0264, 498342.0265]` | `[1681906.0121, 1681906.0122]` | `[3986740.9957, 3986740.9958]` |
+| `1/2` | `[91407.1672, 91407.5337]` | `[47013.1800, 47013.1841]` | `[155096.4369, 155096.4371]` | `[367588.7653, 367588.7654]` | `[1240613.3172, 1240613.3173]` | `[2940713.6523, 2940713.6524]` |
+| `3/4` | `[66363.9329, 66364.1990]` | `[34132.7668, 34132.7698]` | `[112603.9658, 112603.9660]` | `[266878.8117, 266878.8118]` | `[900716.8857, 900716.8858]` | `[2135033.0565, 2135033.0566]` |
+| `1` | `[47383.8021, 47383.9922]` | `[24370.7719, 24370.7740]` | `[80399.1536, 80399.1538]` | `[190551.2868, 190551.2869]` | `[643111.2328, 643111.2329]` | `[1524412.1243, 1524412.1244]` |
+| `6/5` | `[35736.7352, 35736.8785]` | `[18380.3701, 18380.3717]` | `[60636.8238, 60636.8239]` | `[143713.2641, 143713.2642]` | `[485032.7490, 485032.7491]` | `[1149707.4931, 1149707.4932]` |
+
+**Controls (floating, not part of the certificate).** Monte Carlo of `J(k)` from the definition (`2 x 10^5` samples of
+`(v, om, Y)`, seed 2026) and a two-dimensional midpoint quadrature in `(v, om)` with the `Y`-expectation in closed form:
+- `k = 1/6`: Monte Carlo `574856.1 +- 7874.9` (`-1.50` s.e. from the certified midpoint), quadrature `586681.402`.
+- `k = 1/2`: Monte Carlo `301413.4 +- 238.0` (`-1.39` s.e. from the certified midpoint), quadrature `301744.426`.
+- `k = 3/4`: Monte Carlo `995377.2 +- 43.1` (`-1.80` s.e. from the certified midpoint), quadrature `995454.613`.
+- `k = 1`: Monte Carlo `2359221.4 +- 58.1` (`-1.23` s.e. from the certified midpoint), quadrature `2359292.961`.
+- `k = 3/2`: Monte Carlo `7962462.0 +- 130.8` (`-1.22` s.e. from the certified midpoint), quadrature `7962621.665`.
+- `k = 2`: Monte Carlo `18874082.4 +- 232.6` (`-1.22` s.e. from the certified midpoint), quadrature `18874366.340`.
+
+The `--check` mode requires the Monte Carlo value within four standard errors and the quadrature within `3 x 10^-4` of the
+certified midpoint, the enclosure widths below `10^-5` relative, `J >= 2359296 k^3` for `k <= 3/4` and `|J / (2359296 k^3) - 1| < 2 x 10^-6`
+for `k >= 1`, the closed-form `R(b)` against a floating evaluation, `R` decreasing on the grid (consecutive enclosures disjoint), the two headline inequalities,
+and the exact arithmetic of [CAP] (3).
+
+## 5. Consequences for `C_cap`, `r_*`, and the size of the cap criterion's slack
+
+All numbers in this section are reference-kernel values (`c_G^{ref}`). For the torus kernel `K_L` in a frame `R` only the
+symbolic statements transfer (`C_cap >= c_G^{L,R}`; `C_cap r^3 < 1` forces `r < (c_G^{L,R})^{-1/3}`); no numerical value of
+`c_G^{L,R}` is claimed, and its deviation from `c_G^{ref}` is not enclosed here (section 6).
+
+1. **Lower bound for every cap-route constant `C_cap`.** If `Q^W(G_r^c) <= C_cap r^3` for all small `r` at `(b, k)`, then
+   `C_cap >= c_G(b, k)` (for the kernel and frame in question). This does not bound a constant `C` of [LP] (1.1) itself: the
+   only relation is `1 - p_r <= Q^W(G_r^c)`, and a coefficient of the larger probability does not bound the constants of the
+   smaller one (logical countermodel of review 5372981851 on Math-#203: one uniform `U`, `F_r = {U <= r^3}` inside
+   `G_r^c = {U <= 2 r^3}`; the cap coefficient is 2 while `C = 1` is admissible for `F_r`). On the band of Math-#195
+   (`B = [0, 1]`, `K = [1/2, 2]`) the grid maximum of `c_G^{ref}` is at `(0, 2)`: `C_cap > 5.32 x 10^6` for the reference kernel. Since `R` is decreasing in `b`, the band maximum
+   over `b` is at `b = 0` for every `k`; the dependence on `k` is `~ k^3`.
+2. **Where the cap route can be nontrivial.** `C_cap r^3 < 1` forces `r < c_G(b,k)^{-1/3}`: `5.7 x 10^-3` at `(0, 2)`,
+   `3.4 x 10^-2` at `(1, 1/2)` (the band's smallest `c_G` among the grid points), `3.0 x 10^-2` at `(6/5, 1/6)`. Any `r_*`
+   for which the cap bound [LP] (7.8) is informative on the band is below `6 x 10^-3` at the band's corner, whatever the proof
+   of (7.8). A proof of [LP] (1.1) that does not pass through `Q^W(G_r^c)` is not constrained by this item.
+3. **The recorded constant.** [CAP] (2)–(3) give, for the fixed-axis SIDE24 law (`b = 6/5`, `k = 1/6`),
+   `C3_new + C4_new/20 ≈ 2.36 x 10^23` (`< 2.4 x 10^23`; exact rational arithmetic, replayed in `--check`), against the
+   reference-kernel value `c_G^{ref}(6/5, 1/6) = 35736.8`: a factor above `6 x 10^18` between the recorded cap-route constant
+   and the reference-kernel sharp cap coefficient (`--check` verifies it with `C3_new` alone against the upper end of the
+   enclosure; the comparison with SIDE24's own finite-`L` coefficient is not enclosed). The slack is in the moment bounds
+   (`||T||_8 < 320` for a supremum over a set of diameter `<= 6r`, `||M_4||_8 < 340`), not in the geometry.
+4. **The criterion against the truth.** The actual elder-pairing failure has coefficient `alpha_1 + alpha_2` ([S] (S2)), and
+   by [NUM] `alpha_j = (p_b(0)/z_0) J_j(k)` with the same `b`-factor as `c_G` (`R(b) = 36 k^2 p_b(0)/z_0`), so the ratio is `b`-free:
+
+       c_G / (alpha_1 + alpha_2) = 36 k^2 J(k) / (J_1(k) + J_2(k)),
+
+   with `J_1 + J_2` from [NUM] (GH60: `36.785`, `172.87`, `1112.3` at `k = 1/2, 1, 2`): `7.4 x 10^4`, `4.9 x 10^5`, `2.4 x 10^6`.
+   (At `b = 0`: `alpha_1 + alpha_2 = 1.153, 1.355, 2.179` against `c_G = 8.5 x 10^4, 6.7 x 10^5, 5.3 x 10^6`.) The sufficient
+   condition `lambda > (4/(3k)) r M_3^2` with `M_3 >= 12k` forced fails whenever `lambda <= 192 k r`, a set of `w`-probability of
+   order `192 k r p_w(0)` whose typed weight is of order `(192 k r)^2`, while the actual failure requires the competing saddle
+   geometry of [S]; the ratio grows like `k^3 . 36 k^2 / (J_1 + J_2)(k)`.
+
+## 6. What this does not do
+
+Not a proof or a review of [LP] Theorem A, of [CAP], or of [S]: (5.4), (5.5), (7.6)–(7.7), the coupling (4.2) and the
+uniform moments (4.1) are consumed as statements; [S] (S2) and the [NUM] values are quoted for comparison only. No upper
+bound `C` or `C_cap` and no `r_*` are supplied (C8 stays OPEN on both), and no lower bound for the constant of [LP] (1.1)
+itself (section 5, item 1). No rate in the `o(r^3)`. The certified numbers are for the
+reference kernel; for the torus kernel `K_L` in a frame `R` the coefficient is the same formula with the exact jet law at 0,
+whose deviation from the reference (`O(L^6 e^{-L^2/2})`) is not enclosed here, so every numerical consequence in sections 0
+and 5 is a reference-kernel statement (a certified finite-`L`, all-frame enclosure is a natural successor). Planar only (`d = 2`); the same argument in
+`d >= 3` would need the eigenvalue boundary layer of [LP] section 7 and is not attempted. Nothing on `p_r` beyond the
+comparison in section 5.
+
+## 7. Verification and provenance
+
+`python3 -B -S cap_coefficient.py --check --procs 2` (also `-B -O -S`): recomputes every enclosure and compares the decimal
+records exactly with `RESULTS.json`; about four minutes per `k` on one core (`k = 1/6` first, then the others in parallel).
+Mutants `no-third-jet` (drop `|Y|` from `M`), `sign-lo` (add `I3` instead of subtracting), `variance-yyy` (`Var Y = 2`) exit 1.
+`cap_coefficient.py --procs 4` regenerates `RESULTS.json` (about eight minutes on four cores); `--quick K` certifies one `k`.
+Workflow `.github/workflows/c8-cap-failure-coefficient-planar.yml`: manifest, main-resident pins ([LP], [CAP], [NUM], [S]),
+both modes, mutants, clean tree.
+
+Revisions: v1.1 (Codex review 5372193385 on Math-#203: the `|v| > V_MAX` tail majorant, disjoint monotonicity enclosures,
+reference-kernel qualification, `delta(k)` on the grid only; `RESULTS.json` unchanged). v1.2 (OpenAI review 5372981851,
+confirmed by xAI review 5373266293: the lower bound concerns cap-route constants `C_cap` of [LP] (7.8) only, not the constant
+of [LP] (1.1); the domination step of 2.4 made explicit; no numerical change, `RESULTS.json` and `cap_coefficient.py` unchanged).
+v1.2.1 (Slice C successor read 5974168378 on Math-#203, findings F1–F3, wording only: the unenclosed finite-`L` bound for
+SIDE24 is removed from section 0; the [CAP] comparison is stated against `c_G^{ref}`, with the value `≈ 2.36 x 10^23`; only
+the symbolic statements transfer to a finite torus; no order is asserted for the deviation of `c_G` itself (Remark (c) derives
+it for the jet-law parameters only); `RESULTS.json` and `cap_coefficient.py` unchanged).
+
+Author lane Anthropic / Claude, 30 September 2026. Scientific effect NONE. The author will not merge.
diff --git a/frontiers/c8_cap_failure_coefficient_planar_20260930/README.md b/frontiers/c8_cap_failure_coefficient_planar_20260930/README.md
new file mode 100644
index 000000000..0c0adf6af
--- /dev/null
+++ b/frontiers/c8_cap_failure_coefficient_planar_20260930/README.md
@@ -0,0 +1,31 @@
+# The cap-failure coefficient in the plane (C8: what any cap-route constant `C_cap` must exceed)
+
+**Object:** `CL-C8-CAP-FAILURE-COEFFICIENT-PLANAR-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** author-side
+limit theorem with certified enclosures of its coefficient (exact rational interval arithmetic; standard library only).
+**Scientific effect:** NONE (no register, catalog, GRAPH or STATUS change; C8 stays OPEN: no `C`, no `r_*` supplied).
+
+For the planar six-pin law of [LP] and the good event `G_r = {lambda_min(-A_M) > (4/(3k)) r M_3^2, r M_4 <= 3k/10}` of
+[LP] section 7 / [CAP] (1), Theorem G (`NOTE.md`) gives the exact leading behaviour of the cap-failure probability that the
+proof of Theorem A bounds ([LP] (7.8)):
+
+    Q^W(G_r^c) = c_G(b, k) r^3 + o(r^3),      c_G(b, k) = R(b) J(k),
+
+with `R(b)` in closed form and `J(k)` a three-dimensional Gaussian expectation reduced to one-dimensional integrals and
+certified at `k in {1/6, 1/2, 3/4, 1, 3/2, 2}` for the reference kernel (`J(k) = 2359296 k^3 (1 + delta(k))`, `delta -> 0`;
+the finite-torus jet-law deviation, of order `L^6 e^{-L^2/2}`, is not enclosed). Reference-kernel consequences: every
+cap-route constant `C_cap` (any constant with `Q^W(G_r^c) <= C_cap r^3`) at `(b, k)` is at least `c_G(b, k)`; on the
+Math-#195 band `C_cap > 5.3 x 10^6` (not a lower bound for the constant of Theorem A's `1 - p_r <= C r^3`, which bounds a
+smaller probability); the cap route is informative only below `r = c_G^{-1/3}` (`6 x 10^-3` at the band's corner `(0, 2)`);
+the recorded cap-route constant `≈ 2.36 x 10^23` of [CAP] at `(6/5, 1/6)` exceeds the reference-kernel sharp cap coefficient
+`c_G^{ref}(6/5, 1/6)` by more than `6 x 10^18` (the comparison with SIDE24's own finite-`L` coefficient is not enclosed); the criterion overstates the true failure rate `alpha_1 + alpha_2` by `10^4`–`10^6`.
+
+- `NOTE.md` — statement, proof of Theorem G (coupling, regression on `f_yy(M)`, dominated convergence, parity), the
+  one-dimensional reduction, certification method, tables, consequences, non-claims.
+- `cap_coefficient.py` — `--check --procs N` recomputes every enclosure and compares exactly with `RESULTS.json` (about four
+  minutes per `k` on one core); `--mutant {no-third-jet, sign-lo, variance-yyy}` must exit 1; no flags regenerates
+  `RESULTS.json`; `--quick K` certifies one `k`.
+- `RESULTS.json`, `SOURCE_MAP.json` (pins [LP], [CAP], [NUM], [S] on `main`), `SOURCE_FILES.json`.
+- Workflow `.github/workflows/c8-cap-failure-coefficient-planar.yml` replays manifest, pins, both interpreter modes and the
+  six mutant runs on every pull request.
+
+Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge.
diff --git a/frontiers/c8_cap_failure_coefficient_planar_20260930/RESULTS.json b/frontiers/c8_cap_failure_coefficient_planar_20260930/RESULTS.json
new file mode 100644
index 000000000..e3a8e25a4
--- /dev/null
+++ b/frontiers/c8_cap_failure_coefficient_planar_20260930/RESULTS.json
@@ -0,0 +1,394 @@
+{
+ "J": [
+  {
+   "I1_E_M6": [
+    "3437.9998534153",
+    "3438.0136333034"
+   ],
+   "I2_E_v4M2": [
+    "9.4118754485",
+    "9.4118817190"
+   ],
+   "I3_E_F_lo": [
+    "-1.4413227303",
+    "-1.4412321736"
+   ],
+   "J": [
+    "586678.1211613003",
+    "586680.4730695802"
+   ],
+   "J_over_lead": [
+    "53.71198619",
+    "53.71220152"
+   ],
+   "J_width": "2.351908279790",
+   "k": "1/6",
+   "lead_2359296_k3": "32768/3"
+  },
+  {
+   "I1_E_M6": [
+    "47738.1342147530",
+    "47738.1382032075"
+   ],
+   "I2_E_v4M2": [
+    "27.1261695254",
+    "27.1261697283"
+   ],
+   "I3_E_F_lo": [
+    "-8.5543347518",
+    "-8.5541096317"
+   ],
+   "J": [
+    "301744.4364470241",
+    "301744.4616663935"
+   ],
+   "J_over_lead": [
+    "1.02316771",
+    "1.02316780"
+   ],
+   "J_width": "0.025219369345",
+   "k": "1/2",
+   "lead_2359296_k3": "294912"
+  },
+  {
+   "I1_E_M6": [
+    "531511.2852574257",
+    "531511.2855777997"
+   ],
+   "I2_E_v4M2": [
+    "60.7520277408",
+    "60.7520277468"
+   ],
+   "I3_E_F_lo": [
+    "-28.2140373386",
+    "-28.2136502281"
+   ],
+   "J": [
+    "995454.6123240485",
+    "995454.6129283207"
+   ],
+   "J_over_lead": [
+    "1.00012720",
+    "1.00012721"
+   ],
+   "J_width": "0.000604272143",
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+ ],
+ "kind": "certified enclosures of the limit coefficient c_G(b, k) = R(b) J(k) of Theorem G (NOTE.md); the limit theorem itself is author-side",
+ "mutant": null,
+ "object": "CL-C8-CAP-FAILURE-COEFFICIENT-PLANAR-20260930-v1",
+ "parameters": {
+  "CELL": "1/512",
+  "LEAD": 2359296,
+  "PBITS": 160,
+  "T_SPAN": "30",
+  "V_MAX": "8",
+  "b_grid": [
+   "0",
+   "1/4",
+   "1/2",
+   "3/4",
+   "1",
+   "6/5"
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+  "k_grid": [
+   "1/6",
+   "1/2",
+   "3/4",
+   "1",
+   "3/2",
+   "2"
+  ]
+ },
+ "schema": 1,
+ "scientific_effect": "NONE",
+ "statement": "Q^W(G_r^c) = c_G(b, k) r^3 + o(r^3) as r -> 0 for the planar six-pin law of [LP] with the good event G_r of [LP] section 7 / [CAP] (1); reference-kernel jet law"
+}
diff --git a/frontiers/c8_cap_failure_coefficient_planar_20260930/SOURCE_FILES.json b/frontiers/c8_cap_failure_coefficient_planar_20260930/SOURCE_FILES.json
new file mode 100644
index 000000000..51cee3e6c
--- /dev/null
+++ b/frontiers/c8_cap_failure_coefficient_planar_20260930/SOURCE_FILES.json
@@ -0,0 +1,40 @@
+{
+ "schema": 1,
+ "scientific_effect": "NONE",
+ "certified": true,
+ "review_status": "AUTHOR_SIDE_LIMIT_THEOREM_WITH_CERTIFIED_COEFFICIENT",
+ "object": "CL-C8-CAP-FAILURE-COEFFICIENT-PLANAR-20260930-v1",
+ "files": [
+  {
+   "path": "NOTE.md",
+   "bytes": 29358,
+   "sha256": "b3e8d67f7c9f0ee3204e0cfc2e9ce134087f3ff56e26ec4b8a9268040e24a8dc",
+   "git_blob": "2838b76b21a884bd77d18f294d1730780b3ed3ab"
+  },
+  {
+   "path": "README.md",
+   "bytes": 2744,
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+   "git_blob": "0c0adf6af489dffde48d1d22f4da4c27aa1d10cd"
+  },
+  {
+   "path": "RESULTS.json",
+   "bytes": 6427,
+   "sha256": "d3de1a20793fbbcbb051b5740af45628627b905827fe2e0d4758ac9e37c36088",
+   "git_blob": "e3a8e25a49fb572f0e3fb95c88f070ef9fa9d731"
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+   "git_blob": "bb978458dc78895d8289dab6a8b6c47219ce7fd1"
+  },
+  {
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+  }
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diff --git a/frontiers/c8_cap_failure_coefficient_planar_20260930/SOURCE_MAP.json b/frontiers/c8_cap_failure_coefficient_planar_20260930/SOURCE_MAP.json
new file mode 100644
index 000000000..bb978458d
--- /dev/null
+++ b/frontiers/c8_cap_failure_coefficient_planar_20260930/SOURCE_MAP.json
@@ -0,0 +1,120 @@
+{
+  "schema": 1,
+  "object": "CL-C8-CAP-FAILURE-COEFFICIENT-PLANAR-20260930-v1",
+  "scientific_effect": "NONE",
+  "certified": true,
+  "review_status": "AUTHOR_SIDE_LIMIT_THEOREM_WITH_CERTIFIED_COEFFICIENT",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "main_at_creation": "3e0a91b69892b9717564c592e7a966492a81566a",
+  "note": "Theorem G: for the planar six-pin law of [LP] and the good event G_r of [LP] section 7 / [CAP] (1), Q^W(G_r^c) = c_G(b, k) r^3 + o(r^3) with c_G = R(b) J(k) explicit (R closed form; J a Gaussian expectation reduced to one-dimensional integrals and certified on a k-grid). Consumes [LP] (4.1), (4.2), (5.4), (5.5), (7.6)-(7.7) and the definitions of Q, W_r, Z_r, G_r as statements; [CAP] for D, M_3, M_4 and its recorded constant; [NUM] for the exact contact structure and the alpha values; [S] (S2) for the comparison only. Supplies neither C nor r_*: C8 stays OPEN; it shows every cap-route constant C_cap (Q^W(G_r^c) <= C_cap r^3, the bound [LP] (7.8)) on the Math-#195 band exceeds 5.3e6 and locates the cap route's informative range below r = c_G^(-1/3); it gives no lower bound for the constant of [LP] (1.1), which bounds the smaller probability 1 - p_r (v1.2, review 5372981851).",
+  "sources": [
+    {
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "id": "LP",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "sections 1, 3, 4, 5, 7: the law Q, the weight W_r, the normalizer Z_r, the preconditioned contact frame (3.1)-(3.4), the coupling (4.2) and the uniform moments (4.1), the limit (5.4) and the floor (5.5), the good event G_r of section 7 and the bounds (7.5)-(7.8); Theorem A; section 16 lists a numerical r_*, C among the items not supplied",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
+      "bytes": 15160,
+      "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc",
+      "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1",
+      "id": "CAP",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "section 1: the cylinder D = [-2r, 2r] x ball(0, 2r), the partial-block norms M_3, M_4 (coordinate partials in d = 2), condition (1); the probabilistic corollary (2)-(3) with C3_new, C4_new for the fixed-axis SIDE24 law (b = 6/5, k = 1/6): the only explicit C on record, compared with c_G(6/5, 1/6) here",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json",
+      "bytes": 4095,
+      "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb",
+      "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f",
+      "id": "NUM",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "exact_structure: the planar contact law (f_yy | even pins ~ N(-b, 2); odd 3-jet | odd pins ~ N(0, diag(2, 2, 6))) used in Theorem G's identification of (w, v, om, Y); prefactor: the values alpha_1, alpha_2 (GH60) quoted in section 5 for the comparison with the true failure coefficient (Math-#168, merged)",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/local_elder_geometry_20260930/PROOF.md",
+      "bytes": 39722,
+      "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b",
+      "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113",
+      "id": "S",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "Theorem S, (S2): r^-3 (1 - p_r) -> alpha_1 + alpha_2 (the actual planar elder-failure coefficient), quoted for the comparison in section 5 only; not consumed by Theorem G (Math-#170, merged)",
+      "current_required": true,
+      "on_main": true
+    }
+  ],
+  "catalog_reference": {
+    "path": "reviews/candidates_pending_20260928/CANDIDATES.md",
+    "entry": "C8. Numerical constants (still open as of 2026-09-29): explicit finite C, r_*, z_*, c_{B,K}, c_{d,L} on a declared compact-mark / torus band; disposition OPEN",
+    "effect_here": "none; this packet supplies a lower bound for every cap-route constant C_cap (not for the constant of (1.1)) and no upper bound; C8 stays OPEN on C and r_*"
+  },
+  "related_records": [
+    "Math-#195 (z_*)",
+    "Math-#197 (c_{B,K})",
+    "Math-#199, Math-#200 (c_{d,L}, d = 4, 5)"
+  ],
+  "nonauthor_reads": [
+    {
+      "comment": "5372193385",
+      "pr": "Math-#203",
+      "lane": "OpenAI / Codex (automated Codex review)",
+      "scope": "review of 8968f5ef82; findings applied in v1.1",
+      "kind": "pull_request_review"
+    },
+    {
+      "comment": "5372981851",
+      "pr": "Math-#203",
+      "lane": "OpenAI (as recorded in the v1.2 revision note)",
+      "scope": "P1: lower bound concerns cap-route constants only; applied in v1.2",
+      "kind": "pull_request_review"
+    },
+    {
+      "comment": "5373266293",
+      "pr": "Math-#203",
+      "lane": "xAI / Grok team",
+      "scope": "Slice C at 683ed30: P1 confirmed",
+      "kind": "pull_request_review"
+    },
+    {
+      "comment": "5373441664",
+      "pr": "Math-#203",
+      "lane": "nonauthor (lane not stated in the review)",
+      "scope": "P1 amendment verified at a388580: scoped PASS_TECHNICAL",
+      "kind": "pull_request_review"
+    },
+    {
+      "comment": "5378711140",
+      "pr": "Math-#203",
+      "lane": "xAI / Grok team",
+      "scope": "bounded Slice A (sections 1-2) and reduction note at a388580; not a certificate acceptance",
+      "kind": "pull_request_review"
+    },
+    {
+      "comment": "5402161720",
+      "pr": "Math-#203",
+      "lane": "OpenAI / GPT-5.6 Sol",
+      "scope": "Slice B (sections 3-4, cap_coefficient.py, RESULTS.json): PASS_TECHNICAL_SCOPED / ACCEPT_SCOPED at a388580",
+      "kind": "pull_request_review"
+    },
+    {
+      "comment": "5974168378",
+      "pr": "Math-#203",
+      "lane": "Grok Bot agent 14 (Grok support, routed by the Chief of Staff 5974102560)",
+      "scope": "R7 Slice C successor read of the v1.2 wording at a388580, no execution: ACCEPT_WITH_FINDINGS; F1 (required) and F2-F3 (recommended) applied in v1.2.1, wording only",
+      "kind": "issue_comment"
+    }
+  ],
+  "nonauthor_acceptance": false,
+  "governing_status_changed": false,
+  "main_at_update": "3e0a91b69892b9717564c592e7a966492a81566a"
+}
diff --git a/frontiers/c8_cap_failure_coefficient_planar_20260930/cap_coefficient.py b/frontiers/c8_cap_failure_coefficient_planar_20260930/cap_coefficient.py
new file mode 100644
index 000000000..cfdaa4d7c
--- /dev/null
+++ b/frontiers/c8_cap_failure_coefficient_planar_20260930/cap_coefficient.py
@@ -0,0 +1,728 @@
+#!/usr/bin/env python3
+"""Exact leading coefficient of the cap-failure probability of [LP] Theorem A in the plane.
+
+G_r = {lambda_min(-A_M) > (4/(3k)) r M_3^2, r M_4 <= 3k/10} is the good event of [LP] section 7 / [CAP] (1), with M_3, M_4 the
+partial-block derivative norms on the cylinder D = [-2r, 2r] x [-2r, 2r] and Q^W = W_r Q / Z_r the typed pair law.  NOTE.md
+Theorem G:  Q^W(G_r^c) = c_G(b, k) r^3 + o(r^3),  c_G = R(b) J(k),
+
+    R(b) = p_w(0) / E[w^2 1{w < 0}],  w ~ N(-b, 2)  (closed form in Phi, phi),
+    J(k) = (216 k^3)^(-1) E[ integral_{max(a, c)}^{8 M^2} (s - a)(s - c) ds ],
+    a = v^2,  c = 6k om - v^2,  M = max(12k, 2|v|, |om|, |Y|),  v ~ N(0, 1/2), om ~ N(0, 2), Y ~ N(0, 6) independent
+
+(the contact jets -f_xxy/2, f_xyy, f_yyy at the pins' midpoint given the six pins, reference kernel).  By symmetry in om the
+expectation reduces to three one-dimensional integrals (NOTE.md section 3):
+
+    216 k^3 J(k) = (512/3) I1 - 8 I2 - I3,
+    I1 = E[M^6],   I2 = E[v^4 M^2],   I3 = E[F(max(a, c))],   F(s) = s^3/3 - (a + c) s^2/2 + a c s.
+
+Certification: exact rational interval arithmetic (outward rounding to 2^-160; exp by alternating series, pi by Machin,
+Phi by a fixed-point series with an integer rounding budget); each one-dimensional integral by cells with the mean-value
+enclosure  integral_cell f  in  h f(mid) +- (h^2/8) width(f'(cell)),  f' from forward-mode interval differentiation, and
+closed-form Gaussian tails.  Standard library only; deterministic.  The interval toolkit is the one of
+frontiers/c8_normalizer_floor_planar_20260930/floor.py (Math-#195).
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import random
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-third-jet', 'sign-lo', 'variance-yyy')
+MUT = None
+PBITS = 160
+K_GRID = (Fr(1, 6), Fr(1, 2), Fr(3, 4), Fr(1), Fr(3, 2), Fr(2))
+B_GRID = (Fr(0), Fr(1, 4), Fr(1, 2), Fr(3, 4), Fr(1), Fr(6, 5))
+CELL = Fr(1, 512)        # cell width of the certified quadratures
+T_SPAN = Fr(30)          # integrate t on [12k, 12k + T_SPAN]; the rest is a closed-form tail
+V_MAX = Fr(8)            # integrate v on [0, V_MAX]; the rest is a closed-form tail
+LEAD = 2359296           # J(k) ~ LEAD k^3 = (512/3)(12k)^6 / (216 k^3): the pinned f_xxx forces M >= 12k
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from floor.py, Math-#195)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+def phi_iv(x):
+    """Standard normal density on an interval (via exp(-x^2/2) monotone in x^2)."""
+    return exp_neg(x.sq() * Fr(1, 2)) / SQRT2PI
+
+
+# ----------------------------------------------------------------------------- forward-mode interval differentiation
+class DV:
+    """Pair (value interval, derivative interval) of a function of one real variable, both rigorous on a cell."""
+    __slots__ = ('v', 'd')
+
+    def __init__(self, v, d):
+        self.v = v if isinstance(v, IV) else IV(v)
+        self.d = d if isinstance(d, IV) else IV(d)
+
+    @staticmethod
+    def var(x):
+        return DV(x, IV(1))
+
+    @staticmethod
+    def const(c):
+        return DV(c, IV(0))
+
+    def __add__(self, o):
+        o = o if isinstance(o, DV) else DV.const(o)
+        return DV(self.v + o.v, self.d + o.d)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return DV(-self.v, -self.d)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, DV) else DV.const(o)))
+
+    def __rsub__(self, o):
+        return DV.const(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, DV) else DV.const(o)
+        return DV(self.v * o.v, self.d * o.v + self.v * o.d)
+    __rmul__ = __mul__
+
+    def scale(self, c):
+        c = Fr(c)
+        return DV(self.v * c, self.d * c)
+
+    def sq(self):
+        return DV(self.v.sq(), self.d * self.v * 2)
+
+    def pow(self, n):
+        out = DV.const(1)
+        for _ in range(n):
+            out = out * self
+        return out
+
+
+def Phi_dv(x):
+    """Phi(x) with derivative phi(x) x'."""
+    return DV(Phi(x.v), phi_iv(x.v) * x.d)
+
+
+def phi_dv(x):
+    """phi(x) with derivative -x phi(x) x'."""
+    p = phi_iv(x.v)
+    return DV(p, -(x.v * p) * x.d)
+
+
+def upper_moment(j, x):
+    """M_j(x) = integral_x^infinity u^j phi(u) du (x >= 0), as DV in x (derivative -x^j phi(x) x'); by the recursion
+    M_0 = 1 - Phi, M_1 = phi, M_j = x^(j-1) phi(x) + (j - 1) M_(j-2)."""
+    p = phi_iv(x.v)
+    P = Phi(x.v)
+    vals = [IV(1) - P, p]
+    for m in range(2, j + 1):
+        xp = IV(x.v.lo ** (m - 1), x.v.hi ** (m - 1))
+        vals.append(xp * p + vals[m - 2] * (m - 1))
+    xj = IV(x.v.lo ** j, x.v.hi ** j) if j > 0 else IV(1)
+    return DV(vals[j], -(xj * p) * x.d)
+
+
+def abs_moment_between(j, sigma2, lo, hi):
+    """E[|X|^j 1{lo < |X| < hi}] for X ~ N(0, sigma2), lo, hi DV in the integration variable (0 <= lo <= hi)."""
+    sig = isqrt_iv(IV(sigma2))
+    sigj = IV(sig.lo ** j, sig.hi ** j)
+    inv = sig.inv()
+    a = DV(lo.v * inv, lo.d * inv)
+    b = DV(hi.v * inv, hi.d * inv)
+    return (upper_moment(j, a) - upper_moment(j, b)) * DV.const(sigj * 2)
+
+
+def abs_moment_tail(j, sigma2, lo):
+    """E[|X|^j 1{|X| > lo}] for X ~ N(0, sigma2), lo rational >= 0 (interval)."""
+    sig = isqrt_iv(IV(sigma2))
+    sigj = IV(sig.lo ** j, sig.hi ** j)
+    return upper_moment(j, DV.const(IV(lo) * sig.inv())).v * sigj * 2
+
+
+# ----------------------------------------------------------------------------- certified one-dimensional quadrature
+def integrate(f, a, b, h, nf=1):
+    """Enclosures of integral_a^b f_i for f: DV -> (DV, ..., DV) (nf components, rigorous on intervals), by cells of
+    width h with the mean-value enclosure  integral_cell f_i in h f_i(mid) +- (h^2 / 8) width(f_i'(cell))  (pair t and
+    2 mid - t: f(t) + f(2 mid - t) - 2 f(mid) = (t - mid)(f'(xi_1) - f'(xi_2)) is bounded by |t - mid| width(f'(cell)))."""
+    a, b, h = Fr(a), Fr(b), Fr(h)
+    n = int(math.ceil((b - a) / h))
+    tot = [IV(0) for _ in range(nf)]
+    t0 = a
+    for i in range(n):
+        t1 = min(b, a + (i + 1) * h)
+        w = t1 - t0
+        mid = (t0 + t1) / 2
+        fm = f(DV.var(IV(mid)))
+        cell = f(DV.var(IV(t0, t1)))
+        for j in range(nf):
+            err = cell[j].d.width() * w * w / 8
+            tot[j] = tot[j] + fm[j].v * w + IV(-err, err)
+        t0 = t1
+    return tot
+
+
+# ----------------------------------------------------------------------------- the model
+SIG2_V = Fr(1, 2)       # v = -f_xxy(0)/2
+SIG2_OM = Fr(2)         # om = f_xyy(0)
+SIG2_Y = Fr(6)          # Y = f_yyy(0)
+SIG2_2V = Fr(2)         # 2|v| has the law of |N(0, 2)|, as |om|
+
+
+def sig2_y():
+    return Fr(2) if MUT == 'variance-yyy' else SIG2_Y
+
+
+def F_abs(t, sigma2):
+    """P(|N(0, sigma2)| <= t) = 2 Phi(t / sigma) - 1 as DV."""
+    inv = isqrt_iv(IV(sigma2)).inv()
+    return Phi_dv(DV(t.v * inv, t.d * inv)).scale(2) - 1
+
+
+def F3(t):
+    if MUT == 'no-third-jet':
+        return DV.const(1)
+    return F_abs(t, sig2_y())
+
+
+def tail_t5(T):
+    """integral_T^infinity 6 t^5 (1 - F1(t)^2 F3(t)) dt <= sum over the union bound 2(1 - F1) + (1 - F3), each
+    integral_T^inf 6 t^5 2(1 - Phi(t/sigma)) dt <= 12 (sigma^6 / 6) M_6(T / sigma)."""
+    tot = IV(0)
+    for mult, s2 in ((2, SIG2_2V), (1, sig2_y())):
+        if MUT == 'no-third-jet' and s2 == sig2_y():
+            continue
+        sig = isqrt_iv(IV(s2))
+        tot = tot + upper_moment(6, DV.const(IV(T) * sig.inv())).v * IV(sig.lo ** 6, sig.hi ** 6) * (2 * mult)
+    return IV(0, tot.hi)
+
+
+def tail_t1(T):
+    """integral_T^infinity 2 t (1 - F1(t) F3(t)) dt <= sum_i integral_T^inf 2 t 2(1 - Phi(t/sigma_i)) dt <= sum_i 2 sigma_i^2 M_2(T/sigma_i)."""
+    tot = IV(0)
+    for s2 in (SIG2_2V, sig2_y()):
+        if MUT == 'no-third-jet' and s2 == sig2_y():
+            continue
+        sig = isqrt_iv(IV(s2))
+        tot = tot + upper_moment(2, DV.const(IV(T) * sig.inv())).v * sig.sq() * 2
+    return IV(0, tot.hi)
+
+
+def certify_k(k):
+    """Enclosures of I1, I2, I3 and J(k) at a rational k > 0."""
+    k = Fr(k)
+    m = 12 * k
+    T = m + T_SPAN
+    six_k = 6 * k
+
+    def F1(t):
+        return F_abs(t, SIG2_2V)
+
+    # one pass over t in [m, T] for the three t-integrands
+    #   f1 = 6 t^5 (1 - F1^2 F3)                   -> I1 = E[M^6] = m^6 + integral_m^inf f1
+    #   h1 = 2 t (1 - F1 F3)                       -> H = E[max(m, |om|, |Y|)^2] = m^2 + integral_m^inf h1
+    #   f2 = h1 E[v^4 1{6k < |v| < t/2}]           -> the t-integral in I2 = E[v^4 M^2]
+    def fv(t):
+        a1 = F1(t)
+        a3 = F3(t)
+        f1 = t.pow(5).scale(6) * (1 - a1.sq() * a3)
+        h1 = t.scale(2) * (1 - a1 * a3)
+        B = abs_moment_between(4, SIG2_V, DV.const(six_k), t.scale(Fr(1, 2)))
+        return (f1, h1, h1 * B)
+    q1, qh, q2 = integrate(fv, m, T, CELL, 3)
+    I1 = IV(m ** 6) + q1 + tail_t5(T)
+    H = IV(m * m) + qh + tail_t1(T)
+    # I2 = E[v^4 M^2] = H E[v^4 1{|v| <= 6k}] + 4 E[v^6 1{|v| > 6k}] + integral_m^inf 2t (1 - F1 F3) E[v^4 1{6k < |v| < t/2}] dt
+    Ev4_in = abs_moment_between(4, SIG2_V, DV.const(0), DV.const(six_k)).v
+    Ev6_out = abs_moment_tail(6, SIG2_V, six_k)
+    Ev4 = IV(3 * SIG2_V * SIG2_V)     # E v^4 = 3 sigma^4 = 3/4
+    I2 = H * Ev4_in + Ev6_out * 4 + q2 + tail_t1(T) * Ev4
+
+    # I3 = E[F(max(a, c))], F(max) = -a^3/6 + a^2 c/2 on {c <= a} (om <= v^2/(3k)), = -c^3/6 + a c^2/2 on {c > a}
+    sig_om = isqrt_iv(IV(SIG2_OM))
+    sig_v = isqrt_iv(IV(SIG2_V))
+    inv_sv = sig_v.inv()
+
+    def G(v):
+        # om-expectations at threshold theta = v^2/(3k): u = theta / sigma_om
+        theta = v.sq().scale(1 / (3 * k))
+        u = DV(theta.v * sig_om.inv(), theta.d * sig_om.inv())
+        P = Phi_dv(u)
+        p = phi_dv(u)
+        E0le = P
+        E1le = -(p * DV.const(sig_om))
+        E0gt = 1 - P
+        E1gt = p * DV.const(sig_om)
+        E2gt = (u * p + 1 - P) * DV.const(sig_om.sq())
+        E3gt = (u.sq() * p + p.scale(2)) * DV.const(sig_om.sq() * sig_om)
+        v2 = v.sq()
+        v4 = v2.sq()
+        v6 = v4 * v2
+        piece1 = -(v6 * E0le).scale(Fr(2, 3)) + (v4 * E1le).scale(3 * k)
+        # c = 6k om - v^2:  c^2 = 36k^2 om^2 - 12k v^2 om + v^4 ;  c^3 = 216k^3 om^3 - 108k^2 v^2 om^2 + 18k v^4 om - v^6
+        c2 = E2gt.scale(36 * k * k) - (v2 * E1gt).scale(12 * k) + v4 * E0gt
+        c3 = E3gt.scale(216 * k ** 3) - (v2 * E2gt).scale(108 * k * k) + (v4 * E1gt).scale(18 * k) - v6 * E0gt
+        piece2 = ((v2 * c2).scale(3) - c3).scale(Fr(1, 6))
+        return piece1 + piece2
+
+    def f3(v):
+        rho = phi_dv(DV(v.v * inv_sv, v.d * inv_sv)) * DV.const(inv_sv)
+        return G(v) * rho
+    # tail |v| > V: |F(max)| <= (2/3) v^6 + 3k v^4 |om| + [3 v^2 c^2 + |c|^3] / 6 with |c| <= 6k|om| + v^2; expanding
+    # (6k|om| + v^2)^2 and (6k|om| + v^2)^3 and collecting powers of |v|:
+    #   <= (4/3) v^6 + 12k |om| v^4 + 36k^2 om^2 v^2 + 36k^3 |om|^3        (v^6: 2/3 + 1/2 + 1/6; v^4 |om|: 3k + 6k + 3k;
+    #                                                                       v^2 om^2: 18k^2 + 18k^2; |om|^3: 216k^3/6)
+    # with E|om| <= sqrt2, E om^2 = 2, E|om|^3 <= sqrt(E om^6) = sqrt(120) < 11
+    Eom1, Eom2, Eom3 = IV(SQRT2.hi), IV(2), IV(11)
+    coef = {6: Fr(4, 3), 4: 12 * k * Eom1.hi, 2: 36 * k * k * Eom2.hi, 0: 36 * k ** 3 * Eom3.hi}
+    tail3 = IV(0)
+    for j, cj in coef.items():
+        tail3 = tail3 + abs_moment_tail(j, SIG2_V, V_MAX) * cj
+    I3 = integrate(lambda v: (f3(v),), 0, V_MAX, CELL)[0] * 2 + IV(-tail3.hi, tail3.hi)
+
+    sign = 1 if MUT == 'sign-lo' else -1
+    J = (I1 * Fr(512, 3) - I2 * 8 + I3 * sign) / (216 * k ** 3)
+    return {'k': str(k), 'I1': I1, 'I2': I2, 'I3': I3, 'J': J}
+
+
+def R_of_b(b):
+    """R(b) = [phi(beta) / sqrt(2)] / [(2 + b^2) Phi(beta) + sqrt(2) b phi(beta)], beta = b / sqrt(2)."""
+    b = Fr(b)
+    beta = IV(b) / SQRT2
+    ph = phi_iv(beta)
+    P = Phi(beta)
+    num = ph / SQRT2
+    den = P * (2 + b * b) + ph * SQRT2 * b
+    return num / den
+
+
+# ----------------------------------------------------------------------------- floating controls (not part of the certificate)
+def _Phi_f(x):
+    return 0.5 * (1 + math.erf(x / math.sqrt(2)))
+
+
+def _phi_f(x):
+    return math.exp(-x * x / 2) / math.sqrt(2 * math.pi)
+
+
+def float_I(k, v, om, Y):
+    a = v * v
+    c = 6 * k * om - v * v
+    M = max(12 * k, 2 * abs(v), abs(om), abs(Y))
+    top = 8 * M * M
+    lo = max(a, c)
+
+    def F(s):
+        return s ** 3 / 3 - (a + c) * s * s / 2 + a * c * s
+    return (F(top) - F(lo)) / (6 * k)
+
+
+def float_mc(k, n, seed):
+    rng = random.Random(seed)
+    s = s2 = 0.0
+    for _ in range(n):
+        v = rng.gauss(0, math.sqrt(0.5))
+        om = rng.gauss(0, math.sqrt(2))
+        Y = rng.gauss(0, math.sqrt(6))
+        x = float_I(k, v, om, Y) / (36 * k * k)
+        s += x
+        s2 += x * x
+    mean = s / n
+    return mean, math.sqrt(max(s2 / n - mean * mean, 0.0) / n)
+
+
+def float_quad2d(k, n=240, V=7.0, O=11.0):
+    """Midpoint rule in (v, om) with the Y-expectation in closed form."""
+    sig_y = math.sqrt(6)
+    tot = 0.0
+    sv, so = math.sqrt(0.5), math.sqrt(2)
+    for i in range(n):
+        v = -V + (i + 0.5) * (2 * V / n)
+        wv = _phi_f(v / sv) / sv * (2 * V / n)
+        for j in range(n):
+            om = -O + (j + 0.5) * (2 * O / n)
+            wo = _phi_f(om / so) / so * (2 * O / n)
+            a = v * v
+            c = 6 * k * om - a
+            lo = max(a, c)
+            m0 = max(12 * k, 2 * abs(v), abs(om))
+            coeffs = [-(lo ** 3 / 3 - (a + c) * lo * lo / 2 + a * c * lo), 8 * a * c, -(a + c) * 32, 512 / 3]
+            t = m0 / sig_y
+            A = [1 - _Phi_f(t)]
+            for jj in range(1, 4):
+                A.append(t ** (2 * jj - 1) * _phi_f(t) + (2 * jj - 1) * A[jj - 1])
+            P_in = 2 * _Phi_f(t) - 1
+            ey = sum(cj * (m0 ** (2 * jj) * P_in + 2 * sig_y ** (2 * jj) * A[jj]) for jj, cj in enumerate(coeffs))
+            tot += wv * wo * ey / (6 * k)
+    return tot / (36 * k * k)
+
+
+def float_R(b):
+    be = b / math.sqrt(2)
+    return (_phi_f(be) / math.sqrt(2)) / ((2 + b * b) * _Phi_f(be) + math.sqrt(2) * b * _phi_f(be))
+
+
+# ----------------------------------------------------------------------------- records
+def dec_down(x, digits=10):
+    x = Fr(x)
+    n = x.numerator * 10 ** digits // x.denominator
+    s_ = str(abs(n)).rjust(digits + 1, '0')
+    return ('-' if n < 0 else '') + s_[:-digits] + '.' + s_[-digits:]
+
+
+def dec_up(x, digits=10):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** digits) // x.denominator)
+    s_ = str(abs(n)).rjust(digits + 1, '0')
+    return ('-' if n < 0 else '') + s_[:-digits] + '.' + s_[-digits:]
+
+
+def enclosure(iv, digits=10):
+    return [dec_down(iv.lo, digits), dec_up(iv.hi, digits)]
+
+
+def k_record(k):
+    rec = certify_k(k)
+    lead = LEAD * Fr(k) ** 3
+    out = {'k': str(Fr(k)), 'J': enclosure(rec['J']), 'J_width': dec_up(rec['J'].width(), 12),
+           'I1_E_M6': enclosure(rec['I1']), 'I2_E_v4M2': enclosure(rec['I2']), 'I3_E_F_lo': enclosure(rec['I3']),
+           'lead_2359296_k3': str(lead), 'J_over_lead': enclosure(rec['J'] / lead, 8)}
+    return out, rec
+
+
+def controls_k(k):
+    kf = float(Fr(k))
+    mc, se = float_mc(kf, 200000, 2026)
+    q = float_quad2d(kf)
+    return {'k': str(Fr(k)), 'J_mc': mc, 'J_mc_se': se, 'J_quad2d': q}
+
+
+def worker(k):
+    out, rec = k_record(k)
+    return out
+
+
+def full_run(procs):
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(worker, list(K_GRID))
+        ctrls = pool.map(controls_k, list(K_GRID))
+    Rs = {str(b): enclosure(R_of_b(b), 12) for b in B_GRID}
+    cg = {}
+    for b in B_GRID:
+        for rec in recs:
+            J = IV(Fr(rec['J'][0]), Fr(rec['J'][1]))
+            cg['b=%s,k=%s' % (b, rec['k'])] = enclosure(R_of_b(b) * J, 4)
+    res = {
+        'schema': 1,
+        'object': 'CL-C8-CAP-FAILURE-COEFFICIENT-PLANAR-20260930-v1',
+        'scientific_effect': 'NONE',
+        'certified': True,
+        'kind': 'certified enclosures of the limit coefficient c_G(b, k) = R(b) J(k) of Theorem G (NOTE.md); the limit theorem itself is author-side',
+        'statement': 'Q^W(G_r^c) = c_G(b, k) r^3 + o(r^3) as r -> 0 for the planar six-pin law of [LP] with the good event G_r of [LP] section 7 / [CAP] (1); reference-kernel jet law',
+        'parameters': {'PBITS': PBITS, 'CELL': str(CELL), 'T_SPAN': str(T_SPAN), 'V_MAX': str(V_MAX), 'LEAD': LEAD,
+                       'k_grid': [str(k) for k in K_GRID], 'b_grid': [str(b) for b in B_GRID]},
+        'J': recs,
+        'R': Rs,
+        'c_G': cg,
+        'float_controls': ctrls,
+        'mutant': MUT,
+    }
+    return res
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        json.dump(res, fh, indent=1, sort_keys=True)
+        fh.write('\n')
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def check_run(procs):
+    ref = load_results()
+    require(ref['parameters'] == {'PBITS': PBITS, 'CELL': str(CELL), 'T_SPAN': str(T_SPAN), 'V_MAX': str(V_MAX), 'LEAD': LEAD,
+                                  'k_grid': [str(k) for k in K_GRID], 'b_grid': [str(b) for b in B_GRID]}, 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json was produced under a mutant')
+    # constants
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19) and Phi(IV(1)).width() < Fr(1, 10 ** 30), 'Phi(1) enclosure')
+    require(abs(exp_neg(IV(1)).mid() - Fr('0.367879441171442321595')) < Fr(1, 10 ** 19) and exp_neg(IV(1)).width() < Fr(1, 10 ** 30), 'exp(-1) enclosure')
+    # the six certified values, recomputed and compared exactly with the stored decimal strings
+    stored = {rec['k']: rec for rec in ref['J']}
+    recs = {}
+
+    def take(k, out):
+        require(out == stored[str(k)], 'certified record for k = %s differs from RESULTS.json' % k)
+        J = IV(Fr(out['J'][0]), Fr(out['J'][1]))
+        lead = LEAD * Fr(k) ** 3
+        # J(k) = LEAD k^3 (1 + delta(k)): delta positive and large for small k (the free third derivatives exceed 12k),
+        # within 2e-6 of zero for k >= 1 (both signs occur: the -8 E[v^4 M^2] term is negative)
+        if Fr(k) <= Fr(3, 4):
+            require(J.lo >= lead, 'J(k) below %s k^3 at k = %s' % (LEAD, k))
+        else:
+            require(abs(J.mid() / lead - 1) < Fr(2, 10 ** 6), 'J(k) not within 2e-6 of %s k^3 at k = %s' % (LEAD, k))
+        require(J.width() / J.lo < Fr(1, 10 ** 5), 'J(k) enclosure wider than 1e-5 relative at k = %s' % k)
+        ctrl = [c for c in ref['float_controls'] if c['k'] == str(k)][0]
+        Jf = float(J.mid())
+        require(abs(ctrl['J_mc'] - Jf) <= 4 * ctrl['J_mc_se'] + 1e-9 * Jf, 'Monte Carlo control off by more than four standard errors at k = %s' % k)
+        require(abs(ctrl['J_quad2d'] / Jf - 1) < 3e-4, 'two-dimensional quadrature control off by more than 3e-4 at k = %s' % k)
+        recs[str(k)] = out
+    take(K_GRID[0], worker(K_GRID[0]))          # first value sequentially: a mutant fails here
+    with multiprocessing.Pool(procs) as pool:
+        for k, out in zip(K_GRID[1:], pool.map(worker, list(K_GRID[1:]))):
+            take(k, out)
+    for b in B_GRID:
+        Rb = R_of_b(b)
+        require(enclosure(Rb, 12) == ref['R'][str(b)], 'R(b) differs at b = %s' % b)
+        require(abs(float_R(float(b)) - float(Rb.mid())) < 1e-12, 'R(b) float control at b = %s' % b)
+        for k in K_GRID:
+            J = IV(Fr(stored[str(k)]['J'][0]), Fr(stored[str(k)]['J'][1]))
+            require(enclosure(Rb * J, 4) == ref['c_G']['b=%s,k=%s' % (b, k)], 'c_G differs at (%s, %s)' % (b, k))
+    # monotonicity of R on the grid and the two headline numbers
+    Rvals = [Fr(ref['R'][str(b)][0]) for b in B_GRID]
+    require(all(Fr(ref['R'][str(B_GRID[i])][0]) > Fr(ref['R'][str(B_GRID[i + 1])][1]) for i in range(len(B_GRID) - 1)), 'R not decreasing on the grid (disjoint enclosures required)')
+    require(Fr(ref['c_G']['b=0,k=2'][0]) > 5300000, 'c_G(0, 2) headline')
+    require(Fr(ref['c_G']['b=6/5,k=1/6'][0]) > 35000, 'c_G(6/5, 1/6) headline')
+    # [CAP] (3): C3_new + C4_new / 20 < 2.4e23, exact; and the recorded constant is at least 6e18 times c_G(6/5, 1/6)
+    C3 = Fr(10, 51) * (Fr(32 ** 3, 3) * 320 ** 8 + Fr(3 * 32 ** 2, 2) * 320 ** 7)
+    C4 = Fr(23, 5) * (10240 ** 4 + 6800 ** 4)
+    require(C3 + C4 / 20 < Fr(24, 10) * 10 ** 23, '[CAP] (3) arithmetic')
+    require(C3 / Fr(ref['c_G']['b=6/5,k=1/6'][1]) > 6 * 10 ** 18, 'ratio of the recorded constant to c_G(6/5, 1/6)')
+    print(json.dumps({'check': 'ok', 'k_values': len(K_GRID), 'c_G(0,2)': ref['c_G']['b=0,k=2'], 'c_G(6/5,1/6)': ref['c_G']['b=6/5,k=1/6'], 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--quick', type=str, default=None, help='certify one k (rational) and print the record')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.quick:
+        import time
+        t0 = time.time()
+        out, rec = k_record(Fr(args.quick))
+        print(json.dumps(out), 'time', round(time.time() - t0, 1))
+        print('mc', float_mc(float(Fr(args.quick)), 100000, 1), 'quad2d', float_quad2d(float(Fr(args.quick)), 160))
+        return
+    if args.check:
+        check_run(args.procs)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({r['k']: r['J'] for r in res['J']}))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/NOTE.md b/frontiers/c8_elder_failure_coefficient_20261001/NOTE.md
new file mode 100644
index 000000000..9903dd6f6
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/NOTE.md
@@ -0,0 +1,411 @@
+# Certified planar near cluster coefficients `alpha_1`, `alpha_2` and the elder-failure constant `C_fail^{B,K}`
+
+**Object:** CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1.
+**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 1 October 2026.
+**Disposition:** certified numerical enclosures. Binary64 interval arithmetic with outward rounding encloses explicit
+finite-dimensional Gaussian integrals that the sources define. Author-side. Nonauthor reads at `b54ea71` (OpenAI /
+Codex): Slices A and C ACCEPT (review 5378890193), Slice B ACCEPT (review 5379043497); v1.3 applies their three
+non-blocking clarifications (the `chi^2` coordinates, outward comparison brackets, the `Phi`-series bound).
+**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX`, `GRAPH`, claim, catalog, prize or source-body change. Same GitHub
+account as every lane; zero organizational-independence credit.
+
+## 0. Statement
+
+Planar model, reference (continuum) kernel `K(z) = exp(-|z|^2/2)`. `(a, beta, c) ~ N(0, diag(2, 2, 6))` are the odd
+contact jets and `I_1`, `I_2` the near failure integrals of [NUM] §§2–3 (recalled in §1). Write `J_j(k) = E[I_j]` and
+`J_fail = J_1 + J_2`. By [NUM] §2, from [CL] Theorem N or [CUB] Theorem F,
+
+    alpha_j(b, k) = (p_b(0)/z_0) J_j(k),      p_b(0)/z_0 = phi(b/sqrt2) / (sqrt2 36 k^2 m_(2,b)),
+    m_(2,b) = (b^2 + 2) Phi(b/sqrt2) + sqrt2 b phi(b/sqrt2),
+
+with `nu(2) = alpha_2` ([NUM] §1) and `a_fail = alpha_1 + alpha_2` the selector-failure coefficient of [S] (S2).
+
+**Theorem (certified enclosures).** The following intervals contain the exact values for the reference kernel. By Lemma T
+(§6a) they also contain them for the torus model of [LP] and [S], with every side `L >= 10` and every frame. The torus
+enclosures in `RESULTS.json` agree with these at the displayed precision. The certificate's own intervals (`RESULTS.json`,
+10 digits) are rounded further outward here.
+
+| `k` | `J_fail = J_1 + J_2` | `J_1` | `J_2` | `J_2 / J_fail` | near Palm excess `2J_2/(J_1 + 2J_2)` |
+|---|---|---|---|---|---|
+| `1/2` | `[36.783058, 36.786801]` | `[35.673609, 35.680652]` | `[1.10614, 1.10945]` | `[0.030069, 0.030162]` | `[0.058377, 0.058563]` |
+| `1` | `[172.84636, 172.86021]` | `[169.58395, 169.60709]` | `[3.25312, 3.26241]` | `[0.018819, 0.018875]` | `[0.036941, 0.037052]` |
+| `2` | `[1111.9822, 1112.0308]` | `[1103.1296, 1103.2057]` | `[8.82508, 8.85263]` | `[0.0079360, 0.0079612]` | `[0.015746, 0.015797]` |
+
+| `k` | `b` | `p_b(0)/z_0` | `alpha_1` | `alpha_2 = nu(2)` | `a_fail = alpha_1 + alpha_2` |
+|---|---|---|---|---|---|
+| `1/2` | `0` | `[0.031343862, 0.031343869]` | `[1.118148, 1.118370]` | `[0.034670, 0.034775]` | `[1.152923, 1.153041]` |
+| `1/2` | `1` | `[0.0089740288, 0.0089740307]` | `[0.3201360, 0.3201993]` | `[0.0099266, 0.0099563]` | `[0.3300922, 0.3301259]` |
+| `1` | `0` | `[0.0078359656, 0.0078359673]` | `[1.328854, 1.329036]` | `[0.025491, 0.025565]` | `[1.354418, 1.354527]` |
+| `1` | `1` | `[0.0022435072, 0.0022435077]` | `[0.3804628, 0.3805148]` | `[0.0072984, 0.0073193]` | `[0.3877820, 0.3878132]` |
+| `2` | `0` | `[0.0019589914, 0.0019589919]` | `[2.161021, 2.161171]` | `[0.017288, 0.017343]` | `[2.178363, 2.178459]` |
+| `2` | `1` | `[0.00056087680, 0.00056087692]` | `[0.6187198, 0.6187626]` | `[0.0049497, 0.0049653]` | `[0.6236851, 0.6237124]` |
+
+**Corollary 1 (the elder-failure constant).** On `B x K = [0, 1] x [1/2, 2]`, the coefficient of [S] §9 for the reference
+kernel,
+
+    C_fail^{B,K} = integral_(B x K x S^1) A_0^ref a_fail / (3 k^(5/3)) db dk dsigma(u) = const I_b I_k,
+    const = 4 / (sqrt2 (2 pi)^(5/2) sqrt12),   I_b = integral_0^1 exp(-b^2) db,
+    I_k = integral_(1/2)^2 exp(-12 k^2) k^(-5/3) J_fail(k) dk,
+
+satisfies
+
+    I_k in [0.4421366, 0.4432234],      C_fail^{B,K} in [0.00272445, 0.00273116].
+
+By Lemma T the same interval holds for the torus model with every `L >= 10`. That is the model of [S] §9, which states
+`nu_rej^{B,K}(ell) ~ C_fail^{B,K} ell^(2/3)` and `E N_rej^{B,K}(0, t] ~ (3/5) C_fail^{B,K} t^(5/3)`.
+
+**Corollary 2 (`d = 3`, conditional on Math-#175 and Math-#184).**
+- Math-#175 (OpenAI/Codex, open, AUTHOR_SIDE / HOLD) is [FIB]:
+  - its Theorem F, (S3), identifies `(1 - p_r)/r^3 -> a_1 + a_2` in fixed `d >= 3`;
+  - its (L1) is `C_fail^{B,K} = integral_(B x K x S^(d-1)) A_0 a_fail / (3k^(5/3))`.
+- Math-#184 (open) is [HD]. Its Theorem 1 gives `a_j^(d) = R_d(b) alpha_j` with `R_d = N_d m_(2,b) / m_(d,b)`.
+
+In `d = 3`, for the reference kernel, the derivation of §1 gives
+`A_0^ref,(3) = 432 (2 pi)^-4 12^(-1/2) k^2 exp(-12k^2) exp(-3b^2/4) m_(3,b)`: the transverse jet `f_xy2` adds a factor
+`(2 pi)^(-1/2)`, `p_G(0) = (2 pi)^(-3/2)` and `z_0 = 36 k^2 m_(3,b)`. Both `m_(3,b)` and `m_(2,b)` cancel. With
+`N_3(b) = pi (2 pi)^(-1/2) g(b)` ([HD] (2), `m = 2`) we have `exp(-3b^2/4) phi(b/sqrt2) N_3(b) = exp(-b^2) g(b) / 2`, so
+
+    C_fail^(3),{B,K} = 4 pi (4/sqrt2) (2 pi)^-4 12^(-1/2) I_b^(3) I_k,      I_b^(3) = (1/2) integral_0^1 exp(-b^2) g(b) db,
+
+with `g(b) = (b^3 + 6b) Phi(b/sqrt2) + sqrt2 (b^2 + 4) phi(b/sqrt2)`. The certificate checks `N_3(0) = 2 sqrt2` ([HD] (3)) and
+gives
+
+    I_b^(3) in [1.513649, 1.513684],      C_fail^(3),{B,K} in [0.00440581, 0.00441675].
+
+At `b = 0`, `R_3(0) = (32 + 28 sqrt2)/17`:
+
+| `k` | `alpha_1^(3)` | `alpha_2^(3) = nu^(3)(2)` | `a_fail^(3)` |
+|---|---|---|---|
+| `1/2` | `[4.709246, 4.710177]` | `[0.14602, 0.14646]` | `[4.855704, 4.856199]` |
+| `1` | `[5.596663, 5.597427]` | `[0.10736, 0.10767]` | `[5.704330, 5.704788]` |
+| `2` | `[9.101458, 9.102086]` | `[0.072812, 0.073040]` | `[9.174497, 9.174898]` |
+
+**Consequences.**
+- **`nu(2)` and `a_fail` are certified.** At `(b, k) = (0, 1)`: `alpha_2 = nu(2)` is in `[0.025491, 0.025565]` and `a_fail` in
+  `[1.354418, 1.354527]`. The two-point share `J_2/J_fail` decreases from
+  `3.01%` at `k = 1/2` to `0.79%` at `k = 2`.
+- **These are the model's constants.** By Lemma T the planar enclosures hold for the torus model of [LP] and [S] with
+  every `L >= 10` and every frame, not only for the reference kernel.
+- **[NUM]'s open digit is settled.** At `k = 2`, [NUM] observed that Gauss–Hermite and Monte Carlo disagree on `J_2` by
+  2.6% and said the third digit of `alpha_2(2)` was not established. The certified `J_2(2)` is in `[8.82508, 8.85263]`. It
+  excludes the GH60 value `8.641` and is consistent with the Monte Carlo value `8.875 +- 0.073`. The Gauss–Hermite
+  sequence had not converged there.
+- **The cap route overstates the lifetime difference by about `1.5 x 10^5`.** Math-#215 (open) certifies
+  `nu_cand - nu_eld <= C_{B,K}(r_*) ell^(2/3)` through the cap criterion, with `C_{B,K}(1/4096) = 402.5297`. As
+  `r_* -> 0`, that route can give no less than `Gamma_2 = 307.51` (floating). The leading constant of the difference is
+  `C_fail^{B,K}`, about `0.00273`: `1.47 x 10^5` times smaller than `C_{B,K}(1/4096)` and `1.13 x 10^5` times
+  smaller than `Gamma_2`.
+- **The same holds in `d = 3`, conditionally.** `C_fail^(3),{B,K}` is about `0.00441`. That is `2.73 x 10^5` times
+  smaller than Math-#215's `C_{B,K}^(3)(1/4096) = 1205.7089` and `1.23 x 10^5` times smaller than
+  `Gamma_3 = 540.97`.
+
+## 1. What is evaluated
+
+[NUM] §3 (from [CUB] (C2)–(C3) and [CL] (3.12)) defines, with `B = beta - a^2/(12k)`, `D = (c - a beta/(4k) +
+a^3/(72k^2))/2`, the typed domain `s < -|B|/2`, the weight `w_0 = 9k^2 (4s^2 - B^2)` and
+
+    g(s) = -2s^3 + 3Bs^2 - B^3 - 12k D^2,
+
+the classification: `n = 2` iff `s > B` and `g(s) > 0`; `n = 1` iff `g(s) < 0`; `n = 0` iff `s <= B` and `g(s) > 0`.
+Then `I_j = integral_(s < -|B|/2) w_0 1{n = j} ds`. [NUM] §2 gives the prefactor `p_b(0)/z_0`. Here `p_b(0)` is the density
+at `0` of `A = f_zz | (f, f_xx, f_xz) = (b, 0, 0) ~ N(-b, 2)`, and `z_0 = 36 k^2 m_(2,b)`.
+
+The reference integrand of [S] §9 is `A_0^ref = 12 pi_0 z_0`. [LP] §15 factorizes `pi_0 = p_(f,V_u)(b, 0) p_G(0)
+phi_tau(12k)`. With the jet covariances of [NUM] §2 these factors are:
+- `(f, f_xx)` has covariance `[[1, -1], [-1, 3]]`, and `f_xz` is independent with variance `1`. So
+  `p_(f,V_u)(b, 0) = exp(-3b^2/4) / (2 pi sqrt2 sqrt(2 pi))`.
+- `p_G(0) = 1/(2 pi)`.
+- `tau^2 = Var(f_xxx | grad f = 0) = 15 - 9 = 6`.
+
+Hence `pi_0 = (2 pi)^-3 12^(-1/2) exp(-3b^2/4 - 12k^2)` and
+
+    A_0^ref = 432 (2 pi)^-3 12^(-1/2) k^2 exp(-12 k^2) exp(-3b^2/4) m_(2,b).
+
+In `A_0^ref a_fail / (3 k^(5/3))`, `m_(2,b)` and `k^2` cancel, `exp(-3b^2/4) phi(b/sqrt2) / sqrt2 = exp(-b^2) / (2 sqrt pi)`,
+and the circle gives `2 pi`. This leaves `const I_b I_k` with the `const` above.
+
+## 2. Reduction to one root (Lemma 1)
+
+Put `x = -s` and `P(x) = (x + B)^2 (2x - B)`. Expanding, `g(s) = P(x) - T` with `T = 12k D^2`. The typed domain is
+`x > |B|/2`.
+
+- **`B >= 0`.** `P` increases on `x >= B/2` from `P(B/2) = 0`. So `n = 1` exactly on `(B/2, x_b)`, and `n = 2` never
+  occurs (it needs `x < -B`).
+- **`B < 0`.** `P` decreases on `(|B|/2, |B|)` from `|B|^3/2` to `0`, then increases.
+  - If `T < |B|^3/2`: `n = 2` on `(|B|/2, x_m)` and `n = 1` on `(x_m, x_b)`.
+  - Otherwise: `n = 1` on `(|B|/2, x_b)`.
+
+Here `x_b >= max(-B, B/2)` is the **big root** and `x_m in [|B|/2, |B|]` the **middle root** of `P = T`. In every case
+the failure set `{n >= 1}` is the single interval `|B|/2 < x < x_b` (up to null sets). With the antiderivative
+`9k^2 (4x^3/3 - B^2 x)`:
+
+    I_fail = I_1 + I_2 = 3k^2 H,     H   = |B|^3 + 4 x_b^3 - 3B^2 x_b = 2(B_-)^3 + 2T + V,   V = -3B(2x_b - B)(x_b + B),
+    I_2 = 3k^2 H_2,                  H_2 = |B|^3 + 4 x_m^3 - 3B^2 x_m   on {B < 0, T < |B|^3/2}, 0 elsewhere.
+
+The identity `2T + V = (x + B)(2x - B)^2 = 4x^3 - 3B^2 x + B^3` holds at either root. The controls `(0, -2k, 0)` (`H_2 = 16k^3`,
+`I_2 = 48k^5`) and `(0, 0, c)` (`H = 2T = 24k D^2`, `I_1 = 72k^3 D^2`) agree with [NUM] §3.
+
+**Scaled coordinates.** `a^ = a / sqrt k ~ N(0, 2/k)`, `c^ = sqrt(k) c ~ N(0, 6k)`. Then `B = beta - a^2/12` and
+`T = 3u^2` with `u = c^ + d` and `d = -a^ beta/4 + a^3/72`, so `J_fail(k) = 3k^2 E[H]` and `J_2(k) = 3k^2 E[H_2]` with
+`k`-free integrands. Exactly, `E[2T] = 6 E[u^2] = 6 (6k + 1/(4k) + 5/(216 k^3))`. Also `E[2(B_-)^3] = 4 integral_0^inf
+phi_(sqrt(2/k))(a) g(a^2/12) da` with `g(m) = E[(m - beta)_+^3]`.
+
+## 3. Monotonicity and the `(B, w)`-calculus (Lemma 2)
+
+Let `r = B/x`, so `r in [-1, 2]` on the big root and `r in [-2, -1]` on the middle root. Let `w = |u| = (T/3)^(1/2)`.
+Implicit differentiation of `P(x) = T` gives:
+- `x_T = 1/(6x(x + B))` (positive on the big root, negative on the middle root);
+- `x_B = -(x - B)/(2x)`;
+- `r_B = (2x - B)(x + B)/(2x^3)`;
+- `H_T = (4 - r^2)/(2(1 + r))`;
+- `V_T = -r(r + 4)/(2(1 + r))`;
+- `H_B / x^2 = -(3/2)(r - 2)^2 (r + 1)` for `B > 0` and `-(3/2)(r + 2)(r^2 - r + 2)` for `B < 0`;
+- `V_B = -(3/2) x^2 (r - 2)^2 (r + 1)`.
+
+Therefore:
+- `H` is non-increasing in `B` and non-decreasing in `T`.
+- `V` is non-increasing in `B`. In `T` it is non-increasing for `B > 0` and non-decreasing for `B < 0`.
+- `H_2` is non-increasing in both.
+- On the big root, `x` increases in `T` and, for fixed `T`, is unimodal in `B` with minimum `x = B` at `B = (T/4)^(1/3)`.
+  `r` is non-decreasing in `B`, increasing in `T` for `B < 0` and decreasing for `B > 0`.
+- On the middle root, `x` and `r` are non-increasing in both `B` and `T`.
+
+In `(B, w)` with `sigma = +1` on the big root and `-1` on the middle root:
+
+    V_w  = -sigma sqrt3 x^(3/2) r (4 + r)(2 - r)^(1/2)
+    V_BB = x p(r),           p(r) = 6 + 3r - 9r^2/2 - 3r^3/4 + 3r^4/4
+    V_Bw = -sigma (sqrt3/2) x^(1/2) e(r),    e(r) = (2 - r)^(3/2)(4 + 2r + r^2)    (decreasing on r <= 2)
+    V_ww = g(r) = -r(4 - r + r^2)                                                    (strictly decreasing)
+
+- The `T`-derivatives blow up as `T -> 0` for `B < 0` (`V_TT ~ T^(-3/2)`). The `(B, w)`-Hessian is bounded: its entries are
+  `x p(r)`, `x^(1/2) e(r)` and `g(r)` with `r` in a bounded range. At the triple point `(0, 0)` it is bounded but
+  discontinuous, and `V` is `C^(1,1)` there, which is all the integral remainder needs.
+- `V` depends on `u` only through `w`, with a kink at `u = 0` when `B < 0` (`V_w(B, 0) = 9|B|^(3/2)`).
+- On the support boundary `T = |B|^3/2` the middle root is `|B|/2` (`r = -2`). There `H_2 = 0`,
+  `H_2,B = -(3/2) x^2 (r + 2)(r^2 - r + 2) = 0` and `H_2,T = (4 - r^2)/(2(1 + r)) = 0`. So the zero extension of `H_2` is
+  `C^1` and piecewise `C^2`.
+
+All formulas were checked against finite differences of the root map (relative agreement `6e-6` at step `1e-4`) and
+symbolically.
+
+## 4. Box enclosures (Lemma 3)
+
+For a box `Q` of jet coordinates, let `[B_l, B_h] x [T_l, T_h]` contain its image. The ranges of `B` and `d` are exact
+up to outward rounding: `d` is linear in `beta`, and its `a`-extremes are at the ends or at `a = +-sqrt(6 beta)`. The
+`u`-range is the `c`-range plus the `d`-range. Two enclosures of `integral_Q F dmu` (`F = V`, `H`, `H_2`) are formed and intersected; an empty
+intersection would raise.
+
+- **First order.** `I_1 = mu(Q) [min F, max F]`. The extremes sit at rectangle corners by Lemma 2, so the corner roots
+  suffice (four big roots, or two middle roots).
+- **Second order** (only if the `u`-range of `Q` avoids `0`, so `w = s u` with a fixed sign `s`). Take a float point
+  `(B*, w*)` of the rectangle and Taylor's formula with integral remainder along the segment to `(B, w)` (the rectangle
+  is convex):
+
+      F = F* + F_B* dB + F_w* dw + 1/2 (h_BB dB^2 + 2 h_Bw dB dw + h_ww dw^2).
+
+  - Each averaged Hessian entry `h` lies in the interval range `S` of the second derivative over the rectangle,
+    computed from the corner ranges of `x` and `r`, with `p` evaluated in mean-value form. For `H_2` on boxes meeting
+    the support boundary, the ranges are hulled with `0`.
+  - Hence `integral_Q F dmu` lies in
+
+        F* m + F_B* E[dB] + s F_w* E[du] + 1/2 (S_BB E[dB^2] + S_ww E[du^2]) + s c E[dB du] +- delta (E[dB^2] E[du^2])^(1/2),
+
+    where `c` and `delta` are the centre and radius of `S_Bw`.
+  - The six moments are exact Gaussian polynomial moments (orders `a <= 6`, `beta, c <= 2`). The second-order part
+    has width `O(diam(S) h^2 mu(Q))`, so the scheme is third order.
+- **Partial moments.** `M_n = (n - 1) s^2 M_(n-2) + s^2 (x_0^(n-1) phi(x_0) - x_1^(n-1) phi(x_1))`. Cell masses are
+  differences of upper tails `Q(t)` on the side away from `0`, so they keep relative precision far out.
+- **Roots.** The roots are bracketed by Newton iteration from a Cardano / trigonometric start. The bracket is then
+  verified: `P(a) - T < 0 < P(b) - T` for the big root, reversed for the middle root. The sign test is a binary64
+  evaluation of `(x + B)^2 (2x - B) - T` with an a priori rounding bound.
+- **Refinement.** An initial `8^3` grid covers `[-8 sigma, 8 sigma]^3`. Boxes are bisected depth-first along their
+  standardized widest side until the enclosure width is at most `10^-8`. Partial sums are rounded outward, and the
+  initial cells are dealt to processes in a fixed pattern.
+
+## 5. Exact and one-dimensional parts; tails
+
+- `E[2T]` is the exact rational above.
+- `E[2(B_-)^3]` is a one-dimensional integral over `4000` cells, each by second-order Taylor expansion about the
+  midpoint.
+  - `G(a) = g(a^2/12)` has `G'' = g''(m) a^2/36 + g'(m)/6`.
+  - `g' = 3[(m^2 + 2) Phi(m/sqrt2) + sqrt2 m phi(m/sqrt2)]` and `g'' = 6[m Phi(m/sqrt2) + sqrt2 phi(m/sqrt2)]` are increasing,
+    so `G''` is bounded by its values at the cell ends.
+  - The tail `a > 12 sigma` uses `g(m) <= m^3 + 6m + sqrt2 (m^2 + 4)/sqrt(2 pi)`.
+- **Outside the box.** Lemma 1 and `x_b <= |B| + (T/2)^(1/3)` (since `P(|B| + t) >= 2t^3`) give
+  `|V|, H, H_2 <= 17|B|^3 + 8T`. This is a polynomial in `|a|, |beta|, |c|`, integrated over the complement by the union
+  bound with exact tail moments.
+
+These are expectations over the scaled law. `J_fail = 3k^2 (E[2(B_-)^3] + E[2T] + E[V])` and `J_2 = 3k^2 E[H_2]`. The
+tail bound enters `E[V]` with both signs and `E[H_2]` from above.
+
+| `k` | `E[2(B_-)^3]` | `E[2T]` exact | `E[V]` boxes | `E[H_2]` boxes | tail bound |
+|---|---|---|---|---|---|
+| `1/2` | `[8.3574312330, 8.3574329380]` | `199/9` (`22.11111111`) | `[18.5755374, 18.5805207]` | `[1.4748651, 1.4792648]` | `1.5E-9` |
+| `1` | `[5.8760207620, 5.8760219460]` | `1355/36` (`37.63888888`) | `[14.1005477, 14.1051535]` | `[1.0843739, 1.0874697]` | `3.3E-10` |
+| `2` | `[5.0934267220, 5.0934277450]` | `20957/288` (`72.76736111`) | `[14.8044089, 14.8084356]` | `[0.73542405, 0.73771834]` | `2.1E-10` |
+
+## 6. The `k`-integral
+
+`I_k` is a four-dimensional box integral in the original coordinates `(a, beta, c, k)`, with weight
+`W(k) = 3k^(1/3) exp(-12k^2)` (that is, `3k^2 exp(-12k^2) k^(-5/3)`):
+
+    B = beta - a^2/(12k),   u = sqrt(k) c - a beta/(4 sqrt k) + a^3/(72 k^(3/2)).
+
+- Each 4D box maps into a box of scaled coordinates, so the rectangle machinery of §4 applies unchanged.
+- The Taylor moments are polynomials in `(a, beta, c)` times `k^(p/2)`, `p = -6..2`. Their `k`-factors
+  `K_p = integral W(k) k^(p/2) dk` are enclosed on the `4096` elementary dyadic cells of `[1/2, 2]` by midpoint (below)
+  and trapezoid (above) rules on `24` sub-cells. This is valid because every `f(k) = k^q exp(-12k^2)` with
+  `q in [-8/3, 4/3]` is convex on `[1/2, 2]`: `f''/f = (24k - q/k)^2 - q/k^2 - 24 >= 87 - 16/3 - 24` for `q >= 0` and
+  `>= 120` for `q < 0`.
+- `k`-cells are dyadic and never finer than elementary.
+- The tail is bounded uniformly in `k` by `17|B|^3 + 8T <= 68|beta|^3 + (68/216 + 576/5184) a^6 + 144 c^2 + 9 a^2 beta^2`.
+
+Values:
+- `int_K W(k) dk` is in `[0.00907557755, 0.00907557942]`;
+- `const` is in `[0.008250963314, 0.008250964965]`;
+- `I_b = sqrt(pi)(Phi(sqrt2) - 1/2)` is in `[0.7468240581, 0.7468242075]`;
+- `I_k` (boxes plus tail) is in `[0.4421366, 0.4432234]`.
+
+`I_b^(3)` is integrated on `100000` cells of `[0, 1]`. On each cell `exp(-b^2)` decreases and `g` increases (both positive),
+so the cell integral lies between `h exp(-b_1^2) g(b_0)` and `h exp(-b_0^2) g(b_1)`.
+
+## 6a. Lemma T: transfer to the torus
+
+[LP] and [S] work on the torus `R^2/(L Z^2)` with
+`K_L(z) = sum_n exp(-|z + Ln|^2/2) / sum_n exp(-|Ln|^2/2)` ([LP] §2, by Poisson summation). Everything certified in the
+plane is a function of **one-site** jet covariances of order at most `6`:
+- the odd conditional law of `(a, beta, c)` given `(f_x, f_z, f_xxx) = (0, 0, 12k)`;
+- the even conditional law of `A = f_zz` given `(f, f_xx, f_xz) = (b, 0, 0)`, which gives `p_b(0)` and `z_0`;
+- the factors of `pi_0` ([LP] §15).
+
+`K_L` is even, so odd and even jets stay exactly uncorrelated.
+
+**Lemma T.** For every side `L >= 10` and every frame:
+1. **Covariances.** Every one-site covariance of order `<= 6` differs from its reference value by at most
+   `delta <= 2.86E-14`. In a frame `R`,
+   `d^g K_L(0) = sum_n (d^g G)(R^T L n)/Theta_L`, `|R^T L n| = L|n|`, and `|He_g1(y1) He_g2(y2)| <= Hh_g1(|y|) Hh_g2(|y|)` with
+   `Hh_j(t) = sum_k j!/(k!(j-2k)! 2^k) t^(j-2k)`. The shell `|n|_inf = j` has `8j` points with `|n| >= j`. Each product
+   `Hh_a Hh_c e^(-t^2/2)` decreases for `t^2 > a + c`, so the bound at `L = 10` holds for every `L >= 10`.
+2. **Regressions.** Interval regressions with every entry widened by `delta` give the torus conditional laws:
+   - odd: `N(12k v, S_c)` with `||diag(2,2,6)^(-1/2)(S_c - diag(2,2,6))diag(2,2,6)^(-1/2)|| <= eps = 1.30e-12`;
+   - even: `N(c_1 b, s^2)`;
+   - the ratio `(pi_0 p_b(0))^L/(pi_0 p_b(0))^ref` lies in `1 +- 8.8e-12`, uniformly in `(b, k)` in the band;
+   - `z_0` cancels in `A_0 a_fail` on the torus as well.
+3. **Expectations.** Let `S_c = S_0^(1/2)(I + E)S_0^(1/2)` with eigenvalues `e_i` of `E`, and `mu~ = S_0^(-1/2) mu`.
+   Coordinate-free, `1 + chi^2 = det(I - E^2)^(-1/2) exp(mu~^T (I - E)^(-1) mu~)`. In an orthonormal eigenbasis of `E`,
+   with `mu~_i` the coordinates of `mu~` in that basis (not the original jet coordinates), this reads
+
+       1 + chi^2 = prod (1 - e_i^2)^(-1/2) exp(sum mu~_i^2/(1 - e_i)).
+
+   For `eps^2 <= 0.006` we have `(1 - eps^2)^(-3/2) <= 1.01` and `(1 - eps^2)^(-3/2) - 1 <= 2 eps^2`. With
+   `e^m - 1 <= m e^m`, this gives `chi^2 <= 1.01 m e^m + 2 eps^2`, where `m = |mu~|^2/(1 - eps)`. The certificate checks
+   `eps^2 <= 0.006` and obtains `chi^2 <= 2.88E-23`. By Cauchy–Schwarz, `|J^L - J| <= (E[I^2] chi^2)^(1/2)`, using
+   `I = 3k^2 H <= 12 Pi_4` (the tail polynomial of §6, `k <= 2`), with `E[I^2] <= 5.405e+08`. This bounds `|J^L - J|` by
+   `1.25E-7` for every `k` in `[1/2, 2]`, for `J_fail`
+   and for `J_2` (`0 <= I_2 <= I_fail`).
+4. **Constants.** Hence `C_fail^L` lies in `(1 +- 8.8e-12)(C_fail^ref +- const I_b K_(-4)/3 . |J^L - J|)`, with
+   `K_(-4)/3 = integral e^(-12k^2) k^(-5/3) dk`. The torus `alpha_j`, `a_fail` at the grid points use the torus
+   `p_b(0)/z_0`.
+
+`torus.py` computes all of this in interval arithmetic. `RESULTS.json` lists the torus enclosures under "torus model".
+Lemma T is planar: Corollary 2 stays a reference-kernel statement.
+
+## 7. Arithmetic
+
+Every `+, -, *, /, sqrt` is IEEE binary64 round-to-nearest, widened outward by one ulp. No libm transcendental is
+trusted:
+- `pi` comes from Machin's series with a rational remainder, and `ln 2` from `sum 1/(j 2^j)`.
+- `exp(-y)`: `y = n ln2 + r` with `r in [0, 0.75]`, a degree-18 Taylor polynomial with the a priori bound
+  `(56u + 10^-19) e^0.75`, then exact scaling by `2^-n`. The observed relative width is below `4e-13` for `y <= 700`.
+- `Phi(t)` for `t < 2` is `1/2 + phi(t) sum t^(2n+1)/(2n+1)!!`, with an a priori summation bound and a geometric tail.
+- `Q(t) = 1 - Phi(t)` for `t >= 2` is `phi(t) R(t)`, where `R` is Laplace's continued fraction
+  `1/(t + 1/(t + 2/(t + ...)))`. It has positive elements, so consecutive approximants bracket `R`.
+
+libm `pow`, `acos` and `cos` only seed iterations whose results are verified.
+
+## 8. Verification
+
+- **`--check`** (CI, under `-B -S` and `-B -O -S`) runs two stages:
+  - a self-test: every box enclosure, and its first- and second-order parts separately, on `13` fixed 3D boxes and
+    `2` 4D boxes contains a pure-Python Gauss–Legendre value (`12`–`16` points per axis; relative slack `1e-9`, or
+    `1e-6` on the `C^1` support boundary of `H_2`), and the `k`-moments contain Gauss–Legendre values;
+  - the full replay.
+- The replayed enclosures must lie inside the published ones and be at most `2 x 10^-7` relatively (plus one unit of
+  the 10th digit) tighter. The leaf counts must agree:
+  - `V`: `1,209,983`, `1,096,831`, `951,074` at `k = 1/2, 1, 2`;
+  - `H_2`: `1,131,985`, `799,321`, `583,149`;
+  - `I_k` (4D): `997,266`.
+- A full replay takes about 10 minutes on four cores.
+- **Mutants** exit 1 at the self-test:
+  - `vw-sign` flips `V_w`;
+  - `ww-drop` drops `V_ww`;
+  - `corner-swap` swaps the monotonicity corners of `V`;
+  - `mid-branch` gives `H_2` the big-root sign of `V_w`;
+  - `k-weight` drops `k^(1/3)` from `W`.
+- **Development checks** (not in the packet):
+  - `1,200` random 3D boxes and `250` random 4D boxes, `0` failures against `numpy` Gauss–Legendre;
+  - the full-`H` integrand, boxed directly, gives an enclosure of `J_fail(1/2)` that intersects the split one
+    (`controls.py`).
+
+## 9. Comparisons (floating controls, `controls.py`)
+
+Floating values of [NUM] (`RESULTS.json`, GH40 / GH60 Gauss–Hermite, Monte Carlo with `10^6` samples) against the
+certified enclosures. "in" marks a value inside the enclosure; the last column is the Monte Carlo deviation from
+the enclosure midpoint in Monte Carlo standard errors.
+
+| `k` | | certified | GH40 | GH60 | MC `+-` s.e. | (MC - mid)/s.e. |
+|---|---|---|---|---|---|---|
+| `1/2` | `J_1` | `[35.67361, 35.68065]` | `35.652` (-0.069%) | `35.676` in (-0.004%) | `35.5595 +- 0.0856` | `-1.37` |
+| `1/2` | `J_2` | `[1.106149, 1.109449]` | `1.1211` (+1.205%) | `1.1085` in (+0.065%) | `1.11467 +- 0.00619` | `+1.11` |
+| `1` | `J_1` | `[169.584, 169.6071]` | `169.63` (+0.023%) | `169.61` (+0.009%) | `169.374 +- 0.283` | `-0.78` |
+| `1` | `J_2` | `[3.253122, 3.262409]` | `3.2494` (-0.256%) | `3.2554` in (-0.073%) | `3.28057 +- 0.0222` | `+1.03` |
+| `2` | `J_1` | `[1103.13, 1103.206]` | `1104.2` (+0.097%) | `1103.7` (+0.046%) | `1102.63 +- 1.69` | `-0.32` |
+| `2` | `J_2` | `[8.825089, 8.85262]` | `8.4211` (-4.726%) | `8.6412` (-2.236%) | `8.87486 +- 0.0727` | `+0.50` |
+
+Every Monte Carlo value is within `1.4` standard errors of the certified value. GH60 is inside the enclosure for
+`J_1(1/2)`, `J_2(1/2)` and `J_2(1)`, within `0.05%` of it for `J_1(1)` and `J_1(2)`, and `2.2%` low for `J_2(2)`.
+
+Other controls (`controls.py`):
+- the floating Simpson value of `E[2(B_-)^3]` lies inside its enclosure at all three `k`;
+- the direct box enclosure of `3k^2 E[H]` at `k = 1/2`, `[36.778523711, 36.792370494]` (`444127` leaves, tolerance
+  `1e-7`), intersects the split enclosure of `J_fail(1/2)`;
+- with the displayed numerators, `C_{B,K}(1/4096) / C_fail^{B,K}` lies in `[1.473, 1.478] x 10^5` and
+  `Gamma_2 / C_fail^{B,K}` in `[1.125, 1.129] x 10^5`. These are rounded outward from the certified `C_fail` interval;
+  `Gamma_2 = 307.51` is itself a floating value, so these are comparisons, not certified ratios.
+- `d = 3`:
+  - likewise `C_{B,K}^(3)(1/4096) / C_fail^(3),{B,K}` lies in `[2.729, 2.737] x 10^5`, and
+    `Gamma_3 / C_fail^(3),{B,K}` in `[1.224, 1.228] x 10^5` (outward; `Gamma_3 = 540.97` is floating);
+  - Math-#184's floating `a_fail^(3)(0, k)` (`4.8558` at `k = 1/2`, `5.705` at `k = 1`) agree with the certified values to
+    `0.003%` and `0.008%`.
+- Lemma T: direct floating lattice sums of the torus deviation of six one-site covariance entries at `L = 10`, in three
+  frames, are at most `3.3e-16`, below the certified `delta = 2.86e-14`.
+
+## 10. What this does not do
+
+- **Torus sides `L >= 10` only.** Lemma T covers every side `L >= 10` and every frame; `L < 10` is not covered. Corollary 2
+  (`d = 3`) is for the reference kernel only.
+- **Conditional identification.**
+  - The certified numbers are the Gaussian integrals that [NUM] (from [CL] Theorem N, [CUB] Theorems C and F) and
+    [S] §9 define.
+  - That `a_fail` is the selector-failure coefficient and `C_fail^{B,K}` the lifetime constant are statements of
+    [S]: AUTHOR_SIDE / HOLD, consumed, not reviewed.
+  - `nu(1) = alpha_1 + k integral_X Lambda` also contains the remote part of [RM], which is not evaluated here.
+  - Corollary 2 is conditional in addition on [FIB] (Math-#175, open, AUTHOR_SIDE / HOLD) and [HD] (Math-#184,
+    open). Its pointwise values are given only at `b = 0`; `R_3(1)` would need `m_(3,1)`, which is not certified here.
+- **Coverage.**
+  - Pointwise coefficients only at `k = 1/2, 1, 2` and `b = 0, 1`, plus the band integral `C_fail^{B,K}`.
+  - No other band and no `k -> 0`. `d = 3` is covered only through Corollary 2, and there is nothing for `d >= 4`.
+- No rate, no finite-`r` statement.
+- Not a review of [NUM], [S], [CL], [CUB], [LP] or Math-#215.
+
+## 11. Provenance
+
+- **Sources:** `SOURCE_MAP.json` pins [NUM], [S], [LP], [CL], [CUB] and the [NUM] results on `main` `3e0a91b`.
+  Math-#215 (`d748c2a`, open) is pinned for comparison constants only. Math-#175 (`701dff4`) and Math-#184 (`d37ff5d`),
+  both open, are pinned for Corollary 2.
+- **Packet files:**
+  - `certificate.py` (drivers, exact parts, tails, assembly, self-test, replay), `boxes.py` (roots, Taylor boxes),
+    `torus.py` (Lemma T) and `ia.py` (interval arithmetic), all standard-library Python;
+  - `controls.py` (floating controls);
+  - `RESULTS.json`, the certificate's output.
+- **Manifest:** `SOURCE_FILES.json`.
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/README.md b/frontiers/c8_elder_failure_coefficient_20261001/README.md
new file mode 100644
index 000000000..68ac15054
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/README.md
@@ -0,0 +1,30 @@
+# Certified planar near cluster coefficients and the elder-failure constant
+
+Certified interval enclosures, for the reference planar kernel, of the near cluster coefficients `alpha_1`, `alpha_2`
+(`nu(2) = alpha_2`) of [NUM] at `k = 1/2, 1, 2`, `b = 0, 1`, and of the constant `C_fail^{B,K}` of [S] §9 on
+`[0, 1] x [1/2, 2]`. Details and the method are in [NOTE.md](NOTE.md).
+
+- `J_fail(1/2)` is in `[36.783058, 36.786801]`, `J_fail(1)` in `[172.84636, 172.86021]` and `J_fail(2)` in
+  `[1111.9822, 1112.0308]`.
+- `J_2(1/2)` is in `[1.1061, 1.1095]`, `J_2(1)` in `[3.2531, 3.2625]` and `J_2(2)` in
+  `[8.8250, 8.8527]`. The last settles the Gauss–Hermite / Monte Carlo disagreement that [NUM] reported at
+  `k = 2`.
+- `C_fail^{B,K}` is in `[0.0027244, 0.0027312]`, about `1.5 x 10^5` below the cap-route constant
+  `C_{B,K}(1/4096) = 402.53` of Math-#215.
+- In `d = 3`, conditionally on Math-#175 and Math-#184, `C_fail^(3),{B,K}` is in `[0.0044058, 0.0044168]`.
+
+The planar values hold for the reference kernel and, by a torus-transfer lemma, for the torus model of [LP] and [S] with
+every side `L >= 10` and every frame.
+
+**Method.** A third-order box scheme:
+- second-order Taylor expansion in `(B, w = |u|)`, where the Hessian of the failure integrand is bounded and explicit;
+- interval Hessian ranges times exact Gaussian moments, intersected with first-order monotone bounds;
+- `k` as a fourth box coordinate for `C_fail`.
+
+**Arithmetic.** Standard-library Python and binary64 interval arithmetic; no transcendental library value is trusted.
+
+**Replay.** `python3 -B -S certificate.py --check` (CI in both interpreter modes, with five mutants). Floating controls:
+`python3 -B certificate.py --controls`.
+
+**Scientific effect:** NONE. Author-side; conditional on the sources' identifications ([NUM] from [CL]/[CUB], [S]).
+Same GitHub account as every lane; zero organizational-independence credit.
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/RESULTS.json b/frontiers/c8_elder_failure_coefficient_20261001/RESULTS.json
new file mode 100644
index 000000000..670297d08
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/RESULTS.json
@@ -0,0 +1,533 @@
+{
+ "object": "CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1",
+ "scientific_effect": "NONE",
+ "certified": true,
+ "arithmetic": "binary64 interval arithmetic, outward one-ulp widening after every operation; published intervals widened by 1e-07 relatively and rounded outward to 10 significant digits",
+ "parameters": {
+  "k": [
+   0.5,
+   1.0,
+   2.0
+  ],
+  "tol3": {
+   "V": 1e-08,
+   "H2": 1e-08
+  },
+  "tol4": 2.5e-09,
+  "box_half_width_sigma": 8.0,
+  "initial_cells": 8,
+  "initial_k_cells": 8,
+  "k_split_scale": 0.8,
+  "k_elementary_level": 12
+ },
+ "leaves": {
+  "V k=0.5": 1209983,
+  "H2 k=0.5": 1131985,
+  "V k=1": 1096831,
+  "H2 k=1": 799321,
+  "V k=2": 951074,
+  "H2 k=2": 583149,
+  "I_k 4D": 997266
+ },
+ "results": {
+  "k=0.5": {
+   "J_fail": [
+    "36.78305814",
+    "36.78680021"
+   ],
+   "J_1": [
+    "35.67360980",
+    "35.68065110"
+   ],
+   "J_2": [
+    "1.106148894",
+    "1.109448563"
+   ],
+   "share_J2_over_Jfail": [
+    "0.03006918240",
+    "0.03016194161"
+   ],
+   "near_palm_excess": [
+    "0.05837775723",
+    "0.05856276965"
+   ],
+   "E[2T] exact": "199/9",
+   "parts": {
+    "E[2(B_-)^3]": [
+     "8.357431233",
+     "8.357432938"
+    ],
+    "E[2T]": [
+     "22.11110889",
+     "22.11111333"
+    ],
+    "E[V] boxes": [
+     "18.57553740",
+     "18.58052069"
+    ],
+    "E[H_2] boxes": [
+     "1.474865193",
+     "1.479264750"
+    ],
+    "tail bound": [
+     "0",
+     "1.475969601E-9"
+    ]
+   },
+   "coefficients": {
+    "b=0": {
+     "p_b(0)/z_0": [
+      "0.03134386261",
+      "0.03134386889"
+     ],
+     "alpha_1": [
+      "1.118148836",
+      "1.118369538"
+     ],
+     "alpha_2": [
+      "0.03467098246",
+      "0.03477440682"
+     ],
+     "a_fail = alpha_1 + alpha_2": [
+      "1.152923236",
+      "1.153040528"
+     ]
+    },
+    "b=1": {
+     "p_b(0)/z_0": [
+      "0.008974028845",
+      "0.008974030641"
+     ],
+     "alpha_1": [
+      "0.3201360354",
+      "0.3201992242"
+     ],
+     "alpha_2": [
+      "0.009926613082",
+      "0.009956224403"
+     ],
+     "a_fail = alpha_1 + alpha_2": [
+      "0.3300922578",
+      "0.3301258393"
+     ]
+    }
+   }
+  },
+  "k=1": {
+   "J_fail": [
+    "172.8463609",
+    "172.8602041"
+   ],
+   "J_1": [
+    "169.5839525",
+    "169.6070816"
+   ],
+   "J_2": [
+    "3.253121786",
+    "3.262409093"
+   ],
+   "share_J2_over_Jfail": [
+    "0.01881938142",
+    "0.01887461645"
+   ],
+   "near_palm_excess": [
+    "0.03694156081",
+    "0.03705188358"
+   ],
+   "E[2T] exact": "1355/36",
+   "parts": {
+    "E[2(B_-)^3]": [
+     "5.876020762",
+     "5.876021946"
+    ],
+    "E[2T]": [
+     "37.63888512",
+     "37.63889266"
+    ],
+    "E[V] boxes": [
+     "14.10054776",
+     "14.10515341"
+    ],
+    "E[H_2] boxes": [
+     "1.084373928",
+     "1.087469698"
+    ],
+    "tail bound": [
+     "0",
+     "3.232088888E-10"
+    ]
+   },
+   "coefficients": {
+    "b=0": {
+     "p_b(0)/z_0": [
+      "0.007835965654",
+      "0.007835967222"
+     ],
+     "alpha_1": [
+      "1.328854160",
+      "1.329035399"
+     ],
+     "alpha_2": [
+      "0.02549135313",
+      "0.02556412816"
+     ],
+     "a_fail = alpha_1 + alpha_2": [
+      "1.354418283",
+      "1.354526758"
+     ]
+    },
+    "b=1": {
+     "p_b(0)/z_0": [
+      "0.002243507211",
+      "0.002243507661"
+     ],
+     "alpha_1": [
+      "0.3804628584",
+      "0.3805147487"
+     ],
+     "alpha_2": [
+      "0.007298402918",
+      "0.007319239058"
+     ],
+     "a_fail = alpha_1 + alpha_2": [
+      "0.3877820960",
+      "0.3878131531"
+     ]
+    }
+   }
+  },
+  "k=2": {
+   "J_fail": [
+    "1111.982273",
+    "1112.030781"
+   ],
+   "J_1": [
+    "1103.129655",
+    "1103.205691"
+   ],
+   "J_2": [
+    "8.825088654",
+    "8.852620061"
+   ],
+   "share_J2_over_Jfail": [
+    "0.007936011924",
+    "0.007961115375"
+   ],
+   "near_palm_excess": [
+    "0.01574666826",
+    "0.01579686122"
+   ],
+   "E[2T] exact": "20957/288",
+   "parts": {
+    "E[2(B_-)^3]": [
+     "5.093426722",
+     "5.093427745"
+    ],
+    "E[2T]": [
+     "72.76735383",
+     "72.76736839"
+    ],
+    "E[V] boxes": [
+     "14.80440892",
+     "14.80843560"
+    ],
+    "E[H_2] boxes": [
+     "0.7354240545",
+     "0.7377183383"
+    ],
+    "tail bound": [
+     "0",
+     "2.004587614E-10"
+    ]
+   },
+   "coefficients": {
+    "b=0": {
+     "p_b(0)/z_0": [
+      "0.001958991413",
+      "0.001958991806"
+     ],
+     "alpha_1": [
+      "2.161021739",
+      "2.161170692"
+     ],
+     "alpha_2": [
+      "0.01728827462",
+      "0.01734220843"
+     ],
+     "a_fail = alpha_1 + alpha_2": [
+      "2.178363944",
+      "2.178458970"
+     ]
+    },
+    "b=1": {
+     "p_b(0)/z_0": [
+      "0.0005608768028",
+      "0.0005608769151"
+     ],
+     "alpha_1": [
+      "0.6187198961",
+      "0.6187625424"
+     ],
+     "alpha_2": [
+      "0.004949788004",
+      "0.004965229734"
+     ],
+     "a_fail = alpha_1 + alpha_2": [
+      "0.6236851249",
+      "0.6237123314"
+     ]
+    }
+   }
+  },
+  "d=3 (conditional on Math-#175 (L1) and Math-#184 Theorem 1)": {
+   "R_3(0) = (32 + 28 sqrt2)/17": [
+    "4.211645446",
+    "4.211646289"
+   ],
+   "const_3": [
+    "0.006583316240",
+    "0.006583317557"
+   ],
+   "I_b^(3)": [
+    "1.513649011",
+    "1.513683082"
+   ],
+   "C_fail^(3),{B,K}": [
+    "0.004405817806",
+    "0.004416745237"
+   ],
+   "k=0.5, b=0": {
+    "alpha_1^(3)": [
+     "4.709246927",
+     "4.710176443"
+    ],
+    "alpha_2^(3) = nu^(3)(2)": [
+     "0.1460219000",
+     "0.1464574868"
+    ],
+    "a_fail^(3)": [
+     "4.855704384",
+     "4.856198372"
+    ]
+   },
+   "k=1, b=0": {
+    "alpha_1^(3)": [
+     "5.596663133",
+     "5.597426446"
+    ],
+    "alpha_2^(3) = nu^(3)(2)": [
+     "0.1073605521",
+     "0.1076670548"
+    ],
+    "a_fail^(3)": [
+     "5.704330166",
+     "5.704787020"
+    ]
+   },
+   "k=2, b=0": {
+    "alpha_1^(3)": [
+     "9.101458278",
+     "9.102085610"
+    ],
+    "alpha_2^(3) = nu^(3)(2)": [
+     "0.07281209039",
+     "0.07303924044"
+    ],
+    "a_fail^(3)": [
+     "9.174497503",
+     "9.174897715"
+    ]
+   }
+  },
+  "torus model, every L >= 10, every frame (Lemma T)": {
+   "jet covariance deviation delta (order <= 6)": [
+    "0",
+    "2.850266693E-14"
+   ],
+   "chi^2 bound": [
+    "0",
+    "2.878353120E-23"
+   ],
+   "|J^L - J| bound (all k, J_fail and J_2)": [
+    "0",
+    "1.247240771E-7"
+   ],
+   "(pi_0 p_b(0))^L / (pi_0 p_b(0))^ref": [
+    "0.9999998999",
+    "1.000000101"
+   ],
+   "k=0.5": {
+    "J_fail": [
+     "36.78305802",
+     "36.78680034"
+    ],
+    "J_2": [
+     "1.106148770",
+     "1.109448688"
+    ],
+    "b=0": {
+     "p_b(0)/z_0": [
+      "0.03134386261",
+      "0.03134386889"
+     ],
+     "alpha_1": [
+      "1.118148828",
+      "1.118369546"
+     ],
+     "alpha_2": [
+      "0.03467097855",
+      "0.03477441073"
+     ],
+     "a_fail": [
+      "1.152923232",
+      "1.153040532"
+     ]
+    },
+    "b=1": {
+     "p_b(0)/z_0": [
+      "0.008974028845",
+      "0.008974030641"
+     ],
+     "alpha_1": [
+      "0.3201360332",
+      "0.3201992265"
+     ],
+     "alpha_2": [
+      "0.009926611963",
+      "0.009956225523"
+     ],
+     "a_fail": [
+      "0.3300922567",
+      "0.3301258404"
+     ]
+    }
+   },
+   "k=1": {
+    "J_fail": [
+     "172.8463608",
+     "172.8602042"
+    ],
+    "J_2": [
+     "3.253121661",
+     "3.262409218"
+    ],
+    "b=0": {
+     "p_b(0)/z_0": [
+      "0.007835965654",
+      "0.007835967222"
+     ],
+     "alpha_1": [
+      "1.328854158",
+      "1.329035401"
+     ],
+     "alpha_2": [
+      "0.02549135216",
+      "0.02556412914"
+     ],
+     "a_fail": [
+      "1.354418282",
+      "1.354526759"
+     ]
+    },
+    "b=1": {
+     "p_b(0)/z_0": [
+      "0.002243507211",
+      "0.002243507661"
+     ],
+     "alpha_1": [
+      "0.3804628579",
+      "0.3805147493"
+     ],
+     "alpha_2": [
+      "0.007298402638",
+      "0.007319239338"
+     ],
+     "a_fail": [
+      "0.3877820958",
+      "0.3878131534"
+     ]
+    }
+   },
+   "k=2": {
+    "J_fail": [
+     "1111.982273",
+     "1112.030781"
+    ],
+    "J_2": [
+     "8.825088529",
+     "8.852620186"
+    ],
+    "b=0": {
+     "p_b(0)/z_0": [
+      "0.001958991413",
+      "0.001958991806"
+     ],
+     "alpha_1": [
+      "2.161021738",
+      "2.161170692"
+     ],
+     "alpha_2": [
+      "0.01728827438",
+      "0.01734220867"
+     ],
+     "a_fail": [
+      "2.178363944",
+      "2.178458970"
+     ]
+    },
+    "b=1": {
+     "p_b(0)/z_0": [
+      "0.0005608768028",
+      "0.0005608769151"
+     ],
+     "alpha_1": [
+      "0.6187198960",
+      "0.6187625426"
+     ],
+     "alpha_2": [
+      "0.004949787934",
+      "0.004965229804"
+     ],
+     "a_fail": [
+      "0.6236851248",
+      "0.6237123315"
+     ]
+    }
+   },
+   "C_fail^{B,K}": [
+    "0.002724454687",
+    "0.002731151043"
+   ]
+  },
+  "C_fail": {
+   "I_k": [
+    "0.4421366818",
+    "0.4432233938"
+   ],
+   "I_k boxes": [
+    "0.4421366818",
+    "0.4432233938"
+   ],
+   "I_k tail bound": [
+    "0",
+    "1.296653887E-11"
+   ],
+   "int_K W(k) dk": [
+    "0.009075577556",
+    "0.009075579412"
+   ],
+   "const": [
+    "0.008250963314",
+    "0.008250964965"
+   ],
+   "I_b": [
+    "0.7468240581",
+    "0.7468242075"
+   ],
+   "C_fail^{B,K}": [
+    "0.002724454695",
+    "0.002731151035"
+   ]
+  }
+ }
+}
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/SOURCE_FILES.json b/frontiers/c8_elder_failure_coefficient_20261001/SOURCE_FILES.json
new file mode 100644
index 000000000..1efc243fa
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/SOURCE_FILES.json
@@ -0,0 +1,69 @@
+{
+  "schema": 1,
+  "scientific_effect": "NONE",
+  "certified": true,
+  "review_status": "AUTHOR_SIDE_CERTIFIED_NUMERICS",
+  "object": "CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1",
+  "files": [
+    {
+      "path": "NOTE.md",
+      "bytes": 27438,
+      "sha256": "de50468afe0cae0bd2147a79e1b5b45a9f57ade5a81a9a99fd95aaaccc826c7c",
+      "git_blob": "9903dd6f6b2514069bef78399b3d90201280911e"
+    },
+    {
+      "path": "README.md",
+      "bytes": 1906,
+      "sha256": "63313f05a155a5b93db70c82c16eca7ab98c4f1311ddc770dffc5ccb2db85031",
+      "git_blob": "68ac15054b26ab4e0b9021859727be3385b6dbbd"
+    },
+    {
+      "path": "RESULTS.json",
+      "bytes": 9040,
+      "sha256": "73bd816e0b48d9740c02d58f32fda99a110e4634222760e17c9c98df64b4d870",
+      "git_blob": "670297d08e516b5c3456fef278abb8ce454cfd79"
+    },
+    {
+      "path": "SOURCE_MAP.json",
+      "bytes": 6451,
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+    {
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+      "git_blob": "0cbecb81eefceb11499bce4d78386a46d879cd51"
+    },
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+    },
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+      "git_blob": "cf4bbfa16f577fe65f4ab5fb3621943fa63cf9fa"
+    },
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+      "git_blob": "4a448b2f30e8f1157b758bc9ae5c9c258e112db8"
+    },
+    {
+      "path": "test_replay_metadata.py",
+      "bytes": 2803,
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+    }
+  ]
+}
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/SOURCE_MAP.json b/frontiers/c8_elder_failure_coefficient_20261001/SOURCE_MAP.json
new file mode 100644
index 000000000..f48177918
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/SOURCE_MAP.json
@@ -0,0 +1,111 @@
+{
+  "schema": 1,
+  "object": "CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1",
+  "scientific_effect": "NONE",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "main_at_creation": "3e0a91b69892b9717564c592e7a966492a81566a",
+  "note": "Certified interval enclosures of explicit Gaussian integrals that [NUM] (from [CL]/[CUB]) and [S] define, for the reference (continuum) kernel. The identifications of alpha_1, alpha_2, a_fail and C_fail^{B,K} with model quantities are those of the sources and are not re-proved or accepted here. [LDC] is an open pull request, used only for comparison constants. The d = 3 corollary is conditional in addition on [FIB] (Math-#175) and [HD] (Math-#184), both open.",
+  "sources": [
+    {
+      "path": "frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md",
+      "bytes": 11018,
+      "sha256": "f20c1ac96b00edc071e428bd2e5dc4c91b8546a3ee0693367b783109b26aa518",
+      "git_blob": "845881d40520d4acbe15ee196a1ba0d5412eafa1",
+      "id": "NUM",
+      "commit": "e27b4c80a4f2b11227a7139f819b3e9ba4de8493",
+      "role": "section 2: the jet covariances, the even/odd regression, A ~ N(-b, 2), p_b(0) = phi(b/sqrt2)/sqrt2, z_0 = 36 k^2 m_(2,b), (a, beta, c) ~ N(0, diag(2, 2, 6)) and alpha_j = [phi(b/sqrt2)/(sqrt2 36 k^2 m_(2,b))] J_j(k); section 3: B, D, the typed domain s < -|B|/2, the weight 9k^2 (4s^2 - B^2), the classifier (n = 2 iff s > B and g(s) > 0; n = 1 iff g(s) < 0) and I_j; the floating values compared in section 9",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/local_elder_geometry_20260930/PROOF.md",
+      "bytes": 39722,
+      "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b",
+      "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113",
+      "id": "S",
+      "commit": "79f18f0cb318fceccd5bdf4e6dad29815165bc7d",
+      "role": "Theorem S, (S2): r^-3 (1 - p_r) -> a_fail = alpha_1 + alpha_2; section 9: C_fail^{B,K} = int_{B x K x S^1} A_0 a_fail/(3k^(5/3)), nu_rej ~ C_fail ell^(2/3), E N_rej(0, t] ~ (3/5) C_fail t^(5/3). AUTHOR_SIDE / HOLD in its own README; consumed as a statement, not reviewed",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "id": "LP",
+      "commit": "b6282ecc44cf63664a989efd055e473aa1abc50b",
+      "role": "section 2: the kernel; (10.2)-(10.3): A_r = 12 pi_r Z_r / r^2 -> A_0 = 12 pi_0 z_0; section 15: pi_0 = p_(f,V_u)(b, 0) p_G(0) phi_tau_u(12k), z_0 = (6k)^2 E[(det A_u)^2 1{A_u < 0} | f = b, V_u = 0]",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c6_cluster_law_20260929/PROOF.md",
+      "bytes": 70131,
+      "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38",
+      "git_blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5",
+      "id": "CL",
+      "commit": "4807bfffb880ce9e4ae3b04a3faf0dd777a9945c",
+      "role": "Theorem N, (1.3) and (3.11): the near coefficients alpha_1, alpha_2 and nu(2) = alpha_2 (through [NUM]); nonauthor-accepted (OpenAI review 5360192822) and merged",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md",
+      "bytes": 19889,
+      "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0",
+      "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758",
+      "id": "CUB",
+      "commit": "325c5ebdc2ca0165b483cddce8972ee2d0252c79",
+      "role": "Theorem C (the classifier of the sheared cubic), Theorem F (alpha_1, alpha_2 finite and positive), Theorem B (parity factorization), through [NUM]",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json",
+      "bytes": 4095,
+      "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb",
+      "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f",
+      "id": "NUM-R",
+      "commit": "0b35e76ff05676e6ad5c89171d65807668c8de15",
+      "role": "floating Gauss-Hermite and Monte Carlo values of J_1, J_2; read only by controls.py for the comparison table (not part of the certificate)",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c8_lifetime_difference_constant_20261001/NOTE.md",
+      "bytes": 30864,
+      "sha256": "d77c9ab1792f1cba55c512828ad0601216068167b56bc4c912f9e8e7d985ae3e",
+      "git_blob": "0f6c7fb79988b52e26c1e2f1456ab196f075ecf0",
+      "id": "LDC",
+      "commit": "d748c2abee2cbfbad6504b5700c70bbd83ba5785",
+      "role": "Math-#215 (open, not on main): C_{B,K}(1/4096) = 402.5297, Gamma_2 = 307.51 (floating) and Theorem L (eps_2 <= 3.96e-10); comparison and scope remarks only",
+      "current_required": false,
+      "on_main": false
+    },
+    {
+      "path": "frontiers/concave_fibre_elder_20260930/PROOF.md",
+      "bytes": 23416,
+      "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e",
+      "git_blob": "923d3236b5291a29bd162937f172190db84dcef5",
+      "id": "FIB",
+      "commit": "701dff49119897932ebe1891657126eafe395711",
+      "role": "Math-#175 (OpenAI/Codex, open, AUTHOR_SIDE/HOLD): Theorem F (S3), (1 - p_r)/r^3 -> a1 + a2 in fixed d >= 3, and (L1), C_fail^{B,K} = int_{B x K x S^(d-1)} A0 a_fail/(3k^(5/3)); consumed as statements for the d = 3 corollary only",
+      "current_required": false,
+      "on_main": false
+    },
+    {
+      "path": "frontiers/c6_hard_direction_factorization_20260930/PROOF.md",
+      "bytes": 22456,
+      "sha256": "26743f82bf4b86d4ae4c7dfb085d4d15bf36014aa627dee5edfc86a75ca64c05",
+      "git_blob": "91d22fce0fd7d4630bfe2e2c93eab289babc97f2",
+      "id": "HD",
+      "commit": "d37ff5d0315a4f71453d37f0cae2da1f40fdef3c",
+      "role": "Math-#184 (Anthropic/Claude, open; OpenAI nonauthor ACCEPTs 5366497205 (Slices A/B) and 5366261768 (Slice C)): Theorem 1 and Corollary 1, a_j^(d) = R_d(b) a_j^(2), R_d = N_d m_(2,b)/m_(d,b), N_3(b) = pi (2 pi)^(-1/2) int_0^inf h^3 phi_2(h - b) dh, R_3(0) = (32 + 28 sqrt2)/17; consumed for the d = 3 corollary only",
+      "current_required": false,
+      "on_main": false
+    }
+  ],
+  "nonauthor_acceptance": false,
+  "governing_status_changed": false
+}
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/boxes.py b/frontiers/c8_elder_failure_coefficient_20261001/boxes.py
new file mode 100644
index 000000000..f6b42e2a0
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/boxes.py
@@ -0,0 +1,734 @@
+"""Box enclosures of the failure integrands (certified engine; standard library only).
+
+Jet coordinates.  (a, beta, c) ~ N(0, diag(2, 2, 6)) are the odd jets of [NUM] section 2.  For a gap mark k the scaled
+coordinates a^ = a / sqrt k ~ N(0, 2/k), beta ~ N(0, 2), c^ = sqrt(k) c ~ N(0, 6k) give
+    B = beta - a^2/12,   u = c^ + d,   d = -a^ beta/4 + a^3/72,   w = |u|,   T = 3 w^2  ( = 12 k D^2 of [NUM] section 3).
+Roots.  P(x) = (x + B)^2 (2x - B).  The big root x(B, T) >= xm = max(-B, B/2) solves P(x) = T (P increasing and convex on
+[xm, inf), P(xm) = 0).  For B < 0 and T <= |B|^3/2 the middle root solves P(x) = T in [|B|/2, |B|] (P decreasing there).
+Integrands.  With r = B/x:
+    H   = |B|^3 + 4x^3 - 3B^2 x = 2(B_-)^3 + 2T + V,   V = -3B(2x - B)(x + B)             (big root)
+    H_2 = |B|^3 + 4x^3 - 3B^2 x = 2|B|^3 + 2T + V                                    (middle root, inside its support)
+    V_B  = x^2 (-6 + 9r^2/2 - 3r^3/2),     V_w = -sigma sqrt3 x^(3/2) r (4 + r)(2 - r)^(1/2)
+    V_BB = x p(r),  p(r) = 6 + 3r - 9r^2/2 - 3r^3/4 + 3r^4/4
+    V_Bw = -sigma (sqrt3/2) x^(1/2) e(r),  e(r) = (2 - r)^(3/2)(4 + 2r + r^2)
+    V_ww = g(r) = -r(4 - r + r^2)
+with sigma = +1 on the big root and -1 on the middle root (derivatives in (B, w) at fixed w, resp. fixed B).
+Per box:  I1 = Gaussian mass x pointwise range (monotonicity at the corners of the image rectangle), and, on one side of
+u = 0, I2 = second-order Taylor expansion in (B, w) about a float point (B*, w*) of the rectangle with exact Gaussian
+polynomial moments and the integral-form remainder, whose averaged Hessian entries lie in the interval ranges of the
+second derivatives over the rectangle.  The box integral lies in I1 and in I2; the intersection is returned (an empty
+intersection raises)."""
+import math
+from ia import (dn, up, add, sub, neg, mul, div, scal, divc, sqr, pw, isqrt, exp_neg, Q_point, SQRT2PI)
+
+MUTANT = None                       # set by certificate.py --mutant (the certificate must then fail)
+SQRT3 = isqrt((3.0, 3.0))
+HSQRT3 = scal(0.5, SQRT3)
+_U = 2.0 ** -53
+
+
+# ================================================================ 1D Gaussian partial moments
+def phis(x, s, s2):
+    """phi_sigma(x) for a float x, sigma interval s, sigma^2 interval s2."""
+    return div(exp_neg(div(sqr((x, x)), scal(2.0, s2))), mul(s, SQRT2PI))
+
+
+def _Qiv(t):
+    """Q = 1 - Phi over an interval t >= 0 (decreasing)."""
+    return (Q_point(t[1])[0], Q_point(t[0])[1])
+
+
+def cell_mass(x0, x1, s):
+    """Gaussian mass of [x0, x1], computed on the side away from 0 as a difference of upper tails."""
+    t0, t1 = div((x0, x0), s), div((x1, x1), s)
+    if x0 >= 0.0:
+        return sub(_Qiv(t0), _Qiv(t1))
+    if x1 <= 0.0:
+        return sub(_Qiv(neg(t1)), _Qiv(neg(t0)))
+    return sub(sub((1.0, 1.0), _Qiv(t1)), _Qiv(neg(t0)))
+
+
+def moments(x0, x1, s, s2, nmax):
+    """M_n = int_{x0}^{x1} x^n phi_sigma(x) dx, n = 0..nmax:  M_1 = s^2 (phi(x0) - phi(x1)),
+    M_n = (n - 1) s^2 M_{n-2} + s^2 (x0^(n-1) phi(x0) - x1^(n-1) phi(x1))."""
+    p0, p1 = phis(x0, s, s2), phis(x1, s, s2)
+    m0 = cell_mass(x0, x1, s)
+    M = [(max(0.0, m0[0]), m0[1])]
+    if nmax >= 1:
+        M.append(mul(s2, sub(p0, p1)))
+    x0i, x1i = (x0, x0), (x1, x1)
+    for n in range(2, nmax + 1):
+        t = sub(mul(pw(x0i, n - 1), p0), mul(pw(x1i, n - 1), p1))
+        M.append(add(mul(scal(float(n - 1), s2), M[n - 2]), mul(s2, t)))
+    return M
+
+
+class Moments:
+    """Cache of the partial moments of one coordinate."""
+    def __init__(self, sig, sig2, nmax):
+        self.sig, self.sig2, self.nmax, self.c = sig, sig2, nmax, {}
+
+    def get(self, x0, x1):
+        key = (x0, x1)
+        r = self.c.get(key)
+        if r is None:
+            r = moments(x0, x1, self.sig, self.sig2, self.nmax)
+            self.c[key] = r
+        return r
+
+
+# ================================================================ verified roots
+def _sgn_f(x, B, T):
+    """Rigorous sign of P(x) - T at floats x, B, T >= 0 (+1, -1, or 0 if undecided).  The binary64 error of
+    p = x + B, q = 2x - B (2x exact), p*p*q and the final subtraction is below 5u (p^2 |q|) + u |f|; the test uses
+    8u (p^2 |q| + T) + 4u |f| + 1e-300."""
+    p = x + B
+    q = 2.0 * x - B
+    r = p * p * q
+    f = r - T
+    e = 8.0 * _U * (abs(r) + T) + 4.0 * _U * abs(f) + 1e-300
+    if f > e:
+        return 1
+    if f < -e:
+        return -1
+    return 0
+
+
+def _guess_big(B, T):
+    aB = abs(B)
+    if B < 0 and T < aB ** 3:
+        c = -1.0 + 2.0 * T / aB ** 3
+        ph = math.acos(max(-1.0, min(1.0, c)))
+        return aB * (math.cos(ph / 3.0) + 0.5)
+    h = B ** 3 / 8.0 + T / 4.0
+    sd = math.sqrt(max(0.0, T * (B ** 3 + T) / 16.0))
+    A, C = h + sd, h - sd
+    y = math.copysign(abs(A) ** (1.0 / 3.0), A) + math.copysign(abs(C) ** (1.0 / 3.0), C)
+    return y - B / 2.0
+
+
+def _guess_mid(B, T):
+    aB = -B
+    c = -1.0 + 2.0 * T / aB ** 3
+    ph = math.acos(max(-1.0, min(1.0, c)))
+    return aB * (math.cos((2.0 * math.pi - ph) / 3.0) + 0.5)
+
+
+def _newton_big(B, T, xm):
+    lo = xm
+    hi = xm + 2.0 * (T / 2.0) ** (1.0 / 3.0) + abs(B) + 1.0      # starting bracket only (the result is verified)
+    x = _guess_big(B, T)
+    if not (lo < x < hi):
+        x = 0.5 * (lo + hi)
+    for _ in range(60):
+        fx = (x + B) * (x + B) * (2 * x - B) - T
+        d = 6 * x * (x + B)
+        if fx > 0:
+            hi = min(hi, x)
+        else:
+            lo = max(lo, x)
+        nx = x - fx / d if d > 0 else 0.5 * (lo + hi)
+        if not (lo < nx < hi):
+            nx = 0.5 * (lo + hi)
+        if abs(nx - x) <= 2e-15 * max(1.0, abs(x)):
+            return nx
+        x = nx
+    return x
+
+
+def xbig(B, Tlo, Thi=None):
+    """Interval containing the big root x(B, T) for every T in [Tlo, Thi]: the lower end a has a = xm or P(a) < Tlo, the
+    upper end b has P(b) > Thi (P increasing on [xm, inf))."""
+    if Thi is None:
+        Thi = Tlo
+    xm = -B if B < 0 else B / 2.0
+    if Thi <= 0.0:
+        return (xm, xm)
+    x = _newton_big(B, 0.5 * (Tlo + Thi), xm)
+    delta = 8e-16 * max(1.0, abs(x))
+    for _ in range(60):
+        a, b = max(xm, x - delta), x + delta
+        if (a <= xm or _sgn_f(a, B, Tlo) == -1) and _sgn_f(b, B, Thi) == 1:
+            return (a, b)
+        delta *= 4.0
+    raise RuntimeError('big-root bracket failed: B=%r T=[%r, %r]' % (B, Tlo, Thi))
+
+
+def xmid(B, Tlo, Thi=None):
+    """Interval containing the middle root x(B, T) in [|B|/2, |B|] for every T in [Tlo, Thi] (B < 0,
+    Thi <= |B|^3/2): lower end a = |B|/2 or P(a) > Thi, upper end b = |B| or P(b) < Tlo (P decreasing there)."""
+    if Thi is None:
+        Thi = Tlo
+    aB = -B
+    lo0, hi0 = aB / 2.0, aB
+    if Thi <= 0.0:
+        return (hi0, hi0)
+    T = 0.5 * (Tlo + Thi)
+    lo, hi = lo0, hi0
+    x = _guess_mid(B, T)
+    if not (lo < x < hi):
+        x = 0.5 * (lo + hi)
+    for _ in range(80):
+        fx = (x + B) * (x + B) * (2 * x - B) - T
+        if fx > 0:
+            lo = x
+        else:
+            hi = x
+        d = 6 * x * (x + B)
+        nx = x - fx / d if d < 0 else 0.5 * (lo + hi)
+        if not (lo < nx < hi):
+            nx = 0.5 * (lo + hi)
+        if abs(nx - x) <= 2e-15 * max(1.0, abs(x)):
+            x = nx
+            break
+        x = nx
+    delta = 8e-16 * max(1.0, abs(x))
+    for _ in range(60):
+        a, b = max(lo0, x - delta), min(hi0, x + delta)
+        if (a <= lo0 or _sgn_f(a, B, Thi) == 1) and (b >= hi0 or _sgn_f(b, B, Tlo) == -1):
+            return (a, b)
+        delta *= 4.0
+    raise RuntimeError('middle-root bracket failed: B=%r T=[%r, %r]' % (B, Tlo, Thi))
+
+
+def _cbrt_lo(y):
+    """A float c >= 0 with c^3 <= y (y >= 0)."""
+    if y <= 0:
+        return 0.0
+    c = y ** (1.0 / 3.0)
+    while pw((c, c), 3)[1] > y:
+        c = dn(c)
+    return c
+
+
+# ================================================================ pointwise pieces
+def _nonneg(a):
+    return (max(0.0, a[0]), max(0.0, a[1]))
+
+
+def _hull0(a):
+    return (min(0.0, a[0]), max(0.0, a[1]))
+
+
+def V_iv(Bi, xi):
+    """V = -3B(2x - B)(x + B)."""
+    return mul(scal(-3.0, Bi), mul(sub(scal(2.0, xi), Bi), add(xi, Bi)))
+
+
+def VB_iv(Bi, xi):
+    """V_B = -6x^2 + 9B^2/2 - 3B^3/(2x) (either root, x > 0)."""
+    return add(add(scal(-6.0, sqr(xi)), scal(4.5, sqr(Bi))), neg(div(scal(1.5, pw(Bi, 3)), xi)))
+
+
+def Vw_big_iv(Bi, xi):
+    """V_w on the big root: -sqrt3 B (4x + B)(2x - B)^(1/2)/x  (the middle root has the opposite sign)."""
+    q = _nonneg(sub(scal(2.0, xi), Bi))
+    v = neg(div(mul(mul(SQRT3, mul(Bi, add(scal(4.0, xi), Bi))), isqrt(q)), xi))
+    return neg(v) if MUTANT == 'vw-sign' else v
+
+
+_P = (6.0, 3.0, -4.5, -0.75, 0.75)          # p(r), increasing degree
+_DP = (3.0, -9.0, -2.25, 3.0)               # p'(r)
+
+
+def _horner(coef, R):
+    acc = (coef[-1], coef[-1])
+    for c in reversed(coef[:-1]):
+        acc = add(mul(acc, R), (c, c))
+    return acc
+
+
+def p_range(R):
+    """Mean-value enclosure of p over the interval R."""
+    rc = 0.5 * (R[0] + R[1])
+    return add(_horner(_P, (rc, rc)), mul(_horner(_DP, R), sub(R, (rc, rc))))
+
+
+def _g_pt(r):
+    ri = (r, r)
+    return neg(mul(ri, add(sub((4.0, 4.0), ri), sqr(ri))))
+
+
+def g_range(R):
+    """g(r) = -r(4 - r + r^2) is strictly decreasing (g' = -4 + 2r - 3r^2 < 0)."""
+    if MUTANT == 'ww-drop':
+        return (0.0, 0.0)
+    return (_g_pt(R[1])[0], _g_pt(R[0])[1])
+
+
+def _e_pt(r):
+    t = _nonneg(sub((2.0, 2.0), (r, r)))
+    ri = (r, r)
+    return mul(mul(t, isqrt(t)), add(add((4.0, 4.0), scal(2.0, ri)), sqr(ri)))
+
+
+def e_range(R):
+    """e(r) = (2 - r)^(3/2)(4 + 2r + r^2) is decreasing on r <= 2 (e' = (2 - r)^(1/2)(-2 - r - 7r^2/2) < 0)."""
+    return (_e_pt(R[1])[0], _e_pt(R[0])[1])
+
+
+# ================================================================ image rectangles
+def d_iv(ai, bi):
+    return add(divc(neg(mul(ai, bi)), 4.0), divc(pw(ai, 3), 72.0))
+
+
+def d_range(a0, a1, b0, b1):
+    """Range of d = -a beta/4 + a^3/72 over a rectangle: linear in beta (extremes at beta = b0, b1); for fixed beta
+    the a-extremes are at a0, a1 or at a = +-sqrt(6 beta) (beta > 0), where d = -+ beta^(3/2)/sqrt6."""
+    cands = []
+    for b in (b0, b1):
+        bi = (b, b)
+        for a in (a0, a1):
+            cands.append(d_iv((a, a), bi))
+        if b > 0:
+            r = isqrt(scal(6.0, bi))
+            val = div(mul(bi, isqrt(bi)), isqrt((6.0, 6.0)))
+            if r[0] < a1 and a0 < r[1]:
+                cands.append(neg(val))
+            if -r[1] < a1 and a0 < -r[0]:
+                cands.append(val)
+    return (min(c[0] for c in cands), max(c[1] for c in cands))
+
+
+def box_geometry(box):
+    """(Bl, Bh, ul, uh): ranges of B and u over a box of scaled coordinates."""
+    (a0, a1), (b0, b1), (c0, c1) = box
+    A2 = sqr((a0, a1))
+    Bl = dn(b0 - up(A2[1] / 12.0))
+    Bh = up(b1 - dn(A2[0] / 12.0))
+    dl, dh = d_range(a0, a1, b0, b1)
+    return Bl, Bh, dn(c0 + dl), up(c1 + dh)
+
+
+def wT(ul, uh):
+    """(s, wl, wh, Tl, Th): s = sign of u on the box (0 if the u-range meets 0), ranges of w = |u| and T = 3w^2."""
+    if ul > 0.0:
+        s, wl, wh = 1, ul, uh
+    elif uh < 0.0:
+        s, wl, wh = -1, -uh, -ul
+    else:
+        s, wl, wh = 0, 0.0, max(-ul, uh)
+    Tl = max(0.0, dn(3.0 * max(0.0, dn(wl * wl)))) if wl > 0.0 else 0.0
+    Th = up(3.0 * up(wh * wh))
+    return s, wl, wh, Tl, Th
+
+
+def half_cube(B):
+    """|B|^3/2 for a float B < 0."""
+    return scal(0.5, pw((-B, -B), 3))
+
+
+# ================================================================ second-order pieces
+def taylor_moments3(A, Bm, C, Bs, us):
+    """(m, int dB, int du, int dB^2, int du^2, int dB du) over a 3D box, dB = B - Bs, du = u - us (exact polynomial
+    moments; A to order 6, Bm and C to order 2)."""
+    ab = lambda i, j: mul(A[i], Bm[j])
+    ab00, ab01, ab02 = ab(0, 0), ab(0, 1), ab(0, 2)
+    ab11, ab12 = ab(1, 1), ab(1, 2)
+    ab20, ab21, ab22 = ab(2, 0), ab(2, 1), ab(2, 2)
+    ab30, ab31 = ab(3, 0), ab(3, 1)
+    ab40, ab41 = ab(4, 0), ab(4, 1)
+    ab50, ab60 = ab(5, 0), ab(6, 0)
+    Bsi, usi = (Bs, Bs), (us, us)
+    gB0 = sub(sub(ab01, divc(ab20, 12.0)), mul(Bsi, ab00))                       # int (B - Bs) over (a, beta)
+    gBB = add(add(add(ab02, divc(ab40, 144.0)), mul(sqr(Bsi), ab00)),
+              add(sub(divc(mul(Bsi, ab20), 6.0), divc(ab21, 6.0)), scal(-2.0, mul(Bsi, ab01))))
+    Id1 = add(divc(neg(ab11), 4.0), divc(ab30, 72.0))                            # int d
+    Id2 = add(add(divc(ab22, 16.0), divc(neg(ab41), 144.0)), divc(ab60, 5184.0))   # int d^2
+    IBd = sub(add(add(divc(neg(ab12), 4.0), divc(scal(5.0, ab31), 144.0)), divc(neg(ab50), 864.0)), mul(Bsi, Id1))
+    C0, C1, C2 = C[0], C[1], C[2]
+    Cs1 = sub(C1, mul(usi, C0))                                                  # int (c - us)
+    Cs2 = add(sub(C2, scal(2.0, mul(usi, C1))), mul(sqr(usi), C0))                # int (c - us)^2
+    m = mul(ab00, C0)
+    E1B = mul(gB0, C0)
+    E1u = add(mul(ab00, Cs1), mul(Id1, C0))
+    QB = mul(gBB, C0)
+    Qu = add(add(mul(ab00, Cs2), scal(2.0, mul(Id1, Cs1))), mul(Id2, C0))
+    QBu = add(mul(gB0, Cs1), mul(IBd, C0))
+    return m, E1B, E1u, _nonneg(QB), _nonneg(Qu), QBu
+
+
+def _quad(SBB, SBw, Sww, QB, Qu, QBu, s):
+    """Remainder integral 1/2 int (h_BB dB^2 + 2 h_Bw dB dw + h_ww dw^2) with averaged Hessian entries h in the ranges
+    S: in 1/2 (SBB QB + Sww Qu) + s c QBu +- d sqrt(QB Qu), c = centre and d = radius of SBw (dw = s du)."""
+    c = 0.5 * (SBw[0] + SBw[1])
+    dBw = up(max(up(SBw[1] - c), up(c - SBw[0])))
+    cross = scal(float(s), mul((c, c), QBu))
+    rad = up(dBw * up(math.sqrt(up(QB[1] * Qu[1]))) * (1.0 + 4 * _U))
+    q = add(add(scal(0.5, mul(SBB, QB)), scal(0.5, mul(Sww, Qu))), cross)
+    return (dn(q[0] - rad), up(q[1] + rad))
+
+
+def _expansion_point(ac, bc, cc, Bl, Bh, s, wl, wh):
+    """Float point (B*, w*) of the image rectangle near the image of the box centre (a^, beta, c^) = (ac, bc, cc)."""
+    Bs = min(max(bc - ac * ac / 12.0, Bl), Bh)
+    ws = min(max(s * (cc - ac * bc / 4.0 + ac ** 3 / 72.0), wl), wh)
+    return Bs, ws
+
+
+def _intersect(I1, I2, what):
+    lo, hi = max(I1[0], I2[0]), min(I1[1], I2[1])
+    if lo > hi:
+        raise RuntimeError('inconsistent %s enclosures: %r vs %r' % (what, I1, I2))
+    return (lo, hi)
+
+
+class Rect:
+    """Image rectangle of a box with its corner roots (big root)."""
+    def __init__(self, Bl, Bh, ul, uh):
+        self.Bl, self.Bh, self.ul, self.uh = Bl, Bh, ul, uh
+        self.s, self.wl, self.wh, self.Tl, self.Th = wT(ul, uh)
+        self._c = None
+
+    def corners(self):
+        if self._c is None:
+            Bl, Bh, Tl, Th = self.Bl, self.Bh, self.Tl, self.Th
+            self._c = (xbig(Bl, Tl), xbig(Bl, Th), xbig(Bh, Tl), xbig(Bh, Th))
+        return self._c
+
+
+def V_range(R):
+    """[min V, max V] over the rectangle: V non-increasing in B; in T non-increasing for B > 0, non-decreasing for
+    B < 0; V(0, T) = 0."""
+    xLL, xLH, xHL, xHH = R.corners()
+    Bl, Bh = R.Bl, R.Bh
+    if MUTANT == 'corner-swap':
+        xHH, xHL = xHL, xHH
+    vmin = V_iv((Bh, Bh), xHH)[0] if Bh > 0 else (V_iv((Bh, Bh), xHL)[0] if Bh < 0 else 0.0)
+    vmax = V_iv((Bl, Bl), xLH)[1] if Bl < 0 else (V_iv((Bl, Bl), xLL)[1] if Bl > 0 else 0.0)
+    return (vmin, vmax)
+
+
+def _Bneg3(Bi):
+    lo = pw((max(0.0, -Bi[1]),) * 2, 3)[0]
+    hi = pw((max(0.0, -Bi[0]),) * 2, 3)[1]
+    return (max(0.0, lo), hi)
+
+
+def _Bneg(Bi):
+    return (max(0.0, -Bi[1]), max(0.0, -Bi[0]))
+
+
+def _H_at(B, T, xi):
+    return add(add(scal(2.0, _Bneg3((B, B))), (dn(2.0 * T), up(2.0 * T))), V_iv((B, B), xi))
+
+
+def H_range(R):
+    """[min H, max H]: H is non-increasing in B and non-decreasing in T."""
+    xLL, xLH, xHL, xHH = R.corners()
+    return (max(0.0, _H_at(R.Bh, R.Tl, xHL)[0]), _H_at(R.Bl, R.Th, xLH)[1])
+
+
+def big_hessian(R, with_H):
+    """Interval ranges (S_BB, S_Bw, S_ww) of the (B, w)-Hessian of V (or of H if with_H) over the rectangle (one side of
+    u = 0).  x is increasing in T and, for fixed T, unimodal in B with minimum x = B at B = (T/4)^(1/3); r = B/x is
+    non-decreasing in B, increasing in T for B < 0 and decreasing for B > 0; r in [-1, 2]."""
+    xLL, xLH, xHL, xHH = R.corners()
+    Bl, Bh, Tl = R.Bl, R.Bh, R.Tl
+    XL, XH = min(xLL[0], xHL[0]), max(xLH[1], xHH[1])
+    if Bh > 0:
+        XL = min(XL, _cbrt_lo(max(0.0, dn(Tl / 4.0))))
+    if XL <= 0.0:
+        Rr = (-1.0, 2.0)
+        XL = 0.0
+    else:
+        rmin = div((Bl, Bl), xLL)[0] if Bl < 0 else (div((Bl, Bl), xLH)[0] if Bl > 0 else 0.0)
+        rmax = div((Bh, Bh), xHL)[1] if Bh > 0 else (div((Bh, Bh), xHH)[1] if Bh < 0 else 0.0)
+        Rr = (max(-1.0, rmin), min(2.0, rmax))
+    X = (XL, XH)
+    SBB = mul(X, p_range(Rr))
+    if with_H:
+        SBB = add(SBB, scal(12.0, _Bneg((Bl, Bh))))
+    SBw = neg(mul(HSQRT3, mul(isqrt(X), e_range(Rr))))
+    Sww = g_range(Rr)
+    if with_H:
+        Sww = add(Sww, (12.0, 12.0))
+    return SBB, SBw, Sww
+
+
+def big_center(Bs, ws, with_H):
+    """(value, d/dB, d/dw) of V (or H) at the float point (B*, w*), w* >= 0."""
+    w2 = (max(0.0, dn(ws * ws)), up(ws * ws))
+    T2 = (dn(3.0 * w2[0]), up(3.0 * w2[1]))
+    xs = xbig(Bs, T2[0], T2[1])
+    if xs[0] <= 0.0:
+        return None
+    Bsi = (Bs, Bs)
+    v0, vB, vw = V_iv(Bsi, xs), VB_iv(Bsi, xs), Vw_big_iv(Bsi, xs)
+    if with_H:
+        v0 = add(add(scal(2.0, _Bneg3(Bsi)), scal(2.0, T2)), v0)
+        vB = sub(vB, scal(6.0, sqr(_Bneg(Bsi))))
+        vw = add(vw, scal(12.0, (ws, ws)))
+    return v0, vB, vw
+
+
+def second_order(R, Bs, ws, mom, with_H):
+    """Second-order enclosure (None if the expansion point has x = 0)."""
+    c = big_center(Bs, ws, with_H)
+    if c is None:
+        return None
+    v0, vB, vw = c
+    SBB, SBw, Sww = big_hessian(R, with_H)
+    m, E1B, E1u, QB, Qu, QBu = mom
+    lin = add(add(mul(v0, m), mul(vB, E1B)), scal(float(R.s), mul(vw, E1u)))
+    return add(lin, _quad(SBB, SBw, Sww, QB, Qu, QBu, R.s))
+
+
+# ================================================================ 3D box integrals (scaled coordinates, fixed k)
+def _box3_common(box, Ma, Mb, Mc):
+    (a0, a1), (b0, b1), (c0, c1) = box
+    A, Bm, C = Ma.get(a0, a1), Mb.get(b0, b1), Mc.get(c0, c1)
+    R = Rect(*box_geometry(box))
+    return A, Bm, C, R
+
+
+def big_box3(box, Ma, Mb, Mc, with_H=False, second=True, parts=False):
+    """Enclosure of int_box V dmu (with_H: of H) over a 3D box of scaled coordinates."""
+    A, Bm, C, R = _box3_common(box, Ma, Mb, Mc)
+    m0 = mul(mul(A[0], Bm[0]), C[0])
+    I1 = mul(m0, H_range(R) if with_H else V_range(R))
+    I2 = None
+    if second and R.s != 0:
+        (a0, a1), (b0, b1), (c0, c1) = box
+        Bs, ws = _expansion_point(0.5 * (a0 + a1), 0.5 * (b0 + b1), 0.5 * (c0 + c1), R.Bl, R.Bh, R.s, R.wl, R.wh)
+        I2 = second_order(R, Bs, ws, taylor_moments3(A, Bm, C, Bs, R.s * ws), with_H)
+    I = I1 if I2 is None else _intersect(I1, I2, 'H' if with_H else 'V')
+    return (I, I1, I2) if parts else I
+
+
+def H2_point(B, T):
+    """H_2(B, T) = 2|B|^3 + 2T + V(B, T; middle root) on {B < 0, T < |B|^3/2}, 0 outside; near the boundary
+    0 <= H_2(B, T) <= H_2(B, T') for T' <= T inside (H_2 is non-increasing in T)."""
+    if B >= 0.0:
+        return (0.0, 0.0)
+    half = half_cube(B)
+    if T >= half[1]:
+        return (0.0, 0.0)
+    if T > half[0]:
+        return (0.0, H2_point(B, dn(dn(half[0]) * (1.0 - 1e-12)))[1])
+    v = add(add(scal(2.0, pw((-B, -B), 3)), (dn(2 * T), up(2 * T))), V_iv((B, B), xmid(B, T)))
+    return (max(0.0, v[0]), max(0.0, v[1]))
+
+
+def _H2_center(Bs, ws, A, Bm, C, s):
+    w2 = (max(0.0, dn(ws * ws)), up(ws * ws))
+    T2 = (dn(3.0 * w2[0]), up(3.0 * w2[1]))
+    if Bs < 0.0 and T2[1] < half_cube(Bs)[0]:
+        xs = xmid(Bs, T2[0], T2[1])
+        Bsi = (Bs, Bs)
+        h0 = add(add(scal(-2.0, pw(Bsi, 3)), scal(2.0, T2)), V_iv(Bsi, xs))
+        hB = add(scal(-6.0, sqr(Bsi)), VB_iv(Bsi, xs))
+        vw = neg(Vw_big_iv(Bsi, xs))                     # middle root: opposite sign
+        if MUTANT == 'mid-branch':
+            vw = neg(vw)
+        hw = add(scal(12.0, (ws, ws)), vw)
+        return h0, hB, hw
+    if Bs >= 0.0 or T2[0] > half_cube(Bs)[1]:
+        return (0.0, 0.0), (0.0, 0.0), (0.0, 0.0)        # outside the support: value and gradient 0
+    return None
+
+
+def H2_box3(box, Ma, Mb, Mc, second=True, parts=False):
+    """Enclosure of int_box H_2 dmu.  Inside the support the middle root x and r = B/x are non-increasing in B and T,
+    r in [-2, -1]; on boxes meeting the support boundary the zero extension of H_2 is C^1 with zero value and gradient
+    on {T = |B|^3/2} and piecewise C^2, so the Taylor remainder holds with Hessian ranges hulled with 0, the ranges being
+    taken over rectangle and support, where |B|/2 <= x <= x(Bl, Tl) and r in [-2, r(Bl, Tl)]."""
+    (a0, a1), (b0, b1), (c0, c1) = box
+    Bl, Bh, ul, uh = box_geometry(box)
+    zero = ((0.0, 0.0),) * 3
+    if Bl >= 0.0:
+        return zero if parts else (0.0, 0.0)
+    s, wl, wh, Tl, Th = wT(ul, uh)
+    if Tl >= half_cube(Bl)[1]:
+        return zero if parts else (0.0, 0.0)
+    A, Bm, C = Ma.get(a0, a1), Mb.get(b0, b1), Mc.get(c0, c1)
+    m0 = mul(mul(A[0], Bm[0]), C[0])
+    inside = Bh < 0.0 and Th < half_cube(Bh)[0]
+    I2 = None
+    if inside:
+        xLL, xHH = xmid(Bl, Tl), xmid(Bh, Th)
+        hmin = add(add(scal(2.0, pw((-Bh, -Bh), 3)), (dn(2 * Th), up(2 * Th))), V_iv((Bh, Bh), xHH))[0]
+        hmax = add(add(scal(2.0, pw((-Bl, -Bl), 3)), (dn(2 * Tl), up(2 * Tl))), V_iv((Bl, Bl), xLL))[1]
+        I1 = mul(m0, (max(0.0, hmin), max(0.0, hmax)))
+        X = (xHH[0], xLL[1])
+        Rr = (max(-2.0, div((Bh, Bh), xHH)[0]), min(-1.0, div((Bl, Bl), xLL)[1]))
+        Bhe = Bh
+    else:
+        I1 = mul(m0, (H2_point(Bh, Th)[0], H2_point(Bl, Tl)[1]))
+        if Tl < half_cube(Bl)[0]:
+            Bhe = min(Bh, 0.0)
+            xLL = xmid(Bl, Tl)
+            X = (max(0.0, -Bhe / 2.0), xLL[1])
+            Rr = (-2.0, min(-1.0, div((Bl, Bl), xLL)[1]))
+        else:
+            X = None
+    if second and s != 0 and X is not None:
+        SBB = add(scal(-12.0, (Bl, Bhe)), mul(X, p_range(Rr)))
+        SBw = mul(HSQRT3, mul(isqrt(X), e_range(Rr)))
+        Sww = add((12.0, 12.0), g_range(Rr))
+        if not inside:
+            SBB, SBw, Sww = _hull0(SBB), _hull0(SBw), _hull0(Sww)
+        Bs, ws = _expansion_point(0.5 * (a0 + a1), 0.5 * (b0 + b1), 0.5 * (c0 + c1), Bl, Bh, s, wl, wh)
+        cen = _H2_center(Bs, ws, A, Bm, C, s)
+        if cen is not None:
+            h0, hB, hw = cen
+            m, E1B, E1u, QB, Qu, QBu = taylor_moments3(A, Bm, C, Bs, s * ws)
+            lin = add(add(mul(h0, m), mul(hB, E1B)), scal(float(s), mul(hw, E1u)))
+            I2 = add(lin, _quad(SBB, SBw, Sww, QB, Qu, QBu, s))
+    I = I1 if I2 is None else _intersect(I1, I2, 'H2')
+    return (I, I1, I2) if parts else I
+
+
+# ================================================================ 4D: the gap mark k as a box coordinate
+K_LO, K_HI = 0.5, 2.0
+PMIN, PMAX = -6, 2
+KLEVEL = 12                          # elementary dyadic cells of [1/2, 2]: width 3 * 2^-13
+
+
+def _cbrt_iv(x):
+    """Enclosure of x^(1/3) for a float x > 0."""
+    c = x ** (1.0 / 3.0)
+    lo, hi = c, c
+    while pw((lo, lo), 3)[1] > x:
+        lo = dn(lo)
+    while pw((hi, hi), 3)[0] < x:
+        hi = up(hi)
+    return (lo, hi)
+
+
+def kweights(k):
+    """f_p(k) = 3 k^(1/3) k^(p/2) e^(-12 k^2), p = PMIN..PMAX, at a float k > 0  (3k^2 x e^(-12k^2) k^(-5/3) x k^(p/2))."""
+    third = _cbrt_iv(k)
+    if MUTANT == 'k-weight':
+        third = (1.0, 1.0)
+    base = mul(scal(3.0, third), exp_neg((dn(12.0 * dn(k * k)), up(12.0 * up(k * k)))))
+    sk = isqrt((k, k))
+    isk = div((1.0, 1.0), sk)
+    out = {}
+    v = base
+    for p in range(0, PMAX + 1):
+        out[p] = v
+        v = mul(v, sk)
+    v = mul(base, isk)
+    for p in range(-1, PMIN - 1, -1):
+        out[p] = v
+        v = mul(v, isk)
+    return out
+
+
+def elementary_kmoments(nsub=24):
+    """K_p = int f_p over each elementary cell.  Every f_p is convex on [1/2, 2]: f''/f = (24k - q/k)^2 - q/k^2 - 24 with
+    q = 1/3 + p/2 in [-8/3, 4/3] is >= 87 - 16/3 - 24 > 0 (q >= 0) and >= 144 - 24 > 0 (q < 0).  Hence on every sub-cell
+    h f(mid) <= int f <= h (f(l) + f(r))/2.  All cell and sub-cell ends are exact dyadic floats."""
+    n = 2 ** KLEVEL
+    width = (K_HI - K_LO) / n
+    cells = []
+    prev = kweights(K_LO)
+    for i in range(n):
+        k0 = K_LO + width * i
+        acc = {p: (0.0, 0.0) for p in range(PMIN, PMAX + 1)}
+        for j in range(nsub):
+            l = k0 + width * j / nsub
+            r = k0 + width * (j + 1) / nsub
+            h = (dn(r - l), up(r - l))
+            mid = kweights(0.5 * (l + r))
+            fr = kweights(r)
+            for p in acc:
+                lo = mul(h, mid[p])[0]
+                hi = mul(h, scal(0.5, add(prev[p], fr[p])))[1]
+                acc[p] = (dn(acc[p][0] + lo), up(acc[p][1] + hi))
+            prev = fr
+        cells.append(acc)
+    return cells
+
+
+class KMoments:
+    """K_p on dyadic k-cells (unions of elementary cells), cached."""
+    def __init__(self, elem):
+        self.elem, self.c = elem, {}
+        self.width = (K_HI - K_LO) / len(elem)
+
+    def get(self, k0, k1):
+        key = (k0, k1)
+        r = self.c.get(key)
+        if r is None:
+            i0 = round((k0 - K_LO) / self.width)
+            i1 = round((k1 - K_LO) / self.width)
+            if K_LO + i0 * self.width != k0 or K_LO + i1 * self.width != k1:
+                raise ValueError('k-cell is not a union of elementary cells: %r' % (key,))
+            r = {}
+            for p in range(PMIN, PMAX + 1):
+                lo = hi = 0.0
+                for i in range(i0, i1):
+                    e = self.elem[i][p]
+                    lo, hi = dn(lo + e[0]), up(hi + e[1])
+                r[p] = (lo, hi)
+            self.c[key] = r
+        return r
+
+
+def _integrate4(terms, A, Bm, C, K):
+    acc = (0.0, 0.0)
+    for (cf, i, j, l, p) in terms:
+        acc = add(acc, mul(cf, mul(mul(A[i], Bm[j]), mul(C[l], K[p]))))
+    return acc
+
+
+def _prod4(t1, t2):
+    out = {}
+    for (c1, i1, j1, l1, p1) in t1:
+        for (c2, i2, j2, l2, p2) in t2:
+            key = (i1 + i2, j1 + j2, l1 + l2, p1 + p2)
+            c = mul(c1, c2)
+            out[key] = add(out[key], c) if key in out else c
+    return [(c,) + key for key, c in sorted(out.items())]
+
+
+_C12 = div((1.0, 1.0), (12.0, 12.0))
+_C72 = div((1.0, 1.0), (72.0, 72.0))
+
+
+def taylor_moments4(A, Bm, C, K, Bs, us):
+    """The six Taylor moments over a 4D box against W(k) dk x Gaussian, in the original coordinates:
+    dB = beta - a^2 k^-1/12 - Bs,  du = c k^(1/2) - a beta k^(-1/2)/4 + a^3 k^(-3/2)/72 - us
+    (terms: coefficient, powers of a, beta, c, and p for k^(p/2))."""
+    dB = [((1.0, 1.0), 0, 1, 0, 0), (neg(_C12), 2, 0, 0, -2), ((-Bs, -Bs), 0, 0, 0, 0)]
+    du = [((1.0, 1.0), 0, 0, 1, 1), ((-0.25, -0.25), 1, 1, 0, -1), (_C72, 3, 0, 0, -3), ((-us, -us), 0, 0, 0, 0)]
+    m = _integrate4([((1.0, 1.0), 0, 0, 0, 0)], A, Bm, C, K)
+    E1B = _integrate4(dB, A, Bm, C, K)
+    E1u = _integrate4(du, A, Bm, C, K)
+    QB = _integrate4(_prod4(dB, dB), A, Bm, C, K)
+    Qu = _integrate4(_prod4(du, du), A, Bm, C, K)
+    QBu = _integrate4(_prod4(dB, du), A, Bm, C, K)
+    return m, E1B, E1u, _nonneg(QB), _nonneg(Qu), QBu
+
+
+def _scaled_range(x0, x1, f0, f1):
+    p = (x0 * f0, x0 * f1, x1 * f0, x1 * f1)
+    return dn(min(p)), up(max(p))
+
+
+def scaled_box(box):
+    """Outer box of the scaled coordinates (a/sqrt k, beta, sqrt(k) c) over a 4D box (a, beta, c, k)."""
+    (a0, a1), (b0, b1), (c0, c1), (k0, k1) = box
+    sk0, sk1 = isqrt((k0, k0)), isqrt((k1, k1))
+    isk = (div((1.0, 1.0), sk1)[0], div((1.0, 1.0), sk0)[1])
+    sk = (sk0[0], sk1[1])
+    return (_scaled_range(a0, a1, isk[0], isk[1]), (b0, b1), _scaled_range(c0, c1, sk[0], sk[1]))
+
+
+def H_box4(box, Ma, Mb, Mc, Km, second=True, parts=False):
+    """Enclosure of int_box W(k) H dmu dk over a 4D box (a, beta, c, k), W(k) = 3 k^(1/3) e^(-12k^2)."""
+    (a0, a1), (b0, b1), (c0, c1), (k0, k1) = box
+    A, Bm, C, K = Ma.get(a0, a1), Mb.get(b0, b1), Mc.get(c0, c1), Km.get(k0, k1)
+    R = Rect(*box_geometry(scaled_box(box)))
+    m0 = mul(mul(A[0], Bm[0]), mul(C[0], K[0]))
+    I1 = mul(m0, H_range(R))
+    I2 = None
+    if second and R.s != 0:
+        ac, bc, cc, kc = 0.5 * (a0 + a1), 0.5 * (b0 + b1), 0.5 * (c0 + c1), 0.5 * (k0 + k1)
+        skc = math.sqrt(kc)
+        Bs, ws = _expansion_point(ac / skc, bc, cc * skc, R.Bl, R.Bh, R.s, R.wl, R.wh)
+        I2 = second_order(R, Bs, ws, taylor_moments4(A, Bm, C, K, Bs, R.s * ws), True)
+    I = I1 if I2 is None else _intersect(I1, I2, 'H (4D)')
+    return (I, I1, I2) if parts else I
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/certificate.py b/frontiers/c8_elder_failure_coefficient_20261001/certificate.py
new file mode 100644
index 000000000..0cbecb81e
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/certificate.py
@@ -0,0 +1,737 @@
+#!/usr/bin/env python3
+"""Certificate for CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1 (standard library only).
+
+    python3 -B -S certificate.py --write [--procs N]     full run, writes RESULTS.json
+    python3 -B -S certificate.py --check [--procs N]     self-test, then full replay; every replayed enclosure must lie
+                                                         inside the published one (and the published one must not be
+                                                         looser than the replay by more than the publication margin),
+                                                         leaf counts must agree
+    python3 -B -S certificate.py --check --mutant NAME   must exit 1
+    python3 -B certificate.py --controls                 floating-point cross-checks (not part of the certificate)
+
+Certified quantities (planar near cluster coefficients of [NUM], reference kernel):
+    J_fail(k) = J_1 + J_2 = 3k^2 (E[2(B_-)^3] + E[2T] + E[V]),   J_2(k) = 3k^2 E[H_2]       (k = 1/2, 1, 2)
+    I_k = int_{1/2}^{2} e^(-12k^2) k^(-5/3) J_fail(k) dk,   C_fail^{B,K} = const * I_b * I_k,
+    const = 4 / (sqrt2 (2 pi)^(5/2) sqrt12),   I_b = int_0^1 e^(-b^2) db = sqrt(pi) (Phi(sqrt2) - 1/2).
+d = 3 (conditional on Math-#175 (L1) and Math-#184 Theorem 1): a_j^(3) = R_3(b) alpha_j, R_3(0) = (32 + 28 sqrt2)/17, and
+    C_fail^(3) = const_3 I_b^(3) I_k,  const_3 = 4 pi (4/sqrt2) (2 pi)^-4 12^(-1/2),  I_b^(3) = (1/2) int_0^1 e^(-b^2) g(b) db,
+    g(b) = E[(b + sqrt2 Z)_+^3] = (b^3 + 6b) Phi(b/sqrt2) + sqrt2 (b^2 + 4) phi(b/sqrt2)."""
+import sys, os, json, time, math, argparse
+from decimal import Decimal, ROUND_FLOOR, ROUND_CEILING
+from fractions import Fraction as Fr
+from multiprocessing import Pool
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+import ia
+from ia import (dn, up, add, sub, neg, mul, div, scal, divc, sqr, pw, isqrt, exp_neg, Phi, Phi_point, SQRT2PI,
+                from_frac, PI)
+import boxes as BX
+import torus as TOR
+
+PACKET = 'CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1'
+KS = (0.5, 1.0, 2.0)
+TOL3 = {'V': 1e-8, 'H2': 1e-8}      # per-box width tolerance (unscaled expectations)
+TOL4 = 2.5e-9
+L_BOX = 8.0                         # boxes cover [-8 sigma, 8 sigma] in every jet coordinate
+N0, NK0 = 8, 8                      # initial grid: 8 cells per jet coordinate, 8 dyadic k-cells
+SK = 0.8                            # k-scale used only to choose the split direction
+MAXDEPTH3, MAXDEPTH4 = 60, 80
+MARGIN = 1e-7                       # publication: widen relatively by MARGIN, then round outward to 10 digits
+DIGITS = 10
+SQRT2 = isqrt((2.0, 2.0))
+
+
+# ================================================================ drivers
+def sig3(k):
+    """Scaled-law standard deviations for k in KS (all variances exact floats)."""
+    v = ((2.0 / k), 2.0, 6.0 * k)
+    if v[0] * k != 2.0 or v[2] / k != 6.0:
+        raise ValueError('k must make 2/k and 6k exact')
+    return [(isqrt((x, x)), (x, x)) for x in v]
+
+
+_ST = {}
+
+
+def _init3(k):
+    S = sig3(k)
+    _ST['M'] = (BX.Moments(S[0][0], S[0][1], 6), BX.Moments(S[1][0], S[1][1], 2), BX.Moments(S[2][0], S[2][1], 2))
+    _ST['sd'] = tuple(s[0][1] for s in S)
+
+
+def _work3(args):
+    which, tol, cells, mutant = args
+    BX.MUTANT = mutant
+    Ma, Mb, Mc = _ST['M']
+    sd = _ST['sd']
+    fn = BX.H2_box3 if which == 'H2' else (lambda b, x, y, z: BX.big_box3(b, x, y, z, with_H=(which == 'H')))
+    lo = hi = 0.0
+    nleaf = 0
+    for cell in cells:
+        stack = [(cell, 0)]
+        while stack:
+            box, dep = stack.pop()
+            I = fn(box, Ma, Mb, Mc)
+            if I[1] - I[0] > tol and dep < MAXDEPTH3:
+                w = [(box[i][1] - box[i][0]) / sd[i] for i in range(3)]
+                d = w.index(max(w))
+                l_, h_ = box[d]
+                mid = 0.5 * (l_ + h_)
+                for half in ((l_, mid), (mid, h_)):
+                    nb = list(box)
+                    nb[d] = half
+                    stack.append((tuple(nb), dep + 1))
+            else:
+                lo, hi = dn(lo + I[0]), up(hi + I[1])
+                nleaf += 1
+    return lo, hi, nleaf
+
+
+def run3(k, which, tol, procs, mutant=None):
+    """Sum of box enclosures of E[which] over [-L sigma, L sigma]^3 (which in V, H2, H)."""
+    S = sig3(k)
+    edges = [[-L_BOX * s[0][1] + 2 * L_BOX * s[0][1] * i / N0 for i in range(N0 + 1)] for s in S]
+    cells = [((edges[0][i], edges[0][i + 1]), (edges[1][j], edges[1][j + 1]), (edges[2][l], edges[2][l + 1]))
+             for i in range(N0) for j in range(N0) for l in range(N0)]
+    chunks = [(which, tol, cells[p::procs], mutant) for p in range(procs)]
+    with Pool(procs, initializer=_init3, initargs=(k,)) as pool:
+        res = pool.map(_work3, chunks)
+    lo = hi = 0.0
+    n = 0
+    for r in res:
+        lo, hi, n = dn(lo + r[0]), up(hi + r[1]), n + r[2]
+    return (lo, hi), n, edges
+
+
+def _init4(elem):
+    S = sig3(1.0)                                    # original jet law N(0, diag(2, 2, 6))
+    _ST['M'] = (BX.Moments(S[0][0], S[0][1], 6), BX.Moments(S[1][0], S[1][1], 2), BX.Moments(S[2][0], S[2][1], 2))
+    _ST['K'] = BX.KMoments(elem)
+    _ST['sd'] = tuple(s[0][1] for s in S) + (SK,)
+
+
+def _work4(args):
+    tol, cells, mutant = args
+    BX.MUTANT = mutant
+    Ma, Mb, Mc = _ST['M']
+    Km = _ST['K']
+    sd = _ST['sd']
+    lo = hi = 0.0
+    nleaf = 0
+    for cell in cells:
+        stack = [(cell, 0)]
+        while stack:
+            box, dep = stack.pop()
+            I = BX.H_box4(box, Ma, Mb, Mc, Km)
+            if I[1] - I[0] > tol and dep < MAXDEPTH4:
+                w = [(box[i][1] - box[i][0]) / sd[i] for i in range(4)]
+                if box[3][1] - box[3][0] <= Km.width:
+                    w[3] = -1.0
+                d = w.index(max(w))
+                l_, h_ = box[d]
+                mid = 0.5 * (l_ + h_)
+                for half in ((l_, mid), (mid, h_)):
+                    nb = list(box)
+                    nb[d] = half
+                    stack.append((tuple(nb), dep + 1))
+            else:
+                lo, hi = dn(lo + I[0]), up(hi + I[1])
+                nleaf += 1
+    return lo, hi, nleaf
+
+
+def run4(tol, procs, elem, mutant=None):
+    S = sig3(1.0)
+    edges = [[-L_BOX * s[0][1] + 2 * L_BOX * s[0][1] * i / N0 for i in range(N0 + 1)] for s in S]
+    kw = (BX.K_HI - BX.K_LO) / NK0
+    kedges = [BX.K_LO + kw * j for j in range(NK0 + 1)]
+    cells = [((edges[0][i], edges[0][i + 1]), (edges[1][j], edges[1][j + 1]), (edges[2][l], edges[2][l + 1]),
+              (kedges[q], kedges[q + 1])) for q in range(NK0) for i in range(N0) for j in range(N0) for l in range(N0)]
+    chunks = [(tol, cells[p::procs], mutant) for p in range(procs)]
+    with Pool(procs, initializer=_init4, initargs=(elem,)) as pool:
+        res = pool.map(_work4, chunks)
+    lo = hi = 0.0
+    n = 0
+    for r in res:
+        lo, hi, n = dn(lo + r[0]), up(hi + r[1]), n + r[2]
+    return (lo, hi), n
+
+
+# ================================================================ exact and one-dimensional parts, tails
+def E2T(k):
+    """E[2T] = 6 E[u^2] = 6 (6k + 1/(4k) + 5/(216 k^3)) (exact rational)."""
+    k = Fr(k)
+    return 6 * (6 * k + Fr(1) / (4 * k) + Fr(5) / (216 * k ** 3))
+
+
+def _gs(mi):
+    """g = E[(m - beta)_+^3], g' = 3 E[(m - beta)_+^2], g'' = 6 E[(m - beta)_+] for beta ~ N(0, 2), t = m/sqrt2:
+    g = (m^3 + 6m) Phi(t) + sqrt2 (m^2 + 4) phi(t),  g' = 3 [(m^2 + 2) Phi(t) + sqrt2 m phi(t)],
+    g'' = 6 [m Phi(t) + sqrt2 phi(t)]; all increasing in m."""
+    t = div(mi, SQRT2)
+    P = Phi(t)
+    ph = div(exp_neg(divc(sqr(mi), 4.0)), SQRT2PI)
+    g = add(mul(add(pw(mi, 3), scal(6.0, mi)), P), mul(mul(SQRT2, add(sqr(mi), (4.0, 4.0))), ph))
+    g1 = scal(3.0, add(mul(add(sqr(mi), (2.0, 2.0)), P), mul(mul(SQRT2, mi), ph)))
+    g2 = scal(6.0, add(mul(mi, P), mul(SQRT2, ph)))
+    return g, g1, g2
+
+
+def tail_moments(L, s, s2, nmax):
+    """int_L^inf x^n phi_s(x) dx, n = 0..nmax, L >= 0."""
+    pL = BX.phis(L, s, s2)
+    q = BX._Qiv(div((L, L), s))
+    M = [q, mul(s2, pL)]
+    for n in range(2, nmax + 1):
+        M.append(add(mul(scal(float(n - 1), s2), M[n - 2]), mul(s2, mul(pw((L, L), n - 1), pL))))
+    return M
+
+
+def E2B(k, L=12.0, N=4000):
+    """E[2(B_-)^3] = 4 int_0^inf phi_sa(a) G(a) da with G(a) = g(a^2/12), sa^2 = 2/k: second-order Taylor expansion on each
+    of N cells about the midpoint (G'' = g''(m) a^2/36 + g'(m)/6 is increasing in a >= 0, bounded by its values at the
+    cell ends), plus the tail a > L sa with g(m) <= m^3 + 6m + sqrt2 (m^2 + 4)/sqrt(2 pi)."""
+    s, s2 = sig3(k)[0]
+    top = L * s[1]
+    acc = (0.0, 0.0)
+    for i in range(N):
+        a0, a1 = top * i / N, top * (i + 1) / N
+        M = BX.moments(a0, a1, s, s2, 2)
+        ac = 0.5 * (a0 + a1)
+        aci = (ac, ac)
+        g, g1, _ = _gs(divc(sqr(aci), 12.0))
+        _, g1l, g2l = _gs(divc(sqr((a0, a0)), 12.0))
+        _, g1h, g2h = _gs(divc(sqr((a1, a1)), 12.0))
+        G2lo = add(divc(mul(g2l, sqr((a0, a0))), 36.0), divc(g1l, 6.0))[0]
+        G2hi = add(divc(mul(g2h, sqr((a1, a1))), 36.0), divc(g1h, 6.0))[1]
+        M1c = sub(M[1], mul(aci, M[0]))
+        M2c = BX._nonneg(add(sub(M[2], scal(2.0, mul(aci, M[1]))), mul(sqr(aci), M[0])))
+        cell = add(add(mul(g, M[0]), mul(mul(g1, divc(aci, 6.0)), M1c)), scal(0.5, mul((G2lo, G2hi), M2c)))
+        acc = add(acc, cell)
+    T = tail_moments(top, s, s2, 6)
+    tail = add(add(divc(T[6], 1728.0), divc(T[2], 2.0)),
+               mul(div(SQRT2, SQRT2PI), add(divc(T[4], 144.0), scal(4.0, T[0]))))
+    return (scal(4.0, acc)[0], scal(4.0, add(acc, (0.0, tail[1])))[1])
+
+
+def abs_moments(s, s2, nmax):
+    """E|X|^n, X ~ N(0, s^2): 1, 2 s^2 phi_s(0), then (n - 1) s^2 E|X|^(n-2)."""
+    M = [(1.0, 1.0), scal(2.0, mul(s2, BX.phis(0.0, s, s2)))]
+    for n in range(2, nmax + 1):
+        M.append(mul(scal(float(n - 1), s2), M[n - 2]))
+    return M
+
+
+def _tail_union(S, L, terms):
+    """Upper bound of E[sum_terms c |X1|^i |X2|^j |X3|^l 1{some |X_m| > L sigma_m}] (union bound over m; independence)."""
+    full = [abs_moments(S[m][0], S[m][1], 6) for m in range(3)]
+    tl = [[scal(2.0, t) for t in tail_moments(L * S[m][0][1], S[m][0], S[m][1], 6)] for m in range(3)]
+    tot = (0.0, 0.0)
+    for m in range(3):
+        for cf, e in terms:
+            v = cf
+            for j in range(3):
+                v = mul(v, tl[j][e[j]] if j == m else full[j][e[j]])
+            tot = add(tot, v)
+    return tot[1]
+
+
+def tail3(k):
+    """Outside the box, |V|, H and H_2 are at most 17|B|^3 + 8T <= 68|beta|^3 + (68/1728 + 96/5184) a^6 + 48 c^2
+    + 6 a^2 beta^2 (scaled coordinates; x <= |B| + (T/2)^(1/3), |B|^3 <= 4(|beta|^3 + a^6/1728), T <= 6c^2 + 6d^2,
+    d^2 <= 2(a^2 beta^2/16 + a^6/5184))."""
+    c6 = add(divc((68.0, 68.0), 1728.0), divc((96.0, 96.0), 5184.0))
+    terms = [((68.0, 68.0), (0, 3, 0)), (c6, (6, 0, 0)), ((48.0, 48.0), (0, 0, 2)), ((6.0, 6.0), (2, 2, 0))]
+    return _tail_union(sig3(k), L_BOX, terms)
+
+
+def tail4(Ktot):
+    """Outside the jet box, for every k in [1/2, 2] (original coordinates): H <= 17|B|^3 + 8T <= 68|beta|^3 + 68 a^6/216
+    + 576 a^6/5184 + 144 c^2 + 9 a^2 beta^2 (|B| <= |beta| + a^2/6; u^2 <= 6c^2 + 3a^2 beta^2/8 + 24 a^6/5184); times
+    int W(k) dk = K_0([1/2, 2])."""
+    c6 = add(divc((68.0, 68.0), 216.0), divc((576.0, 576.0), 5184.0))
+    terms = [((68.0, 68.0), (0, 3, 0)), (c6, (6, 0, 0)), ((144.0, 144.0), (0, 0, 2)), ((9.0, 9.0), (2, 2, 0))]
+    return up(_tail_union(sig3(1.0), L_BOX, terms) * Ktot[1] * (1.0 + 4e-16))
+
+
+# ================================================================ assembly
+def m2(b):
+    """m_(2,b) = (b^2 + 2) Phi(b/sqrt2) + sqrt2 b phi(b/sqrt2) (b a float >= 0)."""
+    bi = (b, b)
+    t = div(bi, SQRT2)
+    ph = div(exp_neg(divc(sqr(t), 2.0)), SQRT2PI)
+    return add(mul(add(sqr(bi), (2.0, 2.0)), Phi(t)), mul(mul(SQRT2, bi), ph))
+
+
+def prefactor(b, k):
+    """p_b(0)/z_0 = phi(b/sqrt2) / (sqrt2 36 k^2 m_(2,b))."""
+    t = div((b, b), SQRT2)
+    ph = div(exp_neg(divc(sqr(t), 2.0)), SQRT2PI)
+    return div(ph, mul(mul(SQRT2, (36.0 * k * k, 36.0 * k * k)), m2(b)))
+
+
+def constants():
+    sqrt_pi = isqrt(PI)
+    two_pi = scal(2.0, PI)
+    const = div((4.0, 4.0), mul(mul(SQRT2, mul(sqr(two_pi), isqrt(two_pi))), isqrt((12.0, 12.0))))
+    Ib = mul(sqrt_pi, sub(Phi(SQRT2), (0.5, 0.5)))
+    return const, Ib
+
+
+def constants3():
+    """d = 3: const_3 = 4 pi (4/sqrt2) (2 pi)^-4 12^(-1/2) and R_3(0) = (32 + 28 sqrt2)/17; also checks
+    N_3(0) = pi (2 pi)^(-1/2) g(0) = 2 sqrt2, the identity behind I_b^(3)."""
+    two_pi = scal(2.0, PI)
+    const3 = mul(mul(scal(4.0, PI), div((4.0, 4.0), SQRT2)), div((1.0, 1.0), mul(pw(two_pi, 4), isqrt((12.0, 12.0)))))
+    R30 = divc(add((32.0, 32.0), scal(28.0, SQRT2)), 17.0)
+    N30 = mul(div(PI, isqrt(two_pi)), _gs((0.0, 0.0))[0])
+    two_sqrt2 = scal(2.0, SQRT2)
+    if N30[1] < two_sqrt2[0] or two_sqrt2[1] < N30[0]:
+        raise RuntimeError('N_3(0) = 2 sqrt2 check failed: %r' % (N30,))
+    return const3, R30
+
+
+def Ib3(N=100000):
+    """I_b^(3) = (1/2) int_0^1 e^(-b^2) g(b) db: on each cell e^(-b^2) is decreasing and g increasing (both positive).
+    (The prefactor e^(-3b^2/4) phi(b/sqrt2) pi (2 pi)^(-1/2) of N_3 equals e^(-b^2)/2.)"""
+    lo = hi = 0.0
+    prev_g = _gs((0.0, 0.0))[0]
+    prev_e = (1.0, 1.0)
+    for i in range(1, N + 1):
+        b0, b1 = (i - 1) / N, i / N
+        h = (dn(b1 - b0), up(b1 - b0))
+        g1 = _gs((b1, b1))[0]
+        e1 = exp_neg(sqr((b1, b1)))
+        lo = dn(lo + mul(h, mul(e1, prev_g))[0])
+        hi = up(hi + mul(h, mul(prev_e, g1))[1])
+        prev_g, prev_e = g1, e1
+    return (dn(0.5 * lo), up(0.5 * hi))
+
+
+def iv_frac(q):
+    return from_frac(q)
+
+
+def certify(procs, mutant=None, log=print, only=None):
+    """Run everything; returns the raw results (floats) and leaf counts."""
+    out = {'J': {}, 'leaves': {}}
+    t0 = time.time()
+    for k in KS:
+        if only is not None and k not in only:
+            continue
+        s = 3.0 * k * k
+        EV, nV, _ = run3(k, 'V', TOL3['V'], procs, mutant)
+        EH2, nH2, _ = run3(k, 'H2', TOL3['H2'], procs, mutant)
+        e2b = E2B(k)
+        e2t = iv_frac(E2T(k))
+        tl = tail3(k)
+        EVt = (dn(EV[0] - tl), up(EV[1] + tl))
+        EH2t = (EH2[0], up(EH2[1] + tl))
+        Jf = scal(s, add(add(e2b, e2t), EVt))
+        J2 = scal(s, EH2t)
+        out['J'][k] = {'J_fail': Jf, 'J_2': J2, 'E2B': e2b, 'E2T': e2t, 'EV_boxes': EV, 'EH2_boxes': EH2, 'tail': tl}
+        out['leaves']['V k=%g' % k] = nV
+        out['leaves']['H2 k=%g' % k] = nH2
+        log('k=%g  J_fail in [%.10f, %.10f]  J_2 in [%.10f, %.10f]  (leaves %d, %d; %.0fs)' %
+            (k, Jf[0], Jf[1], J2[0], J2[1], nV, nH2, time.time() - t0))
+    if only is None:
+        elem = BX.elementary_kmoments()
+        Km = BX.KMoments(elem)
+        Ktot = Km.get(BX.K_LO, BX.K_HI)[0]
+        Ik_boxes, n4 = run4(TOL4, procs, elem, mutant)
+        t4 = tail4(Ktot)
+        Ik = (Ik_boxes[0], up(Ik_boxes[1] + t4))
+        const, Ib = constants()
+        C = mul(mul(const, Ib), Ik)
+        out['Ik'] = {'Ik': Ik, 'boxes': Ik_boxes, 'tail': t4, 'const': const, 'Ib': Ib, 'C_fail': C, 'Wtot': Ktot,
+                     'K_m4': Km.get(BX.K_LO, BX.K_HI)[-4]}
+        out['torus'] = TOR.transfer(TOR.E_I2_upper())
+        const3, R30 = constants3()
+        Ib3_ = Ib3()
+        out['d3'] = {'const3': const3, 'R30': R30, 'Ib3': Ib3_, 'C_fail3': mul(mul(const3, Ib3_), Ik)}
+        out['leaves']['I_k 4D'] = n4
+        log('I_k in [%.10f, %.10f]  C_fail in [%.12f, %.12f]  (leaves %d; %.0fs)' % (Ik[0], Ik[1], C[0], C[1], n4,
+                                                                                    time.time() - t0))
+    return out
+
+
+# ================================================================ publication rounding and the result document
+def pub(iv_):
+    """Outward publication of an enclosure: widen by MARGIN relatively, round outward to DIGITS significant digits."""
+    lo, hi = Decimal(iv_[0]), Decimal(iv_[1])
+    lo = lo - abs(lo) * Decimal(MARGIN)
+    hi = hi + abs(hi) * Decimal(MARGIN)
+
+    def rnd(x, mode):
+        if x == 0:
+            return Decimal(0)
+        e = x.adjusted() - (DIGITS - 1)
+        return x.quantize(Decimal(1).scaleb(e), rounding=mode)
+    return [str(rnd(lo, ROUND_FLOOR)), str(rnd(hi, ROUND_CEILING))]
+
+
+def contains_pub(pub_iv, iv_):
+    return Decimal(pub_iv[0]) <= Decimal(iv_[0]) and Decimal(iv_[1]) <= Decimal(pub_iv[1])
+
+
+def not_looser(pub_iv, iv_):
+    """The published interval is within 2 MARGIN + one unit of the last digit of the replayed one."""
+    lo, hi = Decimal(iv_[0]), Decimal(iv_[1])
+    unit = lambda x: Decimal(1).scaleb(x.adjusted() - (DIGITS - 1)) if x != 0 else Decimal(0)
+    lo_min = lo - abs(lo) * Decimal(2 * MARGIN) - unit(lo)
+    hi_max = hi + abs(hi) * Decimal(2 * MARGIN) + unit(hi)
+    return Decimal(pub_iv[0]) >= lo_min and Decimal(pub_iv[1]) <= hi_max
+
+
+def derived(raw):
+    """Published quantities derived from the raw enclosures."""
+    res = {}
+    for k in KS:
+        J = raw['J'][k]
+        Jf, J2 = J['J_fail'], J['J_2']
+        J1 = sub(Jf, J2)
+        share = div(J2, Jf)
+        palm = div(scal(2.0, J2), add(Jf, J2))
+        row = {'J_fail': Jf, 'J_1': J1, 'J_2': J2, 'share_J2_over_Jfail': share, 'near_palm_excess': palm,
+               'E[2T] exact': str(E2T(k)),
+               'parts': {'E[2(B_-)^3]': J['E2B'], 'E[2T]': J['E2T'], 'E[V] boxes': J['EV_boxes'],
+                         'E[H_2] boxes': J['EH2_boxes'], 'tail bound': (0.0, J['tail'])}}
+        coeffs = {}
+        for b in (0.0, 1.0):
+            P = prefactor(b, k)
+            coeffs['b=%g' % b] = {'p_b(0)/z_0': P, 'alpha_1': mul(P, J1), 'alpha_2': mul(P, J2),
+                                 'a_fail = alpha_1 + alpha_2': mul(P, Jf)}
+        row['coefficients'] = coeffs
+        res['k=%g' % k] = row
+    if 'd3' in raw:
+        D = raw['d3']
+        sec = {'R_3(0) = (32 + 28 sqrt2)/17': D['R30'], 'const_3': D['const3'], 'I_b^(3)': D['Ib3'],
+               'C_fail^(3),{B,K}': D['C_fail3']}
+        for k in KS:
+            c = res['k=%g' % k]['coefficients']['b=0']
+            sec['k=%g, b=0' % k] = {'alpha_1^(3)': mul(D['R30'], c['alpha_1']), 'alpha_2^(3) = nu^(3)(2)': mul(D['R30'], c['alpha_2']),
+                                    'a_fail^(3)': mul(D['R30'], c['a_fail = alpha_1 + alpha_2'])}
+        res['d=3 (conditional on Math-#175 (L1) and Math-#184 Theorem 1)'] = sec
+    if 'torus' in raw and 'Ik' in raw:
+        T = raw['torus']
+        dJ = (-T['dJ'], T['dJ'])
+        sec = {'jet covariance deviation delta (order <= 6)': (0.0, T['delta']), 'chi^2 bound': (0.0, T['chi2']),
+               '|J^L - J| bound (all k, J_fail and J_2)': (0.0, T['dJ']),
+               '(pi_0 p_b(0))^L / (pi_0 p_b(0))^ref': T['pi0p_ratio']}
+        for k in KS:
+            J = raw['J'][k]
+            JfL, J2L = add(J['J_fail'], dJ), add(J['J_2'], dJ)
+            row = {'J_fail': JfL, 'J_2': J2L}
+            for b in (0.0, 1.0):
+                mu = mul(T['mean_coeff_A'], (b, b))
+                s = isqrt(T['var_A'])
+                t = div(mu, s)
+                ph = div(exp_neg(divc(sqr(t), 2.0)), SQRT2PI)
+                p = div(ph, s)
+                m2L = sub(mul(add(sqr(mu), T['var_A']), sub((1.0, 1.0), Phi(t))), mul(mul(mu, s), ph))
+                P = div(p, mul((36.0 * k * k, 36.0 * k * k), m2L))
+                row['b=%g' % b] = {'p_b(0)/z_0': P, 'alpha_1': mul(P, sub(JfL, J2L)), 'alpha_2': mul(P, J2L),
+                                   'a_fail': mul(P, JfL)}
+            sec['k=%g' % k] = row
+        I = raw['Ik']
+        dC = mul(mul(mul(I['const'], I['Ib']), divc(I['K_m4'], 3.0)), (T['dJ'], T['dJ']))
+        CL = mul(T['pi0p_ratio'], add(I['C_fail'], (-dC[1], dC[1])))
+        sec['C_fail^{B,K}'] = CL
+        res['torus model, every L >= 10, every frame (Lemma T)'] = sec
+    if 'Ik' in raw:
+        I = raw['Ik']
+        res['C_fail'] = {'I_k': I['Ik'], 'I_k boxes': I['boxes'], 'I_k tail bound': (0.0, I['tail']),
+                         'int_K W(k) dk': I['Wtot'], 'const': I['const'], 'I_b': I['Ib'], 'C_fail^{B,K}': I['C_fail']}
+    return res
+
+
+def _walk(d, f):
+    if isinstance(d, dict):
+        return {k: _walk(v, f) for k, v in d.items()}
+    if isinstance(d, str):
+        return d
+    return f(d)
+
+
+def document(raw):
+    der = derived(raw)
+    return {
+        'object': PACKET,
+        'scientific_effect': 'NONE',
+        'certified': True,
+        'arithmetic': 'binary64 interval arithmetic, outward one-ulp widening after every operation; published '
+                      'intervals widened by %g relatively and rounded outward to %d significant digits' % (MARGIN, DIGITS),
+        'parameters': {'k': list(KS), 'tol3': TOL3, 'tol4': TOL4, 'box_half_width_sigma': L_BOX, 'initial_cells': N0,
+                       'initial_k_cells': NK0, 'k_split_scale': SK, 'k_elementary_level': BX.KLEVEL},
+        'leaves': raw['leaves'],
+        'results': _walk(der, pub),
+    }
+
+
+def compare(doc, raw):
+    """Replay check: every replayed interval inside the published one, not looser by more than the margin."""
+    expected = document(raw)
+    if not isinstance(doc, dict) or set(doc) != set(expected):
+        return [('document', 'publication fields differ')]
+    der = derived(raw)
+    bad = []
+    for key in expected.keys() - {'results', 'leaves'}:
+        # JSON identity also distinguishes booleans from numeric substitutes.
+        if json.dumps(doc[key], sort_keys=True) != json.dumps(expected[key], sort_keys=True):
+            bad.append((key, doc[key], expected[key]))
+
+    def rec(p, r, path):
+        if isinstance(r, dict):
+            if sorted(p) != sorted(r):
+                bad.append(('/'.join(path), 'keys', sorted(p), sorted(r)))
+                return
+            for key in r:
+                rec(p[key], r[key], path + [key])
+        elif isinstance(r, str):
+            if p != r:
+                bad.append(('/'.join(path), p, r))
+        else:
+            if not contains_pub(p, r) or not not_looser(p, r):
+                bad.append(('/'.join(path), p, r))
+    rec(doc['results'], der, [])
+    if doc['leaves'] != raw['leaves']:
+        bad.append(('leaves', doc['leaves'], raw['leaves']))
+    return bad
+
+
+# ================================================================ self-test (pure-Python Gauss-Legendre references)
+def _gl(n):
+    xs, ws = [], []
+    for i in range(1, n + 1):
+        x = math.cos(math.pi * (i - 0.25) / (n + 0.5))
+        for _ in range(100):
+            p0, p1 = 1.0, x
+            for j in range(2, n + 1):
+                p0, p1 = p1, ((2 * j - 1) * x * p1 - (j - 1) * p0) / j
+            dp = n * (x * p1 - p0) / (x * x - 1)
+            dx = p1 / dp
+            x -= dx
+            if abs(dx) < 1e-16:
+                break
+        xs.append(x)
+        ws.append(2.0 / ((1 - x * x) * dp * dp))
+    return xs, ws
+
+
+def _xb_f(B, T):
+    xm = -B if B < 0 else B / 2.0
+    return xm if T <= 0 else BX._newton_big(B, T, xm)
+
+
+def _xm_f(B, T):
+    b = BX.xmid(B, T)
+    return 0.5 * (b[0] + b[1])
+
+
+def _H_f(B, T):
+    x = _xb_f(B, T)
+    return abs(B) ** 3 + 4 * x ** 3 - 3 * B * B * x
+
+
+def _V_f(B, T):
+    x = _xb_f(B, T)
+    return -3 * B * (2 * x - B) * (x + B)
+
+
+def _H2_f(B, T):
+    if B >= 0 or T >= abs(B) ** 3 / 2:
+        return 0.0
+    x = _xm_f(B, T)
+    return abs(B) ** 3 + 4 * x ** 3 - 3 * B * B * x
+
+
+def _ref3(box, k, f, n=12):
+    xs, ws = _gl(n)
+    var = (2.0 / k, 2.0, 6.0 * k)
+    pts = []
+    for (l, h), v in zip(box, var):
+        pts.append([(0.5 * (h - l) * x + 0.5 * (h + l),
+                     0.5 * (h - l) * w * math.exp(-(0.5 * (h - l) * x + 0.5 * (h + l)) ** 2 / (2 * v)) / math.sqrt(2 * math.pi * v))
+                    for x, w in zip(xs, ws)])
+    tot = 0.0
+    for a, wa in pts[0]:
+        for be, wb in pts[1]:
+            B = be - a * a / 12.0
+            d = -a * be / 4.0 + a ** 3 / 72.0
+            for c, wc in pts[2]:
+                u = c + d
+                tot += wa * wb * wc * f(B, 3 * u * u)
+    return tot
+
+
+def _ref4(box, n=10):
+    xs, ws = _gl(n)
+    var = (2.0, 2.0, 6.0)
+    pts = []
+    for (l, h), v in zip(box[:3], var):
+        pts.append([(0.5 * (h - l) * x + 0.5 * (h + l),
+                     0.5 * (h - l) * w * math.exp(-(0.5 * (h - l) * x + 0.5 * (h + l)) ** 2 / (2 * v)) / math.sqrt(2 * math.pi * v))
+                    for x, w in zip(xs, ws)])
+    l, h = box[3]
+    kp = [(0.5 * (h - l) * x + 0.5 * (h + l), 0.5 * (h - l) * w) for x, w in zip(xs, ws)]
+    tot = 0.0
+    for k, wk in kp:
+        Wk = wk * 3 * k ** (1.0 / 3.0) * math.exp(-12 * k * k)
+        sk = math.sqrt(k)
+        for a, wa in pts[0]:
+            for be, wb in pts[1]:
+                B = be - a * a / (12 * k)
+                for c, wc in pts[2]:
+                    u = sk * c - a * be / (4 * sk) + a ** 3 / (72 * k * sk)
+                    tot += Wk * wa * wb * wc * _H_f(B, 3 * u * u)
+    return tot
+
+
+SELFTEST3 = [   # (k, integrand, box in scaled coordinates (a, beta, c)); boxes on one side of u = 0
+    (0.5, 'V', ((0.30, 0.55), (-1.20, -0.95), (0.80, 1.10))),
+    (0.5, 'V', ((-2.10, -1.80), (0.40, 0.70), (-1.60, -1.30))),
+    (0.5, 'V', ((0.05, 0.25), (0.90, 1.20), (2.00, 2.30))),
+    (0.5, 'H', ((1.00, 1.30), (-0.60, -0.30), (0.30, 0.60))),
+    (1.0, 'V', ((0.10, 0.35), (-2.00, -1.70), (0.20, 0.45))),
+    (1.0, 'H', ((-0.80, -0.55), (0.10, 0.40), (-2.60, -2.30))),
+    (2.0, 'V', ((0.40, 0.60), (-0.90, -0.60), (3.00, 3.40))),
+    (0.5, 'H2', ((0.20, 0.40), (-2.40, -2.15), (0.60, 0.80))),      # inside the support
+    (1.0, 'H2', ((-0.30, -0.10), (-3.00, -2.70), (-1.20, -0.95))),   # inside the support
+    (0.5, 'H2', ((0.10, 0.30), (-1.30, -1.10), (0.60, 0.85))),      # meets the support boundary
+    (0.5, 'V', ((0.30, 0.32), (-1.00, -0.98), (0.50, 1.50))),       # elongated in c: the w-curvature dominates
+    (1.0, 'H', ((0.30, 0.32), (0.40, 0.42), (1.00, 2.00))),
+    (0.5, 'H2', ((0.20, 0.22), (-2.40, -2.38), (0.20, 0.90))),
+]
+SELFTEST4 = [
+    ((0.40, 0.70), (-1.20, -0.90), (1.00, 1.40), (0.5, 0.5 + 3 * 2.0 ** -7)),
+    ((-1.50, -1.20), (0.50, 0.80), (-2.00, -1.60), (0.5 + 3 * 2.0 ** -6, 0.5 + 3 * 2.0 ** -5)),
+]
+
+
+def selftest(mutant=None, log=print):
+    """Every box enclosure (and its first- and second-order parts separately) must contain a high-order floating-point
+    Gauss-Legendre value; the k-moments must contain Gauss-Legendre values.  Returns a list of failures."""
+    BX.MUTANT = mutant
+    fails = []
+    for k, which, box in SELFTEST3:
+        S = sig3(k)
+        Ma, Mb, Mc = BX.Moments(S[0][0], S[0][1], 6), BX.Moments(S[1][0], S[1][1], 2), BX.Moments(S[2][0], S[2][1], 2)
+        if which == 'H2':
+            I, I1, I2 = BX.H2_box3(box, Ma, Mb, Mc, parts=True)
+            ref = _ref3(box, k, _H2_f, 16)
+            slack = 1e-6 * abs(ref) + 1e-15        # the zero extension is only C^1 at the support boundary
+        else:
+            try:
+                I, I1, I2 = BX.big_box3(box, Ma, Mb, Mc, with_H=(which == 'H'), parts=True)
+            except RuntimeError as e:
+                fails.append((which, k, box, 'raised: %s' % e))
+                continue
+            ref = _ref3(box, k, _H_f if which == 'H' else _V_f)
+            slack = 1e-9 * abs(ref) + 1e-15
+        for name, J in (('I', I), ('I1', I1), ('I2', I2)):
+            if J is not None and not (J[0] - slack <= ref <= J[1] + slack):
+                fails.append((which, k, box, name, J, ref))
+        if I2 is None:
+            fails.append((which, k, box, 'no second-order enclosure'))
+    S = sig3(1.0)
+    Ma, Mb, Mc = BX.Moments(S[0][0], S[0][1], 6), BX.Moments(S[1][0], S[1][1], 2), BX.Moments(S[2][0], S[2][1], 2)
+    elem = {}
+    for box in SELFTEST4:
+        k0, k1 = box[3]
+        w = (BX.K_HI - BX.K_LO) / 2 ** BX.KLEVEL
+        i0, i1 = round((k0 - BX.K_LO) / w), round((k1 - BX.K_LO) / w)
+        Km = _partial_kmoments(i0, i1)              # elementary k-moments of the needed cells only
+        try:
+            I, I1, I2 = BX.H_box4(box, Ma, Mb, Mc, Km, parts=True)
+        except RuntimeError as e:
+            fails.append(('H4', box, 'raised: %s' % e))
+            continue
+        ref = _ref4(box)
+        slack = 1e-9 * abs(ref) + 1e-15
+        for name, J in (('I', I), ('I1', I1), ('I2', I2)):
+            if J is not None and not (J[0] - slack <= ref <= J[1] + slack):
+                fails.append(('H4', box, name, J, ref))
+        if I2 is None:
+            fails.append(('H4', box, 'no second-order enclosure'))
+        xs, ws = _gl(20)
+        K = Km.get(k0, k1)
+        for p in (BX.PMIN, 0, BX.PMAX):
+            refk = sum(0.5 * (k1 - k0) * wq * 3 * kk ** (1 / 3 + p / 2) * math.exp(-12 * kk * kk)
+                       for kk, wq in ((0.5 * (k1 - k0) * x + 0.5 * (k1 + k0), wq) for x, wq in zip(xs, ws)))
+            if not (K[p][0] * (1 - 1e-12) <= refk <= K[p][1] * (1 + 1e-12)):
+                fails.append(('K', box[3], p, K[p], refk))
+    log('self-test: %d boxes, %d failures' % (len(SELFTEST3) + len(SELFTEST4), len(fails)))
+    return fails
+
+
+def _partial_kmoments(i0, i1, nsub=24):
+    """KMoments object whose elementary cells i0..i1-1 are computed (others are never touched)."""
+    n = 2 ** BX.KLEVEL
+    width = (BX.K_HI - BX.K_LO) / n
+    elem = [None] * n
+    for i in range(i0, i1):
+        k0 = BX.K_LO + width * i
+        acc = {p: (0.0, 0.0) for p in range(BX.PMIN, BX.PMAX + 1)}
+        prev = BX.kweights(k0)
+        for j in range(nsub):
+            l = k0 + width * j / nsub
+            r = k0 + width * (j + 1) / nsub
+            h = (dn(r - l), up(r - l))
+            mid = BX.kweights(0.5 * (l + r))
+            fr = BX.kweights(r)
+            for p in acc:
+                lo = mul(h, mid[p])[0]
+                hi = mul(h, scal(0.5, add(prev[p], fr[p])))[1]
+                acc[p] = (dn(acc[p][0] + lo), up(acc[p][1] + hi))
+            prev = fr
+        elem[i] = acc
+    return BX.KMoments(elem)
+
+
+# ================================================================ main
+def main():
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--write', action='store_true')
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--controls', action='store_true')
+    ap.add_argument('--mutant', default=None)
+    ap.add_argument('--procs', type=int, default=os.cpu_count() or 2)
+    a = ap.parse_args()
+    path = os.path.join(HERE, 'RESULTS.json')
+    if a.controls:
+        import controls
+        controls.main()
+        return 0
+    fails = selftest(a.mutant)
+    if fails:
+        for f in fails[:10]:
+            print('SELF-TEST FAILURE', f)
+        return 1
+    try:
+        raw = certify(a.procs, a.mutant)
+    except RuntimeError as e:
+        print('CERTIFICATE FAILURE', e)
+        return 1
+    if a.write:
+        doc = document(raw)
+        with open(path, 'w') as fh:
+            json.dump(doc, fh, indent=1, sort_keys=False)
+            fh.write('\n')
+        print('wrote', path)
+        return 0
+    if a.check:
+        with open(path) as fh:
+            doc = json.load(fh)
+        bad = compare(doc, raw)
+        if bad:
+            for b in bad[:20]:
+                print('MISMATCH', b)
+            return 1
+        print(json.dumps({'object': PACKET, 'check': 'passed', 'scientific_effect': 'NONE'}))
+        return 0
+    return 0
+
+
+if __name__ == '__main__':
+    sys.exit(main())
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/controls.py b/frontiers/c8_elder_failure_coefficient_20261001/controls.py
new file mode 100644
index 000000000..396138322
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/controls.py
@@ -0,0 +1,106 @@
+"""Floating-point controls for CL-C8-ELDER-FAILURE-COEFFICIENT-20261001-v1 (not part of the certificate).
+
+1. E[2(B_-)^3] by composite Simpson (float) against the certified enclosure.
+2. Consistency of the split J_fail = 3k^2 (E[2(B_-)^3] + E[2T] + E[V]) with a direct box enclosure of E[H] at k = 1/2
+   (same engine, different integrand: the two enclosures must intersect).
+3. The certified J_1, J_2 against the Gauss-Hermite and Monte Carlo values of [NUM] (RESULTS.json on this tree).
+4. C_fail^{B,K} against the cap-route constants of Math-#215 (C_{B,K}(1/4096) = 402.5297, Gamma_2 = 307.51), and the
+   d = 3 constant against C_{B,K}^(3)(1/4096) = 1205.7089, Gamma_3 = 540.97.
+5. The d = 3 near coefficients at b = 0 against Math-#184's floating values.
+6. Lemma T: floating lattice sums of the torus jet-covariance deviation at L = 10 against the certified delta."""
+import os, sys, json, math
+from decimal import Decimal
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+import certificate as C
+from ia import dn, up
+
+NUM = os.path.join(HERE, '..', 'c6_cluster_coefficients_numerics_20260930', 'RESULTS.json')
+
+
+def e2b_float(k, N=200000):
+    sa = math.sqrt(2 / k)
+    def g(m):
+        return (m ** 3 + 6 * m) * 0.5 * (1 + math.erf(m / 2)) + math.sqrt(2) * (m * m + 4) * math.exp(-m * m / 4) / math.sqrt(2 * math.pi)
+    L = 12 * sa
+    h = L / N
+    f = lambda a: math.exp(-a * a / (2 * sa * sa)) / (sa * math.sqrt(2 * math.pi)) * g(a * a / 12)
+    S = f(0) + f(L) + sum((4 if i % 2 else 2) * f(i * h) for i in range(1, N))
+    return 4 * S * h / 3
+
+
+def main():
+    doc = json.load(open(os.path.join(HERE, 'RESULTS.json')))
+    R = doc['results']
+    print('1. E[2(B_-)^3]: float Simpson vs certified')
+    for k in C.KS:
+        e = e2b_float(k)
+        iv = R['k=%g' % k]['parts']['E[2(B_-)^3]']
+        print('   k=%g  float %.12f  certified [%s, %s]  inside: %s' % (k, e, iv[0], iv[1], Decimal(iv[0]) <= Decimal(e) <= Decimal(iv[1])))
+    print('2. direct E[H] boxes at k = 1/2 (tol 1e-7) vs the split')
+    (lo, hi), n, _ = C.run3(0.5, 'H', 1e-7, os.cpu_count() or 2)
+    tl = C.tail3(0.5)
+    s = 0.75
+    J = R['k=0.5']['J_fail']
+    direct = (s * lo, s * (hi + tl))
+    print('   direct 3k^2 E[H] in [%.9f, %.9f] (%d leaves); split J_fail [%s, %s]; intersect: %s' %
+          (direct[0], direct[1], n, J[0], J[1], Decimal(direct[0]) <= Decimal(J[1]) and Decimal(J[0]) <= Decimal(direct[1])))
+    print('3. [NUM] floating values against the certified enclosures')
+    num = json.load(open(NUM))['per_k']
+    for k, key in ((0.5, '1/2'), (1.0, '1'), (2.0, '2')):
+        r = R['k=%g' % k]
+        for j in ('1', '2'):
+            iv = [float(x) for x in r['J_%s' % j]]
+            mid = 0.5 * (iv[0] + iv[1])
+            v = num[key]
+            line = '   k=%g J_%s certified [%.6f, %.6f]' % (k, j, iv[0], iv[1])
+            for lab in ('gh40', 'gh60', 'mc'):
+                x = v['J%s_%s' % (j, lab)]
+                inside = iv[0] <= x <= iv[1]
+                line += '  %s %.5f (%+.3f%%%s)' % (lab, x, 100 * (x - mid) / mid, ', inside' if inside else '')
+            line += '  mc s.e. %.4f -> (mc - mid)/se = %+.2f' % (v['J%s_mc_se' % j], (v['J%s_mc' % j] - mid) / v['J%s_mc_se' % j])
+            print(line)
+    print('4. C_fail^{B,K} against the cap-route constants')
+    Cf = [float(x) for x in R['C_fail']['C_fail^{B,K}']]
+    for name, val in (('C_{B,K}(1/4096), Math-#215', 402.5297), ('Gamma_2 (floating), Math-#215', 307.51)):
+        print('   %s = %g : ratio to C_fail in [%.4g, %.4g]' % (name, val, val / Cf[1], val / Cf[0]))
+    D3 = R['d=3 (conditional on Math-#175 (L1) and Math-#184 Theorem 1)']
+    Cf3 = [float(x) for x in D3['C_fail^(3),{B,K}']]
+    for name, val in (('C_{B,K}^(3)(1/4096), Math-#215', 1205.7089), ('Gamma_3 (floating), Math-#215', 540.97)):
+        print('   d = 3: %s = %g : ratio to C_fail^(3) in [%.4g, %.4g]' % (name, val, val / Cf3[1], val / Cf3[0]))
+    print('5. d = 3 near coefficients at b = 0 against Math-#184 (floating, from [NUM] GH60 times R_3(0))')
+    hd = {0.5: 4.8558, 1.0: 5.705}
+    for k, v in hd.items():
+        iv = [float(x) for x in D3['k=%g, b=0' % k]['a_fail^(3)']]
+        print('   k=%g a_fail^(3) certified [%.6f, %.6f]  Math-#184 %.4f  (%+.3f%%)' % (k, iv[0], iv[1], v, 100 * (v / (0.5 * (iv[0] + iv[1])) - 1)))
+    print('6. Lemma T: direct floating lattice sums of the torus deviation at L = 10 (three frames) against delta')
+    import torus as TOR
+    dlt = TOR.delta()
+    def He(j, t):
+        h0, h1 = 1.0, t
+        if j == 0:
+            return h0
+        for i in range(1, j):
+            h0, h1 = h1, t * h1 - i * h0
+        return h1
+    worst = 0.0
+    for th in (0.0, 0.3, math.pi / 4):
+        c, s_ = math.cos(th), math.sin(th)
+        for g in ((6, 0), (3, 3), (4, 2), (2, 2), (5, 1), (0, 0)):
+            tot = 0.0
+            theta = 0.0
+            for n1 in range(-3, 4):
+                for n2 in range(-3, 4):
+                    if n1 == 0 and n2 == 0:
+                        continue
+                    y1, y2 = 10.0 * (c * n1 + s_ * n2), 10.0 * (-s_ * n1 + c * n2)
+                    e = math.exp(-(y1 * y1 + y2 * y2) / 2)
+                    tot += (-1) ** (g[0] + g[1]) * He(g[0], y1) * He(g[1], y2) * e
+                    theta += e
+            d0 = TOR._D(g)
+            dev = abs((tot - d0 * theta) / (1.0 + theta))      # (d0 + tot)/(1 + theta) - d0 without cancellation
+            worst = max(worst, dev)
+    print('   max floating deviation %.3e  <=  delta %.3e: %s' % (worst, dlt, worst <= dlt))
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/ia.py b/frontiers/c8_elder_failure_coefficient_20261001/ia.py
new file mode 100644
index 000000000..cf4bbfa16
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/ia.py
@@ -0,0 +1,279 @@
+"""Interval arithmetic on IEEE-754 binary64: every +, -, *, / and sqrt is correctly rounded to nearest and the result is
+widened outward by one ulp (math.nextafter), so each interval contains the exact real result.  No libm transcendental is
+trusted: pi and ln 2 come from exact rational series, exp(-y) from a reduced Taylor polynomial with an a priori error
+bound, Phi from its positive power series and Q = 1 - Phi (t >= 2) from Laplace's continued fraction for the Mills ratio.
+libm values (pow, acos, cos) are used only as starting points of iterations whose results are then verified.
+An interval is a pair (lo, hi) of floats with lo <= hi."""
+import math
+from fractions import Fraction as Fr
+
+INF = math.inf
+_na = math.nextafter
+
+
+def dn(x):
+    return _na(x, -INF)
+
+
+def up(x):
+    return _na(x, INF)
+
+
+def iv(x):
+    """Exact float point as an interval."""
+    return (x, x)
+
+
+def from_frac(q):
+    """Tightest float interval containing the rational q."""
+    f = float(q)
+    fq = Fr(f)
+    if fq == q:
+        return (f, f)
+    return (f, up(f)) if fq < q else (dn(f), f)
+
+
+def add(a, b):
+    return (dn(a[0] + b[0]), up(a[1] + b[1]))
+
+
+def sub(a, b):
+    return (dn(a[0] - b[1]), up(a[1] - b[0]))
+
+
+def neg(a):
+    return (-a[1], -a[0])
+
+
+def mul(a, b):
+    p = (a[0] * b[0], a[0] * b[1], a[1] * b[0], a[1] * b[1])
+    return (dn(min(p)), up(max(p)))
+
+
+def div(a, b):
+    if b[0] <= 0.0 <= b[1]:
+        raise ZeroDivisionError('interval division by an interval containing 0')
+    p = (a[0] / b[0], a[0] / b[1], a[1] / b[0], a[1] / b[1])
+    return (dn(min(p)), up(max(p)))
+
+
+def scal(c, a):
+    """c * a for an exact float c."""
+    if c >= 0:
+        return (dn(c * a[0]), up(c * a[1]))
+    return (dn(c * a[1]), up(c * a[0]))
+
+
+def divc(a, c):
+    """a / c for an exact float c > 0."""
+    return (dn(a[0] / c), up(a[1] / c))
+
+
+def sqr(a):
+    if a[0] >= 0:
+        return (max(0.0, dn(a[0] * a[0])), up(a[1] * a[1]))
+    if a[1] <= 0:
+        return (max(0.0, dn(a[1] * a[1])), up(a[0] * a[0]))
+    m = max(-a[0], a[1])
+    return (0.0, up(m * m))
+
+
+def pw(a, n):
+    """a ** n for an integer n >= 0 (monotone pieces)."""
+    if n == 0:
+        return (1.0, 1.0)
+    if n == 1:
+        return a
+    if n % 2 == 1:
+        # odd powers are increasing: bound each endpoint by repeated outward multiplication
+        return (_pw_pt(a[0], n, False), _pw_pt(a[1], n, True))
+    r = pw(sqr(a), n // 2)
+    return (max(0.0, r[0]), r[1])
+
+
+def _pw_pt(x, n, upper):
+    """Outward bound of x ** n at a point (odd n): upper -> rounded up, else rounded down."""
+    r = (x, x)
+    for _ in range(n - 1):
+        r = mul(r, (x, x))
+    return r[1] if upper else r[0]
+
+
+def hull(*ivs):
+    return (min(x[0] for x in ivs), max(x[1] for x in ivs))
+
+
+def absmax(a):
+    return max(-a[0], a[1])
+
+
+def isqrt(a):
+    if a[0] < 0:
+        raise ValueError('sqrt of a negative interval')
+    lo = math.sqrt(a[0])
+    hi = math.sqrt(a[1])
+    return (max(0.0, dn(lo)), up(hi))
+
+
+def contains(a, x):
+    return a[0] <= x <= a[1]
+
+
+# ---------------------------------------------------------------- constants (exact rationals, then outward floats)
+def _pi_frac(digits=40):
+    """pi by Machin's formula with an explicit rational enclosure."""
+    def arctan_inv(n, terms):
+        s = Fr(0)
+        for k in range(terms):
+            s += Fr((-1) ** k, (2 * k + 1) * n ** (2 * k + 1))
+        return s
+    T = 40
+    lo = 16 * arctan_inv(5, T) - 4 * arctan_inv(239, T)
+    err = Fr(16, (2 * T + 1) * 5 ** (2 * T + 1)) + Fr(4, (2 * T + 1) * 239 ** (2 * T + 1))
+    return lo - err, lo + err
+
+
+_PI_LO, _PI_HI = _pi_frac()
+PI = (from_frac(_PI_LO)[0], from_frac(_PI_HI)[1])
+SQRT2PI = isqrt(scal(2.0, PI))
+SQRT2 = isqrt((2.0, 2.0))
+INV_SQRT2PI = div((1.0, 1.0), SQRT2PI)
+
+
+# ---------------------------------------------------------------- exp(-y) for y >= 0
+_INVFACT = [1.0]
+for _i in range(1, 19):
+    _INVFACT.append(_INVFACT[-1] / _i)
+_U = 2.0 ** -53
+
+
+def _ln2_frac():
+    """ln 2 = sum_{j>=1} 1/(j 2^j), with the tail after J terms below 2/(J 2^J)."""
+    J = 80
+    s = Fr(0)
+    for j in range(1, J + 1):
+        s += Fr(1, j * 2 ** j)
+    return s, s + Fr(2, J * 2 ** J)
+
+
+_L2LO, _L2HI = _ln2_frac()
+LN2 = (from_frac(_L2LO)[0], from_frac(_L2HI)[1])
+
+
+def _poly_exp_neg(z):
+    """Binary64 Horner value p of the degree-18 Taylor polynomial of exp(-z) at a float z with |z| <= 0.75, and a bound
+    of |p - exp(-z)|: Horner and coefficient rounding (coefficients 1/i! carry relative error <= 18u) give at most
+    (2*18 + 18 + 2) u e^|z| (1 + 1e-12), truncation at most |z|^19/19! e^|z| < 1e-19 e^|z|; exp(-z) >= e^-|z| >= 0.47."""
+    p = _INVFACT[18]
+    for i in range(17, -1, -1):
+        p = _INVFACT[i] - z * p
+    ez = 2.12                                   # >= e^0.75
+    err = (56.0 * _U * (1.0 + 1e-12) + 1e-19) * ez
+    return p, err
+
+
+def _exp_neg_point(y):
+    """Enclosure of exp(-y) for a float y >= 0: y = n ln2 + r with n = floor(y / ln2) and r in an interval inside
+    [-1e-9, 0.75] (from the rational enclosure of ln 2), exp(-y) = 2^-n exp(-r) (exact power-of-two scaling)."""
+    if y == 0.0:
+        return (1.0, 1.0)
+    if y > 700.0:
+        return (0.0, math.ldexp(1.0, -1000))     # exp(-700) < 2^-1000
+    n = int(y / 0.6931471805599453)
+    nL = mul((float(n), float(n)), LN2)
+    r = sub((y, y), nL)
+    if not (-1e-9 <= r[0] <= r[1] <= 0.75):
+        raise RuntimeError('exp range reduction failed at y=%r' % y)
+    plo, elo = _poly_exp_neg(r[1])               # exp(-r) is decreasing: lower end at r[1]
+    phi_, ehi = _poly_exp_neg(r[0])
+    lo = dn(plo - elo)
+    hi = up(phi_ + ehi)
+    return (max(0.0, math.ldexp(lo, -n)), math.ldexp(hi, -n))
+
+
+def exp_neg(a):
+    """Enclosure of exp(-y) for y in the interval a, a[0] >= 0 (decreasing)."""
+    if a[0] < 0:
+        # a lower bound that is negative only by underflow-level rounding (y >= -1e-300): exp(-y) <= exp(1e-300) < up(1)
+        if a[0] < -1e-300:
+            raise ValueError('exp_neg needs y >= 0')
+        return (_exp_neg_point(max(0.0, a[1]))[0], up(1.0))
+    return (_exp_neg_point(a[1])[0], _exp_neg_point(a[0])[1])
+
+
+def phi_std(t):
+    """Standard normal density at the interval t."""
+    return mul(exp_neg(divc(sqr(t), 2.0)), INV_SQRT2PI)
+
+
+def _Phi_pos_point(t):
+    """Enclosure of Phi(t) for a float 0 <= t < 2: Phi(t) = 1/2 + phi(t) S(t), S(t) = sum t^(2n+1)/(2n+1)!! (positive
+    terms), summed in binary64.  On 0 < t < 2 the loop stops by n = 24 (there term_n/S <= 4^n/(2n+1)!! < 5e-18), so the
+    accumulated relative error is at most (3 * 24 + 3) u < 1e-14, inside the 1e-13 allowance; the cap n <= 300 only
+    guards termination.  The geometric tail bound applies once the term ratio t^2/(2n+3) is below 1/2."""
+    if t == 0.0:
+        return (0.5, 0.5)
+    if t < 1e-100:
+        return (0.5, up(0.5 + t))
+    if not t < 2.0:
+        raise ValueError('series branch is for t < 2')
+    t2 = t * t
+    term = t
+    S = t
+    n = 0
+    while True:
+        n += 1
+        if n > 300:
+            raise RuntimeError('Phi series did not terminate at t=%r' % t)
+        term = term * t2 / (2 * n + 1)
+        S += term
+        ratio = t2 * (1.0 + 4 * _U) / (2 * n + 3)
+        if ratio < 0.5 and term < 1e-17 * S:
+            tail = term * ratio / (1.0 - ratio) * (1.0 + 1e-12)
+            break
+    Slo, Shi = dn(S * (1.0 - 1e-13)), up((S + tail) * (1.0 + 1e-13))
+    r = add((0.5, 0.5), mul(phi_std((t, t)), (Slo, Shi)))
+    return (r[0], min(r[1], 1.0))
+
+
+def _mills(t):
+    """Enclosure of the Mills ratio R(t) = Q(t)/phi(t), t >= 2 float: Laplace's continued fraction
+    R(t) = 1/(t + 1/(t + 2/(t + 3/(t + ...)))) has positive partial numerators and denominators, so consecutive
+    approximants bracket R(t) (Stieltjes); the hull of the depth-n and depth-(n+1) approximants is returned once its
+    relative width is below 1e-14 (each approximant evaluated in outward interval arithmetic)."""
+    ti = (t, t)
+    n = 16
+    while True:
+        res = []
+        for depth in (n, n + 1):
+            v = ti
+            for j in range(depth, 0, -1):
+                v = add(ti, div((float(j), float(j)), v))
+            res.append(div((1.0, 1.0), v))
+        h = (min(res[0][0], res[1][0]), max(res[0][1], res[1][1]))
+        if h[1] - h[0] <= 1e-14 * h[0] or n > 4000:
+            return h
+        n *= 2
+
+
+def Q_point(t):
+    """Enclosure of the upper tail Q(t) = 1 - Phi(t) for a float t >= 0, with relative precision for large t."""
+    if t >= 2.0:
+        return mul(phi_std((t, t)), _mills(t))
+    r = _Phi_pos_point(t)
+    return (max(0.0, dn(1.0 - r[1])), up(1.0 - r[0]))
+
+
+def Phi_point(t):
+    if t >= 0:
+        if t >= 2.0:
+            q = Q_point(t)
+            return (dn(1.0 - q[1]), min(1.0, up(1.0 - q[0])))
+        return _Phi_pos_point(t)
+    q = Q_point(-t)
+    return q
+
+
+def Phi(a):
+    """Normal CDF over an interval argument (increasing)."""
+    return (Phi_point(a[0])[0], Phi_point(a[1])[1])
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/test_replay_metadata.py b/frontiers/c8_elder_failure_coefficient_20261001/test_replay_metadata.py
new file mode 100644
index 000000000..f943d6a1f
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/test_replay_metadata.py
@@ -0,0 +1,65 @@
+"""Reject false publication metadata without rerunning the expensive cubature."""
+import copy
+import json
+from pathlib import Path
+import unittest
+
+import certificate as C
+
+
+class ReplayMetadataTests(unittest.TestCase):
+    def setUp(self):
+        # Small raw enclosures exercise the real derivation and comparator.
+        self.raw = {'leaves': {'fixture': 1}, 'J': {
+            k: {'J_fail': (4.0, 4.1), 'J_2': (1.0, 1.1),
+                'E2B': (1.0, 1.1), 'E2T': (2.0, 2.1),
+                'EV_boxes': (0.1, 0.2), 'EH2_boxes': (0.2, 0.3),
+                'tail': 0.01}
+            for k in (0.5, 1.0, 2.0)}}
+        self.doc = C.document(self.raw)
+        published = json.loads(Path(C.HERE, 'RESULTS.json').read_text())
+        for key in ('object', 'scientific_effect', 'certified', 'arithmetic', 'parameters'):
+            self.doc[key] = copy.deepcopy(published[key])
+
+    def test_published_metadata_accepts_valid_replay(self):
+        self.assertEqual(C.compare(self.doc, self.raw), [])
+
+    def test_changed_header_is_rejected(self):
+        for key, value in [('object', 'different packet'), ('scientific_effect', 'ACCEPT'),
+                           ('certified', False), ('certified', 1), ('arithmetic', 'exact')]:
+            with self.subTest(key=key, value=value):
+                doc = copy.deepcopy(self.doc)
+                doc[key] = value
+                self.assertTrue(C.compare(doc, self.raw))
+
+    def test_changed_parameters_are_rejected(self):
+        for key, value in [('k', [0.5, 1.0]), ('tol3', {'V': 1e-2, 'H2': 1e-8}),
+                           ('tol4', 1e-2), ('box_half_width_sigma', 9.0),
+                           ('initial_cells', 4), ('initial_k_cells', 4),
+                           ('k_split_scale', 0.9), ('k_elementary_level', 4)]:
+            with self.subTest(key=key):
+                doc = copy.deepcopy(self.doc)
+                doc['parameters'][key] = value
+                self.assertTrue(C.compare(doc, self.raw))
+
+    def test_missing_or_extra_header_is_rejected(self):
+        for key in ('object', 'scientific_effect', 'certified', 'arithmetic', 'parameters'):
+            with self.subTest(missing=key):
+                doc = copy.deepcopy(self.doc)
+                del doc[key]
+                self.assertTrue(C.compare(doc, self.raw))
+        doc = copy.deepcopy(self.doc)
+        doc['unverified_claim'] = True
+        self.assertTrue(C.compare(doc, self.raw))
+
+    def test_result_and_leaf_checks_remain_active(self):
+        doc = copy.deepcopy(self.doc)
+        doc['leaves']['fixture'] = 2
+        self.assertTrue(C.compare(doc, self.raw))
+        doc = copy.deepcopy(self.doc)
+        doc['results']['k=0.5']['J_fail'] = ['0', '1']
+        self.assertTrue(C.compare(doc, self.raw))
+
+
+if __name__ == '__main__':
+    unittest.main()
diff --git a/frontiers/c8_elder_failure_coefficient_20261001/torus.py b/frontiers/c8_elder_failure_coefficient_20261001/torus.py
new file mode 100644
index 000000000..4a448b2f3
--- /dev/null
+++ b/frontiers/c8_elder_failure_coefficient_20261001/torus.py
@@ -0,0 +1,219 @@
+"""Lemma T (torus transfer) for CL-C8-ELDER-FAILURE-COEFFICIENT-20261001: the planar coefficients of the torus model of
+[LP] / [S] (side L >= 10, any frame) against the reference-kernel ones.
+
+One-site jet covariances.  Cov(d^a f, d^b f) = (-1)^|b| d^(a+b) K(0).  For the torus kernel
+K_L(z) = sum_n G(z + Ln) / Theta_L, G = exp(-|z|^2/2), Theta_L = sum_n exp(-L^2 |n|^2/2) ([LP] section 2), in a rotated
+frame d^g K_L(0) = sum_n (d^g G)(R^T L n) / Theta_L with |R^T L n| = L|n|, and |d^g G(y)| = |He_g1(y1) He_g2(y2)| G(y)
+<= Hh_g1(|y|) Hh_g2(|y|) G(y), Hh_j(t) = sum_k j!/(k!(j-2k)! 2^k) t^(j-2k).  Grouping n != 0 by |n|_inf = j (8j points,
+|n| >= j) and using that t -> Hh_a(t) Hh_c(t) e^(-t^2/2) decreases for t^2 > a + c, every entry of order |a| + |b| <= 6
+differs from the reference by at most delta(10) for every L >= 10 and every frame.
+
+Propagation.  Exact interval regressions with every entry widened by delta give: the odd conditional law
+N(12k v, S_c) of (f_xxz, f_xzz, f_zzz) given (f_x, f_z, f_xxx) = (0, 0, 12k) (reference: N(0, diag(2, 2, 6))); the even
+conditional law N(c1 b, s^2) of f_zz given (f, f_xx, f_xz) = (b, 0, 0) (reference N(-b, 2)); and the factors of
+pi_0 = p_(f,V_u)(b, 0) p_G(0) phi_tau(12k) ([LP] section 15).  Parity (K_L even) keeps odd and even jets uncorrelated.
+
+Expectations.  With S_c = S0^(1/2) (I + E) S0^(1/2), ||E|| <= eps, mu~ = S0^(-1/2) mu:
+    1 + chi^2(N(mu, S_c) || N(0, S0)) = prod (1 - e_i^2)^(-1/2) exp(sum mu~_i^2/(1 - e_i)) <= (1 - eps^2)^(-3/2) exp(|mu~|^2/(1 - eps)),
+and for eps^2 <= 0.006, where (1 - eps^2)^(-3/2) <= 1.01 and (1 - eps^2)^(-3/2) - 1 <= 2 eps^2, with e^m - 1 <= m e^m,
+    chi^2 <= 1.01 m e^m + 2 eps^2      (m = |mu~|^2/(1 - eps)),
+and |E_L[I] - E_ref[I]| <= (E_ref[I^2] chi^2)^(1/2)
+(Cauchy-Schwarz), with I = 3k^2 H <= 12 Pi_4 for k <= 2 (Pi_4 the tail polynomial of certificate.tail4)."""
+import math
+from fractions import Fraction as Fr
+from functools import reduce
+from ia import (dn, up, add, sub, neg, mul, div, scal, divc, sqr, pw, isqrt, exp_neg, from_frac, absmax, Phi, SQRT2PI)
+
+L_MIN = 10.0
+ORDER = 6
+
+
+def _Hh(j):
+    return {j - 2 * k: Fr(math.factorial(j), math.factorial(k) * math.factorial(j - 2 * k) * 2 ** k) for k in range(j // 2 + 1)}
+
+
+def _poly(coef, t):
+    acc = (0.0, 0.0)
+    for p, c in coef.items():
+        acc = add(acc, mul(from_frac(c), pw(t, p)))
+    return acc
+
+
+def _M(t):
+    """max over a + c <= ORDER of Hh_a(t) Hh_c(t) (upper end), t >= 1."""
+    best = 0.0
+    for m in range(ORDER + 1):
+        for a in range(m + 1):
+            best = max(best, mul(_poly(_Hh(a), t), _poly(_Hh(m - a), t))[1])
+    return best
+
+
+def delta(L=L_MIN, J=6):
+    """Bound on |c_L - c_ref| for every one-site covariance entry of order <= ORDER, every side >= L and every frame."""
+    S = 0.0
+    term = 0.0
+    for j in range(1, J + 1):
+        t = (L * j, L * j)
+        term = up(8.0 * j * up(_M(t) * exp_neg(divc(sqr(t), 2.0))[1]))
+        S = up(S + term)
+    S = up(S + 2.0 * term)                  # shells j > J: consecutive bounds decrease by far more than 1/2
+    theta_minus_1 = S                       # sum_{n != 0} e^(-L^2 |n|^2/2) <= S since Hh products are >= 1
+    return up(S + up(15.0 * theta_minus_1)) # |d^g G(0)| <= 15 for |g| <= 6; |1/Theta - 1| <= Theta - 1
+
+
+def _dfact(n):
+    r = 1
+    while n > 1:
+        r, n = r * n, n - 2
+    return r
+
+
+def _D(g):
+    if g[0] % 2 or g[1] % 2:
+        return 0
+    return (-1) ** ((g[0] + g[1]) // 2) * _dfact(g[0] - 1) * _dfact(g[1] - 1)
+
+
+def cov_ref(a, b):
+    """Reference covariance of the jets d^a f and d^b f at one site (multi-indices (x-order, z-order))."""
+    return (-1) ** (b[0] + b[1]) * _D((a[0] + b[0], a[1] + b[1]))
+
+
+def _icov(rows, cols, d):
+    return [[(dn(cov_ref(a, b) - d), up(cov_ref(a, b) + d)) for b in cols] for a in rows]
+
+
+def _det2(M):
+    return sub(mul(M[0][0], M[1][1]), mul(M[0][1], M[1][0]))
+
+
+def _det3(M):
+    return sub(add(mul(M[0][0], sub(mul(M[1][1], M[2][2]), mul(M[1][2], M[2][1]))),
+                   mul(M[0][2], sub(mul(M[1][0], M[2][1]), mul(M[1][1], M[2][0])))),
+               mul(M[0][1], sub(mul(M[1][0], M[2][2]), mul(M[1][2], M[2][0]))))
+
+
+def _inv3(M):
+    d = _det3(M)
+    adj = [[None] * 3 for _ in range(3)]
+    for i in range(3):
+        for j in range(3):
+            r = [x for x in range(3) if x != i]
+            c = [x for x in range(3) if x != j]
+            m = sub(mul(M[r[0]][c[0]], M[r[1]][c[1]]), mul(M[r[0]][c[1]], M[r[1]][c[0]]))
+            adj[j][i] = m if (i + j) % 2 == 0 else neg(m)
+    return [[div(adj[i][j], d) for j in range(3)] for i in range(3)]
+
+
+def _inv2(M):
+    d = _det2(M)
+    return [[div(M[1][1], d), div(neg(M[0][1]), d)], [div(neg(M[1][0]), d), div(M[0][0], d)]]
+
+
+def _mm(A, B):
+    return [[reduce(add, [mul(A[i][t], B[t][j]) for t in range(len(B))]) for j in range(len(B[0]))] for i in range(len(A))]
+
+
+def _T(A):
+    return [list(r) for r in zip(*A)]
+
+
+def _expiv(x):
+    """exp over a small interval x (|x| < 1) from the verified exp_neg of ia: exp(t) = exp_neg(-t) for t <= 0 and
+    1/exp_neg(t) for t > 0."""
+    def e(t):
+        return exp_neg((-t, -t)) if t <= 0 else div((1.0, 1.0), exp_neg((t, t)))
+    return (e(x[0])[0], e(x[1])[1])
+
+
+def transfer(E_I2_upper):
+    """Lemma T bounds.  E_I2_upper: an upper bound of E_ref[I^2], I = 3k^2 H, uniform over k in [1/2, 2].
+    Returns a dict of bounds (all floats, upper bounds unless stated)."""
+    d = delta()
+    # ---- odd block: Y = (f_x, f_z, f_xxx) pinned at (0, 0, 12k); X = (f_xxz, f_xzz, f_zzz)
+    Y = [(1, 0), (0, 1), (3, 0)]
+    X = [(2, 1), (1, 2), (0, 3)]
+    SYY, SXY, SXX = _icov(Y, Y, d), _icov(X, Y, d), _icov(X, X, d)
+    Bc = _mm(SXY, _inv3(SYY))
+    BS = _mm(Bc, _T(SXY))
+    Sc = [[sub(SXX[i][j], BS[i][j]) for j in range(3)] for i in range(3)]
+    S0 = (2.0, 2.0, 6.0)
+    eps2 = (0.0, 0.0)                              # Frobenius bound of E = S0^(-1/2) (S_c - S0) S0^(-1/2)
+    for i in range(3):
+        for j in range(3):
+            dev = sub(Sc[i][j], ((S0[i] if i == j else 0.0),) * 2)
+            e = div((absmax(dev),) * 2, isqrt((S0[i] * S0[j],) * 2))
+            eps2 = add(eps2, sqr(e))
+    eps2 = eps2[1]
+    eps = isqrt((eps2, eps2))[1]
+    mu2 = (0.0, 0.0)                               # |mu~|^2, mu = 12k Bc[:, 2] with k <= 2
+    for i in range(3):
+        m_i = scal(24.0, (absmax(Bc[i][2]),) * 2)
+        mu2 = add(mu2, divc(sqr(m_i), S0[i]))
+    mu2 = mu2[1]
+    if not eps2 <= 0.006:                          # the range of the chi^2 bound (module docstring)
+        raise RuntimeError('torus transfer: perturbation too large')
+    m = div((mu2, mu2), sub((1.0, 1.0), (eps, eps)))[1]
+    chi2 = add(mul(from_frac(Fr(101, 100)), mul((m, m), _expiv((m, m)))), scal(2.0, (eps2, eps2)))[1]
+    dJ = isqrt(mul((E_I2_upper, E_I2_upper), (chi2, chi2)))[1]
+    # ---- even block: W = (f, f_xx, f_xz) pinned at (b, 0, 0); A = f_zz
+    W = [(0, 0), (2, 0), (1, 1)]
+    A = [(0, 2)]
+    SWW, SAW, SAA = _icov(W, W, d), _icov(A, W, d), _icov(A, A, d)
+    Wi = _inv3(SWW)
+    cA = _mm(SAW, Wi)[0]                           # regression row: mean of A = cA[0] b
+    sA2 = sub(SAA[0][0], _mm(_mm(SAW, Wi), _T(SAW))[0][0])
+    # p_b(0)^L / p_b(0)^ref = (sqrt2/s_L) exp(-b^2 (cA0^2/(2 s_L^2) - 1/4)), b in [0, 1]
+    x = sub(div(sqr(cA[0]), scal(2.0, sA2)), (0.25, 0.25))
+    ax = absmax(x)
+    rp = mul(div(isqrt((2.0, 2.0)), isqrt(sA2)), _expiv((-ax, ax)))
+    # pi_0 factors: p_W(b,0,0) ratio, p_G(0) ratio, phi_tau(12k) ratio
+    det_ref_W = (2.0, 2.0)                         # det [[1,-1,0],[-1,3,0],[0,0,1]]
+    y = sub(Wi[0][0], (1.5, 1.5))
+    ay = absmax(y)
+    rW = mul(isqrt(div(det_ref_W, _det3(SWW))), _expiv((-0.5 * ay, 0.5 * ay)))
+    G = [(1, 0), (0, 1)]
+    SGG = _icov(G, G, d)
+    rG = isqrt(div((1.0, 1.0), _det2(SGG)))
+    T3 = [(3, 0)]
+    S3G, S33 = _icov(T3, G, d), _icov(T3, T3, d)
+    tau2 = sub(S33[0][0], _mm(_mm(S3G, _inv2(SGG)), _T(S3G))[0][0])
+    z = sub(div((1.0, 1.0), tau2), div((1.0, 1.0), (6.0, 6.0)))
+    az = up(288.0 * absmax(z))                     # 72 k^2 |1/tau_L^2 - 1/6|, k <= 2
+    rT = mul(isqrt(div((6.0, 6.0), tau2)), _expiv((-az, az)))
+    rho = mul(mul(mul(rW, rG), rT), rp)            # (pi_0 p_b(0))^L / (pi_0 p_b(0))^ref, uniform in (b, k)
+    return {'delta': d, 'eps': eps, 'mu2': mu2, 'chi2': chi2, 'dJ': dJ, 'mean_coeff_A': cA[0], 'var_A': sA2,
+            'p_ratio': rp, 'pi0p_ratio': rho, 'tau2': tau2, 'S_c': Sc}
+
+
+
+def abs_moment(s2, n):
+    """E|X|^n for X ~ N(0, s2) with s2 an exact float: n even (n-1)!! s2^(n/2); n odd 2^(n/2) Gamma((n+1)/2)/sqrt(pi) s^n,
+    i.e. (n-1)!! s^n sqrt(2/pi) (enclosed)."""
+    if n % 2 == 0:
+        return pw(scal(float(_dfact(n - 1)), (1.0, 1.0)), 1) if n == 0 else mul((float(_dfact(n - 1)),) * 2, pw((s2, s2), n // 2))
+    from ia import PI
+    s = isqrt((s2, s2))
+    return mul(mul((float(_dfact(n - 1)),) * 2, pw(s, n)), isqrt(div((2.0, 2.0), PI)))
+
+
+def E_I2_upper():
+    """E_ref[I^2] <= 144 E[Pi_4^2], I = 3k^2 H <= 12 Pi_4 for k in [1/2, 2], (a, beta, c) ~ N(0, diag(2, 2, 6)),
+    Pi_4 = 68|beta|^3 + c6 a^6 + 144 c^2 + 9 a^2 beta^2, c6 = 68/216 + 576/5184."""
+    c6 = add(divc((68.0, 68.0), 216.0), divc((576.0, 576.0), 5184.0))
+    terms = [((68.0, 68.0), (0, 3, 0)), (c6, (6, 0, 0)), ((144.0, 144.0), (0, 0, 2)), ((9.0, 9.0), (2, 2, 0))]
+    var = (2.0, 2.0, 6.0)
+    tot = (0.0, 0.0)
+    for c1, e1 in terms:
+        for c2, e2 in terms:
+            v = mul(c1, c2)
+            for j in range(3):
+                v = mul(v, abs_moment(var[j], e1[j] + e2[j]))
+            tot = add(tot, v)
+    return up(144.0 * tot[1])
+
+
+if __name__ == '__main__':
+    t = transfer(E_I2_upper())
+    for kk in ('delta', 'eps', 'mu2', 'chi2', 'dJ', 'mean_coeff_A', 'var_A', 'p_ratio', 'pi0p_ratio', 'tau2'):
+        print(kk, t[kk])
diff --git a/frontiers/c8_lifetime_difference_constant_20261001/NOTE.md b/frontiers/c8_lifetime_difference_constant_20261001/NOTE.md
new file mode 100644
index 000000000..845112c90
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/NOTE.md
@@ -0,0 +1,557 @@
+# Theorem B of [LP] with explicit constants on the C8 band (`d = 2, 3`): the radial ledger to second order, two-sided lifetime densities, and the difference constant `C_{B,K}`
+
+**Object:** `CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1`. **Author lane:** Anthropic / Claude.
+
+**Kind.** An author-side theorem with explicit constants, certified by exact rational interval arithmetic (standard library
+only). It uses Taylor models in the radius for the pin law, and the box-bound engines of Math-#206 and Math-#212 for the cap
+route.
+
+**Scientific effect:** NONE. Catalog entry C8 (`reviews/candidates_pending_20260928/CANDIDATES.md`) stays OPEN. No STATUS,
+PROOF_INDEX, GRAPH or catalog edit. Same GitHub account as every lane; zero organizational-independence credit. The author will
+not merge. Scope claim: Math-#206 comment 5924133127.
+
+**Sources** (pinned in `SOURCE_MAP.json`):
+- [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`:
+  - section 1 (the model in every `d`, Theorem B with (1.2));
+  - section 3 (the preconditioned pins (3.1)–(3.3));
+  - section 8 (`1 - p_r <= Q^W(G_r^c)`);
+  - sections 9–11 (the weighted Kac–Rice interface, the radial ledger (10.1)–(10.2), the lifetime pushforward (11.1)–(11.3));
+  - section 12 ((12.1), (12.3)).
+  Sections 8–12 are consumed as statements; nothing else of [LP] is used.
+- [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md`, section 1 (the good event `G_r`), through the
+  engines.
+- **The corrected parent reading set.** [LP] is read together with three mandatory amendments, all on `main` and pinned
+  `current_required`. The parent object is P + CAP + E1 + E2 + REC, as REC section 1 requires of every consumer (review C61):
+  - [E1] `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`: the preconditioning congruence is `diag(r^(-1/2), I)`;
+  - [E2] `reviews/d1_section9_borel_repair_20260925/REPAIR.md`: [LP] section 9 is replaced by the marked Kac–Rice identity
+    for the Borel elder indicator (finite-measure equality on a cylinder algebra, then a monotone-class extension);
+  - [REC] `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`: the reading rule. It uses joint convergence in law
+    and the embedding condition `r < L/(4 sqrt 2)`, which `r <= 1/64` and `L >= 10` satisfy.
+- [E] Math-#206 at `13f932c` (v1.2): the planar certificate `theorem_a.py`, carried byte for byte as `engine_e2.py`.
+- [E3] Math-#212 at `250b907` (v1.1): the `d = 3` certificate `theorem_a3.py`, carried byte for byte as `engine_e3.py`.
+- Read-only cross-checks and comparisons:
+  - [W] Math-#197: `c_{B,K}` in the plane.
+  - [W3] Math-#205: `c_{B,K}` in `d = 3`.
+  - [G] Math-#203: `c_G = R(b) J(k)`.
+  - [G3] Math-#208: `c_G^(3)`.
+
+## 0. Statement
+
+**Setting** ([LP] sections 1, 10, 11 with `d in {2, 3}`, `m = d - 1`).
+- The torus `X = (R / L Z)^d` with the periodized Gaussian kernel `K_L`, or the reference kernel `exp(-|z|^2/2)`.
+- For an orthonormal frame `R` with axial vector `u`: the pins `M = -(r/2) u`, `S = (r/2) u`, `f(M) = b`, `f(S) = b - k r^3`,
+  `grad f(M) = grad f(S) = 0`, the regression law `Q`, the typed weight `W_r`, and `Z_r = E_Q W_r`.
+- [LP] (10.2) writes the candidate intensity per unit volume as `r A_r(b, k, u) dr db dk dsigma(u)` with
+
+      A_r(b, k, u) = 12 pi_r(R; v_r) Z_r / r^2,
+
+  where `pi_r(R; v_r)` is the density of the preconditioned pins (3.1) at their prescribed value (3.3).
+- The band is `B = [0, 1]`, `K = [1/2, 2]`.
+- The **reference contact integrand** is the closed form
+
+      A_0^ref(b, k) = 432 (2 pi)^-(d+1) 12^(-1/2) k^2 exp(-12 k^2) exp(-3 b^2/4) m_d(b),
+
+      m_2(b) = (b^2 + 2) Phi(b/sqrt2) + sqrt2 b phi(b/sqrt2),          m_3(b) = E[det(B)^2 1{B > 0}],  B = b I + GOE_2.
+
+  This is the integrand of [LP] (11.3) for the reference kernel: `pi_0 = (2 pi)^-(d+1) 12^(-1/2) exp(-3b^2/4 - 12k^2)` and
+  `z_0 = 36 k^2 m_d(b)`. `m_3` has the closed form of [E3] section 6.
+
+**Theorem L (the radial ledger to second order).** For `d = 2, 3`, every torus side `L >= 10`, every frame, every `b in B`,
+`k in K`, every `R` in the table and every `0 < r <= R`,
+
+    -(a_dn(R) r^2 + eps_d)  <=  A_r(b, k, u) / A_0^ref(b, k) - 1  <=  a_up(R) r^2 + eps_d.
+
+The same holds for the reference kernel. The constants are exact rationals (`RESULTS.json`), shown here rounded up:
+
+| `R` | `a_up`, `d = 2` | `a_dn`, `d = 2` | `a_up`, `d = 3` | `a_dn`, `d = 3` |
+|---|---|---|---|---|
+| `1/4096` | `13.822900` | `3.637234` | `25.666574` | `25.516405` |
+| `1/2048` | `13.825546` | `3.643346` | `26.256537` | `26.108063` |
+| `1/1024` | `13.830901` | `3.655573` | `27.446808` | `27.301701` |
+| `1/512` | `13.841872` | `3.680046` | `29.869053` | `29.730582` |
+| `1/256` | `13.864854` | `3.729062` | `34.883076` | `34.757367` |
+| `1/128` | `13.915029` | `3.827402` | `45.611590` | `45.508415` |
+| `1/64` | `14.032636` | `4.025510` | `70.060400` | `69.981330` |
+
+with `eps_2 <= 3.96 x 10^-10` and `eps_3 <= 1.79 x 10^-8`. These collect the torus, truncation and rounding remainders and
+do not depend on `r`.
+
+What Theorem L gives, and what it does not:
+- **It gives** the second-order coefficient of the deviation of the ledger from the reference contact integrand, `a(R)`,
+  up to the radius-independent `eps_d`. [LP] proves the convergence ((10.3)) and states in section 11 that "Convergence of
+  `A_r` to `A_0` was not assigned a numerical rate."
+- **It does not give** a vanishing-remainder rate. `eps_d` does not tend to `0` with `r`. On the torus, Theorem L
+  therefore does not prove `A_r = A_0 (1 + O(r^2))` for the torus limit `A_0`. With [LP] (10.3) it gives only
+  `|A_0 / A_0^ref - 1| <= eps_d` for that limit. A vanishing-remainder rate would need the torus corrections tracked as
+  functions of `r`, which the certificate does not do.
+
+There is no first-order term:
+- for the Gaussian kernel, the law of the transverse Hessian at `M` is the contact law at every `r` (Lemma 1);
+- the first-order parts of `Z_r / r^2` cancel exactly (Lemma 3).
+
+In the plane `a_up` is attained at `k = 2`, where the genuine coefficient is about `13.2`–`13.4` (Monte Carlo, section 9).
+
+**Corollary L1 (lifetime densities to second order).** Let `nu_cand`, `nu_eld` be the compact-window densities of [LP]
+Theorem B for pairs at distance at most `r_pop`, in the versions (11.2). For `0 < ell < min(r_pop, R)^3 / 2`:
+
+    | ell^(1/3) nu_cand(ell) - c^ref_{B,K} |  <=  c'_up(R) ell^(2/3) + eps'_d,    (upper and lower sides with c'_up, c'_dn)
+    ell^(1/3) nu_cand(ell) - C_{B,K}(R) ell  <=  ell^(1/3) nu_eld(ell)  <=  ell^(1/3) nu_cand(ell).
+
+- `c^ref_{B,K} = |S^(d-1)| int_{B x K} A_0^ref / (3 k^(2/3))` is certified. Its enclosures contain the values of Math-#197 and
+  Math-#205:
+  - `d = 2`: `[0.002386938000079198, 0.002386938059531751]`;
+  - `d = 3`: `[0.000974382342936858, 0.000974382390951253]`.
+- `c'_up(R) = |S^(d-1)| sum_boxes a_up,box int_box A_0^ref / (3 k^(4/3))` is the weighted second-order constant, and `c'_dn`
+  is the same with `a_dn`. As ratios to `c^ref`:
+
+  | `R` | `c'_up/c`, `d = 2` | `c'_dn/c`, `d = 2` | `c'_up/c`, `d = 3` | `c'_dn/c`, `d = 3` |
+  |---|---|---|---|---|
+  | `1/4096` | `6.2826` | `4.4646` | `23.154` | `22.428` |
+  | `1/512` | `6.3221` | `4.5056` | `27.378` | `26.662` |
+  | `1/64` | `6.6431` | `4.8364` | `67.061` | `66.397` |
+
+  (Rounded up. All seven `R` are in `RESULTS.json`.)
+- `eps'_2 <= 1.2 x 10^-13` and `eps'_3 <= 2.5 x 10^-12`, absolute.
+
+Hence, up to an additive `eps'_d ell^(-1/3)`, both compact-window densities are `c^ref_{B,K} ell^(-1/3) + O(ell^(1/3))`
+with explicit constants. [LP] remarks that (1.2) "is an O(ell^(2/3)) difference, not a claim that either density separately
+has a second-order expansion with that remainder".
+- **What Corollary L1 is.** A two-sided expansion of each density about the reference constant.
+- **What it is not.** Its remainder `eps'_d ell^(-1/3)` does not vanish as `ell -> 0`. For the torus constant `c_{B,K}`
+  (any `L >= 10`) it gives `|c_{B,K} - c^ref_{B,K}| <= eps'_d`, not a second-order expansion about `c_{B,K}` with a
+  vanishing remainder.
+
+For example, in the plane, for `ell < 2^-19`:
+
+    -4.84 ell^(2/3) - 10^-10  <=  ell^(1/3) nu_cand(ell) / c_{B,K} - 1  <=  6.65 ell^(2/3) + 10^-10,
+
+which is a relative error below `1.1 x 10^-3`.
+
+**Corollary L2 (explicit [LP] (1.2), (12.1), (12.3)).** For `0 < ell, t < min(r_pop, r_*)^3 / 2` and `q > -5/3`:
+
+    0 <= nu_cand(ell) - nu_eld(ell) <= C_{B,K}(r_*) ell^(2/3),
+    0 <= E[N_cand(0, t] - N_eld(0, t]] <= (3/5) C_{B,K}(r_*) t^(5/3),
+    E sum_{candidate, not selected, ell <= t} ell^q <= C_{B,K}(r_*) t^(q + 5/3) / (q + 5/3),
+
+and the selected count satisfies
+
+    (3/2)(c^ref - eps') t^(2/3) - (3/4) c'_dn t^(4/3) - (3/5) C_{B,K} t^(5/3)  <=  E N_eld(0, t]  <=  (3/2)(c^ref + eps') t^(2/3) + (3/4) c'_up t^(4/3).
+
+The certified difference constants (exact rationals in `RESULTS.json`, rounded up), with the relative size of the bound
+at `ell = r_*^3/2`, `C_{B,K} ell / c^ref_{B,K}` (rounded up):
+
+| `r_*` | `C_{B,K}`, `d = 2` | `(3/5) C_{B,K}` | `C ell/c` at `r_*^3/2` | `C_{B,K}`, `d = 3` | `(3/5) C_{B,K}` | `C ell/c` at `r_*^3/2` |
+|---|---|---|---|---|---|---|
+| `1/4096` | `402.5297` | `241.5178` | `1.3e-06` | `1205.7089` | `723.4254` | `9.1e-06` |
+| `1/2048` | `406.7323` | `244.0394` | `1.0e-05` | `1223.4762` | `734.0857` | `7.4e-05` |
+| `1/1024` | `413.9359` | `248.3615` | `8.1e-05` | `1270.6952` | `762.4171` | `6.1e-04` |
+| `1/512` | `428.0180` | `256.8108` | `6.7e-04` | `2488.0639` | `1492.8383` | `9.6e-03` |
+| `1/256` | `3384.7882` | `2030.8729` | `4.3e-02` | `1807674.3925` | `1084604.6355` | `5.6e+01` |
+| `1/128` | `178525.1111` | `107115.0667` | `1.8e+01` | `13273148.1212` | `7963888.8728` | `3.3e+03` |
+| `1/64` | `198574.3344` | `119144.6007` | `1.6e+02` | `2950751318.4742` | `1770450791.0845` | `5.8e+06` |
+
+[LP] section 11 gives (1.2) in the form `C ell^(2/3) integral A^*/(3 k^(5/3))` with no value for either factor, and its
+section 16 lists "a numerical r_* or C on a prescribed band" among the items not supplied.
+
+**What the refinement and the cap route contribute** (`r_* = 1/4096`):
+- **Box resolution.** On their own boxes, the per-band bounds of [E] and [E3] give `2325.98` in the plane and `4999.57` in
+  `d = 3`. The refined boxes improve these by factors `5.8` and `4.1`. [E] has `28` boxes, the coarsest `k`-box being
+  `[1/2, 1]`; [E3] has `39`.
+- **A single constant.** Using one sup constant `C(r_*)` of [E] or [E3] times the total weight gives `22467.8` and `38970.6`.
+- **The reference-kernel cap-route benchmark.** `Gamma_{B,K} = |S^(d-1)| int A_0^ref c_G / (3 k^(5/3))` is built from
+  the reference-kernel cap-failure coefficients `c_G` of [G] and [G3]. It is the reference-kernel asymptotic lower
+  benchmark for any cap-route bound that is uniform over the present family. That family contains the reference
+  kernel, so as `r_* -> 0` such a bound can give no less than `Gamma_{B,K}`. In floating point, `Gamma_2 = 307.5` and
+  `Gamma_3 = 541` (section 9).
+  - `C_{B,K}(1/4096)` is within a factor `1.31` of that benchmark in the plane and `2.23` in `d = 3`. These floating
+    ratios compare our finite-radius bounds with the benchmark. They do not identify the exact cap-route limit of each
+    finite torus, which would use the torus `A_0^L` and `c_G^(L,u)`; [G] says the torus deviation from `c_G^ref` is not
+    enclosed.
+  - The `d = 3` excess comes from the [E3] box bounds near `k = 1/2`, which are about `2.2 c_G^(3)` there.
+- **The cap route itself is loose.** By [G], the cap criterion overstates the actual elder-pairing failure rate by
+  `7.4 x 10^4` at `k = 1/2`. `C_{B,K}` bounds the [LP] quantity as [LP] defines it; it is not an estimate of the true
+  difference.
+- **Above `r_* = 1/512`** the certified per-band bounds of [E] and [E3] grow quickly; their own `C(r_*)` tables show the same
+  jump. The useful range is `r_* <= 1/512`.
+
+## 1. The radial ledger and what is consumed
+
+[LP] sections 8–11 are consumed as statements:
+- **Section 9, as repaired by [E2].** The weighted Kac–Rice identity on the near-diagonal pair domain.
+- **Section 10, the ledger (10.1)–(10.2).** The candidate intensity is `r A_r dr db dk dsigma(u)`. The selected intensity
+  carries the extra factor `p_r`. `A_r` does not depend on the transverse frame.
+- **Section 11, the pushforward (11.1)–(11.2).** For `ell < k_- r_pop^3`,
+
+      nu_cand(ell) = ell^(-1/3) int_{B x K x S^(d-1)} A_{(ell/k)^(1/3)}(b, k, u) / (3 k^(2/3)) db dk dsigma(u),
+
+  and `nu_eld` has the extra factor `p_r`.
+- **Section 8.** `0 <= 1 - p_r <= Q^W(G_r^c)`.
+
+Theorem L is a statement about `A_r` alone. The corollaries follow by integration.
+- **Corollary L1.**
+  - At fixed `ell < R^3/2`, every `k in K` has `r = (ell/k)^(1/3) <= (2 ell)^(1/3) < R`, and `r^2 = ell^(2/3) k^(-2/3)`.
+  - Integrate Theorem L box by box against `A_0^ref / (3 k^(2/3))`. This gives `c^ref + ell^(2/3) c'_up + eps'` above and the
+    mirror image below.
+  - `A_0^ref` does not depend on `u`, so the sphere contributes `|S^(d-1)|`.
+  - The integrals of `ell^(-1/3)`, `ell^(1/3)` and `ell^(2/3)` over `(0, t]` give the bounds on `E N_eld(0, t]`.
+- **Corollary L2.**
+  - `nu_cand - nu_eld = ell^(-1/3) int A_r (1 - p_r) / (3 k^(2/3))`.
+  - On each cap box `1 - p_r <= Q^W(G_r^c) <= C_box r^3 = C_box ell / k`, so the difference is at most
+    `ell^(2/3) sum_box C_box int_box A_r / (3 k^(5/3))`.
+  - Theorem L bounds `A_r` by `(1 + a_up r_*^2 + eps) A_0^ref`.
+  - Integrating over `(0, t]` gives (12.1) and (12.3).
+
+## 2. The laws: Taylor models in `r` (Lemmas 1–2)
+
+**Taylor models.** Each covariance of two preconditioned functionals is the exact series of [E] / [E3] section 2,
+`sum_gamma kappa_gamma pi_gamma r^(g_x - w)`. Added to it are:
+- the truncation tail beyond `g_x = 60`;
+- the torus remainder, by Cauchy estimates on the polydisc: `M_1(10) < 4.46 x 10^-15` in the plane and `< 1.94 x 10^-13`
+  in `d = 3`.
+
+The certificate keeps each one on `[0, 1/64]` as a **Taylor model**:
+- `f(r) = sum_{j <= 8} c_j r^j + theta e`, with `|theta| <= 1`;
+- the coefficients are fixed-point integers with `288` fractional bits;
+- every rounding, every truncated power `r^j <= 64^-j`, both tails and the torus remainder go into `e`.
+
+Products, sums and reciprocals (a geometric series with an explicit tail) stay in this class. Gaussian elimination without
+pivoting then gives Taylor models for the following, all exact in their polynomial parts up to `2^-288`:
+- `det Sigma_r`;
+- the pin energies `e_bb`, `e_bk`, `e_kk`, with `v_r' Sigma_r^-1 v_r = e_bb b^2 + 2 e_bk b k + e_kk k^2`;
+- the regression coefficients `alpha` and `beta` (mean `= b alpha + k beta`) of the transverse Hessian `A = D_y^2 f(M)` and of
+  `Omega' = (D_y^2 f(S) - D_y^2 f(M))/r`;
+- their covariances;
+- in `d = 3`, the regression matrix of `Omega'` on `B = -A`.
+
+This removes the dependence loss of interval elimination. For example, the band laws of [E] give the `k`-coefficient of the
+mean of `A` on `[0, r]` only as `+-3 r`, while the Taylor model gives `0` up to `10^-11`.
+
+**Lemma 1 (the transverse Hessian is exactly contact-distributed).** For the reference kernel, every `r > 0`, every `b` and
+every `k`, the `Q`-law of `A = D_y^2 f(M)` is that of `-b I_m + G`. Here `G` is the `GOE_m` of [G3], with `N(0, 2)` diagonal
+and `N(0, 1)` off-diagonal entries; the law does not depend on `r` or `k`.
+
+*Proof.* The kernel factorizes, `exp(-|z|^2/2) = exp(-x^2/2) exp(-|y|^2/2)`. Every pin is either axial (`partial_x^a f` on the
+axis) or a first transverse derivative on the axis.
+- For an axial functional `g`, `Cov(f_{y_i y_j}(x, 0), g) = -delta_ij Cov(f(x, 0), g)`, because
+  `partial_{y_i} partial_{y_j} exp(-|y|^2/2) = -delta_ij` at `y = 0`.
+- For a transverse pin, `Cov(f_{y_i y_j}(x, 0), f_{y_l}(x', 0)) = Cov(f(x, 0), f_{y_l}(x', 0)) = 0`, because odd
+  `y`-derivatives of the kernel vanish at `y = 0`.
+
+So `A + f(M) I` is uncorrelated with every pin, hence independent of all of them. Under `Q`, `f(M) = b`, so
+`A = -b I + (A + f(M) I)` with `A + f(M) I` keeping its unconditional law:
+- `Var(f_yy + f) = 3 - 2 + 1 = 2`;
+- `Cov(f_{y_1 y_1} + f, f_{y_2 y_2} + f) = 1 - 1 - 1 + 1 = 0`;
+- `Var f_{y_1 y_2} = 1`.
+
+∎
+
+The certificate checks Lemma 1 on the Taylor models:
+- every polynomial coefficient of `alpha_A + 1`, `beta_A` and `Cov(A) - Cov(GOE)` vanishes to within `10^-60`;
+- the code requires this, and only the remainder survives: at most `1.9 x 10^-12` in the plane and `8.1 x 10^-11` in `d = 3`.
+
+So on the torus the law is the contact law up to these remainders.
+
+**Lemma 2 (pin law, energies, `Omega'`).** For the reference kernel, the Taylor models give (coefficients exact up to
+`2^-288`):
+
+    d = 2:  det Sigma_r = 12 - 18 r^2 + 14.4 r^4 + O(r^6),       d = 3:  det Sigma_r = 12 - 24 r^2 + 25.4 r^4 + O(r^6),
+    both:   e_bb = 3/2 + r^2/8 - r^4/96 + ...,   e_bk = -(3/4) r^3 + ...,   e_kk = 24 - 6 r^2 + (6/5) r^4 + ...
+
+For `Omega'`:
+- **Mean.** `alpha = 0`, and `beta = r^2` on the diagonal entries.
+- **Covariance with the Hessian.**
+  - Plane: `Cov(om, w) = -r + r^3/4`.
+  - `d = 3`: `Cov(A_ii, O_ii) = -r + r^3/4` and `Cov(A_12, O_12) = -r/2 + r^3/8`; all other cross-covariances vanish.
+  - Hence in `d = 3`, `E[Omega' | B] = mu_Omega + beta(r) (B - mu_B)` with `beta(r) = (r - r^3/4)/2 + ...`, the same scalar for
+    all three entries.
+- **Covariance.** `Var om = 2 - r^2/2 + r^4/12`; `Cov(O_11, O_22) = 0`; `Var O_12 = 1 - r^2/4 + ...`.
+
+At `r = 0` this is the contact law of [LP] section 15:
+- `det = 12 = det Cov(f, f_xx) det Cov(f_x, f_xxx) = 2 . 6`;
+- the energy is `3b^2/2 + 24 k^2`, which gives `pi_0 ∝ exp(-3b^2/4 - 12 k^2)`.
+
+## 3. The pin-density ratio
+
+    pi_r / pi_0^ref = (12 Theta^n / det S_r)^(1/2) exp(-(E_r - E_0)/2),
+
+with:
+- `n = 2(d + 1)`;
+- `S_r` the unnormalized pin covariance and `Theta in [1, 1 + theta_1(10)]` the kernel normalization;
+- `E_r = Theta (e_bb b^2 + 2 e_bk b k + e_kk k^2)` and `E_0 = 3 b^2/2 + 24 k^2`.
+
+The Taylor models give `det S_r - 12 in [d_lo r^2 - e, d_hi r^2 + e]`, and the same for each energy coefficient. On a box
+`b in [b_0, b_1]`, `k in [k_0, k_1]` (`b, k >= 0`), `log(pi_r/pi_0^ref)` lies between
+
+    -max(0, d_hi) r^2/24 - max(0, E_hi/2) r^2 - (remainders)    and    max(0, -d_lo) r^2 / (24 (1 - x_m)) + max(0, -E_lo/2) r^2 + (remainders),
+
+using `log(1 + x) >= x/(1 + x)` and `log(1 + x) <= x`. The dominant genuine term is `+3 k^2 r^2` from `e_kk = 24 - 6 r^2`. At
+`k = 2` this is `12 r^2`, the largest part of `a_up` in the plane.
+
+## 4. The normalizer: exact expansion and the first-order cancellation (Lemma 3)
+
+**Plane.** Use the notation of [E] section 1: `lambda = -w`, `c1 = 6k - rT`, `c2 = 6k + rT + r^2 tau`, `e = (r nu - v)^2`.
+The typed event is `{c1 > 0, P1 > 0, P2 > 0}`, which lies inside `{lambda > 0}`, and on it `W_r = r^2 P1 P2`. Expanding
+exactly,
+
+    P1 P2 = c1 c2 lambda^2 - r c1 c2 lambda om + r^2 lambda xi + r^2 v^2 (c2 om - e),
+
+    c1 c2 = 36 k^2 + r^2 kappa,        kappa = 6 k tau - T^2 - r T tau,
+    xi    = 6k (r nu^2 - 2 nu v) - T (r^2 nu^2 - 2 r nu v + 2 v^2) - r tau v^2.
+
+Three cancellations remove every first-order term:
+- **In `c1 c2`.** The first-order parts of `c1` and `c2` cancel.
+- **Between `P1` and `P2`.** The terms `r c1 lambda e` (from `P2`) and `-r c2 lambda v^2` (from `P1`) cancel, because
+  `e - v^2 = r^2 nu^2 - 2 r nu v`.
+- **The `om` term.** The remaining first-order term `-36 k^2 r E[lambda_+ om]` has an `O(r)` coefficient. Regress `om` on `w`:
+  `om = om_g + bw (w - mu_w)` with `om_g` independent of `w`. Then
+
+      E[lambda_+ om] = mu_om E[lambda_+] - bw (E[lambda_+^2] - mu_lambda E[lambda_+]),        mu_lambda = -mu_w,
+
+  and Lemma 2 gives `mu_om = k r^2` and `bw = -r/2 + O(r^3)`.
+
+So `Z_r / (36 k^2 r^2) = E[lambda_+^2] + O(r^2)`, and by Lemma 1 `E[lambda_+^2] = m_2(b)` exactly for the reference kernel.
+
+The second-order terms are bounded by Hölder's inequality with the Gaussian norms of the band law of [E] on `[0, R]`. The one
+exception is `kappa`. Since `E[tau | w] = mu_tau + bw_tau (w - mu_w)` with `mu_tau = -2k + O(r^2)`, the term
+`6k E[tau lambda_+^2] = 6k mu_tau E[lambda_+^2] + ...` is negative. The certificate therefore bounds the `kappa` term with its
+sign:
+- above by `max(0, 6k mu_tau F + |side terms|)`;
+- below by `6k mu_tau F - E[T^2 lambda_+^2] - |side terms|`.
+
+**`d = 3`.** Write `D = det B` (`B = -A`), `A* = adj B`, `B_S = B - r Omega'`, and `e = r nu - v` (a vector). [E3] (I2) gives
+`P1 = c1 D - r v' A* v` and `P2 = c2 det B_S + r e' adj(B_S) e`. In two dimensions `det B_S = D - r tr(A* Omega') + r^2 det Omega'`
+and `adj(B_S) = A* - r adj(Omega')`, so
+
+    P1 P2 = c1 c2 D^2 - r c1 c2 D tr(A* Omega') + r^2 c1 c2 D det Omega' + r^2 D xi_3 - r^2 c1 D e' adj(Omega') e
+            + r^2 c2 tr(A* Omega') v' A* v - r^3 c2 det Omega' v' A* v - r^2 (v' A* v)(e' A* e) + r^3 (v' A* v)(e' adj(Omega') e),
+
+    xi_3 = -(2T + r tau) v' A* v - 2 c1 nu' A* v + r c1 nu' A* nu.
+
+Again `c1 e' A* e - c2 v' A* v = r xi_3`. The main term is `36 k^2 E[D^2 1{B > 0}]`, which equals `36 k^2 m_3(b)` for the
+reference kernel (Lemma 1). On the torus the Loewner coupling of [E3] section 6 compares it with `m_3`, at the same `b` and with
+shift `eps + 12 eta` (remainders only).
+
+Four terms whose expectations are of lower order than their absolute values are treated through conditional expectations:
+- **The first-order term `-36 k^2 r E[D tr(A* Omega') 1]`.** By Lemma 2, `E[Omega' | B] = mu_Omega + beta (B - mu_B) + DY`, so
+
+      E[D tr(A* Omega') 1] = beta (2 E[D^2 1] - b E[D tr B 1]) + R_1,
+
+  with `|R_1| <= ||mu_Omega|| E[D tr B 1] + ||DG|| ||D tr B 1||_2 (tr Cb)^(1/2)` (`DY = DG (B - mu_B)`, remainder only). Since
+  `beta = r/2 + O(r^3)`, write `T_Omega = -36 k^2 r E[D tr(A* Omega') 1]`, `M = E[D^2 1{B > 0}]` and
+  `J = E[D tr B 1{B > 0}]`. `T_Omega` is a term of `E[P1 P2] = Z_r / r^2`, so its raw `Z_r` contribution is `r^2 T_Omega`, and
+  after division by `36 k^2 r^2` it contributes `T_Omega / (36 k^2) = -r beta (2M - bJ) - r R_1` to `Z_r / (36 k^2 r^2)`. For
+  the reference kernel `beta = a(r) = (1 - e^(-r^2/2))/r`, `mu_Omega = k r^2 I` and `DY = 0`, so this is exactly
+
+      T_Omega / (36 k^2) = -r a(r) (2M - bJ) - k r^3 J = -(1/2) r^2 (2M - bJ) + O(r^3),
+
+  uniformly on the band (OA-215-A-01, successor read 5934120553). Relative to the main term `M = m_3`, divide by `M`. It is
+  signed and enters `a_dn`, with only `O(r)` slack in `a_up`.
+- **`r^2 E[c1 c2 D det Omega' 1]`.** `E[det Omega' | B] = det(E[Omega' | B]) + E[det X']`, where `X' = Omega' - E[Omega' | B]`
+  is independent of `B`. Here `E[det X'] = Cov(X'_11, X'_22) - Var X'_12` is about `-1`, and `det(E[Omega' | B]) = O(r^2)`.
+- **`r^2 E[c2 tr(A* Omega') v' A* v 1]` and `r^2 E[c1 D e' adj(Omega') e 1]`.** Condition on `zeta = (B, v, nu)`. Then
+  `E[Omega' | zeta] = mu_Omega + Y_K`, and the explained part `Y_K` has `E||Y_K||^2 <= sum_O c_O' D^-1 c_O / (1 - ||E||)`, which is
+  `O(r^2)`. So both terms are `O(r^3)`.
+- **The `kappa` term**, with its sign as in the plane.
+
+The moments of `B` are the closed-form contact moments `E[det^a tr^c 1{B > 0}]` at the shifted mean, plus outside-ball parts.
+
+## 5. The thin typed shell (Lemma 4)
+
+The typed event differs from `{lambda > 0}` (plane) or `{B > 0}` (`d = 3`) only where the smallest eigenvalue is `O(r)`. On
+`G_0 = {r |T| <= 3k/2, r^2 |tau| <= 3k/2}` one has `c1 >= 9k/2` and `3k <= c2 <= 9k`.
+
+**Plane.**
+- `{lambda > 0}` minus the typed event lies in `{0 < lambda <= r h}`, with `h = 2 v^2/(9k) + |om|`.
+- There `|P1 P2| <= r^2 (15 k h/2 + v^2)(9k (h + |om|) + e)`.
+- Given `(v, om, nu)`, `lambda` is Gaussian. Its variance is at least `1/(1/Var w + bw' Cg^-1 bw)` (Schur complement), with
+  `Cg^-1 <= D^-1/(1 - ||E||_F)` by diagonal scaling.
+- So `E[G 1{0 < lambda <= r h}] <= r pbar E[G h]`, and the shell contributes `O(r^3)`.
+
+**`d = 3`.**
+- The shell is `{0 < lambda_1 <= r h}`, with `h = 2 ||v||^2/(9k) + ||Omega'||_F`.
+- There `|P1 P2| <= r^2 lambda_2 (lambda_2 + r ||Omega'||)(15 k h/2 + ||v||^2)(9k (h + ||Omega'||) + ||e||^2)`.
+- Given `Y = (v, Omega', nu)`, `B` is Gaussian with covariance `C' >= cl I`, where `1/cl <= 1/lambda_min(Cb) + ||Bt' Cg^-1 Bt||`.
+- In eigenvalue coordinates (`dB = sqrt2 (lambda_2 - lambda_1) d lambda_1 d lambda_2 d theta`, `theta in [0, pi)`):
+
+      E[lambda_2 (lambda_2 + r Omega) 1{0 < lambda_1 <= s} | Y] <= s KB' J(||mu'||, r Omega),
+
+  where:
+  - `KB' = pi sqrt2 (2 pi)^(-3/2) cl^(-3/2)`;
+  - `J(m, x) = int_0^inf lambda^2 (lambda + x) exp(-(lambda - m)_+^2/(2 c_hi)) d lambda`, a polynomial in `m` and `x` built from
+    half-Gaussian moments;
+  - `||mu' - mu_B|| <= ||C_BY C_YY^(-1/2)|| chi_7`.
+- The rows of `Bt` for `Omega'` come from the Taylor models and are `O(r)`; those for `v` and `nu` are below `10^-11`.
+
+**Both.** The complement of `G_0` has probability below `10^-50` for `R <= 1/64`. It enters through
+`||P1 P2||_2 P(G_0^c)^(1/2)`.
+
+The sweep evaluates these bounds on `96` boxes (`b`-step `1/8`, `k`-step `1/8`) for each `R`. The box constants feed the
+weighted `c'`; their maxima are Theorem L's `a_up` and `a_dn`.
+
+## 6. Reference integrals
+
+`A_0^ref / (3 k^p) = K_d k^(2-p) exp(-12 k^2) exp(-3b^2/4) m_d(b)`, with `K_d = |S^(d-1)| 144 (2 pi)^-(d+1) / sqrt12`.
+
+**The `k`-integrals** are incomplete gamma functions,
+
+    int_{k0}^{k1} k^q exp(-12 k^2) dk = (1/2) 12^(-(q+1)/2) [gamma((q+1)/2, 12 k1^2) - gamma((q+1)/2, 12 k0^2)],
+
+with the powers `x^a` from exact integer roots and the series closed by a geometric tail.
+
+**The `b`-integrals** use the composite midpoint rule on `4096` cells:
+- the error per cell is at most `h^3 |g''|_max / 24`;
+- `|g''|` is bounded per block of `16` cells through `m`, `m'`, `m''`, all increasing in `b`;
+- plane: `m' = 2 E[(b + sqrt2 z)_+]` and `m'' = 2 Phi(b/sqrt2) <= 2`;
+- `d = 3`: `m_3' = 2 E[det tr 1]` and `m_3'' = 2 E[tr^2 1] + 4 E[det 1]` (contact moments).
+
+**Cross-checks** (exact, in the certificate): the enclosures of `c^ref_{B,K}` contain the values of Math-#197
+(`0.0023869380211529483`, Owen-T closed form) and Math-#205 (`0.00097438236602425`).
+
+## 7. The cap sweep and `C_{B,K}`
+
+The cap route needs `sup_{r <= r_*} Q^W(G_r^c)/r^3` on boxes that resolve `k` near `1/2`. There the weight
+`A_0^ref / k^(5/3) ∝ k^(1/3) e^(-12 k^2)` concentrates, and `c_G` grows like `k^3`.
+
+The engines of [E] and [E3] (`box_bound`, with their first parameter pair `(X_0, theta)`) are run on:
+- every band of [E] (`153`) and of [E3] (`121`);
+- `56` boxes per band: `b`-boxes `[i/4, (i+1)/4]`, and `k`-boxes of width `1/32` on `[1/2, 3/4]`, width `1/16` on `[3/4, 1]`,
+  then `[1, 3/2]` and `[3/2, 2]`.
+
+The certified per-box bounds are stored to four decimals, rounded up. For each `r_*`, `C_box(r_*)` is the maximum over the bands
+with `r_1 <= r_*`, and
+
+    C_{B,K}(r_*) = (1 + a_up(r_*) r_*^2 + eps) sum_box C_box(r_*) |S^(d-1)| int_box A_0^ref / (3 k^(5/3)).
+
+**Reviewed disposition of `C_{B,K}` (C61, review 5386924911 at `b41eb28`).** C61 accepts the planar and `d = 3` weighted
+cap engines at this caller, `(X_0, theta) = (1/64, 1/32)`. It then validates all fourteen published `C_{B,K}` and `C_121`
+upper constants by a different, cellwise transfer. That transfer uses Math-#227's reviewed finite-radius rate (C56–C59) and
+C60's direct contact comparison in place of the factor `(1 + a_up r_*^2 + eps)` above. The published constants dominate the
+alternative bound.
+
+The cap-derived candidate-minus-selected-elder error estimates no longer rest on those unaccepted positive-radius parts;
+C61 does not accept the older individual-density expansions or the selected-count bounds that also use `c'` and `eps'`.
+C61's scope is
+the compact window `b in [0, 1]`, `k in [1/2, 2]`, `L >= 10`, `d = 2, 3`, and the corrected parent set above. It gives
+`0 <= nu_cand - nu_eld <= C_{B,K}(R) ell^(2/3)` and the cumulative `C_121(R) t^(5/3)` for `ell, t < min(r_pop, R)^3/2`.
+
+C61 does not claim:
+- that every short bar is a candidate;
+- an unrestricted difference rate;
+- the shrinking multiple-witness node;
+- C8.
+
+## 8. Arithmetic
+
+- Exact rationals throughout.
+- The Taylor models are fixed-point integers with explicit rounding.
+- Interval enclosures of `exp`, `Phi` and the Gaussian and contact moments come from the engines.
+- Fractional powers use integer `p`-th roots.
+- Every stored number is rounded outward to a decimal and replayed exactly.
+
+## 9. Controls (floating point; not part of the certificate)
+
+**Monte Carlo of the genuine coefficient.** `theorem_b.py --controls` estimates `(A_r / A_0^ref - 1) / r^2` for the reference
+kernel at `r = 1/8` and `1/16` and at the four corners of the band:
+- the law at `r` comes from the exact series, which is valid at every `r` (the Taylor models stop at `1/64`);
+- the pin ratio is exact;
+- `Z_r / r^2` is sampled as `P1 P2 1{typed}` (seed `20261001`; `2 x 10^5` samples per point in the plane, `10^5` in `d = 3`).
+
+Values are shown with one standard error:
+
+| `d` | `r` | `(b, k) = (0, 1/2)` | `(0, 2)` | `(1, 1/2)` | `(1, 2)` |
+|---|---|---|---|---|---|
+| `2` | `1/8` | `0.08 ± 0.32` | `13.19 ± 0.39` | `0.64 ± 0.22` | `13.43 ± 0.26` |
+| `2` | `1/16` | `0.16 ± 1.28` | `12.74 ± 1.34` | `1.09 ± 0.86` | `13.35 ± 0.90` |
+| `3` | `1/8` | `2.24 ± 1.61` | `10.89 ± 1.75` | `0.84 ± 0.73` | `12.88 ± 0.87` |
+| `3` | `1/16` | `-1.95 ± 5.71` | `17.29 ± 6.16` | `2.64 ± 2.99` | `17.45 ± 3.15` |
+
+The certified `a_up` at `R = 1/4096` on the four corner boxes (`b`- and `k`-width `1/8`):
+
+| `d` | `(0, 1/2)` | `(0, 2)` | `(1, 1/2)` | `(1, 2)` |
+|---|---|---|---|---|
+| `2` | `4.818760` | `13.822900` | `3.918911` | `13.742115` |
+| `3` | `25.666574` | `16.876052` | `12.525134` | `14.762963` |
+
+What the comparison shows:
+- **Plane, `k = 2`.** The coefficient is `13.2`–`13.4`, and `a_up = 13.823` is attained here, about `4%` above it.
+  - Most of `a_up` is the exact pin-density limit `12.75 r^2` (`0.75` from `det`, `12` from `e_kk`).
+  - The `Z`-part is small and negative in the Monte Carlo (about `-0.7`); the certificate bounds it above by `1.07`.
+- **Plane, `k = 1/2`.** The coefficient is `0.1`–`1.1`, against box constants `3.9`–`4.8`.
+  - Here the `Z`-part is about `-0.8` to `-1.4`, while the certificate bounds it above by `2.0`–`2.9`.
+  - This region carries most of the weight of `c'`, so `c'_up` is several times the genuine weighted coefficient.
+- **`d = 3`.** At `k = 2` the coefficient is about `11`–`17`, against box constants `14.8`–`16.9`.
+  - At `(0, 1/2)` the coefficient is about `0`–`2`, against the box constant `25.67`, which is the global `a_up`.
+  - There `23.5` of that constant is the `Z`-part. It consists mostly of the Hölder bounds of `r^2 E[D xi_3 1]` (`13.9`) and
+    `r^2 E[(v' A* v)(e' A* e) 1]` (`8.7`), relative to `36 k^2 m_3`.
+
+**Reference-kernel cap-route benchmarks** (section 0). In floating point, `Gamma_{B,K} = |S^(d-1)| int A_0^ref c_G /
+(3 k^(5/3))` is:
+- `Gamma_2 = 307.51`, with the planar `c_G = R(b) J(k)` of [G] (`J(k) = 2359296 k^3 (1 + delta(k))`);
+- `Gamma_3 = 540.97`, with the `d = 3` ratios of [G3].
+
+So `C_{B,K}(1/4096)` is `1.31 Gamma_2` and `2.23 Gamma_3`. These are comparisons with the reference-kernel benchmark,
+not with the exact cap-route limit of a finite torus.
+
+## 10. What this does not do
+
+- **`d = 2, 3` and the band `B x K` only.**
+  - The method needs a compact `K` away from `0`; the shell bound uses `k >= 1/2`.
+  - The unrestricted densities of [LP] Theorem C are not addressed. There the constant term `B_{d,L}` (Math-#191, Math-#211)
+    comes from pairs this window excludes.
+- **No vanishing-remainder rate on the torus.** The remainders `eps_d` and `eps'_d` depend on neither `r` nor `ell`.
+  Theorem L and Corollary L1 are expansions about the reference contact integrand and constant. They give no rate for
+  `A_r -> A_0`, nor for `ell^(1/3) nu -> c_{B,K}`, on the torus.
+- **The Kac–Rice interface is consumed, not reviewed.** [LP] sections 8–12 are used as statements.
+- **The cap route only for `C_{B,K}`.**
+  - `C_{B,K}` bounds `nu_cand - nu_eld` through `1 - p_r <= Q^W(G_r^c)`.
+  - By [G] the cap criterion overstates the pairing failure by `7.4 x 10^4` at `k = 1/2`, so the true difference coefficient
+    is far smaller.
+  - No lower bound for it is given.
+- **Precision** (section 9).
+  - In the plane `a_up` is within about `4%` at `k = 2`, where it is attained. Near `k = 1/2`, where most of the weight of
+    `c'` lies, the box constants exceed the genuine coefficient by about `3`–`5`.
+  - In `d = 3`, `a_up` and `a_dn` are attained near `(b, k) = (0, 1/2)`, where the genuine coefficient is about `0`–`2`. The
+    excess comes from Hölder bounds of `r^2 E[D xi_3 1]` and `r^2 E[(v' A* v)(e' A* e) 1]`. These are large relative to
+    `36 k^2 m_3` at small `k`, and conditional moments would sharpen them further.
+  - At `k = 2` the `d = 3` box constants (`14.8`–`16.9`) are comparable to the Monte Carlo values (`11`–`17`).
+- **Not a review** of [LP], [CAP], [E] or [E3]. The engines are used as certified box-bound and law functions; their
+  correctness is the subject of Math-#206 and Math-#212.
+- **Status.** C8 stays OPEN.
+- **Independence.** Same GitHub account as every lane; zero organizational-independence credit. The author will not merge.
+
+## 11. Verification
+
+- `python3 -B -S theorem_b.py --check --procs N` replays exactly:
+  - the Taylor-model laws and their recorded coefficients;
+  - the rate sweep (`96` boxes, seven `R`, both dimensions);
+  - the reference integrals and the cross-checks against Math-#197 and Math-#205;
+  - a sample of the cap bands (the band ending at each `r_*`, at `2^-20`, and the last band);
+  - the assembled tables.
+- `--check` takes about 4 minutes on four cores.
+- `--check-full` replays all `274` cap bands, in about 28 minutes on four cores.
+- Mutants `no-om-term`, `no-bad-set`, `no-det`, `no-torus` and `cap-half` make the replay fail.
+- The workflow checks:
+  - the manifest;
+  - the main-resident pins ([LP], [CAP], [E1], [E2], [REC]);
+  - the engine copies against `SOURCE_MAP.json`;
+  - `--check-full` and `--check` in the two interpreter modes;
+  - the mutants;
+  - a clean tree.
+
+## 12. Revisions
+
+**v1.5 (documentation only).** It records review C61 and the corrected parent set:
+- [E1], [E2] and [REC] become `current_required` pins (§0), and §1 reads [LP] section 9 through [E2].
+- §7 records C61's disposition of `C_{B,K}`.
+- Two clarifications in `theorem_b.py`:
+  - `wq_bound`'s docstring keeps the nonnegative diagonal surplus `H` (C58);
+  - a comment at the `Fae` line says that `(3/2) sb >= 1` is all the variance multiplier needs, and that it is not a claim
+    `Phi <= 3/4`, which is false at `mu = 1, sigma^2 = 2` (C60).
+- The engine copies follow Math-#206 v1.2 and Math-#212 v1.1, which are docstring-only: `sqrt_up`'s bound is now `2^-198`
+  (C59-DOC-01), and `box_bound` now states its proved parameter domain `0 < x0 < 2/9`, `0 < theta < 1` (C61).
+
+In every changed Python file the AST with docstrings removed is unchanged, and `RESULTS.json` and every constant are unchanged.
+
+**v1.5.1 (wording; C63, 5943465973).** §7 is narrowed. C61's transfer covers the cap-derived candidate-minus-selected-elder
+error estimates. It covers neither Corollary L1's individual-density expansions nor Corollary L2's selected-count bounds,
+both of which also use `c'` and `eps'` from the full positive-radius Theorem L.
diff --git a/frontiers/c8_lifetime_difference_constant_20261001/README.md b/frontiers/c8_lifetime_difference_constant_20261001/README.md
new file mode 100644
index 000000000..d14fafdfa
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/README.md
@@ -0,0 +1,42 @@
+# Theorem B of [LP] with explicit constants (C8; `d = 2, 3`): the radial ledger to second order, two-sided lifetime densities, and `C_{B,K}`
+
+**Object:** `CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1`. **Author lane:** Anthropic / Claude. **Kind:** author-side theorem
+with explicit constants, certified by exact rational interval arithmetic (standard library only). **Scientific effect:** NONE: no
+register, catalog, GRAPH or STATUS change, and catalog entry C8 stays OPEN.
+
+**Setting.** The compact-window lifetime densities of [LP] Theorem B on the band `b in [0, 1]`, `k in [1/2, 2]`, uniformly over
+frames and torus sides `L >= 10` (and the reference kernel), through the radial ledger `A_r = 12 pi_r(R; v_r) Z_r / r^2` of [LP]
+(10.2) and the pushforward (11.2), which are consumed as statements.
+
+**Results** (`NOTE.md` Theorem L and Corollaries L1, L2):
+- **The ledger to second order.** For `0 < r <= R`, `|A_r / A_0^ref - 1| <= a(R) r^2 + eps` with the closed-form contact
+  integrand `A_0^ref`; `eps` is a radius-independent remainder (torus, truncation and rounding), so this is not a rate for
+  `A_r -> A_0` on the torus. In the plane `a_up(1/4096) = 13.823`, `a_dn = 3.638`, `eps <= 4.0 x 10^-10`; in `d = 3`
+  `a_up(1/4096) = 25.67`, `a_dn = 25.52`, `eps <= 1.8 x 10^-8`. [LP] assigns no rate to `A_r -> A_0`.
+- **Densities to second order.** For `ell < min(r_pop, R)^3 / 2`,
+  `|ell^(1/3) nu_cand(ell) / c^ref_{B,K} - 1| <= 6.29 ell^(2/3) + 5 x 10^-11` in the plane and
+  `<= 23.2 ell^(2/3) + 3 x 10^-9` in `d = 3` (`R = 1/4096` constants). The same holds for `nu_eld` up to
+  `C_{B,K} ell / c^ref_{B,K}`. So both densities are `c^ref_{B,K} ell^(-1/3) + O(ell^(1/3))` up to an additive
+  `eps'_d ell^(-1/3)` that does not vanish as `ell -> 0` (`eps'_d <= 1.2 x 10^-13` in the plane, `2.5 x 10^-12` in
+  `d = 3`). For the torus constant `c_{B,K}` this gives `|c_{B,K} - c^ref_{B,K}| <= eps'_d`, not a vanishing-remainder
+  expansion.
+- **The difference constant.** `0 <= nu_cand - nu_eld <= C_{B,K}(r_*) ell^(2/3)` for `ell < min(r_pop, r_*)^3 / 2`.
+  - `C_{B,K}(1/4096) = 402.53` in the plane and `1205.71` in `d = 3`.
+  - At `r_* = 1/512` the values are `428.02` and `2488.07` (rounded up).
+  - With these, (12.1) holds with `(3/5) C_{B,K}` and (12.3) with `C_{B,K} / (q + 5/3)`.
+  - [LP] gives (1.2) with no value.
+- `c^ref_{B,K}` is certified in both dimensions; the enclosures contain the values of Math-#197 and Math-#205.
+
+**Files.**
+- `NOTE.md`: statement; the ledger and what is consumed; Lemma 1 (for the Gaussian kernel the transverse Hessian at `M` is
+  exactly contact-distributed at every `r`); the Taylor-model laws; the pin-density ratio; the exact expansion of `P1 P2` and
+  the first-order cancellation; the thin typed shell; reference integrals; the cap sweep; controls; non-claims.
+- `theorem_b.py`: the certificate. `--check --procs N` replays the Taylor models, the rate sweep, the reference integrals, a
+  sample of cap bands and the tables exactly against `RESULTS.json`; `--check-full` replays every cap band;
+  `--mutant {no-om-term, no-bad-set, no-det, no-torus, cap-half}` must fail; `--controls` prints floating controls.
+- `engine_e2.py`, `engine_e3.py`: byte-identical copies of the certificates of Math-#206 (`0cb8ad3`) and Math-#212
+  (`c4b8ec3`), used as libraries (band laws, contact moments, box bounds); pinned in `SOURCE_MAP.json`.
+- `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`.
+- Workflow `.github/workflows/c8-lifetime-difference-constant.yml`.
+
+Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge.
diff --git a/frontiers/c8_lifetime_difference_constant_20261001/RESULTS.json b/frontiers/c8_lifetime_difference_constant_20261001/RESULTS.json
new file mode 100644
index 000000000..894fae2e4
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/RESULTS.json
@@ -0,0 +1,10501 @@
+{
+ "certified": true,
+ "dims": {
+  "2": {
+   "cap_boxes": [
+    [
+     "0",
+     "1/4",
+     "1/2",
+     "17/32"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "1/2",
+     "17/32"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "1/2",
+     "17/32"
+    ],
+    [
+     "3/4",
+     "1",
+     "1/2",
+     "17/32"
+    ],
+    [
+     "0",
+     "1/4",
+     "17/32",
+     "9/16"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "17/32",
+     "9/16"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "17/32",
+     "9/16"
+    ],
+    [
+     "3/4",
+     "1",
+     "17/32",
+     "9/16"
+    ],
+    [
+     "0",
+     "1/4",
+     "9/16",
+     "19/32"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "9/16",
+     "19/32"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "9/16",
+     "19/32"
+    ],
+    [
+     "3/4",
+     "1",
+     "9/16",
+     "19/32"
+    ],
+    [
+     "0",
+     "1/4",
+     "19/32",
+     "5/8"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "19/32",
+     "5/8"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "19/32",
+     "5/8"
+    ],
+    [
+     "3/4",
+     "1",
+     "19/32",
+     "5/8"
+    ],
+    [
+     "0",
+     "1/4",
+     "5/8",
+     "21/32"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "5/8",
+     "21/32"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "5/8",
+     "21/32"
+    ],
+    [
+     "3/4",
+     "1",
+     "5/8",
+     "21/32"
+    ],
+    [
+     "0",
+     "1/4",
+     "21/32",
+     "11/16"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "21/32",
+     "11/16"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "21/32",
+     "11/16"
+    ],
+    [
+     "3/4",
+     "1",
+     "21/32",
+     "11/16"
+    ],
+    [
+     "0",
+     "1/4",
+     "11/16",
+     "23/32"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "11/16",
+     "23/32"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "11/16",
+     "23/32"
+    ],
+    [
+     "3/4",
+     "1",
+     "11/16",
+     "23/32"
+    ],
+    [
+     "0",
+     "1/4",
+     "23/32",
+     "3/4"
+    ],
+    [
+     "1/4",
+     "1/2",
+     "23/32",
+     "3/4"
+    ],
+    [
+     "1/2",
+     "3/4",
+     "23/32",
+     "3/4"
+    ],
+    [
+     "3/4",
+     "1",
+     "23/32",
+     "3/4"
+    ],
+    [
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diff --git a/frontiers/c8_lifetime_difference_constant_20261001/SOURCE_FILES.json b/frontiers/c8_lifetime_difference_constant_20261001/SOURCE_FILES.json
new file mode 100644
index 000000000..b34dd8cf2
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/SOURCE_FILES.json
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+ "review_status": "AUTHOR_SIDE_THEOREM_WITH_CERTIFIED_CONSTANTS",
+ "object": "CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1",
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+   "head": "d748c2abee2cbfbad6504b5700c70bbd83ba5785",
+   "scope": "ACCEPT the d748c2a amendment (additive-remainder scope only); requested the reference-kernel benchmark wording applied in v1.2"
+  }
+ ]
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diff --git a/frontiers/c8_lifetime_difference_constant_20261001/SOURCE_MAP.json b/frontiers/c8_lifetime_difference_constant_20261001/SOURCE_MAP.json
new file mode 100644
index 000000000..2118c7ae5
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/SOURCE_MAP.json
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+{
+ "schema": 1,
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+ "source_repository": "d6g8k5htny-coder/Math-",
+ "main_at_creation": "3e0a91b69892b9717564c592e7a966492a81566a",
+ "note": "current_required sources ([LP], [CAP], the mandatory parent amendments [E1], [E2], [REC], and the read-only comparison [G3]) are verified on the tree by the workflow; packet_copy sources ([E], [E3]) are verified against the packet's byte-identical engine copies; the remaining sources ([W], [W3], [G]) are read-only cross-checks and comparisons pinned for provenance",
+ "sources": [
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+   "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
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+   "current_required": true,
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+  },
+  {
+   "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
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+   "on_main": true,
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+  },
+  {
+   "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md",
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+   "role": "mandatory erratum to [LP] (corrected parent reading set P + CAP + E1 + E2 + REC, per REC section 1): the preconditioning congruence is diag(r^(-1/2), I)"
+  },
+  {
+   "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md",
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+   "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449",
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+   "role": "mandatory repair of [LP] section 9: the marked Kac-Rice identity for the Borel (not necessarily continuous) elder indicator, by equality of finite measures on a generating cylinder algebra and a monotone-class extension; it replaces LP section 9 wherever sections 9-11 are consumed"
+  },
+  {
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+  },
+  {
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+  },
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+   "id": "W3",
+   "commit": "c0d8e925955d7a23c58221006510f804ef3631fe",
+   "pull_request": 205,
+   "current_required": false,
+   "on_main": false,
+   "role": "read-only cross-check: the reference d = 3 c_{B,K} = 0.00097438236602425 on B x K; Math-#205 has since merged at 044d42d with a wording-only NOTE revision (blob 549dc7cc), and the cross-check value is unchanged"
+  },
+  {
+   "path": "frontiers/c8_cap_failure_coefficient_planar_20260930/NOTE.md",
+   "bytes": 28432,
+   "sha256": "0cb75475d9ee387a4045ff01ab9edb43a73c8c4edb6fc78d30518ced5bb02c5a",
+   "git_blob": "0b64f1c2828dfae7e6c104bac044004fc72c221d",
+   "id": "G",
+   "commit": "a388580eee356a88ce31aeed3999ab3fdc302312",
+   "pull_request": 203,
+   "current_required": false,
+   "on_main": false,
+   "role": "read-only comparison: c_G = R(b) J(k) with J(k) = 2359296 k^3 (1 + delta(k)), for the floating reference-kernel cap-route benchmark of C_{B,K}"
+  },
+  {
+   "path": "frontiers/c8_cap_failure_coefficient_dims_20261001/RESULTS.json",
+   "bytes": 6837,
+   "sha256": "1406d5138ac3d33bcb0358f6a3005a67e3a4a2dbfc2c43bc3ac2593667550608",
+   "git_blob": "e9fd6f377d1fd9dca9d09cc884d2bad2422e8b17",
+   "id": "G3",
+   "commit": "da483732664d128728950dcad6d4fdbe5181f7cd",
+   "pull_request": 208,
+   "current_required": true,
+   "on_main": true,
+   "role": "read-only comparison: the floating R_3(b) R(b) values and the J^(3)/J ratios, for the floating d = 3 reference-kernel cap-route benchmark; Math-#208 merged at b55b8f3 with this RESULTS.json byte-identical, so the pin is checked on the tree"
+  }
+ ]
+}
diff --git a/frontiers/c8_lifetime_difference_constant_20261001/engine_e2.py b/frontiers/c8_lifetime_difference_constant_20261001/engine_e2.py
new file mode 100644
index 000000000..ae279e901
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/engine_e2.py
@@ -0,0 +1,1249 @@
+#!/usr/bin/env python3
+"""Explicit constants for Theorem A of [LP] in the plane (d = 2), for the torus kernel K_L with L >= 10 in every frame:
+for 0 < r <= r_*, b in [0, 1], k in [1/2, 2],
+
+    1 - p_r(b, k, R)  <=  Q^W(G_r^c)  <=  C(r_*) r^3,
+
+with G_r the good event of [LP] section 7 / [CAP] (1) and C(r_*) certified here (NOTE.md Theorem E); the same at the single
+fixed-axis SIDE24 point (b, k) = (6/5, 1/6) of [CAP] section 1 (Theorem E').  The proof bounds
+Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0:
+
+  * covariances of the preconditioned pin and target functionals are exact power series in r, obtained from the Taylor
+    coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded
+    uniformly in r and in the frame by Cauchy estimates on a polydisc;
+  * the failure event is reduced to {lambda <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded through exact averaging
+    identities of the pinned jets and M_4 through the Taylor series at the midpoint;
+  * the main term is integrated in lambda = -f_yy(M) against its Gaussian density after an exact regression on w = f_yy(M);
+    the remaining expectations are closed-form incomplete Gaussian moments plus Hoelder-controlled perturbations;
+  * the normalizer Z_r is bounded below in closed form; the rare branches are Gaussian tails.
+All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only.
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term')
+MUT = None
+PBITS = 160
+L_MIN = 10
+NX = 60                    # Taylor order in x of the covariance series (exact); the rest is a bounded tail
+NB = 24                    # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded)
+X0 = Fr(1, 64)             # default threshold for the quadratic near-branch analysis (records use COMBOS)
+THETA = Fr(1, 32)          # default splitting parameter in the max-moment lemma (records use COMBOS)
+THETA_E = Fr(1, 16)        # splitting parameter for the (r nu + v)^2 factor
+QS = (2, 4, 8, 16)         # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+BAND_B = (Fr(0), Fr(1))
+BAND_K = (Fr(1, 2), Fr(2))
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+
+
+def phi_pt(x):
+    """phi(x) for rational x (interval)."""
+    x = Fr(x)
+    return exp_neg_point(x * x / 2) / SQRT2PI
+
+
+def Phibar_pt(x):
+    """1 - Phi(x) for rational x (interval)."""
+    p = phi_point(Fr(x)) if abs(Fr(x)) <= 12 else None
+    if p is None:
+        if x > 0:
+            # Mills: Phibar(x) <= phi(x)/x
+            return IV(0, (phi_pt(x) / x).hi)
+        return IV(1 - (phi_pt(-x) / (-x)).hi, 1)
+    return IV(1 - p.hi, 1 - p.lo)
+
+
+def upper_moments(x, J):
+    """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x >= 0 (intervals)."""
+    x = Fr(x)
+    require(x >= 0, 'upper moment at negative point')
+    ph = phi_pt(x)
+    M = [Phibar_pt(x), ph]
+    for j in range(2, J + 1):
+        M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1))
+    return M
+
+
+def dfact(n):
+    """(n)!! for n >= -1."""
+    return math.prod(range(n, 0, -2)) if n > 0 else 1
+
+
+def rup_rel(x, bits=200):
+    """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds)."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh)
+    return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh)
+
+
+def sqrt_up(x):
+    """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root
+    step adds less than 2^-209 and rup_rel(., 200) less than 2^-199.  (The earlier '2^-200' was too sharp: at x = 49/100 the
+    excess is (8/7) 2^-200, review C59-DOC-01.)"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m))
+
+
+def iroot(x, p):
+    """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two."""
+    v = Fr(x)
+    while p > 1:
+        v = sqrt_up(v)
+        p //= 2
+    return v
+
+
+def pow2(p):
+    q = 1
+    while q < p:
+        q *= 2
+    return q
+
+
+ZN = {}
+
+
+def znorm(p):
+    """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5)."""
+    if p == 1:
+        return Fr(4, 5)
+    if p not in ZN:
+        ZN[p] = iroot(Fr(dfact(p - 1)), p)
+    return ZN[p]
+
+
+def habs_moment(j):
+    """E|zeta|^j (interval)."""
+    if j % 2 == 0:
+        return IV(dfact(j - 1))
+    return SQRT2 / isqrt_iv(PI) * (2 ** ((j - 1) // 2) * math.factorial((j - 1) // 2))
+
+
+# ----------------------------------------------------------------------------- functionals and exact covariance series
+HALF = Fr(1, 2)
+XM, XS, X0PT = -HALF, HALF, Fr(0)
+
+
+def term(c, p, a, b, x0):
+    """c r^p d_x^a d_y^b f(x0 r, 0)."""
+    return (Fr(c), p, a, b, Fr(x0))
+
+
+def functionals():
+    pins = {
+        'p1': [term(HALF, 0, 0, 0, XM), term(HALF, 0, 0, 0, XS)],
+        'p2': [term(-1, -1, 0, 0, XM), term(1, -1, 0, 0, XS)],
+        'p3p': [term(6, -2, 1, 0, XM), term(6, -2, 1, 0, XS), term(12, -3, 0, 0, XM), term(-12, -3, 0, 0, XS)],
+        'p4': [term(-1, -1, 1, 0, XM), term(1, -1, 1, 0, XS)],
+        'q1': [term(HALF, 0, 0, 1, XM), term(HALF, 0, 0, 1, XS)],
+        'q2': [term(-1, -1, 0, 1, XM), term(1, -1, 0, 1, XS)],
+    }
+    targets = {
+        'w': [term(1, 0, 0, 2, XM)],
+        'v': [term(1, -1, 1, 1, XM), term(1, -2, 0, 1, XM), term(-1, -2, 0, 1, XS)],
+        'om': [term(1, -1, 0, 2, XS), term(-1, -1, 0, 2, XM)],
+        'Y': [term(1, 0, 0, 3, X0PT)],
+        'T': [term(1, -2, 2, 0, XM), term(4, -3, 1, 0, XM), term(2, -3, 1, 0, XS), term(6, -4, 0, 0, XM), term(-6, -4, 0, 0, XS)],
+        'tau': [term(1, -3, 2, 0, XS), term(-1, -3, 2, 0, XM), term(-6, -4, 1, 0, XM), term(-6, -4, 1, 0, XS),
+                term(-12, -5, 0, 0, XM), term(12, -5, 0, 0, XS)],
+        'nu': [term(1, -2, 1, 1, XM), term(1, -2, 1, 1, XS), term(2, -3, 0, 1, XM), term(-2, -3, 0, 1, XS)],
+        'X40': [term(1, 0, 4, 0, X0PT)], 'X31': [term(1, 0, 3, 1, X0PT)], 'X22': [term(1, 0, 2, 2, X0PT)],
+        'X13': [term(1, 0, 1, 3, X0PT)], 'X04': [term(1, 0, 0, 4, X0PT)],
+    }
+    return pins, targets
+
+
+PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q1', 'q2')
+TARGET_NAMES = ('w', 'v', 'om', 'Y', 'T', 'tau', 'nu', 'X40', 'X31', 'X22', 'X13', 'X04')
+JET4 = ('X40', 'X31', 'X22', 'X13', 'X04')
+JET4_IDX = {'X40': (4, 0), 'X31': (3, 1), 'X22': (2, 2), 'X13': (1, 3), 'X04': (0, 4)}
+
+
+def weight(F):
+    """Scaling weight a - (power of r), common to all terms of a preconditioned functional."""
+    ws = {t[2] - t[1] for t in F}
+    require(len(ws) == 1, 'functional not homogeneous')
+    return ws.pop()
+
+
+def hcoef(n):
+    """d^n/dx^n exp(-x^2/2) at 0."""
+    if n % 2:
+        return 0
+    return (-1) ** (n // 2) * dfact(n - 1)
+
+
+PAIR_CACHE = {}
+
+
+def pair_series(F, G, key):
+    """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG):
+    returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', b+b') for the remainder bounds)."""
+    if key in PAIR_CACHE:
+        return PAIR_CACHE[key]
+    wF, wG = weight(F), weight(G)
+    w = wF + wG
+    pi = {}
+    absterms = []
+    for (cf, pf, af, bf, xf) in F:
+        for (cg, pg, ag, bg, xg) in G:
+            gy = bf + bg
+            sgn = (-1) ** (ag + bg)
+            d = xf - xg
+            absterms.append((abs(cf * cg), af + ag, gy))
+            for gx in range(af + ag, NX + 1):
+                m = gx - af - ag
+                val = cf * cg * sgn * d ** m / math.factorial(m)
+                if val != 0:
+                    pi[(gx, gy)] = pi.get((gx, gy), Fr(0)) + val
+    for (gx, gy), v in pi.items():
+        require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)')
+    ser = {}
+    for (gx, gy), v in pi.items():
+        kap = hcoef(gx) * hcoef(gy)
+        if kap and v:
+            ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v
+    out = ({p: c for p, c in ser.items() if c != 0}, w, absterms)
+    PAIR_CACHE[key] = out
+    return out
+
+
+def torus_M1(L):
+    """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2| <= 1 (every frame, |p_n| = |n| L):
+    exp(1) * sum_{j >= 1} 8 j exp(-(j L - 3/2)^2 / 2)."""
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += 8 * j * exp_neg_point((j * L - Fr(3, 2)) ** 2 / 2).hi
+    tail = 8 * 7 * exp_neg_point((6 * L - Fr(3, 2)) ** 2 / 2).hi * 4      # j >= 6, very crude
+    return Fr(27183, 10000) * (tot + tail)
+
+
+def theta_minus_one(L):
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += 8 * j * exp_neg_point((j * L) ** 2 / 2).hi
+    return tot + 8 * 7 * exp_neg_point((6 * L) ** 2 / 2).hi * 4
+
+
+def cov_band(F, G, key, r0, r1, M1):
+    """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1]."""
+    ser, w, absterms = pair_series(F, G, key)
+    val = IV(0)
+    for p, c in ser.items():
+        val = val + IV(r0 ** p, r1 ** p) * c
+    A = max((t[1] for t in absterms), default=0)
+    # Taylor truncation tail of the reference kernel (orders gx > NX)
+    tail = Fr(0)
+    for (acc, aa, gy) in absterms:
+        tail += acc * abs(hcoef(gy))
+    tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w)
+    # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy! M1 on the unit polydisc
+    tor = Fr(0)
+    if True:
+        for (acc, aa, gy) in absterms:
+            tor += acc * math.factorial(gy) * Fr(max(w, 1)) ** aa
+        tor = tor * M1 / (1 - Fr(27183, 10000) * r1)
+    err = tail + tor
+    return val + IV(-err, err)
+
+
+# ----------------------------------------------------------------------------- interval linear algebra
+def solve_iv(A, B):
+    """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c]."""
+    n = len(A)
+    M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    for col in range(n):
+        p = max(range(col, n), key=lambda i: abs(M[i][col].mid()))
+        M[col], M[p] = M[p], M[col]
+        piv = M[col][col]
+        for i in range(col + 1, n):
+            f = M[i][col] / piv
+            for j in range(col, len(M[i])):
+                M[i][j] = M[i][j] - f * M[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            s = M[i][n + c]
+            for j in range(i + 1, n):
+                s = s - M[i][j] * x[j]
+            x[i] = s / M[i][i]
+        X.append(x)
+    return X
+
+
+def cholesky3(C):
+    """Interval Cholesky of a 3 x 3 SPD interval matrix."""
+    L = [[IV(0)] * 3 for _ in range(3)]
+    for i in range(3):
+        for j in range(i + 1):
+            s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0))
+            if i == j:
+                require(s.lo > 0, 'Cholesky pivot not positive')
+                L[i][j] = isqrt_iv(s)
+            else:
+                L[i][j] = s / L[j][j]
+    return L
+
+
+def absup(x):
+    return max(abs(x.lo), abs(x.hi))
+
+
+# ----------------------------------------------------------------------------- the conditional law on an r-band
+class Law:
+    pass
+
+
+def band_law(r0, r1, L=L_MIN):
+    """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the
+    targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients."""
+    pins, targets = functionals()
+    P = [pins[n] for n in PIN_NAMES]
+    T = [targets[n] for n in TARGET_NAMES]
+    M1 = torus_M1(L)
+    th1 = theta_minus_one(L)
+    nt = len(T)
+    Spp = [[cov_band(P[i], P[j], ('p', i, j), r0, r1, M1) for j in range(6)] for i in range(6)]
+    Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(6)] for i in range(nt)]
+    Stt = [[cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1) for j in range(nt)] for i in range(nt)]
+    r = IV(r0, r1)
+    kvec = [-(r * r * r) * HALF, -(r * r), IV(12), IV(0), IV(0), IV(0)]
+    e1 = [IV(1), IV(0), IV(0), IV(0), IV(0), IV(0)]
+    W = solve_iv(Spp, Stp)
+    xs = solve_iv(Spp, [e1, kvec])
+    law = Law()
+    law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1
+    law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)}
+    law.beta = {n: sum((W[i][j] * kvec[j] for j in range(6)), IV(0)) for i, n in enumerate(TARGET_NAMES)}
+    scale = IV(1 / (1 + th1), 1)          # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1]
+    C = {}
+    for i, a in enumerate(TARGET_NAMES):
+        for j, c in enumerate(TARGET_NAMES):
+            if j < i:
+                continue
+            v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(6)), IV(0))
+            if i != j:
+                v2 = Stt[j][i] - sum((W[j][l] * Stp[i][l] for l in range(6)), IV(0))
+                v = v.meet(v2)
+            C[(a, c)] = C[(c, a)] = v * scale
+    law.C = C
+    escale = IV(1, 1 + th1)
+    law.e_bb = xs[0][0] * escale
+    law.e_bk = sum((kvec[j] * xs[0][j] for j in range(6)), IV(0)) * escale
+    law.e_kk = sum((kvec[j] * xs[1][j] for j in range(6)), IV(0)) * escale
+    # regression on w and the g-parts
+    sw2 = C[('w', 'w')]
+    require(sw2.lo > 0, 'Var w not positive')
+    law.sw2 = sw2
+    law.sw = isqrt_iv(sw2)
+    law.bw = {n: C[(n, 'w')] / sw2 for n in TARGET_NAMES if n != 'w'}
+    Cg = {}
+    for a in TARGET_NAMES:
+        for c in TARGET_NAMES:
+            if a != 'w' and c != 'w':
+                Cg[(a, c)] = C[(a, c)] - C[(a, 'w')] * C[(c, 'w')] / sw2
+    law.Cg = Cg
+    law.L3 = cholesky3([[Cg[(a, c)] for c in ('v', 'om', 'Y')] for a in ('v', 'om', 'Y')])
+    law.sdg = {n: isqrt_iv(IV(max(Cg[(n, n)].lo, Fr(0)), Cg[(n, n)].hi)) if Cg[(n, n)].lo > 0 else IV(0, isqrt_iv(IV(Cg[(n, n)].hi)).hi)
+               for n in TARGET_NAMES if n != 'w'}
+    law.sdQ = {n: isqrt_iv(C[(n, n)]) for n in TARGET_NAMES}
+    # crude higher jets: Taylor-tail coefficients S_alpha(r1) = sum_{beta != 0} sd0_{alpha+beta} (2 r1)^{|beta|} / beta!
+    law.S = {n: taylor_tail_coefficient(JET4_IDX[n], r1, M1) for n in JET4}
+    return law
+
+
+def sd0_bound(gx, gy, M1):
+    """Upper bound of sqrt(Var d^gamma f(0)) = sqrt(|d^{2 gamma} K_L(0)|) (normalized, every frame, L >= L_MIN)."""
+    v = dfact(2 * gx - 1) * dfact(2 * gy - 1) + math.factorial(2 * gx) * math.factorial(2 * gy) * M1
+    return isqrt_iv(IV(v)).hi
+
+
+def taylor_tail_coefficient(alpha, r1, M1):
+    ax, ay = alpha
+    tot = Fr(0)
+    for n in range(1, NB + 1):
+        for bx in range(n + 1):
+            by = n - bx
+            tot += sd0_bound(ax + bx, ay + by, M1) * (2 * r1) ** n / (math.factorial(bx) * math.factorial(by))
+    # tail |beta| > NB: sd0_gamma <= 2^|gamma| gx! gy! sqrt(1 + M1) (binom(2m, m) <= 4^m), so a term is at most
+    # 2^|alpha| (4 + n)^4 (4 r1)^n 2 per multi-index; n + 1 multi-indices of order n
+    q = 4 * r1
+    require(q <= Fr(1, 16), 'Taylor tail ratio too large')
+    tail = Fr(0)
+    base = 2 ** (ax + ay) * 2
+    n = NB + 1
+    # sum_{n > NB} (n + 1)(n + 4)^8 q^n <= (NB + 2)(NB + 5)^8 q^(NB+1) / (1 - 2 q)   (consecutive ratio <= 2q for these n)
+    tail = base * (NB + 2) * Fr(NB + 5) ** 8 * q ** (NB + 1) / (1 - 2 * q)
+    return tot + tail
+
+
+# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k)
+def binom(n, k):
+    return math.comb(n, k)
+
+
+def gauss_norm(mu_abs, sd, p):
+    """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd."""
+    return rup_rel(mu_abs + sd * znorm(pow2(p)))
+
+
+def pos_part_moments(s, c, nmax, extra_max):
+    """P[n][e] = upper bound of E[u^e (u^n - c^n)_+], u = s |zeta|, for 1 <= n <= nmax, 0 <= e <= extra_max."""
+    x = c / s
+    M = upper_moments(x, nmax + extra_max)
+    out = {}
+    for n in range(1, nmax + 1):
+        for e in range(extra_max + 1):
+            out[(n, e)] = rup_rel(((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e) * 2).hi)
+    return out
+
+
+def normalized_moments(q, nmax, theta):
+    """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax:
+      E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmu1/K + (r1/K) Delta1;
+      Tn[n], To[n], Tv[n]   : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2),
+                              y_i <= u_i + delta_i (u_i = s_i |zeta| independent half-normals), threshold c = q['c'].
+    o0 = s3 |z2| and v0^2 = s2^2 z1^2 / 4 are unnormalized."""
+    th = theta
+    cp = q['c'] * (1 - th)                  # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli)
+    PP = {i: pos_part_moments(q['s'][i], cp, nmax, 2) for i in (2, 3, 4)}
+    Eu3 = (habs_moment(1) * q['s3']).hi
+    Eu2sq = (habs_moment(2) * q['s2'] * q['s2']).hi
+    n_o0_2 = q['s3'] * znorm(2)
+    n_u2sq_2 = q['s2'] * q['s2'] * znorm(4) ** 2
+    r3 = q['s3'] / q['s'][3]
+    r2sq = (q['s2'] / q['s'][2]) ** 2
+    E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {}
+    base = 1 + q['cmt1']
+    for n in range(0, nmax + 1):
+        ey = base ** n
+        eo = base ** n * Eu3
+        ev = base ** n * Eu2sq
+        for j in range(1, n + 1):
+            coef = binom(n, j) * base ** (n - j) * q['rK'] ** j
+            ey += coef * q['nD1'](pow2(j)) ** j
+            eo += coef * n_o0_2 * q['nD1'](pow2(2 * j)) ** j
+            ev += coef * n_u2sq_2 * q['nD1'](pow2(2 * j)) ** j
+        E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev / 4)
+        if n == 0:
+            continue
+        f1 = (1 + th) ** (n - 1)
+        f2 = (1 + 1 / th) ** (n - 1)
+        tn = to = tv = Fr(0)
+        for i in (2, 3, 4):
+            P = PP[i]
+            dn = q['nd'](i, pow2(n)) ** n
+            dn2 = q['nd'](i, pow2(2 * n)) ** n
+            tn += f1 * P[(n, 0)] + f2 * dn
+            to += (f1 * r3 * P[(n, 1)] if i == 3 else f1 * Eu3 * P[(n, 0)]) + f2 * n_o0_2 * dn2
+            tv += (f1 * r2sq * P[(n, 2)] if i == 2 else f1 * Eu2sq * P[(n, 0)]) + f2 * n_u2sq_2 * dn2
+        Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv / 4)
+    return E1, E1o, E1v, Tn, To, Tv
+
+
+def elam_pos(m, s):
+    """E[(s z + m)_+] at a point (rational m, s > 0)."""
+    x = Fr(m) / s
+    return m * Phi(IV(x)) + phi_pt(x) * s
+
+
+def elam2_pos(m, s):
+    """E[(s z + m)_+^2] at a point."""
+    x = Fr(m) / s
+    return (m * m + s * s) * Phi(IV(x)) + phi_pt(x) * (m * s)
+
+
+def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None):
+    """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch
+    threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5).  The bound is proved for 0 < x0 < 2/9 and
+    0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and
+    (1 + kappa x0) c^2 <= 1.  No claim is made outside that domain.  Both COMBOS pairs, and the (1/64, 1/32) of Math-#215's
+    caller, lie inside it (review C61)."""
+    x0 = X0 if x0 is None else Fr(x0)
+    theta = THETA if theta is None else Fr(theta)
+    b = IV(bb[0], bb[1])
+    k = IV(kb[0], kb[1])
+    klo, khi = kb[0], kb[1]
+    Klo = 12 * klo
+    mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES}
+    mabs = {n: absup(mu[n]) for n in TARGET_NAMES}
+    E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk
+    require(E.hi > 0, 'energy')
+    sqE = isqrt_iv(IV(E.hi)).hi
+    sw_lo, sw_hi = law.sw.lo, law.sw.hi
+    muw_abs = mabs['w']
+    bw = {n: absup(law.bw[n]) for n in law.bw}
+    t = {n: law.S[n] / sw_lo for n in JET4}
+    a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1)      # the cap constant 4/(3k) (mutant: halved)
+    B = lambda n: bw[n] + t[n]
+    bbars = {1: r1 * (Fr(5, 2) * B('X40') + 2 * B('X31')),
+             2: 2 * bw['v'] + r1 * (Fr(5, 2) * B('X31') + 2 * B('X22')),
+             3: bw['om'] + r1 * (Fr(5, 2) * B('X22') + 2 * B('X13')),
+             4: bw['Y'] + r1 * (2 * B('X13') + 2 * B('X04'))}
+    bbars = {i: rup_rel(v) for i, v in bbars.items()}
+    bbar = max(bbars.values())
+    L3 = law.L3
+    s2, s3, s4 = 2 * L3[0][0].hi, L3[1][1].hi, L3[2][2].hi
+    l21, l31, l32 = absup(L3[1][0]), absup(L3[2][0]), absup(L3[2][1])
+
+    memo = {}
+
+    def nA(n, p):     # ||A_alpha^g||_p, A_alpha^g = |X_alpha,g| + T_alpha^g
+        key = ('A', n, p)
+        if key not in memo:
+            memo[key] = rup_rel(gauss_norm(mabs[n], law.sdg[n].hi, p) + law.S[n] * (sqE + znorm(pow2(p))))
+        return memo[key]
+
+    def nDelta(i, p):
+        key = ('D', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nDelta0(i, p))
+        return memo[key]
+
+    def nDelta0(i, p):
+        if MUT == 'no-delta':
+            return Fr(0)
+        if i == 1:
+            return Fr(5, 2) * nA('X40', p) + 2 * nA('X31', p)
+        if i == 2:
+            return Fr(5, 2) * nA('X31', p) + 2 * nA('X22', p)
+        if i == 3:
+            return Fr(5, 2) * nA('X22', p) + 2 * nA('X13', p)
+        return 2 * nA('X13', p) + 2 * nA('X04', p)
+
+    def nd(i, p):
+        key = ('d', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nd0(i, p))
+        return memo[key]
+
+    def nd0(i, p):     # ||delta_i||_p / Klo, delta_i including beta_i |mu_w|
+        zp = znorm(pow2(p))
+        cm = bbars[i] * muw_abs
+        if i == 2:
+            v = 2 * mabs['v'] + cm + r1 * nDelta(2, p)
+        elif i == 3:
+            v = mabs['om'] + l21 * zp + cm + r1 * nDelta(3, p)
+        else:
+            v = mabs['Y'] + (l31 + l32) * zp + cm + r1 * nDelta(4, p)
+        return v / Klo
+
+    kappa = 3 / (1 - 3 * x0)
+    Fk = 1 + kappa * x0                      # (1 + kappa x_i) <= Fk on the event {x_i <= x0}
+    cth = 1 - kappa * x0 / 2                 # <= Fk^{-1/2}
+    NMAX = 24
+    q = {'cmt1': bbars[1] * muw_abs / Klo, 'rK': r1 / Klo, 'nD1': lambda p: nDelta(1, p),
+         's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth}
+    E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta)
+    et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac)     # eta_1 K, k-free
+    aK2_hi = 192 * khi * a_fac                    # a K^2 = 192 k
+
+    # moments of Ut = U / (a K^2) <= max(Z1, Fk y_i^2), Z1 = y1^2 (1 + et1 y1)
+    def EUt(p):                                   # upper bound of E[Ut^p]
+        key = ('U', p)
+        if key not in memo:
+            memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p])
+        return memo[key]
+
+    def EoUt2():
+        return sum(binom(2, l) * et1 ** l * E1o[4 + l] for l in range(3)) + Fk ** 2 * To[4]
+
+    def EvUt2():
+        return sum(binom(2, l) * et1 ** l * E1v[4 + l] for l in range(3)) + Fk ** 2 * Tv[4]
+
+    def sq(x):
+        return isqrt_iv(IV(x)).hi
+
+    U3_2 = sq(EUt(6))                              # ||Ut^3||_2
+    U2_2 = sq(EUt(4))                              # ||Ut^2||_2
+
+    def nU(p):                                     # ||U||_p <= a K^2 Fk ||max_i y_i||_2p^2 (on the event x_i <= X0)
+        require(2 * p <= NMAX and pow2(p) == p, 'norm order')
+        key = ('nU', p)
+        if key not in memo:
+            memo[key] = rup_rel(aK2_hi * Fk * iroot(E1[2 * p] + Tn[2 * p], p))
+        return memo[key]
+
+    def nbar(n, p):                                # ||F_bar||_p, F_bar = |F_g| + |beta_F| (|mu_w| + r1 U)
+        return gauss_norm(mabs[n], law.sdg[n].hi, p) + bw[n] * (muw_abs + r1 * nU(p))
+
+    six_klo = 6 * klo
+    nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4)
+    epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2
+    nT8, nt8 = nbar('T', 8), nbar('tau', 8)
+    epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2
+    # Term A / (36 k^2)
+    A_n = EUt(3) + epsc2 * U3_2
+    termA = aK2_hi ** 3 / 3 * A_n
+    # Term B / (36 k^2)
+    ndo = lambda p: l21 * znorm(pow2(p)) + mabs['om'] + bw['om'] * (muw_abs + r1 * nU(p))
+    no4 = s3 * znorm(4) + ndo(4)
+    B_n = EoUt2() + (ndo(2) + epsc4 * no4) * U2_2
+    termB = aK2_hi ** 2 / 2 * B_n
+    if MUT == 'no-om-term':
+        termB = Fr(0)
+    # Term C / (36 k^2) = (a K^2)^2 / (12 k) * E[(1 + r T/(6k)) e Ut^2]
+    th = THETA_E
+    dv = lambda p: mabs['v'] + bw['v'] * (muw_abs + r1 * nU(p))
+    ebar4 = (r1 * nbar('nu', 8) + L3[0][0].hi * znorm(8) + dv(8)) ** 2
+    C_n = ((1 + th) ** 2 * EvUt2()
+           + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U2_2
+           + r1 * nT4 / six_klo * ebar4 * U2_2)
+    termC = aK2_hi ** 2 / (12 * klo) * C_n
+    # one more power of U for the density refinement: the same three terms with lambda inserted, divided by r
+    # (int_0^{rU} lambda W dlambda <= r^6 |c1| [|c2| (U^4/4 + om U^3/3) + e U^3/3])
+    U3_2_ = U3_2
+    termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * sq(EUt(8)))
+    EoUt3 = sum(binom(3, l) * et1 ** l * E1o[6 + l] for l in range(4)) + Fk ** 3 * To[6]
+    EvUt3 = sum(binom(3, l) * et1 ** l * E1v[6 + l] for l in range(4)) + Fk ** 3 * Tv[6]
+    termBp = aK2_hi ** 3 / 3 * (EoUt3 + (ndo(2) + epsc4 * no4) * U3_2_)
+    if MUT == 'no-om-term':
+        termBp = Fr(0)
+    termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt3
+                                         + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U3_2_
+                                         + r1 * nT4 / six_klo * ebar4 * U3_2_)
+    # density of lambda = -w: pbar = its supremum on [0, inf); on the failure window [0, rU] also p(lambda) <= p0 + lambda Lp
+    # with p0 = sup over the box of the density at 0 and Lp = sup |p'| <= phi(1)/sigma^2, phi(1) < 0.24198
+    mlo = max(Fr(0), mu['w'].lo)
+    pbar = (phi_pt(mlo / sw_hi) / sw_lo).hi
+    m0 = Fr(0) if mu['w'].lo <= 0 <= mu['w'].hi else min(abs(mu['w'].lo), abs(mu['w'].hi))
+    p0 = (phi_pt(m0 / sw_hi) / sw_lo).hi
+    Lp = Fr(24198, 100000) / (sw_lo * sw_lo)
+    # ---- normalizer floor Z_r / (36 k^2 r^2) >= El2 - corr
+    m_lam = -mu['w']
+    El2 = elam2_pos(m_lam.lo, sw_lo).lo
+    Elp = elam_pos(m_lam.hi, sw_hi).hi
+    lam_n = lambda p: muw_abs + sw_hi * znorm(pow2(p))
+    sdT, sdtau = law.sdQ['T'].hi, law.sdQ['tau'].hi
+    q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1))
+    Eom_g = mabs['om'] + law.sdg['om'].hi * Fr(4, 5)
+    Evg2 = mabs['v'] ** 2 + law.sdg['v'].hi ** 2
+    E_om_lam = Eom_g * Elp + bw['om'] * (muw_abs * lam_n(1) + lam_n(2) ** 2)
+    E_v2_lam = 2 * Evg2 * Elp + 2 * bw['v'] ** 2 * (sw_hi * znorm(4)) ** 2 * lam_n(2)
+    corr = (lam_n(4) ** 2 * iroot(q_bad, 2) if q_bad > 0 else Fr(0)) \
+        + (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) * lam_n(4) ** 2 / six_klo \
+        + r1 * (E_v2_lam / (3 * klo) + E_om_lam)
+    if MUT == 'no-zfloor-correction':
+        corr = Fr(0)
+    zt = El2 - corr
+    require(zt > 0, 'normalizer floor not positive')
+    main = min(pbar * (termA + termB + termC), p0 * (termA + termB + termC) + r1 * Lp * (termAp + termBp + termCp)) / zt
+    # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q) (Hoelder), ||W 1_typed||_q / r^2 <= g1 g2,
+    # for each q in QS; the smallest resulting bound is used
+    six_khi = 6 * khi
+
+    def wnorm(q):
+        p4 = 4 * q
+        nTQ = gauss_norm(mabs['T'], sdT, p4)
+        ntauQ = gauss_norm(mabs['tau'], sdtau, p4)
+        g1 = (six_khi + r1 * nTQ) * lam_n(p4)
+        g2 = (six_khi + r1 * nTQ + r1 * r1 * ntauQ) * (lam_n(p4) + r1 * gauss_norm(mabs['om'], law.sdQ['om'].hi, p4)) \
+            + r1 * (r1 * gauss_norm(mabs['nu'], law.sdQ['nu'].hi, p4) + gauss_norm(mabs['v'], law.sdQ['v'].hi, p4)) ** 2
+        return g1 * g2
+    thr = 3 * klo / (10 * r1)
+    qA2 = Fr(0)
+    args = []
+    for n in JET4:
+        args.append((thr * Fr(7, 8) - mabs[n]) / law.sdQ[n].hi)
+        qA2 += gauss_tail2(mabs[n], law.sdQ[n].hi, thr * Fr(7, 8))
+        qA2 += (law.S[n] * (sqE + znorm(32)) / (thr / 8)) ** 32
+    a_hi = Fr(4, 3) / klo * a_fac
+    qfar = Fr(0)
+    qx = Fr(0)
+    if bbar > 0:
+        lam_far = 1 / (4 * a_hi * r1 * bbar * bbar)
+        args.append((lam_far - muw_abs) / sw_hi)
+        qfar = gauss_tail1(muw_abs, sw_hi, lam_far)
+    for i in (1, 2, 3, 4):                         # x_i > X0  <=>  y_i' > X0 / (a r1 beta_i)
+        if bbars[i] == 0:
+            continue
+        tx = x0 / (a_hi * r1 * bbars[i])
+        if i == 1:
+            cb = 12 * khi + bbars[1] * muw_abs
+            qx += Fr(1) if tx <= cb else (r1 * nDelta(1, 32) / (tx - cb)) ** 32
+        else:
+            s_i = {2: s2, 3: s3, 4: s4}[i]
+            args.append(tx / (2 * s_i))
+            qx += gauss_tail2(0, s_i, tx / 2) + (nd(i, 32) * Klo / (tx / 2)) ** 32
+    qs = [min(Fr(1), x) for x in (qA2, qfar, qx)]
+    tail_num = min(wnorm(q) * sum(pow_tail(x, q) for x in qs) for q in QS)
+    if last_band:
+        # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian tail argument scales at least like 1/r and is >= 3 at R,
+        # so each Gaussian piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32; after the
+        # power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of tail / r^3
+        # over (0, R] is attained on [R/2, R]
+        require(min(args) >= 3, 'last-band tail arguments')
+        # no piece is capped at 1, so the factor-64 decrease of every piece per halving carries over to each sum
+        require(max(qA2, qfar, qx) < 1, 'last-band tail pieces')
+        rdiv = r1 / 2
+    else:
+        rdiv = r0
+    tail = tail_num / (36 * klo * klo * zt * rdiv ** 3)
+    total = main + tail
+    return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'pbar': pbar, 'termA': termA, 'termB': termB,
+            'termC': termC, 'qA2': qA2, 'qfar': qfar, 'qx': qx, 'El2': El2, 'corr': corr, 'min_arg': min(args)}
+
+
+def pow_tail(x, q):
+    """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two)."""
+    if x <= 0:
+        return Fr(0)
+    return iroot(rup_rel(x, 64) ** (q - 1), q)
+
+
+def phibar_upper(x):
+    """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound
+    phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor."""
+    x = Fr(x)
+    if x <= 12:
+        return Phibar_pt(x).hi
+    y = x * x / 2
+    m = 40
+    return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2))      # sqrt(2 pi) > 5/2
+
+
+def gauss_tail2(mu_abs, sd, t):
+    """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd."""
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_tail1(mu_abs, sd, t):
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), phibar_upper((t - mu_abs) / sd))
+
+
+# ----------------------------------------------------------------------------- bands, boxes, runs
+B_BOXES = ((Fr(0), Fr(1, 64)), (Fr(1, 64), Fr(1, 16)), (Fr(1, 16), Fr(1, 4)), (Fr(1, 4), Fr(1)))
+K_BOXES = ((Fr(1, 2), Fr(1)), (Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(7, 4)), (Fr(7, 4), Fr(15, 8)), (Fr(15, 8), Fr(31, 16)),
+           (Fr(31, 16), Fr(63, 32)), (Fr(63, 32), Fr(2)))
+HIGH_OCTAVES = 3           # octaves [2^-(j+1), 2^-j], j = 6..8 (r up to 1/64), split into 8 sub-bands
+TOP_OCTAVES = 4            # octaves j = 9..12, split into 16 sub-bands
+LOW_OCTAVES = 16           # octaves j = 13..28, split into 4 sub-bands
+R_LAST = Fr(1, 2 ** 29)    # last band [0, 2^-29]
+
+
+def bands():
+    """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST]."""
+    out = []
+    for j in range(6, 9 + TOP_OCTAVES + LOW_OCTAVES):
+        hi = Fr(1, 2 ** j)
+        lo = hi / 2
+        m = 8 if j < 9 else (16 if j < 9 + TOP_OCTAVES else 4)
+        for i in range(m):
+            out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m))
+    out.append((Fr(0), R_LAST))
+    require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover')
+    return sorted(out)
+
+
+def up(x, digits=6):
+    """Decimal string of a rational rounded UP to `digits` significant digits (as a string float)."""
+    x = Fr(x)
+    if x <= 0:
+        return '0'
+    e = len(str(x.numerator // x.denominator)) if x >= 1 else -len(str(x.denominator // x.numerator))
+    scale = Fr(10) ** (digits - e)
+    n = -((-x.numerator * scale.numerator) // (x.denominator * scale.denominator))
+    return str(Fr(n) / scale) if scale.denominator == 1 else '%s' % (Fr(n) / scale)
+
+
+def dec_down(x, places=4):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_up(x, places=4):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+BOXES = tuple((bb, kb) for bb in B_BOXES for kb in K_BOXES)      # the order of the per-box lists in a band record
+# (x0, theta) choices; each gives a valid bound and the record keeps the smaller one
+COMBOS = ((Fr(1, 64), Fr(1, 32)), (Fr(1, 1024), Fr(1, 256)))
+# the fixed-axis SIDE24 point of [CAP] section 1 (b = 6/5, kappa = 1/6), a single (b, k) point outside the band
+SIDE24_B, SIDE24_K = (Fr(6, 5), Fr(6, 5)), (Fr(1, 6), Fr(1, 6))
+CG_02_LO, CG_SIDE24_LO = Fr('5324360.4426'), Fr('35736.7352')      # certified lower ends of c_G at (0, 2), (6/5, 1/6) ([G])
+
+
+def best_box(law, bb, kb, r0, r1, last):
+    best = None
+    for x0, theta in COMBOS:
+        res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=x0, theta=theta)
+        if best is None or res['total'] < best['total']:
+            best = res
+    return best
+
+
+def band_record(args):
+    """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down,
+    6 places) for each of the 28 boxes in BOXES order, the largest tail part (rounded up, 12 places), and the same two
+    numbers at the SIDE24 point."""
+    r0, r1 = args
+    law = band_law(r0, r1)
+    last = (r0 == 0)
+    bound, zfloor, tails = [], [], []
+    for bb, kb in BOXES:
+        res = best_box(law, bb, kb, r0, r1, last)
+        bound.append(dec_up(res['total']))
+        zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6))
+        tails.append(res['tail'])
+    s24 = best_box(law, SIDE24_B, SIDE24_K, r0, r1, last)
+    return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12),
+            'side24': [dec_up(s24['total']), dec_down(36 * SIDE24_K[0] ** 2 * s24['zt'], 6)]}
+
+
+def full_run(procs):
+    bl = bands()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, bl)
+    return assemble(recs)
+
+
+def assemble(recs):
+    table = {}
+    for rs in R_STARS:
+        mx, arg, zmin = Fr(0), None, None
+        for rec in recs:
+            if Fr(rec['r'][1]) <= rs:
+                for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']):
+                    v = Fr(bs)
+                    if v > mx:
+                        mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]}
+                    z = Fr(zs)
+                    zmin = z if zmin is None or z < zmin else zmin
+        table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG02': dec_up(mx / CG_02_LO, 6),
+                          'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    side = {}
+    for rs in R_STARS:
+        rows = [rec for rec in recs if Fr(rec['r'][1]) <= rs]
+        mx = max(Fr(rec['side24'][0]) for rec in rows)
+        arg = [rec['r'] for rec in rows if Fr(rec['side24'][0]) == mx][0]
+        zmin = min(Fr(rec['side24'][1]) for rec in rows)
+        side[str(rs)] = {'C': str(mx), 'argmax_r': arg, 'C_over_cG': dec_up(mx / CG_SIDE24_LO, 6),
+                         'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1', 'scientific_effect': 'NONE',
+            'certified': True, 'mutant': MUT, 'L_min': L_MIN,
+            'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]},
+            'side24_point': {'b': str(SIDE24_B[0]), 'k': str(SIDE24_K[0])},
+            'parameters': params(), 'C_table': table, 'C_table_side24': side, 'bands': recs}
+
+
+def params():
+    return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBOS_X0_THETA': [[str(a), str(b)] for a, b in COMBOS], 'THETA_E': str(THETA_E), 'QS': list(QS),
+            'R_STARS': [str(x) for x in R_STARS], 'HIGH_OCTAVES': HIGH_OCTAVES, 'TOP_OCTAVES': TOP_OCTAVES,
+            'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST),
+            'B_BOXES': [[str(a), str(b)] for a, b in B_BOXES], 'K_BOXES': [[str(a), str(b)] for a, b in K_BOXES]}
+
+
+def results_text(res):
+    """RESULTS.json layout: the summary indented, then one line per band record."""
+    head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True)
+    lines = ',\n'.join('  ' + json.dumps(rec, sort_keys=True) for rec in res['bands'])
+    return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n'
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def replay_sample():
+    """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band."""
+    bl = bands()
+    idx = []
+    for rs in R_STARS:
+        j = max(i for i, b in enumerate(bl) if b[1] == rs)
+        idx.append(j)
+    idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))]
+    idx.append(0)
+    return sorted(set(idx))
+
+
+def check_run(procs, full=False):
+    ref = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout')
+    require(ref['parameters'] == params(), 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json produced under a mutant')
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure')
+    bl = bands()
+    require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs')
+    idx = list(range(len(bl))) if full else replay_sample()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, [bl[i] for i in idx])
+    for i, rec in zip(idx, recs):
+        require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r']))
+    # the table is the maximum over the stored records
+    asm = assemble(ref['bands'])
+    require(asm['C_table'] == ref['C_table'] and asm['C_table_side24'] == ref['C_table_side24'],
+            'C tables are not the maxima of the stored bounds')
+    # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G(0, 2) of Math-#203 (a lower bound for every
+    # cap-route constant valid for all L >= 10, since c_G^{L,R} -> c_G^{ref} as L -> infinity)
+    for rs, row in ref['C_table'].items():
+        require(Fr(row['C']) > CG_02_LO, 'C below the sharp cap coefficient')
+    for rs, row in ref['C_table_side24'].items():
+        require(Fr(row['C']) > CG_SIDE24_LO, 'SIDE24 C below the sharp cap coefficient')
+    print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()},
+                      'C_table_side24': {rs: row['C'] for rs, row in ref['C_table_side24'].items()}, 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.mc:
+        rr, bb, kk, nn, sd = args.mc
+        print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd))))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({'band': {rs: row['C'] for rs, row in res['C_table'].items()},
+                      'side24': {rs: row['C'] for rs, row in res['C_table_side24'].items()}}))
+
+
+
+# ----------------------------------------------------------------------------- floating control (not part of the certificate)
+def mc_event(r, b, k, n, seed, deg=8, grid=9):
+    """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel (floating point): the Taylor jet of
+    order deg at the midpoint is sampled under Q, the field near D is rebuilt from it, W_r and G_r (M_3, M_4 on a grid of D)
+    are evaluated directly from the definitions."""
+    import random
+    pins, _ = functionals()
+    P = [pins[nm] for nm in PIN_NAMES]
+    jets = [(i, m - i) for m in range(deg + 1) for i in range(m, -1, -1)]
+    J = [[term(1, 0, a, bb, X0PT)] for (a, bb) in jets]
+
+    def ev(F, G, key):
+        ser, w, _ = pair_series(F, G, key)
+        return sum(float(c) * r ** p for p, c in ser.items())
+    Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(6)] for i in range(6)]
+    Sjp = [[ev(J[i], P[j], ('jp', deg, i, j)) for j in range(6)] for i in range(len(J))]
+    Sjj = [[ev(J[i], J[j], ('jj', deg, i, j)) for j in range(len(J))] for i in range(len(J))]
+
+    def solve(A, bv):
+        m = len(A)
+        M = [A[i][:] + [bv[i]] for i in range(m)]
+        for c in range(m):
+            p = max(range(c, m), key=lambda i: abs(M[i][c]))
+            M[c], M[p] = M[p], M[c]
+            for i in range(c + 1, m):
+                f = M[i][c] / M[c][c]
+                for j in range(c, m + 1):
+                    M[i][j] -= f * M[c][j]
+        x = [0.0] * m
+        for i in reversed(range(m)):
+            x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i]
+        return x
+    vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k, 0, 0, 0]
+    Wt = [solve(Spp, Sjp[i]) for i in range(len(J))]
+    mu = [sum(Wt[i][j] * vr[j] for j in range(6)) for i in range(len(J))]
+    C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(6)) for j in range(len(J))] for i in range(len(J))]
+    # symmetric square root (Jacobi)
+    m = len(J)
+    A = [row[:] for row in C]
+    V = [[float(i == j) for j in range(m)] for i in range(m)]
+    for _ in range(100):
+        off = sum(A[p][q] ** 2 for p in range(m) for q in range(p + 1, m))
+        if off < 1e-26:
+            break
+        for p in range(m - 1):
+            for q in range(p + 1, m):
+                if abs(A[p][q]) < 1e-300:
+                    continue
+                th = 0.5 * math.atan2(2 * A[p][q], A[q][q] - A[p][p])
+                c, s = math.cos(th), math.sin(th)
+                for kk in range(m):
+                    akp, akq = A[kk][p], A[kk][q]
+                    A[kk][p], A[kk][q] = c * akp - s * akq, s * akp + c * akq
+                for kk in range(m):
+                    apk, aqk = A[p][kk], A[q][kk]
+                    A[p][kk], A[q][kk] = c * apk - s * aqk, s * apk + c * aqk
+                for kk in range(m):
+                    vkp, vkq = V[kk][p], V[kk][q]
+                    V[kk][p], V[kk][q] = c * vkp - s * vkq, s * vkp + c * vkq
+    lam = [max(A[i][i], 0.0) for i in range(m)]
+    S = [[sum(V[i][kk] * math.sqrt(lam[kk]) * V[j][kk] for kk in range(m)) for j in range(m)] for i in range(m)]
+    fact = [math.factorial(i) for i in range(deg + 2)]
+    pts = [(-2 * r + 4 * r * i / (grid - 1), -2 * r + 4 * r * j / (grid - 1)) for i in range(grid) for j in range(grid)]
+
+    def deriv(cf, a0, b0, x, y):
+        tot = 0.0
+        for idx, (a, bb) in enumerate(jets):
+            if a >= a0 and bb >= b0:
+                tot += cf[idx] * x ** (a - a0) * y ** (bb - b0) / (fact[a - a0] * fact[bb - b0])
+        return tot
+    rng = random.Random(seed)
+    num = den = 0.0
+    acap = 4 / (3 * k)
+    for _ in range(n):
+        z = [rng.gauss(0, 1) for _ in jets]
+        cf = [mu[i] + sum(S[i][j] * z[j] for j in range(m)) for i in range(m)]
+        hM = [deriv(cf, 2, 0, -r / 2, 0.0), deriv(cf, 1, 1, -r / 2, 0.0), deriv(cf, 0, 2, -r / 2, 0.0)]
+        hS = [deriv(cf, 2, 0, r / 2, 0.0), deriv(cf, 1, 1, r / 2, 0.0), deriv(cf, 0, 2, r / 2, 0.0)]
+        dM = hM[0] * hM[2] - hM[1] ** 2
+        dS = hS[0] * hS[2] - hS[1] ** 2
+        if not (hM[0] < 0 and dM > 0 and dS < 0):
+            continue
+        Wv = abs(dM * dS)
+        den += Wv
+        lamv = -hM[2]
+        M3 = max(abs(deriv(cf, a, 3 - a, x, y)) for (x, y) in pts for a in range(4))
+        M4 = max(abs(deriv(cf, a, 4 - a, x, y)) for (x, y) in pts for a in range(5))
+        if not (lamv > acap * r * M3 * M3 and r * M4 <= 3 * k / 10):
+            num += Wv
+    return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3, 'Z_over_r2': den / n / r ** 2}
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_lifetime_difference_constant_20261001/engine_e3.py b/frontiers/c8_lifetime_difference_constant_20261001/engine_e3.py
new file mode 100644
index 000000000..267890232
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/engine_e3.py
@@ -0,0 +1,1714 @@
+#!/usr/bin/env python3
+"""Explicit cap-route constants for Theorem A of [LP] in dimension d = 3, for the torus kernel K_L with L >= 10 in every
+frame: for 0 < r <= r_*, b in [0, 1], k in [1/2, 2],
+
+    1 - p_r(b, k, R)  <=  Q^W(G_r^c)  <=  C3(r_*) r^3,        Z_r / r^2 >= z3(r_*),
+
+with G_r the good event of [LP] section 7 / [CAP] (1) (partial-block operator norms on the cylinder
+D = [-2r, 2r] x ball(0, 2r)) and C3(r_*), z3(r_*) certified here (NOTE.md Theorem E3).  The proof bounds
+Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0:
+
+  * covariances of the eight preconditioned pins and the 31 targets are exact power series in r, obtained from the Taylor
+    coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded
+    uniformly in r and in the frame by Cauchy estimates on a polydisc in C^3;
+  * the failure event is reduced to {lambda_1 <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded blockwise through exact
+    averaging identities of the pinned jets and M_4 through the Taylor series at the midpoint (Frobenius norms);
+  * the main term is integrated over the transverse Hessian B = -D_y^2 f(M) in eigenvalue coordinates after an exact
+    regression on its three entries, against an angle-free Gaussian majorant of its density; the remaining expectations are
+    closed-form incomplete Gaussian and chi moments plus Hoelder-controlled perturbations;
+  * the normalizer Z_r is bounded below through a Loewner-order coupling with the contact law, whose moments
+    E[det(B)^a tr(B)^c 1{B > 0}] are in closed form; the rare branches are Gaussian and chi tails.
+All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only.
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term', 'no-lambda2')
+MUT = None
+PBITS = 160
+L_MIN = 10
+NX = 60                    # Taylor order in x of the covariance series (exact); the rest is a bounded tail
+NB = 24                    # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded)
+X0 = Fr(1, 64)             # default threshold for the quadratic near-branch analysis (records use COMBOS)
+THETA = Fr(1, 32)          # default splitting parameter in the max-moment lemma (records use COMBOS)
+THETA_E = Fr(1, 16)        # splitting parameter for the ||r nu - v||^2 factor
+QS = (2, 4, 8, 16)         # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used
+R0_COUPLING = 12           # radius of the Loewner coupling of the transverse Hessian with the contact law (floor, section 6)
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+BAND_B = (Fr(0), Fr(1))
+BAND_K = (Fr(1, 2), Fr(2))
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203 / #206)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+def phi_pt(x):
+    """phi(x) for rational x (interval)."""
+    x = Fr(x)
+    return exp_neg_point(x * x / 2) / SQRT2PI
+
+
+def Phibar_pt(x):
+    """1 - Phi(x) for rational x (interval)."""
+    x = Fr(x)
+    if abs(x) <= 12:
+        p = phi_point(x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 0:
+        return IV(0, (phi_pt(x) / x).hi)            # Mills: Phibar(x) <= phi(x)/x
+    return IV(1 - (phi_pt(-x) / (-x)).hi, 1)
+
+
+def dfact(n):
+    """(n)!! for n >= -1."""
+    return math.prod(range(n, 0, -2)) if n > 0 else 1
+
+
+def gauss_moment(j):
+    """E[Z^j] for Z ~ N(0, 1)."""
+    return 0 if j % 2 else dfact(j - 1)
+
+
+def upper_moments(x, J):
+    """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x (intervals).  For x < 0,
+    M_j(x) = E[Z^j] - (-1)^j M_j(-x)."""
+    x = Fr(x)
+    if x < 0:
+        M = upper_moments(-x, J)
+        return [IV(gauss_moment(j)) - M[j] * (-1) ** j for j in range(J + 1)]
+    ph = phi_pt(x)
+    M = [Phibar_pt(x), ph]
+    for j in range(2, J + 1):
+        M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1))
+    return M[:J + 1]
+
+
+def rup_rel(x, bits=200):
+    """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds)."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh)
+    return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh)
+
+
+def rdown_rel(x, bits=200):
+    """Round a rational x >= 0 down to a dyadic with `bits` significant bits."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr((x.numerator << sh) // x.denominator, 1 << sh)
+    return Fr((x.numerator // (x.denominator << -sh)) << -sh)
+
+
+def sqrt_up(x):
+    """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root
+    step adds less than 2^-209 and rup_rel(., 200) less than 2^-199.  (The earlier '2^-200' was too sharp: at x = 49/100 the
+    excess is (8/7) 2^-200, review C59-DOC-01.)"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m))
+
+
+def sqrt_down(x):
+    """Rational lower bound of sqrt(x), x >= 0 rational."""
+    x = Fr(x)
+    if x <= 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rdown_rel(Fr(math.isqrt(nd << (2 * m)), d << m))
+
+
+def iroot(x, p):
+    """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two."""
+    v = Fr(x)
+    while p > 1:
+        v = sqrt_up(v)
+        p //= 2
+    return v
+
+
+def pow2(p):
+    q = 1
+    while q < p:
+        q *= 2
+    return q
+
+
+ZN = {}
+
+
+def znorm(p):
+    """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5)."""
+    if p == 1:
+        return Fr(4, 5)
+    if p not in ZN:
+        ZN[p] = iroot(Fr(dfact(p - 1)), p)
+    return ZN[p]
+
+
+def znorm1(p):
+    """max(1, ||N(0,1)||_p): the factor in ||(|mu| + ||Sigma^(1/2) z||)||_p <= |mu| + sqrt(tr Sigma) znorm1(p)."""
+    return Fr(1) if p <= 2 else znorm(p)
+
+
+def frob_norm(mu_abs, tr_hi, p):
+    """Upper bound of || ||X|| ||_p for a Gaussian vector X with ||E X|| <= mu_abs and tr Cov X <= tr_hi (Minkowski in
+    L^(p/2) on the eigen-decomposition for p >= 2; Lyapunov for p <= 2)."""
+    return rup_rel(mu_abs + sqrt_up(tr_hi) * znorm1(pow2(p)))
+
+
+def gamma_half(n2):
+    """Gamma(n2 / 2) for an integer n2 >= 1, as an interval (sqrt(pi) for odd n2)."""
+    if n2 % 2 == 0:
+        return IV(math.factorial(n2 // 2 - 1))
+    n = (n2 - 1) // 2                                 # Gamma(n + 1/2) = (2n - 1)!! sqrt(pi) / 2^n
+    return isqrt_iv(PI) * Fr(dfact(2 * n - 1), 2 ** n)
+
+
+def chi_moment(nu, j):
+    """E[chi_nu^j] = 2^(j/2) Gamma((nu + j)/2) / Gamma(nu/2) (interval), integer j >= 0."""
+    if j % 2 == 0:
+        return IV(math.prod(nu + 2 * i for i in range(j // 2)))
+    out = gamma_half(nu + j) / gamma_half(nu)
+    for _ in range(j):
+        out = out * SQRT2
+    return out
+
+
+CN = {}
+
+
+def chi_norm(nu, p):
+    """Upper bound of ||chi_nu||_p for p a power of two."""
+    if p == 1:
+        return sqrt_up(Fr(nu))
+    key = (nu, p)
+    if key not in CN:
+        CN[key] = iroot(Fr(math.prod(nu + 2 * i for i in range(p // 2))), p)
+    return CN[key]
+
+
+CC = {}
+
+
+def chi_const(nu):
+    """C_nu sqrt(2 pi) with C_nu = 1 / (2^(nu/2 - 1) Gamma(nu/2)): E[chi_nu^j 1{chi > x}] = C_nu sqrt(2 pi) M_(j+nu-1)(x)."""
+    if nu not in CC:
+        p2 = IV(Fr(2) ** ((nu - 2) // 2)) if nu % 2 == 0 else IV(Fr(2) ** ((nu - 3) // 2)) * SQRT2
+        CC[nu] = SQRT2PI / (p2 * gamma_half(nu))
+    return CC[nu]
+
+
+def chi_upper_moments(nu, x, J):
+    """[E[chi_nu^j 1{chi_nu > x}] for j = 0..J] (intervals), rational x >= 0."""
+    M = upper_moments(x, J + nu - 1)
+    c = chi_const(nu)
+    return [M[j + nu - 1] * c for j in range(J + 1)]
+
+
+def chi_tail(nu, x):
+    """Rational upper bound of P(chi_nu > x) for rational x >= 0 (Mills-type bound beyond the series range)."""
+    x = Fr(x)
+    if x <= 0:
+        return Fr(1)
+    if x <= 12:
+        return min(Fr(1), chi_upper_moments(nu, x, 0)[0].hi)
+    # x > 12: P(chi_nu > x) = c_nu sqrt(2pi) M_(nu-1)(x), M_j(x) <= phi(x) x^(j-1) (1 + 2 (j-1)/x^2) for x^2 >= 2 j, and
+    # phi(x) <= m!/(x^2/2)^m / sqrt(2 pi) (m = 40)
+    y = x * x / 2
+    ph = Fr(math.factorial(40)) / y ** 40 * Fr(2, 5)          # 1/sqrt(2 pi) < 2/5
+    j = nu - 1
+    Mj = ph * (x ** (j - 1) if j >= 1 else 1 / x) * (1 + Fr(2 * max(j - 1, 1)) / (x * x))
+    return min(Fr(1), Mj * Fr(51, 20))                          # c_nu sqrt(2 pi) <= sqrt(2 pi) < 51/20
+
+
+def phibar_upper(x):
+    """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound
+    phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor."""
+    x = Fr(x)
+    if x <= 12:
+        return Phibar_pt(x).hi
+    y = x * x / 2
+    m = 40
+    return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2))      # sqrt(2 pi) > 5/2
+
+
+def gauss_tail2(mu_abs, sd, t):
+    """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd."""
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_tail1(mu_abs, sd, t):
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_norm(mu_abs, sd, p):
+    """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd."""
+    return rup_rel(mu_abs + sd * znorm(pow2(p)))
+
+
+def absup(x):
+    return max(abs(x.lo), abs(x.hi))
+
+
+def pow_tail(x, q):
+    """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two)."""
+    if x <= 0:
+        return Fr(0)
+    return iroot(rup_rel(x, 64) ** (q - 1), q)
+
+
+# ----------------------------------------------------------------------------- functionals and exact covariance series (d = 3)
+HALF = Fr(1, 2)
+XM, XS, X0PT = -HALF, HALF, Fr(0)
+JET4_IDX = {'X%d%d%d' % (a, b1, 4 - a - b1): (a, b1, 4 - a - b1) for a in range(5) for b1 in range(5 - a)}
+JET4 = tuple(sorted(JET4_IDX, key=lambda n: (-JET4_IDX[n][0], -JET4_IDX[n][1])))
+BLOCKS = ((4, 0), (3, 1), (2, 2), (1, 3), (0, 4))           # partial blocks (a, c) of the fourth derivatives, a + c = 4
+
+
+def block_members(a, c):
+    """[(name, Frobenius multiplicity c!/beta!)] of the 4-jet block d_x^a D_y^c f(0)."""
+    out = []
+    for b1 in range(c, -1, -1):
+        b2 = c - b1
+        out.append(('X%d%d%d' % (a, b1, b2), math.factorial(c) // (math.factorial(b1) * math.factorial(b2))))
+    return out
+
+
+def term(c, p, a, b1, b2, x0):
+    """c r^p d_x^a d_y1^b1 d_y2^b2 f(x0 r, 0, 0)."""
+    return (Fr(c), p, a, b1, b2, Fr(x0))
+
+
+def functionals():
+    pins = {
+        'p1': [term(HALF, 0, 0, 0, 0, XM), term(HALF, 0, 0, 0, 0, XS)],
+        'p2': [term(-1, -1, 0, 0, 0, XM), term(1, -1, 0, 0, 0, XS)],
+        'p3p': [term(6, -2, 1, 0, 0, XM), term(6, -2, 1, 0, 0, XS), term(12, -3, 0, 0, 0, XM), term(-12, -3, 0, 0, 0, XS)],
+        'p4': [term(-1, -1, 1, 0, 0, XM), term(1, -1, 1, 0, 0, XS)],
+    }
+    for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))):
+        pins['q1%d' % j] = [term(HALF, 0, 0, e1, e2, XM), term(HALF, 0, 0, e1, e2, XS)]
+        pins['q2%d' % j] = [term(-1, -1, 0, e1, e2, XM), term(1, -1, 0, e1, e2, XS)]
+    targets = {
+        'a11': [term(1, 0, 0, 2, 0, XM)], 'a22': [term(1, 0, 0, 0, 2, XM)], 'a12': [term(1, 0, 0, 1, 1, XM)],
+        'o11': [term(1, -1, 0, 2, 0, XS), term(-1, -1, 0, 2, 0, XM)],
+        'o22': [term(1, -1, 0, 0, 2, XS), term(-1, -1, 0, 0, 2, XM)],
+        'o12': [term(1, -1, 0, 1, 1, XS), term(-1, -1, 0, 1, 1, XM)],
+        'T': [term(1, -2, 2, 0, 0, XM), term(4, -3, 1, 0, 0, XM), term(2, -3, 1, 0, 0, XS), term(6, -4, 0, 0, 0, XM),
+              term(-6, -4, 0, 0, 0, XS)],
+        'tau': [term(1, -3, 2, 0, 0, XS), term(-1, -3, 2, 0, 0, XM), term(-6, -4, 1, 0, 0, XM), term(-6, -4, 1, 0, 0, XS),
+                term(-12, -5, 0, 0, 0, XM), term(12, -5, 0, 0, 0, XS)],
+    }
+    for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))):
+        targets['v%d' % j] = [term(1, -1, 1, e1, e2, XM), term(1, -2, 0, e1, e2, XM), term(-1, -2, 0, e1, e2, XS)]
+        targets['nu%d' % j] = [term(1, -2, 1, e1, e2, XM), term(1, -2, 1, e1, e2, XS), term(2, -3, 0, e1, e2, XM),
+                               term(-2, -3, 0, e1, e2, XS)]
+    for b1 in range(3, -1, -1):
+        targets['y%s' % ('1' * b1 + '2' * (3 - b1))] = [term(1, 0, 0, b1, 3 - b1, X0PT)]
+    for n, (a, b1, b2) in JET4_IDX.items():
+        targets[n] = [term(1, 0, a, b1, b2, X0PT)]
+    return pins, targets
+
+
+PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q11', 'q21', 'q12', 'q22')
+BNAMES = ('a11', 'a22', 'a12')                              # the transverse Hessian A = D_y^2 f(M) = -B
+VNAMES = ('v1', 'v2')
+ONAMES = ('o11', 'o22', 'o12')
+YNAMES = ('y111', 'y112', 'y122', 'y222')
+FREE = VNAMES + ONAMES + YNAMES                              # the free third-derivative blocks, in Cholesky order
+FREE_W = (1, 1, 1, 1, 2, 1, 3, 3, 1)                         # Frobenius multiplicities (Euclidean norms in these weights)
+TARGET_NAMES = BNAMES + FREE + ('T', 'tau', 'nu1', 'nu2') + JET4
+NP = len(PIN_NAMES)
+
+
+def weight(F):
+    """Scaling weight a - (power of r), common to all terms of a preconditioned functional."""
+    ws = {t[2] - t[1] for t in F}
+    require(len(ws) == 1, 'functional not homogeneous')
+    return ws.pop()
+
+
+def hcoef(n):
+    """d^n/dx^n exp(-x^2/2) at 0."""
+    if n % 2:
+        return 0
+    return (-1) ** (n // 2) * dfact(n - 1)
+
+
+PAIR_CACHE = {}
+
+
+def pair_series(F, G, key):
+    """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG), gamma = (gx, gy1, gy2):
+    returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', gy1, gy2) for the remainder bounds)."""
+    if key in PAIR_CACHE:
+        return PAIR_CACHE[key]
+    wF, wG = weight(F), weight(G)
+    w = wF + wG
+    pi = {}
+    absterms = []
+    for (cf, pf, af, bf1, bf2, xf) in F:
+        for (cg, pg, ag, bg1, bg2, xg) in G:
+            g1, g2 = bf1 + bg1, bf2 + bg2
+            sgn = (-1) ** (ag + bg1 + bg2)
+            d = xf - xg
+            absterms.append((abs(cf * cg), af + ag, g1, g2))
+            for gx in range(af + ag, NX + 1):
+                m = gx - af - ag
+                val = cf * cg * sgn * d ** m / math.factorial(m)
+                if val != 0:
+                    pi[(gx, g1, g2)] = pi.get((gx, g1, g2), Fr(0)) + val
+    for (gx, g1, g2), v in pi.items():
+        require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)')
+    ser = {}
+    for (gx, g1, g2), v in pi.items():
+        kap = hcoef(gx) * hcoef(g1) * hcoef(g2)
+        if kap and v:
+            ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v
+    out = ({p: c for p, c in ser.items() if c != 0}, w, absterms)
+    PAIR_CACHE[key] = out
+    return out
+
+
+E_UP = Fr(27183, 10000)                                      # > e
+SQRT3_UP = Fr(7, 4)                                          # > sqrt(3)
+
+
+def torus_M1(L):
+    """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2|, |z_3| <= 1 (every frame, |p_n| = |n| L):
+    exp(3/2) sum_{j >= 1} (24 j^2 + 2) exp(-(j L - sqrt 3)^2 / 2); 24 j^2 + 2 lattice points have sup-norm j."""
+    L = Fr(L)
+    e32 = exp_neg_point(Fr(3, 2)).inv().hi
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += (24 * j * j + 2) * exp_neg_point((j * L - SQRT3_UP) ** 2 / 2).hi
+    # j >= 6: (24 j^2 + 2) exp(-(jL - 7/4)^2/2) decreases by more than 1/2 per step for L >= 10
+    tail = 2 * (24 * 36 + 2) * exp_neg_point((6 * L - SQRT3_UP) ** 2 / 2).hi
+    return e32 * (tot + tail)
+
+
+def theta_minus_one(L):
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += (24 * j * j + 2) * exp_neg_point((j * L) ** 2 / 2).hi
+    return tot + 2 * (24 * 36 + 2) * exp_neg_point((6 * L) ** 2 / 2).hi
+
+
+def cov_band(F, G, key, r0, r1, M1):
+    """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1]."""
+    ser, w, absterms = pair_series(F, G, key)
+    val = IV(0)
+    for p, c in ser.items():
+        val = val + IV(r0 ** p, r1 ** p) * c
+    A = max((t[1] for t in absterms), default=0)
+    # Taylor truncation tail of the reference kernel (orders gx > NX)
+    tail = Fr(0)
+    for (acc, aa, g1, g2) in absterms:
+        tail += acc * abs(hcoef(g1) * hcoef(g2))
+    tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w)
+    # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy1! gy2! M1 on the unit polydisc
+    tor = Fr(0)
+    for (acc, aa, g1, g2) in absterms:
+        tor += acc * math.factorial(g1) * math.factorial(g2) * Fr(max(w, 1)) ** aa
+    tor = tor * M1 / (1 - E_UP * r1)
+    err = tail + tor
+    return val + IV(-err, err)
+
+
+# ----------------------------------------------------------------------------- interval linear algebra
+def solve_iv(A, B):
+    """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c]."""
+    n = len(A)
+    M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    for col in range(n):
+        p = max(range(col, n), key=lambda i: abs(M[i][col].mid()))
+        M[col], M[p] = M[p], M[col]
+        piv = M[col][col]
+        for i in range(col + 1, n):
+            f = M[i][col] / piv
+            for j in range(col, len(M[i])):
+                M[i][j] = M[i][j] - f * M[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            s = M[i][n + c]
+            for j in range(i + 1, n):
+                s = s - M[i][j] * x[j]
+            x[i] = s / M[i][i]
+        X.append(x)
+    return X
+
+
+def cholesky(C):
+    """Interval Cholesky of an n x n SPD interval matrix (pivots must be positive)."""
+    n = len(C)
+    L = [[IV(0)] * n for _ in range(n)]
+    for i in range(n):
+        for j in range(i + 1):
+            s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0))
+            if i == j:
+                require(s.lo > 0, 'Cholesky pivot not positive')
+                L[i][j] = isqrt_iv(s)
+            else:
+                L[i][j] = s / L[j][j]
+    return L
+
+
+def opnorm_lower_tri(L, rows, cols):
+    """Upper bound of the operator norm of the block L[rows][cols] of an interval lower-triangular matrix:
+    max |diagonal| + Frobenius norm of the rest (square diagonal blocks), Frobenius norm otherwise."""
+    if rows == cols:
+        dg = max(absup(L[i][i]) for i in rows)
+        off = sum(absup(L[i][j]) ** 2 for i in rows for j in cols if j < i)
+        return rup_rel(dg + sqrt_up(off))
+    return sqrt_up(sum(absup(L[i][j]) ** 2 for i in rows for j in cols))
+
+
+def sym3_eig_bounds(C):
+    """(lower bound of lambda_min, upper bound of lambda_max) of a symmetric 3 x 3 interval matrix (Gershgorin)."""
+    lo = min(C[i][i].lo - sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3))
+    hi = max(C[i][i].hi + sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3))
+    return lo, hi
+
+
+def det3(C):
+    a, b, c = C[0]
+    d, e, f = C[1]
+    g, h, i = C[2]
+    return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
+
+
+# ----------------------------------------------------------------------------- the conditional law on an r-band
+class Law:
+    pass
+
+
+def band_law(r0, r1, L=L_MIN):
+    """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the
+    targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients; then the
+    regression on the transverse Hessian (three entries) and the g-law of the remaining targets."""
+    pins, targets = functionals()
+    P = [pins[n] for n in PIN_NAMES]
+    T = [targets[n] for n in TARGET_NAMES]
+    M1 = torus_M1(L)
+    th1 = theta_minus_one(L)
+    nt = len(T)
+    Spp = [[None] * NP for _ in range(NP)]
+    for i in range(NP):
+        for j in range(i, NP):
+            Spp[i][j] = Spp[j][i] = cov_band(P[i], P[j], ('p', i, j), r0, r1, M1)
+    Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(NP)] for i in range(nt)]
+    Stt = [[None] * nt for _ in range(nt)]
+    for i in range(nt):
+        for j in range(i, nt):
+            Stt[i][j] = Stt[j][i] = cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1)
+    r = IV(r0, r1)
+    kvec = [-(r * r * r) * HALF, -(r * r), IV(12)] + [IV(0)] * (NP - 3)
+    e1 = [IV(1)] + [IV(0)] * (NP - 1)
+    W = solve_iv(Spp, Stp)
+    xs = solve_iv(Spp, [e1, kvec])
+    law = Law()
+    law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1
+    law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)}
+    law.beta = {n: sum((W[i][j] * kvec[j] for j in range(NP)), IV(0)) for i, n in enumerate(TARGET_NAMES)}
+    scale = IV(1 / (1 + th1), 1)          # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1]
+    C = {}
+    for i, a in enumerate(TARGET_NAMES):
+        for j in range(i, nt):
+            c = TARGET_NAMES[j]
+            v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(NP)), IV(0))
+            C[(a, c)] = C[(c, a)] = v * scale
+    law.C = C
+    escale = IV(1, 1 + th1)
+    law.e_bb = xs[0][0] * escale
+    law.e_bk = sum((kvec[j] * xs[0][j] for j in range(NP)), IV(0)) * escale
+    law.e_kk = sum((kvec[j] * xs[1][j] for j in range(NP)), IV(0)) * escale
+    regress_on_B(law)
+    law.S = {blk: taylor_tail_coefficient(blk, r1, M1) for blk in BLOCKS}
+    law.net_var = {blk: net_max_variance(law.C, blk) for blk in BLOCKS if blk[1] >= 2}
+    return law
+
+
+NET_M = 128                                # rational unit vectors ((1 - t^2), 2 t)/(1 + t^2), t = j/NET_M, and their rotations
+
+
+def circle_net():
+    """Rational unit vectors u_j covering angles [0, pi) with consecutive angular gaps <= 2/NET_M (theta = 2 arctan t has
+    d theta/dt <= 2), together with the rotated copies (-u2, u1)."""
+    pts = []
+    for j in range(NET_M + 1):
+        t = Fr(j, NET_M)
+        pts.append(((1 - t * t) / (1 + t * t), 2 * t / (1 + t * t)))
+    return pts + [(-u2, u1) for (u1, u2) in pts]
+
+
+NET = circle_net()
+
+
+def block_form_weights(blk, u):
+    """w_beta = (c!/beta!) u1^beta1 u2^beta2: d_x^a d_u^c f(0) = sum_beta w_beta X_(a,beta)."""
+    a, c = blk
+    return [(n, m * u[0] ** int(n[2]) * u[1] ** int(n[3])) for n, m in block_members(a, c)]
+
+
+def net_max_variance(C, blk):
+    """Upper bound of max_j Var_Q(d_x^a d_(u_j)^c f(0)) over the net."""
+    best = Fr(0)
+    for u in NET:
+        w = block_form_weights(blk, u)
+        v = sum(abs(wi) * abs(wj) * absup(C[(ni, nj)]) if ni != nj else wi * wi * C[(ni, ni)].hi
+                for ni, wi in w for nj, wj in w)
+        best = max(best, v)
+    return best
+
+
+FROB_B = (IV(1), IV(1), SQRT2)               # (B11, B22, B12) -> Frobenius-isometric coordinates (B11, B22, sqrt2 B12)
+
+
+def regress_on_B(law):
+    """Regression of every target on the three entries of the transverse Hessian, in the Frobenius-isometric coordinates
+    b~ = (B11, B22, sqrt2 B12) of B = -A: F = F_g + beta~_F . (b~ - mu~), with F_g independent of B."""
+    C = law.C
+    # covariance of b~ (B = -A: signs cancel in the covariance)
+    Cb = [[C[(BNAMES[i], BNAMES[j])] * FROB_B[i] * FROB_B[j] for j in range(3)] for i in range(3)]
+    law.Cb = Cb
+    law.detCb = det3(Cb)
+    require(law.detCb.lo > 0, 'transverse Hessian covariance not positive')
+    lo, hi = sym3_eig_bounds(Cb)
+    require(lo > 0, 'transverse Hessian covariance eigenvalue bound')
+    law.sB_lo2, law.sB_hi2 = lo, hi
+    law.sB_lo = sqrt_down(lo)
+    # regression coefficients of F on b~: beta~_F = Cov(F, b~) Cb^-1, Cov(F, b~_i) = -Cov(F, A_i) FROB_B[i]
+    others = [n for n in TARGET_NAMES if n not in BNAMES]
+    rhs = [[-C[(n, BNAMES[i])] * FROB_B[i] for i in range(3)] for n in others]
+    sol = solve_iv(Cb, rhs)
+    law.bt = {n: sol[i] for i, n in enumerate(others)}
+    law.bnorm = {n: sqrt_up(sum(absup(x) ** 2 for x in law.bt[n])) for n in others}
+    Cg = {}
+    for a in others:
+        for c in others:
+            Cg[(a, c)] = C[(a, c)] - sum((law.bt[a][i] * (-C[(c, BNAMES[i])] * FROB_B[i]) for i in range(3)), IV(0))
+    law.Cg = Cg
+    # the free blocks in Frobenius coordinates: v, omega = (o11, o22, sqrt2 o12), Y~ = (y111, sqrt3 y112, sqrt3 y122, y222)
+    fw = [isqrt_iv(IV(w)) if w != 1 else IV(1) for w in FREE_W]
+    law.Lfree = cholesky([[Cg[(FREE[i], FREE[j])] * fw[i] * fw[j] for j in range(9)] for i in range(9)])
+    law.fw = fw
+    law.sdg = {n: sqrt_up(max(Cg[(n, n)].hi, Fr(0))) for n in others}
+    law.sdQ = {n: sqrt_up(C[(n, n)].hi) for n in TARGET_NAMES}
+
+
+def sd0_sq(g, M1):
+    """Upper bound of Var d^g f(0) = |d^(2g) K_L(0)| (normalized kernel, every frame, L >= L_MIN), g = (gx, gy1, gy2):
+    the reference value prod (2 g_i - 1)!! plus the Cauchy bound (2g)! M1 of the torus remainder."""
+    return math.prod(dfact(2 * x - 1) for x in g) + math.prod(math.factorial(2 * x) for x in g) * M1
+
+
+def taylor_tail_coefficient(blk, r1, M1):
+    """S_(a,c)(r1) = sum_{gamma != 0} sigma_(a,c),gamma (2 r1)^|gamma| / gamma!, sigma^2 = sum_beta (c!/beta!) s^2_(a+gx, beta+gy):
+    the Frobenius norm of the Taylor tail of the block d_x^a D_y^c f at the midpoint over D."""
+    a, c = blk
+    tot = Fr(0)
+    for n in range(1, NB + 1):
+        for gx in range(n + 1):
+            for g1 in range(n - gx + 1):
+                g2 = n - gx - g1
+                s2 = Fr(0)
+                for b1 in range(c + 1):
+                    b2 = c - b1
+                    mult = math.factorial(c) // (math.factorial(b1) * math.factorial(b2))
+                    s2 += mult * sd0_sq((a + gx, b1 + g1, b2 + g2), M1)
+                tot += sqrt_up(s2) * (2 * r1) ** n / (math.factorial(gx) * math.factorial(g1) * math.factorial(g2))
+    # tail |gamma| = n > NB: s_delta <= 2^|delta| delta! sqrt(1 + M1) and (alpha + gamma)!/gamma! <= (n + 4)^4, so
+    # sigma (2 r1)^n / gamma! <= 2^(c/2) 2^(4 + n) (n + 4)^4 (2 r1)^n sqrt(1 + M1); (n+1)(n+2)/2 multi-indices of order n;
+    # the ratio of consecutive bounds is below 2 q for these n (q = 4 r1 <= 1/16)
+    q = 4 * r1
+    require(q <= Fr(1, 16), 'Taylor tail ratio too large')
+    n = NB + 1
+    base = 2 * 2 ** 4 * sqrt_up(Fr(2) ** c * (1 + M1))
+    tail = base * Fr((n + 1) * (n + 2), 2) * Fr(n + 4) ** 4 * q ** n / (1 - 2 * q)
+    return rup_rel(tot + tail)
+
+
+# ----------------------------------------------------------------------------- closed forms for the contact transverse Hessian
+# B = m I + GOE_2 (diagonal N(0, 2), off-diagonal N(0, 1)): its eigenvalues are s -+ rho with s = m + (G11 + G22)/2 ~ N(m, 1)
+# and rho = sqrt(((G11 - G22)/2)^2 + G12^2) Rayleigh, independent; det B = s^2 - rho^2, tr B = 2 s, B > 0 iff s > rho.
+def upper_moments_iv(x, J):
+    """[M_0(x), ..., M_J(x)] over an interval x whose interior does not contain 0 (each M_j is monotone there)."""
+    if x.lo == x.hi:
+        return upper_moments(x.lo, J)
+    require(not (x.lo < 0 < x.hi), 'incomplete moment over an interval containing 0')
+    A, B = upper_moments(x.lo, J), upper_moments(x.hi, J)
+    return [IV(min(a.lo, b.lo), max(a.hi, b.hi)) for a, b in zip(A, B)]
+
+
+def es_pos(m, n):
+    """[E[s^j 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m (intervals)."""
+    M = upper_moments(-Fr(m), n)
+    return [sum((M[i] * (math.comb(j, i) * Fr(m) ** (j - i)) for i in range(j + 1)), IV(0)) for j in range(n + 1)]
+
+
+def isqrt2_pow(i):
+    """2^(-i/2) (interval)."""
+    return IV(Fr(1, 2 ** (i // 2))) if i % 2 == 0 else IV(Fr(1, 2 ** (i // 2))) / SQRT2
+
+
+def es_pos_exp(m, n):
+    """[E[s^j exp(-s^2/2) 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m:
+    = exp(-m^2/4)/sqrt2 . E[(m/2 + U/sqrt2)^j 1{U > -m/sqrt2}], U ~ N(0, 1)."""
+    m = Fr(m)
+    M = upper_moments_iv(-(IV(m) / SQRT2), n)
+    pre = exp_neg_point(m * m / 4) / SQRT2
+    out = []
+    for j in range(n + 1):
+        tot = IV(0)
+        for i in range(j + 1):
+            tot = tot + M[i] * (math.comb(j, i) * (m / 2) ** (j - i)) * isqrt2_pow(i)
+        out.append(pre * tot)
+    return out
+
+
+def inner_rho_polys(a):
+    """integral_0^s (s^2 - rho^2)^a rho exp(-rho^2/2) d rho = P_a(s) - exp(-s^2/2) Q_a(s) (exact polynomial coefficient
+    dicts {power: Fraction}); with u = rho^2/2 it is integral_0^(s^2/2) (s^2 - 2u)^a e^(-u) du."""
+    P, Q = {}, {}
+    for i in range(a + 1):
+        c = Fr(math.comb(a, i) * (-2) ** i * math.factorial(i))
+        P[2 * (a - i)] = P.get(2 * (a - i), Fr(0)) + c
+        for l in range(i + 1):
+            pw = 2 * (a - i) + 2 * l
+            Q[pw] = Q.get(pw, Fr(0)) + c / (2 ** l * math.factorial(l))
+    return P, Q
+
+
+def contact_moment(a, c, m):
+    """E[det(B)^a tr(B)^c 1{B > 0}] for B = m I + GOE_2, rational m (interval)."""
+    P, Q = inner_rho_polys(a)
+    n = 2 * a + c
+    E1 = es_pos(m, n)
+    E2 = es_pos_exp(m, n)
+    tot = IV(0)
+    for pw, cf in P.items():
+        tot = tot + E1[pw + c] * (cf * 2 ** c)
+    for pw, cf in Q.items():
+        tot = tot - E2[pw + c] * (cf * 2 ** c)
+    return tot
+
+
+def m3(m):
+    """m_(3,m) = E[det(B)^2 1{B > 0}] = E[(s^4 - 4 s^2 + 8) 1{s > 0}] - 4 sqrt2 exp(-m^2/4) Phi(m/sqrt2)."""
+    return contact_moment(2, 0, m)
+
+
+def gauss_half_moment(j, m, sig):
+    """J_j(m, sig) = integral_0^inf lambda^j exp(-(lambda - m)^2/(2 sig^2)) d lambda for rational m, sig > 0 (interval):
+    sig sqrt(2 pi) E[(m + sig Z)^j 1{m + sig Z > 0}]."""
+    m, sig = Fr(m), Fr(sig)
+    M = upper_moments(-m / sig, j)
+    tot = sum((M[i] * (math.comb(j, i) * m ** (j - i) * sig ** i) for i in range(j + 1)), IV(0))
+    return tot * SQRT2PI * sig
+
+
+# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k)
+def binom(n, k):
+    return math.comb(n, k)
+
+
+def pos_part_moments_chi(s, c, nu, nmax, extra_max):
+    """P[(n, e)] = upper bound of E[u^e (u^n - c^n)_+], u = s chi_nu, for 1 <= n <= nmax, 0 <= e <= extra_max."""
+    x = c / s
+    M = chi_upper_moments(nu, x, nmax + extra_max)
+    out = {}
+    for n in range(1, nmax + 1):
+        for e in range(extra_max + 1):
+            out[(n, e)] = rup_rel((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e).hi)
+    return out
+
+
+NU = {2: 2, 3: 3, 4: 4}                      # chi degrees of freedom of the free blocks v (2), omega (3), Y~ (4)
+
+
+def normalized_moments(q, nmax, theta):
+    """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax, under the product of the g-law and nu:
+      E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmt1 + (r1/Klo) D1;
+      Tn[n], To[n], Tv[n]   : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2),
+                              y_i <= u_i + delta_i (u_i = s_i chi_nu_i independent), threshold c = q['c'].
+    o0 = Klo u_3 and v0^2 = Klo^2 u_2^2 / 4 are unnormalized."""
+    th = theta
+    cp = q['c'] * (1 - th)                  # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli)
+    PP = {i: pos_part_moments_chi(q['s'][i], cp, NU[i], nmax, 2) for i in (2, 3, 4)}
+    Eo0 = (chi_moment(3, 1) * q['s3']).hi
+    Ev0sq = (chi_moment(2, 2) * q['s2'] * q['s2']).hi / 4
+    n_o0_2 = q['s3'] * chi_norm(3, 2)
+    n_v0sq_2 = q['s2'] * q['s2'] * chi_norm(2, 4) ** 2 / 4
+    Klo = q['Klo']
+    E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {}
+    base = rup_rel(1 + q['cmt1'])
+    bp = [Fr(1)]
+    for _ in range(nmax):
+        bp.append(rup_rel(bp[-1] * base))
+    # (r/K) D1 terms: ||(rK D1)^j||_1 <= (rK ||D1||_pow2(j))^j and ||(rK D1)^j||_2 <= (rK ||D1||_pow2(2j))^j
+    d1 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(j))) ** j) for j in range(1, nmax + 1)]
+    d2 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(2 * j))) ** j) for j in range(1, nmax + 1)]
+    for n in range(0, nmax + 1):
+        ey = bp[n]
+        eo = bp[n] * Eo0
+        ev = bp[n] * Ev0sq
+        for j in range(1, n + 1):
+            coef = binom(n, j) * bp[n - j]
+            ey += coef * d1[j]
+            eo += coef * n_o0_2 * d2[j]
+            ev += coef * n_v0sq_2 * d2[j]
+        E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev)
+        if n == 0:
+            continue
+        f1 = (1 + th) ** (n - 1)
+        f2 = (1 + 1 / th) ** (n - 1)
+        tn = to = tv = Fr(0)
+        for i in (2, 3, 4):
+            P = PP[i]
+            dn = rup_rel(q['nd'](i, pow2(n)) ** n)
+            dn2 = rup_rel(q['nd'](i, pow2(2 * n)) ** n)
+            tn += f1 * P[(n, 0)] + f2 * dn
+            to += (f1 * Klo * P[(n, 1)] if i == 3 else f1 * Eo0 * P[(n, 0)]) + f2 * n_o0_2 * dn2
+            tv += (f1 * Klo * Klo / 4 * P[(n, 2)] if i == 2 else f1 * Ev0sq * P[(n, 0)]) + f2 * n_v0sq_2 * dn2
+        Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv)
+    return E1, E1o, E1v, Tn, To, Tv
+
+
+def half_gauss_moments(m, sig, J):
+    """[J_j(m, sig) for j = 0..J], J_j = integral_0^inf l^j exp(-(l - m)^2/(2 sig^2)) dl, rational m >= 0, sig > 0 (intervals):
+    J_0 = sig sqrt(2 pi) Phi(m/sig), J_1 = m J_0 + sig^2 exp(-m^2/(2 sig^2)), J_(j+1) = m J_j + j sig^2 J_(j-1)
+    (integration by parts; every term is positive for m >= 0)."""
+    m, sig = Fr(m), Fr(sig)
+    require(m >= 0 and sig > 0, 'half-line Gaussian moments')
+    s2 = sig * sig
+    out = [SQRT2PI * sig * (1 - Phibar_pt(m / sig))]
+    out.append(out[0] * m + exp_neg_point(m * m / (2 * s2)) * s2)
+    for j in range(1, J):
+        nxt = out[j] * m + out[j - 1] * (j * s2)
+        out.append(IV(rdown_rel(nxt.lo), rup_rel(nxt.hi)))
+    return out
+
+
+def lam2_moments(b_lo, b_hi, eps, sig):
+    """Bounds on the lambda_2-integrals of the density majorant g(l) = exp(-(l - b)^2/(2 sig^2) + eps |l - b|/sig^2),
+    uniformly over b in [b_lo, b_hi] (b >= 0): I_hi[j] >= integral_0^inf l^j g(l) dl (j = 0..JM), I3_lo <= integral l^3 g.
+    For l >= 0, eps |l - b| <= eps (l + b), so g(l) <= exp((4 b eps + eps^2)/(2 sig^2)) exp(-(l - b - eps)^2/(2 sig^2));
+    each J_j(m, sig) increases with m; and g(l) >= exp(-(l - b)^2/(2 sig^2))."""
+    JM = 3 + 64
+    shift = (4 * b_hi * eps + eps * eps) / (2 * sig * sig)
+    fac = exp_neg_point(shift).inv().hi
+    Jh = half_gauss_moments(b_hi + eps, sig, JM)
+    I_hi = [rup_rel(x.hi * fac) for x in Jh]
+    I3_lo = rdown_rel(half_gauss_moments(b_lo, sig, 3)[3].lo)
+    return I_hi, I3_lo
+
+
+DETN = {}
+
+
+def det_norm(p, bpp, muB, sig):
+    """Upper bound of ||det(B) 1{B > 0}||_p (p a power of two) for B = mu_B + Gaussian with Loewner coupling
+    B <= (b'' + eta ||z||) I + GOE (b'' rounded up to a 1/1024 grid): the contact moment E[det^p 1{B > 0}] at b'' on
+    {||z|| <= R0}, plus E[(||B||_F^2/2)^p 1{||z|| > R0}] <= 2^-p E[(||mu_B|| + sig chi_3)^(2p) 1{chi_3 > R0}]."""
+    bg = Fr(math.ceil(bpp * 1024), 1024)
+    key = (p, bg)
+    if key not in DETN:
+        DETN[key] = contact_moment(p, 0, bg).hi
+    cm = DETN[key]
+    R0 = Fr(R0_COUPLING)
+    U = chi_upper_moments(3, R0, 2 * p)
+    out = sum((U[i] * (binom(2 * p, i) * muB ** (2 * p - i) * sig ** i) for i in range(2 * p + 1)), IV(0)).hi / 2 ** p
+    return iroot(rup_rel(cm + out), p)
+
+
+def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None):
+    """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch
+    threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5).  The bound is proved for 0 < x0 < 2/9 and
+    0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and
+    (1 + kappa x0) c^2 <= 1.  No claim is made outside that domain.  COMBO = (1/64, 1/32), also used by Math-#215's caller,
+    lies inside it (review C61)."""
+    x0 = X0 if x0 is None else Fr(x0)
+    theta = THETA if theta is None else Fr(theta)
+    b = IV(bb[0], bb[1])
+    k = IV(kb[0], kb[1])
+    blo, bhi = bb
+    klo, khi = kb[0], kb[1]
+    require(blo >= 0, 'b >= 0 assumed by the density majorant')
+    Klo = 12 * klo
+    mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES}
+    mabs = {n: absup(mu[n]) for n in TARGET_NAMES}
+    E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk
+    require(E.hi > 0, 'energy')
+    sqE = sqrt_up(E.hi)
+    nl2 = MUT != 'no-lambda2'
+    # ---- the transverse Hessian B = -A: mean in Frobenius coordinates, its distance to b I, the density majorant
+    dmu = [b * (-law.alpha['a11'] - 1) + k * (-law.beta['a11']), b * (-law.alpha['a22'] - 1) + k * (-law.beta['a22']),
+           (b * law.alpha['a12'] + k * law.beta['a12']) * SQRT2]
+    eps = sqrt_up(sum(absup(x) ** 2 for x in dmu))                 # sup over the box of ||mu_B - b I||_F
+    muB = rup_rel(bhi * sqrt_up(Fr(2)) + eps)                       # sup ||mu_B||_F
+    sig = sqrt_up(law.sB_hi2)                                       # rational, >= sqrt(lambda_max(Cov b~))
+    sB_lo = law.sB_lo                                               # <= sqrt(lambda_min(Cov b~))
+    KB = rup_rel((PI * SQRT2 / (2 * PI * SQRT2PI) / isqrt_iv(law.detCb)).hi)   # pi sqrt2 (2 pi)^(-3/2) det^(-1/2)
+    s2g = sig * sig
+    ex0 = blo * blo / (2 * s2g) - eps * bhi / s2g          # g(0; b) <= exp(-ex0) over the box
+    g0 = exp_neg_point(ex0).hi if ex0 >= 0 else exp_neg_point(-ex0).inv().hi
+    gbar = exp_neg_point(eps * eps / (2 * s2g)).inv().hi
+    # slope of the linear density bound g(l) <= g0 + l Lg on l >= 0: the global |g'| <= exp(-1/2) gbar / sig, or the chord
+    # of min(gbar, g0 exp(c l)) with c = (b_hi + eps)/sig^2 (g(l; b) <= g(0; b) exp((b + eps) l / sig^2) for l >= 0):
+    # g0 c (x - 1)/log x with log x = log(gbar/g0) = y (rational), (x - 1)/log x increasing in x
+    Lg = rup_rel(exp_neg_point(Fr(1, 2)).hi * gbar / sig)
+    y = (eps * eps + blo * blo - 2 * eps * bhi) / (2 * s2g)
+    if y > 0:
+        cc = (bhi + eps) / s2g
+        Lg = min(Lg, rup_rel(g0 * cc * (exp_neg_point(y).inv().hi - 1) / y))
+    else:
+        Lg = Fr(0)                                    # gbar <= g0: the bound g <= gbar alone is the smaller one
+    I_hi, I3_lo = lam2_moments(blo, bhi, eps, sig)
+    require(I3_lo > 0, 'lambda_2 moment')
+
+    def l2norm(p):                                  # ||lambda_2||_p under nu (and under nu'), p a power of two
+        if not nl2:
+            return Fr(0)
+        return iroot(rup_rel(I_hi[3 + p] / I3_lo), p)
+    # ---- regression on B and the coefficients bar-beta_i (NOTE section 4)
+    bn = law.bnorm
+    bv = sqrt_up(bn['v1'] ** 2 + bn['v2'] ** 2)
+    bo = sqrt_up(bn['o11'] ** 2 + bn['o22'] ** 2 + 2 * bn['o12'] ** 2)
+    by = sqrt_up(bn['y111'] ** 2 + 3 * bn['y112'] ** 2 + 3 * bn['y122'] ** 2 + bn['y222'] ** 2)
+    bnu = sqrt_up(bn['nu1'] ** 2 + bn['nu2'] ** 2)
+    a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1)      # the cap constant 4/(3k) (mutant: halved)
+
+    def Bblk(blk):
+        return rup_rel(sqrt_up(sum(m * bn[n] ** 2 for n, m in block_members(*blk))) + law.S[blk] / sB_lo)
+    Bb = {blk: Bblk(blk) for blk in BLOCKS}
+    bb1 = rup_rel(Fr(5, 2) * Bb[(4, 0)] + 2 * Bb[(3, 1)])           # bar-beta_1 / r
+    bbars = {1: rup_rel(r1 * bb1),
+             2: rup_rel(2 * bv + r1 * (Fr(5, 2) * Bb[(3, 1)] + 2 * Bb[(2, 2)])),
+             3: rup_rel(bo + r1 * (Fr(5, 2) * Bb[(2, 2)] + 2 * Bb[(1, 3)])),
+             4: rup_rel(by + r1 * (2 * Bb[(1, 3)] + 2 * Bb[(0, 4)]))}
+    bbar = max(bbars.values())
+    # ---- free blocks: Cholesky factors in Frobenius coordinates (v | omega | Y~)
+    Lf = law.Lfree
+    Iv, Io, Iy = (0, 1), (2, 3, 4), (5, 6, 7, 8)
+    Lvv, Low, Loo = opnorm_lower_tri(Lf, Iv, Iv), opnorm_lower_tri(Lf, Io, Iv), opnorm_lower_tri(Lf, Io, Io)
+    Lyv, Lyo, Lyy = opnorm_lower_tri(Lf, Iy, Iv), opnorm_lower_tri(Lf, Iy, Io), opnorm_lower_tri(Lf, Iy, Iy)
+    mu_v = sqrt_up(mabs['v1'] ** 2 + mabs['v2'] ** 2)
+    mu_o = sqrt_up(mabs['o11'] ** 2 + mabs['o22'] ** 2 + 2 * mabs['o12'] ** 2)
+    mu_y = sqrt_up(mabs['y111'] ** 2 + 3 * mabs['y112'] ** 2 + 3 * mabs['y122'] ** 2 + mabs['y222'] ** 2)
+    mu_nu = sqrt_up(mabs['nu1'] ** 2 + mabs['nu2'] ** 2)
+    s2, s3, s4 = 2 * Lvv, Loo, Lyy
+
+    memo = {}
+
+    def blk_g_norm(blk, p):                         # || ||X_blk,g||_F ||_p
+        mem = block_members(*blk)
+        m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem))
+        tr = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in mem)
+        return frob_norm(m_, tr, p)
+
+    def nA(blk, p):                                 # ||A^g_blk||_p, A^g = ||X_blk,g||_F + T^g_blk
+        key = ('A', blk, p)
+        if key not in memo:
+            memo[key] = rup_rel(blk_g_norm(blk, p) + law.S[blk] * (sqE + znorm1(pow2(p))))
+        return memo[key]
+
+    def nDelta(i, p):
+        key = ('D', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nDelta0(i, p))
+        return memo[key]
+
+    def nDelta0(i, p):
+        if MUT == 'no-delta':
+            return Fr(0)
+        if i == 1:
+            return Fr(5, 2) * nA((4, 0), p) + 2 * nA((3, 1), p)
+        if i == 2:
+            return Fr(5, 2) * nA((3, 1), p) + 2 * nA((2, 2), p)
+        if i == 3:
+            return Fr(5, 2) * nA((2, 2), p) + 2 * nA((1, 3), p)
+        return 2 * nA((1, 3), p) + 2 * nA((0, 4), p)
+
+    def nd(i, p, lam=True):
+        key = ('d', i, p, lam)
+        if key not in memo:
+            memo[key] = rup_rel(nd0(i, p, lam))
+        return memo[key]
+
+    def nd0(i, p, lam):     # ||delta_i||_p / Klo, delta_i including bar-beta_i (||mu_B|| + lambda_2)
+        cm = bbars[i] * (muB + (l2norm(p) if lam else 0))
+        if i == 2:
+            v = 2 * mu_v + cm + r1 * nDelta(2, p)
+        elif i == 3:
+            v = mu_o + Low * chi_norm(2, pow2(p)) + cm + r1 * nDelta(3, p)
+        else:
+            v = mu_y + Lyv * chi_norm(2, pow2(p)) + Lyo * chi_norm(3, pow2(p)) + cm + r1 * nDelta(4, p)
+        return v / Klo
+
+    kappa = 3 / (1 - 3 * x0)
+    Fk = 1 + kappa * x0                      # (1 + kappa x_i) <= Fk on the event {x_i <= x0}
+    cth = 1 - kappa * x0 / 2                 # <= Fk^{-1/2}
+    NMAX = 24
+    q = {'cmt1': bbars[1] * muB / Klo, 'rK': r1 / Klo,
+         'nD1': lambda p: rup_rel(nDelta(1, p) + bb1 * l2norm(pow2(p))),
+         's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth, 'Klo': Klo}
+    E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta)
+    et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac)     # eta_1 K, k-free
+    aK2_hi = 192 * khi * a_fac                    # a K^2 = 192 k
+
+    def EUt(p):                                   # upper bound of E[Ut^p], Ut = U/(a K^2) <= max(Z1, Fk y_i^2)
+        key = ('U', p)
+        if key not in memo:
+            memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p])
+        return memo[key]
+
+    def EoUt(p):
+        return sum(binom(p, l) * et1 ** l * E1o[2 * p + l] for l in range(p + 1)) + Fk ** p * To[2 * p]
+
+    def EvUt(p):
+        return sum(binom(p, l) * et1 ** l * E1v[2 * p + l] for l in range(p + 1)) + Fk ** p * Tv[2 * p]
+
+    def sq(x):
+        return sqrt_up(x)
+
+    def nUt(m, p):                                # ||Ut^m||_p <= Fk^m ||max_i y_i||_2mp^2m (on the event x_i <= X0)
+        require(2 * m * p <= NMAX and pow2(p) == p, 'norm order')
+        key = ('nUt', m, p)
+        if key not in memo:
+            memo[key] = rup_rel(Fk ** m * iroot(E1[2 * m * p] + Tn[2 * m * p], p))
+        return memo[key]
+
+    def nU(p):
+        return rup_rel(aK2_hi * nUt(1, p))
+
+    def ndel(p):                                   # || ||B - mu_B||_F ||_p <= ||mu_B|| + lambda_1 + lambda_2 <= ||mu_B|| + 2 lambda_2
+        return rup_rel(muB + 2 * l2norm(p))
+
+    def nbar(n, p):                                # ||F_bar||_p, F_bar = |F_g| + ||beta_F|| ||B - mu_B||_F
+        return rup_rel(gauss_norm(mabs[n], law.sdg[n], p) + bn[n] * ndel(p))
+
+    six_klo = 6 * klo
+    nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4)
+    epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2
+    nT8, nt8 = nbar('T', 8), nbar('tau', 8)
+    epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2
+    U2_2, U3_2, U4_2 = sq(EUt(4)), sq(EUt(6)), sq(EUt(8))
+    U2_4, U3_4 = nUt(2, 4), nUt(3, 4)
+    # the Omega majorant: Omega_bar = o0 + d_o, o0 = s3 chi_3 (shares u_3), d_o = rest
+    ndo = lambda p: rup_rel(mu_o + Low * chi_norm(2, pow2(p)) + bo * ndel(p))
+    no4 = rup_rel(s3 * chi_norm(3, 4) + ndo(4))
+    # the ||r nu - v||^2 majorant (Term C)
+    th = THETA_E
+    dv = lambda p: rup_rel(mu_v + bv * ndel(p))
+    nnu = lambda p: rup_rel(frob_norm(mu_nu, law.Cg[('nu1', 'nu1')].hi + law.Cg[('nu2', 'nu2')].hi, p)
+                            + bnu * ndel(p))
+    ebar4 = rup_rel((r1 * nnu(8) + Lvv * chi_norm(2, 8) + dv(8)) ** 2)        # ||e_bar||_4
+    ebar8 = rup_rel((r1 * nnu(16) + Lvv * chi_norm(2, 16) + dv(16)) ** 2)     # ||e_bar||_8
+    cC = (1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nnu(4) ** 2
+    # ---- Term A, B, C (one power of U per lambda_1 integration), normalized by 36 k^2 r^5 and by the lambda_2 weight
+    termA = aK2_hi ** 3 / 3 * (EUt(3) + epsc2 * U3_2)
+    termB = aK2_hi ** 2 / 2 * (EoUt(2) + (ndo(2) + epsc4 * no4) * U2_2)
+    termC = aK2_hi ** 2 / (12 * klo) * ((1 + th) ** 2 * EvUt(2) + cC * U2_2 + r1 * nT4 / six_klo * ebar4 * U2_2)
+    termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * U4_2)
+    termBp = aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2)
+    termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt(3) + cC * U3_2 + r1 * nT4 / six_klo * ebar4 * U3_2)
+    if MUT == 'no-om-term':
+        termB = termBp = Fr(0)
+    # ---- the Jacobian-correction terms: the part r Omega_bar of (lambda_2 + r Omega_bar), integrated against nu'
+    # D  = E[Omega_bar (A + B + C integrands)],  D' = E[Omega_bar (A' + B' + C' integrands)]
+    nT16 = nbar('T', 16)
+    termD = (aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2)
+             + aK2_hi ** 2 / 2 * no4 * no4 * (U2_2 + epsc4 * U2_4)
+             + aK2_hi ** 2 / (12 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U2_4)
+    termDp = (aK2_hi ** 4 / 4 * (EoUt(4) + (ndo(2) + epsc4 * no4) * U4_2)
+              + aK2_hi ** 3 / 3 * no4 * no4 * (U3_2 + epsc4 * U3_4)
+              + aK2_hi ** 3 / (18 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U3_4)
+    if MUT == 'no-om-term':
+        termD = termDp = Fr(0)
+    I2, I3 = I_hi[2], I_hi[3]
+    lead = I3 * (termA + termB + termC) + r1 * I2 * termD
+    leadp = I3 * (termAp + termBp + termCp) + r1 * I2 * termDp
+    # ---- normalizer floor Z_r / (36 k^2 r^2) >= m3(b'_lo) - corrections (NOTE section 6)
+    Lb = cholesky([[IV(x.lo, x.hi) for x in row] for row in law.Cb])
+    eta = sqrt_up(sum(absup(Lb[i][j] - (SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1)))
+    R0 = Fr(R0_COUPLING)
+    bp_lo = blo - eps - eta * R0
+    bp_hi = bhi + eps + eta * R0
+    bp_lo = Fr(math.floor(bp_lo * 2 ** 40), 2 ** 40)
+    bp_hi = Fr(math.ceil(bp_hi * 2 ** 40), 2 ** 40)
+    pR0 = chi_tail(3, R0)
+    m3lo = m3(bp_lo).lo
+    tail_c = sqrt_up(contact_moment(4, 0, bp_lo).hi) * sqrt_up(pR0)
+    # upper bounds of E[h(B)] for Loewner-monotone h through b'' = bp_hi, plus the outside-ball part
+    nBF16 = rup_rel(muB + sig * chi_norm(3, 16))                 # || ||B||_F ||_16
+    out_ball = sqrt_up(pR0)
+
+    def up_moment(a, c):                    # E[det^a tr^c 1{B > 0}] <= contact value at b'' + outside-ball part
+        crude = rup_rel(nBF16 ** (2 * a + c))      # det <= ||B||^2, tr <= sqrt2 ||B|| <= 2 ||B||: crude, times P^(1/2)
+        return rup_rel(contact_moment(a, c, bp_hi).hi + 2 ** c * crude * out_ball)
+    F2 = up_moment(4, 0)                    # E[(det B)^4 1{B > 0}]
+    DT, TDT, TTDT = up_moment(1, 1), up_moment(1, 2), up_moment(1, 3)
+    sdT, sdtau = law.sdQ['T'], law.sdQ['tau']
+    q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1))
+    nF = sqrt_up(F2)
+    corr_G0 = nF * sqrt_up(q_bad) if q_bad > 0 else Fr(0)
+    corr_eps = (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) / six_klo * nF
+    trO = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in zip(ONAMES, (1, 1, 2)))
+    trV = law.Cg[('v1', 'v1')].hi + law.Cg[('v2', 'v2')].hi
+    Eog = rup_rel(mu_o + sqrt_up(trO))
+    corr_om = Eog * DT + bo * (TDT + muB * DT)
+    Evg2 = mu_v ** 2 + trV
+    corr_v = (Fr(17, 16) * Evg2 * DT + 17 * bv ** 2 * (TTDT + 2 * muB * TDT + muB * muB * DT)) / (4 * klo)
+    corr = tail_c + corr_G0 + corr_eps + r1 * (corr_om + corr_v)
+    if MUT == 'no-zfloor-correction':
+        corr = Fr(0)
+    zt = m3lo - corr
+    require(zt > 0, 'normalizer floor not positive')
+    main = KB * min(gbar * lead, g0 * lead + r1 * Lg * leadp) / zt
+    # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q), ||W 1_typed||_q / r^2 <= g1 g2
+    six_khi = 6 * khi
+    trB = sum(x.hi for x in (law.Cb[0][0], law.Cb[1][1], law.Cb[2][2]))
+    trOQ = sum(m * law.C[(n, n)].hi for n, m in zip(ONAMES, (1, 1, 2)))
+    trVQ = law.C[('v1', 'v1')].hi + law.C[('v2', 'v2')].hi
+    trNQ = law.C[('nu1', 'nu1')].hi + law.C[('nu2', 'nu2')].hi
+
+    # W_r / r^2 <= |c1| D [|c2| (D + r Om tr + r^2 Om^2) + r ||e||^2 (tr + r Om)] on the typed support, D = det B 1{B > 0},
+    # tr = tr B, Om = ||Omega'||_F (NOTE section 7)
+    sd_tr = sqrt_up(law.C[('a11', 'a11')].hi + law.C[('a22', 'a22')].hi + 2 * absup(law.C[('a11', 'a22')]))
+    m_tr = absup(mu['a11'] + mu['a22'])
+
+    def wnorm(qq):
+        p4, p8 = 4 * qq, 8 * qq
+        nD = lambda p: det_norm(p, bp_hi, muB, sig)
+        ntr = lambda p: gauss_norm(m_tr, sd_tr, p)
+        no = lambda p: frob_norm(mu_o, trOQ, p)
+        nT = lambda p: gauss_norm(mabs['T'], sdT, p)
+        nc2 = lambda p: six_khi + r1 * nT(p) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, p)
+        ne = lambda p: r1 * frob_norm(mu_nu, trNQ, p) + frob_norm(mu_v, trVQ, p)
+        g1 = (six_khi + r1 * nT(p4)) * nD(p4)
+        g2 = (nc2(p4) * (nD(p4) + r1 * no(p8) * ntr(p8) + r1 * r1 * no(p8) ** 2)
+              + r1 * ne(p8) ** 2 * (ntr(p4) + r1 * no(p4)))
+        return g1 * g2
+    thr = 3 * klo / (10 * r1)
+    qA2 = Fr(0)
+    args = []
+    for blk in BLOCKS:
+        a_, c_ = blk
+        mem = block_members(*blk)
+        m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem))       # |mean of d_x^a d_u^c f(0)| <= ||mean||_F for unit u
+        best = None
+        for frac in (Fr(7, 8), Fr(15, 16)):                        # split t = frac t + (1 - frac) t between jet and Taylor tail
+            if c_ == 0:                                            # scalar block
+                x_ = (thr * frac - m_) / law.sdQ[mem[0][0]]
+                pj = gauss_tail2(m_, law.sdQ[mem[0][0]], thr * frac)
+            elif c_ == 1:                                          # vector block: ||X - mean|| <= sqrt(lambda_max) chi_2
+                lmax = max(sum(absup(law.C[(ni, nj)]) for nj, mj in mem) for ni, mi in mem)
+                x_ = (thr * frac - m_) / sqrt_up(lmax)
+                pj = chi_tail(2, x_) if x_ > 0 else Fr(1)
+            else:                                                  # ||X||_op = max_theta |Q(theta)|, Q a trigonometric
+                rho = 1 - Fr(c_, NET_M)                            # polynomial of degree c: Bernstein |Q'| <= c max|Q| and
+                sdn = sqrt_up(law.net_var[blk])                    # net gaps <= 2/NET_M give max|Q| <= max_j |Q_j| / rho
+                x_ = (thr * frac * rho - m_) / sdn
+                pj = min(Fr(1), len(NET) * gauss_tail2(m_, sdn, thr * frac * rho))
+            pj += (law.S[blk] * (sqE + znorm(32)) / (thr * (1 - frac))) ** 32
+            if best is None or pj < best[0]:
+                best = (pj, x_)
+        args.append(best[1])
+        qA2 += best[0]
+    a_hi = Fr(4, 3) / klo * a_fac
+    qfar = Fr(0)
+    qx = Fr(0)
+    sdB11 = sqrt_up(law.C[('a11', 'a11')].hi)
+    if bbar > 0:
+        lam_far = 1 / (4 * a_hi * r1 * bbar * bbar)
+        args.append((lam_far - absup(mu['a11'])) / sdB11)
+        qfar = gauss_tail1(absup(mu['a11']), sdB11, lam_far)
+
+    def lam2_tail(s):                       # P(lambda_2 > s) <= P(sig chi_3 > s - ||mu_B||)
+        x_ = (s - muB) / sig
+        args.append(x_)
+        return chi_tail(3, x_) if x_ > 0 else Fr(1)
+    for i in (1, 2, 3, 4):                         # x_i > X0  <=>  y_i' + bar-beta_i lambda_2 > X0 / (a r1 bar-beta_i)
+        if bbars[i] == 0:
+            continue
+        tx = x0 / (a_hi * r1 * bbars[i])
+        if i == 1:
+            cb = 12 * khi + bbars[1] * muB
+            if tx <= cb:
+                qx += Fr(1)
+            else:
+                qx += (r1 * nDelta(1, 32) / ((tx - cb) / 2)) ** 32
+                qx += lam2_tail((tx - cb) / (2 * bbars[1]))
+        else:
+            s_i = {2: s2, 3: s3, 4: s4}[i]
+            x_ = tx / (3 * s_i)
+            args.append(x_)
+            qx += chi_tail(NU[i], x_) + (nd(i, 32, lam=False) * Klo / (tx / 3)) ** 32 + lam2_tail(tx / (3 * bbars[i]))
+    qs = [min(Fr(1), x) for x in (qA2, qfar, qx)]
+    tail_num = min(wnorm(qq) * sum(pow_tail(x, qq) for x in qs) for qq in QS)
+    if last_band:
+        # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian or chi tail argument scales at least like 1/r and is
+        # >= 3 at R, so each such piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32;
+        # after the power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of
+        # tail / r^3 over (0, R] is attained on [R/2, R]
+        require(min(args) >= 3, 'last-band tail arguments')
+        require(max(qA2, qfar, qx) < 1, 'last-band tail pieces')
+        rdiv = r1 / 2
+    else:
+        rdiv = r0
+    tail = tail_num / (36 * klo * klo * zt * rdiv ** 3)
+    total = main + tail
+    return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'tail_num': tail_num, 'KB': KB, 'g0': g0, 'gbar': gbar, 'I3': I3,
+            'termA': termA, 'termB': termB, 'termC': termC, 'termD': termD, 'qA2': qA2, 'qfar': qfar, 'qx': qx,
+            'm3lo': m3lo, 'corr': corr, 'eps': eps, 'eta': eta, 'min_arg': min(args)}
+# ----------------------------------------------------------------------------- the (k, b) boxes of the sweep
+F_ = Fr
+GRID = (
+    ((F_(1, 2), F_(3, 4)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(3, 4), F_(1)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(1), F_(5, 4)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(5, 4), F_(3, 2)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(3, 2), F_(13, 8)), (F_(0), F_(243, 512), F_(1))),
+    ((F_(13, 8), F_(7, 4)), (F_(0), F_(309, 1024), F_(899, 1024), F_(1))),
+    ((F_(7, 4), F_(15, 8)), (F_(0), F_(37, 256), F_(105, 256), F_(933, 1024), F_(1))),
+    ((F_(15, 8), F_(31, 16)), (F_(0), F_(37, 512), F_(207, 1024), F_(225, 512), F_(113, 128), F_(1))),
+    ((F_(31, 16), F_(63, 32)), (F_(0), F_(37, 1024), F_(103, 1024), F_(111, 512), F_(439, 1024), F_(421, 512), F_(1))),
+    ((F_(63, 32), F_(2)), (F_(0), F_(1, 512), F_(5, 1024), F_(5, 512), F_(19, 1024), F_(35, 1024), F_(1, 16), F_(29, 256),
+                           F_(209, 1024), F_(377, 1024), F_(685, 1024), F_(1))),
+)
+
+
+# ----------------------------------------------------------------------------- bands, boxes, runs
+TOP_OCTAVES = 7            # octaves [2^-(j+1), 2^-j], j = 6..12 (r up to 1/64), split into 8 sub-bands
+LOW_OCTAVES = 16           # octaves j = 13..28, split into 4 sub-bands
+R_LAST = Fr(1, 2 ** 29)    # last band [0, 2^-29]
+COMBO = (Fr(1, 64), Fr(1, 32))     # (X_0, theta) of NOTE sections 4-5
+
+
+def bands():
+    """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST]."""
+    out = []
+    for j in range(6, 6 + TOP_OCTAVES + LOW_OCTAVES):
+        hi = Fr(1, 2 ** j)
+        lo = hi / 2
+        m = 8 if j < 6 + TOP_OCTAVES else 4
+        for i in range(m):
+            out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m))
+    out.append((Fr(0), R_LAST))
+    require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover')
+    out = sorted(out)
+    require(all(out[i][1] == out[i + 1][0] for i in range(len(out) - 1)), 'bands contiguous')
+    return out
+
+
+def dec_down(x, places=4):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_up(x, places=4):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def grid_boxes():
+    out = []
+    for kb, bg in GRID:
+        require(bg[0] == 0 and bg[-1] == 1 and all(bg[i] < bg[i + 1] for i in range(len(bg) - 1)), 'b grid')
+        for i in range(len(bg) - 1):
+            out.append(((Fr(bg[i]), Fr(bg[i + 1])), (Fr(kb[0]), Fr(kb[1]))))
+    ks = [kb for kb, _ in GRID]
+    require(ks[0][0] == BAND_K[0] and ks[-1][1] == BAND_K[1] and all(ks[i][1] == ks[i + 1][0] for i in range(len(ks) - 1)),
+            'k boxes cover K')
+    return tuple(out)
+
+
+CG3_02_LO = Fr('22424320.655069')     # certified lower end of c_G^(3)(0, 2) (Math-#208)
+
+
+def band_record(args):
+    """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down,
+    6 places) for each box in BOXES order, and the largest tail part (rounded up, 12 places)."""
+    r0, r1 = args
+    law = band_law(r0, r1)
+    last = (r0 == 0)
+    bound, zfloor, tails = [], [], []
+    for bb, kb in BOXES:
+        res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=COMBO[0], theta=COMBO[1])
+        bound.append(dec_up(res['total']))
+        zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6))
+        tails.append(res['tail'])
+    return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12)}
+
+
+def full_run(procs):
+    bl = bands()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, bl)
+    return assemble(recs)
+
+
+def assemble(recs):
+    table = {}
+    for rs in R_STARS:
+        mx, arg, zmin = Fr(0), None, None
+        for rec in recs:
+            if Fr(rec['r'][1]) <= rs:
+                for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']):
+                    v = Fr(bs)
+                    if v > mx:
+                        mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]}
+                    z = Fr(zs)
+                    zmin = z if zmin is None or z < zmin else zmin
+        table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG3_02': dec_up(mx / CG3_02_LO, 6),
+                          'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1', 'scientific_effect': 'NONE',
+            'certified': True, 'mutant': MUT, 'L_min': L_MIN, 'dimension': 3,
+            'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]},
+            'parameters': params(), 'C_table': table, 'bands': recs}
+
+
+def params():
+    return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBO_X0_THETA': [str(COMBO[0]), str(COMBO[1])], 'THETA_E': str(THETA_E),
+            'QS': list(QS), 'R0_COUPLING': R0_COUPLING, 'NET_M': NET_M, 'R_STARS': [str(x) for x in R_STARS],
+            'TOP_OCTAVES': TOP_OCTAVES, 'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST),
+            'GRID': [[[str(kb[0]), str(kb[1])], [str(x) for x in bg]] for kb, bg in GRID]}
+
+
+def results_text(res):
+    """RESULTS.json layout: the summary indented, then one line per band record."""
+    head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True)
+    lines = ',\n'.join('  ' + json.dumps(rec, sort_keys=True) for rec in res['bands'])
+    return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n'
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def replay_sample():
+    """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band."""
+    bl = bands()
+    idx = []
+    for rs in R_STARS:
+        j = max(i for i, b in enumerate(bl) if b[1] == rs)
+        idx.append(j)
+    idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))]
+    idx.append(0)
+    return sorted(set(idx))
+
+
+def check_run(procs, full=False):
+    ref = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout')
+    require(ref['parameters'] == params(), 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json produced under a mutant')
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure')
+    require(abs(m3(0).mid() - (Fr(7, 2) - 2 * SQRT2.mid())) < Fr(1, 10 ** 40), 'm_(3,0) closed form')
+    bl = bands()
+    require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs')
+    idx = list(range(len(bl))) if full else replay_sample()
+    with multiprocessing.Pool(procs) as pool:          # in order; the first differing band exits (and stops the pool)
+        for i, rec in zip(idx, pool.imap(band_record, [bl[i] for i in idx])):
+            require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r']))
+    # the table is the maximum over the stored records
+    asm = assemble(ref['bands'])
+    require(asm['C_table'] == ref['C_table'], 'C table is not the maximum of the stored bounds')
+    # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G^(3)(0, 2) of Math-#208 (a lower bound for every
+    # d = 3 cap-route constant valid at (0, 2) for the reference kernel, which this certificate covers)
+    for rs, row in ref['C_table'].items():
+        require(Fr(row['C']) > CG3_02_LO, 'C below the sharp cap coefficient')
+    print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()},
+                      'z_star': {rs: row['z_star'] for rs, row in ref['C_table'].items()}, 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.mc:
+        rr, bb, kk, nn, sd = args.mc
+        print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd))))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({'C': {rs: row['C'] for rs, row in res['C_table'].items()},
+                      'z': {rs: row['z_star'] for rs, row in res['C_table'].items()}}))
+BOXES = grid_boxes()
+
+
+# ----------------------------------------------------------------------------- floating control (not part of the certificate)
+def mc_event(r, b, k, n, seed, deg=7, nang=24):
+    """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel in d = 3 (floating point): the
+    Taylor jet of order deg at the midpoint is sampled under Q (exact pin series evaluated in floating point), W_r is
+    evaluated from the Hessians at M and S, and M_3, M_4 are the partial-block operator norms ([CAP] (1)) maximized over a
+    grid of the cylinder D (binary forms maximized over nang angles; floating, so slightly below the true suprema)."""
+    import random
+    pins, _ = functionals()
+    P = [pins[nm] for nm in PIN_NAMES]
+    jets = [(a, b1, m - a - b1) for m in range(deg + 1) for a in range(m, -1, -1) for b1 in range(m - a, -1, -1)]
+    J = [[term(1, 0, a, b1, b2, X0PT)] for (a, b1, b2) in jets]
+    nj = len(J)
+
+    def ev(F, G, key):
+        ser, w, _ = pair_series(F, G, key)
+        return sum(float(c) * r ** p for p, c in ser.items())
+    Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(NP)] for i in range(NP)]
+    Sjp = [[ev(J[i], P[j], ('jp3', deg, i, j)) for j in range(NP)] for i in range(nj)]
+    Sjj = [[ev(J[i], J[j], ('jj3', deg, i, j)) for j in range(nj)] for i in range(nj)]
+
+    def solve(A, bv):
+        m = len(A)
+        M = [A[i][:] + [bv[i]] for i in range(m)]
+        for c in range(m):
+            p = max(range(c, m), key=lambda i: abs(M[i][c]))
+            M[c], M[p] = M[p], M[c]
+            for i in range(c + 1, m):
+                f = M[i][c] / M[c][c]
+                for j in range(c, m + 1):
+                    M[i][j] -= f * M[c][j]
+        x = [0.0] * m
+        for i in reversed(range(m)):
+            x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i]
+        return x
+    vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k] + [0.0] * (NP - 3)
+    Wt = [solve(Spp, Sjp[i]) for i in range(nj)]
+    mu = [sum(Wt[i][j] * vr[j] for j in range(NP)) for i in range(nj)]
+    C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(NP)) for j in range(nj)] for i in range(nj)]
+    # Cholesky with clamped pivots (the conditional covariance is positive semidefinite up to rounding)
+    L = [[0.0] * nj for _ in range(nj)]
+    for i in range(nj):
+        for j in range(i + 1):
+            s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j))
+            if i == j:
+                L[i][i] = math.sqrt(s) if s > 1e-300 else 0.0
+            else:
+                L[i][j] = s / L[j][j] if L[j][j] > 0 else 0.0
+    index = {al: i for i, al in enumerate(jets)}
+    fact = [math.factorial(i) for i in range(deg + 2)]
+
+    def deriv(cf, be, p):
+        """d^be f(p) from the jet (Taylor at 0)."""
+        tot = 0.0
+        for al, i in index.items():
+            if al[0] >= be[0] and al[1] >= be[1] and al[2] >= be[2]:
+                e = (al[0] - be[0], al[1] - be[1], al[2] - be[2])
+                tot += cf[i] * (p[0] ** e[0] * p[1] ** e[1] * p[2] ** e[2]) / (fact[e[0]] * fact[e[1]] * fact[e[2]])
+        return tot
+    angs = [(math.cos(math.pi * t / nang), math.sin(math.pi * t / nang)) for t in range(nang)]
+    pts = [(x, 0.0, 0.0) for x in (-2 * r, -r, 0.0, r, 2 * r)]
+    pts += [(x, 2 * r * math.cos(2 * math.pi * t / 6), 2 * r * math.sin(2 * math.pi * t / 6)) for x in (-2 * r, 0.0, 2 * r)
+            for t in range(6)]
+
+    def block_norm(vals, c):
+        """Operator norm of the symmetric c-form with components vals[(b1, b2)] (b1 + b2 = c) over unit vectors."""
+        if c == 0:
+            return abs(vals[(0, 0)])
+        best = 0.0
+        for (u1, u2) in angs:
+            q = sum(math.comb(c, b1) * vals[(b1, c - b1)] * u1 ** b1 * u2 ** (c - b1) for b1 in range(c + 1))
+            best = max(best, abs(q))
+        return best
+
+    def Mj(cf, j):
+        best = 0.0
+        for p in pts:
+            for a in range(j + 1):
+                c = j - a
+                vals = {(b1, c - b1): deriv(cf, (a, b1, c - b1), p) for b1 in range(c + 1)}
+                best = max(best, block_norm(vals, c))
+        return best
+
+    def hess(cf, p):
+        return [[deriv(cf, tuple(int(x == i) + int(x == j) for x in range(3)), p) for j in range(3)] for i in range(3)]
+
+    def det3f(H):
+        return (H[0][0] * (H[1][1] * H[2][2] - H[1][2] * H[2][1]) - H[0][1] * (H[1][0] * H[2][2] - H[1][2] * H[2][0])
+                + H[0][2] * (H[1][0] * H[2][1] - H[1][1] * H[2][0]))
+
+    def neg_def(H):
+        return H[0][0] < 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and det3f(H) < 0
+
+    def index2(H):
+        """Number of negative eigenvalues of a symmetric 3 x 3 matrix is 2: det > 0 and not positive definite."""
+        d = det3f(H)
+        pd = H[0][0] > 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and d > 0
+        return d > 0 and not pd
+    rng = random.Random(seed)
+    num = den = 0.0
+    acap = 4 / (3 * k)
+    for _ in range(n):
+        z = [rng.gauss(0, 1) for _ in range(nj)]
+        cf = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(nj)]
+        HM, HS = hess(cf, (-r / 2, 0.0, 0.0)), hess(cf, (r / 2, 0.0, 0.0))
+        if not (neg_def(HM) and index2(HS)):
+            continue
+        Wv = abs(det3f(HM) * det3f(HS))
+        den += Wv
+        a11, a22, a12 = -HM[1][1], -HM[2][2], -HM[1][2]
+        lam1 = (a11 + a22) / 2 - math.sqrt(((a11 - a22) / 2) ** 2 + a12 * a12)
+        M3 = Mj(cf, 3)
+        if lam1 <= acap * r * M3 * M3:
+            num += Wv
+            continue
+        if r * Mj(cf, 4) > 3 * k / 10:
+            num += Wv
+    return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3 if den else None,
+            'Z_over_r2': den / n / r ** 2}
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py b/frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py
new file mode 100644
index 000000000..36b39dbd3
--- /dev/null
+++ b/frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py
@@ -0,0 +1,1166 @@
+#!/usr/bin/env python3
+"""Explicit constants for [LP] Theorem B on the C8 band, d = 2 and d = 3 (NOTE.md Theorem L and its corollaries).
+
+For the radial ledger A_r = 12 pi_r(R; v_r) Z_r / r^2 of [LP] (10.2) the certificate proves, for 0 < r <= R, every frame,
+every torus side L >= 10 and the reference kernel,
+
+    -(a_dn r^2 + eps) <= A_r(b, k, u) / A_0^ref(b, k) - 1 <= a_up r^2 + eps         (b in [0, 1], k in [1/2, 2]),
+
+with A_0^ref the closed-form contact integrand of the Gaussian kernel, and assembles from it two-sided second-order bounds for
+nu_cand and nu_eld, the difference constant C_{B,K} of [LP] (1.2), and the constants of (12.1) and (12.3).
+
+Standard library only; exact rational interval arithmetic throughout.
+- Taylor models in r (fixed-point integer coefficients, every rounding carried in the remainder) for the pin covariance, its
+  determinant, the pin energies and the law of the transverse Hessian and of Omega' (section 2 of NOTE.md).
+- The interval band laws and the box-bound engines of Math-#206 (d = 2) and Math-#212 (d = 3), as byte-identical pinned
+  copies engine_e2.py and engine_e3.py (section 2; SOURCE_MAP.json).
+
+  python3 -B -S theorem_b.py --procs N              regenerate RESULTS.json
+  python3 -B -S theorem_b.py --check --procs N      replay: Taylor models, rate sweep, reference integrals, a sample of
+                                                    cap bands, and the assembled tables (exact comparison)
+  python3 -B -S theorem_b.py --check-full --procs N replay every cap band as well
+  python3 -B -S theorem_b.py --mutant NAME --check  a semantic mutation; the replay must then fail (exit 1)
+  python3 -B -S theorem_b.py --controls             floating controls (not part of the certificate)
+"""
+import argparse
+import importlib.util
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-om-term', 'no-bad-set', 'no-det', 'no-torus', 'cap-half')
+MUT = None
+
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+TM_S = 6                       # Taylor models on [0, 2^-TM_S] = [0, 1/64]
+TM_N = 8                       # Taylor-model degree
+TM_P = 288                     # fixed-point bits of the Taylor-model coefficients
+R0C = Fr(12)                   # Loewner coupling radius (d = 3)
+TINY = Fr(1, 10 ** 60)         # polynomial coefficients below this are rounding residue of exact zeros
+SUB_B = tuple((Fr(i, 8), Fr(i + 1, 8)) for i in range(8))
+SUB_K = tuple((Fr(1, 2) + Fr(j, 8), Fr(1, 2) + Fr(j + 1, 8)) for j in range(12))
+SUBBOXES = tuple((bb, kb) for kb in SUB_K for bb in SUB_B)
+NH = 4096                      # b-cells of the reference integrals
+BLOCK = 16                     # cells per second-derivative block
+CAP_B = tuple((Fr(i, 4), Fr(i + 1, 4)) for i in range(4))
+CAP_K = tuple((Fr(1, 2) + Fr(i, 32), Fr(1, 2) + Fr(i + 1, 32)) for i in range(8)) \
+    + tuple((Fr(3, 4) + Fr(i, 16), Fr(3, 4) + Fr(i + 1, 16)) for i in range(4)) + ((Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(2)))
+CAP_BOXES = tuple((bb, kb) for kb in CAP_K for bb in CAP_B)
+C_REF_197 = Fr('0.0023869380211529483091019117')    # Math-#197, reference c_{B,K} in d = 2 (cross-check only)
+C_REF_205 = Fr('0.0009743823660242504624710565')    # Math-#205, reference c_{B,K} in d = 3 (cross-check only)
+
+
+def require(cond, msg):
+    if not cond:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def load_engine(name, fn):
+    spec = importlib.util.spec_from_file_location(name, os.path.join(HERE, fn))
+    mod = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    return mod
+
+
+E2 = load_engine('engine_e2', 'engine_e2.py')
+E3 = load_engine('engine_e3', 'engine_e3.py')
+ENG = {2: E2, 3: E3}
+
+
+def exp_minus(M, t):
+    """Enclosure of exp(-t) for rational t of either sign."""
+    t = Fr(t)
+    return M.exp_neg_point(t) if t >= 0 else M.exp_neg_point(-t).inv()
+
+
+def grid_up(x, g=2 ** 40):
+    return Fr(math.ceil(Fr(x) * g), g)
+
+
+def grid_dn(x, g=2 ** 40):
+    return Fr(math.floor(Fr(x) * g), g)
+
+
+def dec_up(x, places):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_down(x, places):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+# ============================================================================= 1. Taylor models in r
+def cdiv(a, b):
+    return -((-a) // b)
+
+
+class TM:
+    """f(r) = sum_{j <= N} (C_j / 2^P) r^j + theta (E / 2^P), |theta| <= 1, for every r in [0, 2^-s]."""
+    __slots__ = ('C', 'E', 's', 'N')
+
+    def __init__(self, C, E, s, N):
+        self.C, self.E, self.s, self.N = C, E, s, N
+
+    @classmethod
+    def from_fracs(cls, coeffs, e, s, N):
+        items = coeffs.items() if isinstance(coeffs, dict) else enumerate(coeffs)
+        C = [0] * (N + 1)
+        E = Fr(e)
+        for p, x in items:
+            x = Fr(x)
+            if x == 0:
+                continue
+            if p <= N:
+                v = round(x * (1 << TM_P))
+                C[p] += v
+                E += abs(x - Fr(v, 1 << TM_P)) / Fr(2) ** (s * p)
+            else:
+                E += abs(x) / Fr(2) ** (s * p)
+        return cls(C, cdiv(E.numerator << TM_P, E.denominator), s, N)
+
+    @classmethod
+    def const(cls, x, s, N):
+        return cls.from_fracs({0: Fr(x)}, 0, s, N)
+
+    def polybound(self):
+        return sum(cdiv(abs(c), 1 << (self.s * j)) for j, c in enumerate(self.C) if c)
+
+    def bound(self):
+        return self.polybound() + self.E
+
+    def frac_coeffs(self):
+        return [Fr(c, 1 << TM_P) for c in self.C]
+
+    def frac_e(self):
+        return Fr(self.E, 1 << TM_P)
+
+    def __add__(self, o):
+        if not isinstance(o, TM):
+            o = TM.const(o, self.s, self.N)
+        return TM([a + b for a, b in zip(self.C, o.C)], self.E + o.E, self.s, self.N)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return TM([-a for a in self.C], self.E, self.s, self.N)
+
+    def __sub__(self, o):
+        if not isinstance(o, TM):
+            o = TM.const(o, self.s, self.N)
+        return self + (-o)
+
+    def scale(self, x):
+        x = Fr(x)
+        C = []
+        E = Fr(self.E, 1 << TM_P) * abs(x)
+        for j, c in enumerate(self.C):
+            v = Fr(c, 1 << TM_P) * x
+            w = round(v * (1 << TM_P))
+            C.append(w)
+            E += abs(v - Fr(w, 1 << TM_P)) / Fr(2) ** (self.s * j)
+        return TM(C, cdiv(E.numerator << TM_P, E.denominator), self.s, self.N)
+
+    def __mul__(self, o):
+        if not isinstance(o, TM):
+            return self.scale(o)
+        N, s = self.N, self.s
+        S = [0] * (2 * N + 1)
+        for i, a in enumerate(self.C):
+            if a:
+                for j, b in enumerate(o.C):
+                    if b:
+                        S[i + j] += a * b
+        C, E = [], 0
+        for k in range(N + 1):
+            v = S[k] >> TM_P                                  # floor: error below one unit, times r^k <= 1
+            C.append(v)
+            if S[k] - (v << TM_P):
+                E += 1
+        for k in range(N + 1, 2 * N + 1):
+            if S[k]:
+                E += cdiv(abs(S[k]), 1 << (TM_P + s * k))
+        E += cdiv(self.E * o.polybound() + o.E * self.polybound() + self.E * o.E, 1 << TM_P)
+        return TM(C, E, s, N)
+    __rmul__ = __mul__
+
+    def inv(self):
+        c0 = Fr(self.C[0], 1 << TM_P)
+        require(c0 != 0, 'Taylor-model inverse at a zero constant term')
+        h = (self - TM([self.C[0]] + [0] * self.N, 0, self.s, self.N)).scale(1 / c0)
+        H = Fr(h.bound(), 1 << TM_P)
+        require(H < Fr(1, 2), 'Taylor-model inverse: |h| too large')
+        acc = TM.const(1, self.s, self.N)
+        p = TM.const(1, self.s, self.N)
+        mh = -h
+        for _ in range(self.N):
+            p = p * mh
+            acc = acc + p
+        tail = H ** (self.N + 1) / (1 - H)
+        acc = TM(acc.C, acc.E + cdiv(tail.numerator << TM_P, tail.denominator), self.s, self.N)
+        return acc.scale(1 / c0)
+
+    def enclose(self, r0, r1):
+        lo = hi = Fr(0)
+        for j, c in enumerate(self.C):
+            if c:
+                x = Fr(c, 1 << TM_P)
+                a, b = Fr(r0) ** j, Fr(r1) ** j
+                lo += x * (a if x >= 0 else b)
+                hi += x * (b if x >= 0 else a)
+        e = self.frac_e()
+        return lo - e, hi + e
+
+
+def cov_tm(M, F, G, key, s, N, M1):
+    """Taylor model on [0, 2^-s] of the covariance of F, G under the unnormalized torus kernel: the engine's exact series,
+    its truncation tail and its torus remainder (as in the engines' cov_band, at r1 = 2^-s)."""
+    R = Fr(1, 2 ** s)
+    ser, w, absterms = M.pair_series(F, G, key)
+    A = max((t[1] for t in absterms), default=0)
+    tail = tor = Fr(0)
+    for t in absterms:
+        acc, aa, gys = t[0], t[1], t[2:]
+        hprod = fprod = 1
+        for g in gys:
+            hprod *= M.hcoef(g)
+            fprod *= math.factorial(g)
+        tail += acc * abs(hprod)
+        tor += acc * fprod * Fr(max(w, 1)) ** aa
+    tail *= 2 * Fr(M.NX + 1) ** A * R ** (M.NX + 1 - w)
+    tor = tor * M1 / (1 - Fr(27183, 10000) * R)
+    return TM.from_fracs(ser, tail + tor, s, N)
+
+
+def solve_tm(A, B):
+    """Gaussian elimination without pivoting (A(0) is positive definite): X[c] solves A X[c] = B[c]; also det A."""
+    n = len(A)
+    Mx = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    det, invs = None, []
+    for col in range(n):
+        piv = Mx[col][col]
+        det = piv if det is None else det * piv
+        ip = piv.inv()
+        invs.append(ip)
+        for i in range(col + 1, n):
+            f = Mx[i][col] * ip
+            for j in range(col, len(Mx[i])):
+                Mx[i][j] = Mx[i][j] - f * Mx[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            acc = Mx[i][n + c]
+            for j in range(i + 1, n):
+                acc = acc - Mx[i][j] * x[j]
+            x[i] = acc * invs[i]
+        X.append(x)
+    return X, det
+
+
+class TMLaw:
+    pass
+
+
+TM_NAMES = {2: ('w', 'om'), 3: ('a11', 'a22', 'a12', 'o11', 'o22', 'o12')}
+
+
+def tm_law(d, s=TM_S, N=TM_N, L=10):
+    """Taylor-model law over r in [0, 2^-s] of the targets TM_NAMES[d] (every frame, every torus side >= L, the reference
+    kernel): alpha, beta (mean = b alpha + k beta), the unnormalized covariance block, the unnormalized pin energies, det."""
+    M = ENG[d]
+    names = TM_NAMES[d]
+    pins, targets = M.functionals()
+    P = [pins[nm] for nm in M.PIN_NAMES]
+    idx = [M.TARGET_NAMES.index(nm) for nm in names]
+    T = [targets[nm] for nm in names]
+    M1 = Fr(0) if MUT == 'no-torus' else M.torus_M1(L)
+    n, nt = len(P), len(T)
+    Spp = [[None] * n for _ in range(n)]
+    for i in range(n):
+        for j in range(i, n):
+            Spp[i][j] = Spp[j][i] = cov_tm(M, P[i], P[j], ('p', i, j), s, N, M1)
+    Stp = [[cov_tm(M, T[a], P[j], ('tp', idx[a], j), s, N, M1) for j in range(n)] for a in range(nt)]
+    Stt = [[None] * nt for _ in range(nt)]
+    for a in range(nt):
+        for b in range(a, nt):
+            i, j = idx[a], idx[b]
+            if i <= j:
+                Stt[a][b] = Stt[b][a] = cov_tm(M, T[a], T[b], ('tt', i, j), s, N, M1)
+            else:
+                Stt[a][b] = Stt[b][a] = cov_tm(M, T[b], T[a], ('tt', j, i), s, N, M1)
+    zero = TM.const(0, s, N)
+    kvec = [TM.from_fracs({3: Fr(-1, 2)}, 0, s, N), TM.from_fracs({2: Fr(-1)}, 0, s, N), TM.const(12, s, N)] + [zero] * (n - 3)
+    e1 = [TM.const(1, s, N)] + [zero] * (n - 1)
+    X, det = solve_tm(Spp, Stp + [e1, kvec])
+    W, xs0, xs1 = X[:nt], X[nt], X[nt + 1]
+    law = TMLaw()
+    law.d, law.M1, law.th1, law.s, law.N = d, M1, (Fr(0) if MUT == 'no-torus' else M.theta_minus_one(L)), s, N
+    law.alpha = {nm: W[a][0] for a, nm in enumerate(names)}
+    law.beta = {nm: sum((W[a][j] * kvec[j] for j in range(n)), zero) for a, nm in enumerate(names)}
+    law.Cu = {}
+    for a, na in enumerate(names):
+        for b, nb in enumerate(names):
+            law.Cu[(na, nb)] = Stt[a][b] - sum((W[a][l] * Stp[b][l] for l in range(n)), zero)
+    law.e_bb = xs0[0]
+    law.e_bk = sum((kvec[j] * xs0[j] for j in range(n)), zero)
+    law.e_kk = sum((kvec[j] * xs1[j] for j in range(n)), zero)
+    law.det = det
+    if d == 3:
+        # E[O - mu_O | B] = sum_l G[O][l] (B_l - mu_B_l) in natural coordinates (B11, B22, B12), B = -A:
+        # G = -Cu(O, A) Cu(A, A)^-1
+        an = ('a11', 'a22', 'a12')
+        Xg, _ = solve_tm([[law.Cu[(x, y)] for y in an] for x in an], [[law.Cu[(o, y)] for y in an] for o in ('o11', 'o22', 'o12')])
+        law.G = {o: [-Xg[c][l] for l in range(3)] for c, o in enumerate(('o11', 'o22', 'o12'))}
+    return law
+
+
+def tm_summary(law):
+    """Exact low-order coefficients (decimal, 30 places) and remainders of the Taylor models, for the record."""
+    def row(t):
+        c = t.frac_coeffs()
+        return {'c': [dec_down(x, 30) if x >= 0 else dec_up(x, 30) for x in c[:5]], 'rem': dec_up(t.frac_e(), 30)}
+    out = {'det': row(law.det), 'e_bb': row(law.e_bb), 'e_bk': row(law.e_bk), 'e_kk': row(law.e_kk)}
+    for nm in law.alpha:
+        out['alpha_' + nm] = row(law.alpha[nm])
+        out['beta_' + nm] = row(law.beta[nm])
+    names = TM_NAMES[law.d]
+    for i, a in enumerate(names):
+        for b in names[i:]:
+            out['C_%s_%s' % (a, b)] = row(law.Cu[(a, b)])
+    return out
+
+
+# ============================================================================= 2. (a, eps) forms and Taylor-model readers
+def ae_add(*xs):
+    return (sum(x[0] for x in xs), sum(x[1] for x in xs))
+
+
+def ae_mul(x, c):
+    return (x[0] * c, x[1] * c)
+
+
+def ae_at(x, R):
+    return x[0] * R * R + x[1]
+
+
+def exp_up(M, x, R):
+    """exp(a0 r^2 + e0) - 1 <= (a0 r^2 + e0) exp(a0 R^2 + e0) on (0, R]."""
+    f = exp_minus(M, -ae_at(x, R)).hi
+    return (x[0] * f, x[1] * f)
+
+
+def tm_dev(t, ref, R):
+    """(a, eps): |t(r) - ref| <= a r^2 + eps for 0 <= r <= R."""
+    c = t.frac_coeffs()
+    eps = abs(c[0] - ref) + abs(c[1]) * R + t.frac_e()
+    a, Rp = Fr(0), Fr(1)
+    for x in c[2:]:
+        a += abs(x) * Rp
+        Rp *= R
+    return (a, eps)
+
+
+def tm_lin(t, R):
+    """(s, eps): |t(r)| <= s r + eps for 0 <= r <= R."""
+    c = t.frac_coeffs()
+    s, Rp = Fr(0), Fr(1)
+    for x in c[1:]:
+        s += abs(x) * Rp
+        Rp *= R
+    return (s, abs(c[0]) + t.frac_e())
+
+
+def tm_shift(t, ref, R):
+    """(lo, hi, eps): t(r) - ref in [lo r^2 - eps, hi r^2 + eps] for 0 <= r <= R."""
+    c = t.frac_coeffs()
+    eps = abs(c[0] - ref) + abs(c[1]) * R + t.frac_e()
+    lo = hi = c[2]
+    Rp = R
+    for x in c[3:]:
+        v = x * Rp
+        lo += min(v, 0)
+        hi += max(v, 0)
+        Rp *= R
+    return (lo, hi, eps)
+
+
+def assert_flat(t, ref, what):
+    c = t.frac_coeffs()
+    require(abs(c[0] - ref) <= TINY and all(abs(x) <= TINY for x in c[1:]),
+            'the reference law of %s is not the contact law for every r' % what)
+
+
+def gersh_min(C):
+    n = len(C)
+    return min(C[i][i].lo - sum(max(abs(C[i][j].lo), abs(C[i][j].hi)) for j in range(n) if j != i) for i in range(n))
+
+
+def wq_bound(M, Cg, rows, vecs):
+    """Upper bound of ||V' Cg^-1 V|| (rows of V given as magnitude lists).  With D the lower endpoints of the diagonal,
+    Cg = D^(1/2)(I + H + E)D^(1/2) exactly, where H >= 0 is diagonal (the actual diagonal's surplus over D) and E has zero
+    diagonal with ||E||_F <= Ef; so Cg >= (1 - Ef) D and Cg^-1 <= D^-1/(1 - Ef) (review C58: the surplus H only helps)."""
+    n = len(rows)
+    D = [Cg[(rows[i], rows[i])].lo for i in range(n)]
+    Ef2 = Fr(0)
+    for i in range(n):
+        for j in range(n):
+            if i != j:
+                c = max(abs(Cg[(rows[i], rows[j])].lo), abs(Cg[(rows[i], rows[j])].hi))
+                Ef2 += c * c / (D[i] * D[j])
+    Ef = M.sqrt_up(Ef2)
+    require(Ef < Fr(1, 2), 'diagonal scaling of the conditional covariance')
+    return sum(sum(x * x for x in vecs[i]) / D[i] for i in range(n)) / (1 - Ef)
+
+
+def half_moments(M, c):
+    """H_j = int_0^inf u^j exp(-u^2/(2c)) du, j = 0..3: sqrt(pi c/2), c, c sqrt(pi c/2), 2 c^2 (upper bounds)."""
+    s = M.isqrt_iv(M.PI * c / 2).hi
+    return [s, Fr(c), c * s, 2 * Fr(c) ** 2]
+
+
+# ============================================================================= 3. the ratio A_r / A_0^ref on a (b, k) box
+def pi_ratio(M, tml, n, Q, R):
+    """log(pi_r / pi_0^ref) in [-dn, up], with dn, up in (a, eps) form; pi_0^ref has det 12 and energy 3b^2/2 + 24k^2."""
+    b0, b1, k0, k1 = Q
+    th1 = tml.th1
+    dlo, dhi, de = tm_shift(tml.det, 12, R)
+    if MUT == 'no-det':
+        dlo = dhi = de = Fr(0)
+    xm = (max(abs(dlo), abs(dhi)) * R * R + de) / 12
+    require(xm < Fr(1, 2), 'pin determinant deviation')
+    half_log_up = (max(Fr(0), -dlo) / (24 * (1 - xm)), de / (24 * (1 - xm)))
+    half_log_dn = (max(Fr(0), dhi) / 24, de / 24)
+    parts = ((tml.e_bb, Fr(3, 2), (b0 * b0, b1 * b1)), (tml.e_bk, Fr(0), (2 * b0 * k0, 2 * b1 * k1)),
+             (tml.e_kk, Fr(24), (k0 * k0, k1 * k1)))
+    Elo = Ehi = Ee = Eu = Fr(0)
+    for t, ref, (wlo, whi) in parts:
+        lo, hi, e = tm_shift(t, ref, R)
+        Elo += min(wlo * lo, whi * lo)
+        Ehi += max(wlo * hi, whi * hi)
+        Ee += whi * e
+        Eu += whi * (abs(ref) + max(abs(lo), abs(hi)) * R * R + e)
+    up = ae_add(half_log_up, (max(Fr(0), -Elo / 2), Ee / 2 + n * th1 / 2))
+    dn = ae_add(half_log_dn, (max(Fr(0), Ehi / 2), (Ee + th1 * Eu) / 2))
+    return up, dn
+
+
+def combine(M, p_up, p_dn, z_up, z_dn, R):
+    pu = exp_up(M, p_up, R)
+    zR = ae_at(z_up, R)
+    up = (pu[0] + z_up[0] + pu[0] * zR, pu[1] + z_up[1] + pu[1] * zR)
+    return up, ae_add(p_dn, z_dn)
+
+
+def bounds_d2(Q, R, tml, il):
+    """(up, dn): A_r/A_0^ref - 1 <= up(r), 1 - A_r/A_0^ref <= dn(r) on the box, for 0 < r <= R (NOTE sections 3-5)."""
+    M = E2
+    b0, b1, k0, k1 = Q
+    th1 = tml.th1
+    assert_flat(tml.alpha['w'], -1, 'alpha_w')
+    assert_flat(tml.beta['w'], 0, 'beta_w')
+    assert_flat(tml.Cu[('w', 'w')], 2, 'Var w')
+    # the law of lambda = -w: mean b + D, variance 2 + S (remainder only)
+    Dae = ae_add(ae_mul(tm_dev(tml.alpha['w'], -1, R), b1), ae_mul(tm_dev(tml.beta['w'], 0, R), k1))
+    aC = tm_dev(tml.Cu[('w', 'w')], 2, R)
+    Sae = (aC[0], aC[1] + (2 + ae_at(aC, R)) * th1)
+    Dm, Sm = ae_at(Dae, R), ae_at(Sae, R)
+    r2 = M.isqrt_iv(M.IV(2))
+    s_lo, s_hi = M.isqrt_iv(M.IV(2 - Sm)).lo, M.isqrt_iv(M.IV(2 + Sm)).hi
+    sa, sb = min(s_lo, r2.lo), max(s_hi, r2.hi)
+    m_lo, m_hi = b0 - Dm, b1 + Dm
+    mmax = max(abs(m_lo), abs(m_hi))
+    Fmin = M.elam2_pos(m_lo, sa).lo
+    Fmax = M.elam2_pos(m_hi, sb).hi
+    F1max = M.elam_pos(m_hi, sb).hi
+    # the variance multiplier (3/4) 2 sb = (3/2) sb only needs to be >= 1 (dF/d(sigma^2) = Phi(mu/sigma) <= 1); it is not
+    # a claim that Phi <= 3/4, which fails at mu = 1, sigma^2 = 2 (review C60)
+    Fae = ae_add(ae_mul(Dae, 2 * F1max / Fmin), ae_mul(Sae, 2 * sb / Fmin * Fr(3, 4)))
+    Fref = M.elam2_pos(b0, r2.lo).lo
+    # secondary law: the engine's interval law over [0, R]
+    b, k = M.IV(b0, b1), M.IV(k0, k1)
+    ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES}
+    gn = lambda nm, p: M.gauss_norm(ma[nm], il.sdQ[nm].hi, p)
+    lam = lambda p: M.gauss_norm(mmax, s_hi, p)
+    T, tau, om, v, nu = (lambda p, x=x: gn(x, p) for x in ('T', 'tau', 'om', 'v', 'nu'))
+    kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p)
+    norm36 = 36 * k0 * k0 * Fref
+    # first order: r E[lambda_+ om] = r [mu_om F_1 - bw_om (F - m F_1)], with mu_om, bw_om = O(r) (Taylor models)
+    sa_, ea_ = tm_lin(tml.alpha['om'], R)
+    sb_, eb_ = tm_lin(tml.beta['om'], R)
+    sc_, ec_ = tm_lin(tml.Cu[('om', 'w')], R)
+    cwlo = 2 - ae_at(aC, R)
+    T2 = (((b1 * sa_ + k1 * sb_) * F1max + sc_ / cwlo * (Fmax + mmax * F1max)) / Fref,
+          R * ((b1 * ea_ + k1 * eb_) * F1max + ec_ / cwlo * (Fmax + mmax * F1max)) / Fref)
+    if MUT == 'no-om-term':
+        T2 = (Fr(0), Fr(0))
+    # second order (explicit r^2). kappa = 6k tau - T^2 - r T tau with E[tau | w] = mu_tau + bw_tau (w - mu_w):
+    # 6k mu_tau F + 6k bw_tau E[(w - mu_w) lambda_+^2] - E[T^2 lambda_+^2] - r E[T tau lambda_+^2] (signed in the leading part)
+    mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(Fmin, Fmax)
+    side = 6 * k1 * M.absup(il.bw['tau']) * s_hi * lam(4) ** 2 + R * T(4) * tau(4) * lam(4) ** 2
+    T3_up = max(Fr(0), mt.hi + side) / norm36
+    T3_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * lam(4) ** 2) / norm36
+    if MUT == 'no-om-term':
+        T3_up = T3_dn = Fr(0)
+    T4 = R * kap(4) * lam(4) * om(2)
+    xi2 = 6 * k1 * (R * nu(4) ** 2 + 2 * nu(4) * v(4)) + T(4) * (R * R * nu(8) ** 2 + 2 * R * nu(8) * v(8) + 2 * v(8) ** 2) \
+        + R * tau(4) * v(8) ** 2
+    T5 = lam(2) * xi2
+    T6 = v(8) ** 2 * (6 * k1 + R * T(4) + R * R * tau(4)) * om(4) + v(4) ** 2 * (R * nu(4) + v(4)) ** 2
+    # the thin set {0 < lambda <= r h} on G_0 (conditional density of lambda given (v, om, nu))
+    Yn = ('v', 'om', 'nu')
+    prec = 1 / (2 - Sm) + wq_bound(M, il.Cg, Yn, [[M.absup(il.bw[nm])] for nm in Yn])
+    pbar = Fr(2, 5) * M.sqrt_up(prec)
+    h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + om(p)
+    bad = R * pbar * h(2) * (Fr(15, 2) * k1 * h(4) + v(8) ** 2) * (9 * k1 * (h(4) + om(4)) + (R * nu(8) + v(8)) ** 2)
+    if MUT == 'no-bad-set':
+        bad = Fr(0)
+    # G_0^c (not r^2-scaled; astronomically small)
+    q = M.gauss_tail2(ma['T'], il.sdQ['T'].hi, 3 * k0 / (2 * R)) + M.gauss_tail2(ma['tau'], il.sdQ['tau'].hi, 3 * k0 / (2 * R * R))
+    P1n = (6 * k1 + R * T(8)) * lam(8) + R * v(8) ** 2
+    P2n = (6 * k1 + R * T(8) + R * R * tau(8)) * (lam(8) + R * om(8)) + R * (R * nu(8) + v(8)) ** 2
+    g0c = P1n * P2n * M.sqrt_up(q) if q > 0 else Fr(0)
+    X = ae_add(T2, ((T4 + T5 + T6 + bad) / norm36, g0c / norm36))
+    p_up, p_dn = pi_ratio(M, tml, 6, Q, R)
+    return combine(M, p_up, p_dn, ae_add(exp_up(M, Fae, R), X, (T3_up, Fr(0))), ae_add(Fae, X, (T3_dn, Fr(0))), R)
+
+
+def bounds_d3(Q, R, tml, il):
+    """(up, dn) in d = 3 (NOTE sections 3-5)."""
+    M = E3
+    b0, b1, k0, k1 = Q
+    th1 = tml.th1
+    names = ('a11', 'a22', 'a12')
+    for nm, ref in (('a11', -1), ('a22', -1), ('a12', 0)):
+        assert_flat(tml.alpha[nm], ref, 'alpha_' + nm)
+        assert_flat(tml.beta[nm], 0, 'beta_' + nm)
+    for (x, y), ref in ((('a11', 'a11'), 2), (('a22', 'a22'), 2), (('a12', 'a12'), 1), (('a11', 'a22'), 0),
+                        (('a11', 'a12'), 0), (('a22', 'a12'), 0)):
+        assert_flat(tml.Cu[(x, y)], ref, 'Cov(%s, %s)' % (x, y))
+    # Loewner coupling with the contact law at the same b: eps = sup ||mu_B - b I||_F, eta = ||chol(Cb) - sqrt2 I||
+    comps = []
+    for nm, ref, f in (('a11', -1, 1), ('a22', -1, 1), ('a12', 0, 2)):
+        comps.append(ae_at(ae_add(ae_mul(tm_dev(tml.alpha[nm], ref, R), b1), ae_mul(tm_dev(tml.beta[nm], 0, R), k1)), R) * f)
+    eps = M.sqrt_up(sum(x * x for x in comps))
+    Fw = (Fr(1), Fr(1), M.SQRT2)
+    Cb = []
+    for i in range(3):
+        row = []
+        for j in range(3):
+            lo, hi = tml.Cu[(names[i], names[j])].enclose(0, R)
+            row.append(M.IV(lo, hi) * Fw[i] * Fw[j] * M.IV(1 / (1 + th1), 1))
+        Cb.append(row)
+    Lb = M.cholesky(Cb)
+    eta = M.sqrt_up(sum(M.absup(Lb[i][j] - (M.SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1)))
+    Dl = M.rup_rel(eps + eta * R0C)
+    cm = M.contact_moment
+    bA, bB = grid_dn(b0 - Dl), grid_up(b1 + Dl)
+    m3b0 = cm(2, 0, b0).lo
+    L3up = 2 * cm(1, 1, bB).hi / m3b0
+    L3dn = 2 * cm(1, 1, b1).hi / cm(2, 0, bA).lo
+    pR0 = M.chi_tail(3, R0C)
+    muB = M.rup_rel(b1 * M.sqrt_up(Fr(2)) + eps)
+    sig = M.sqrt_up(max(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2]))
+                    + 2 * max(M.absup(Cb[i][j]) for i in range(3) for j in range(3) if i != j))
+    U = M.chi_upper_moments(3, R0C, 4)
+    out_D2 = sum((U[i] * (M.binom(4, i) * muB ** (4 - i) * sig ** i) for i in range(5)), M.IV(0)).hi / 4
+    tail_c = M.sqrt_up(cm(4, 0, b1).hi) * M.sqrt_up(pR0)
+    nBF16 = M.rup_rel(muB + sig * M.chi_norm(3, 16))
+    out_ball = M.sqrt_up(pR0)
+
+    def up_moment(a, c):
+        return M.rup_rel(cm(a, c, bB).hi + 2 ** c * M.rup_rel(nBF16 ** (2 * a + c)) * out_ball)
+    ED2, ED4, ED2T2, ET4, DT = up_moment(2, 0), up_moment(4, 0), up_moment(2, 2), up_moment(0, 4), up_moment(1, 1)
+    trCb = sum(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2]))
+    BF = lambda p: M.frob_norm(muB, trCb, p)
+    # secondary law (engine interval law over [0, R])
+    b, k = M.IV(b0, b1), M.IV(k0, k1)
+    ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES}
+    T = lambda p: M.gauss_norm(ma['T'], il.sdQ['T'], p)
+    tau = lambda p: M.gauss_norm(ma['tau'], il.sdQ['tau'], p)
+    mu_o = M.sqrt_up(ma['o11'] ** 2 + ma['o22'] ** 2 + 2 * ma['o12'] ** 2)
+    trO = sum(m * il.C[(nm, nm)].hi for nm, m in zip(M.ONAMES, (1, 1, 2)))
+    Om = lambda p: M.frob_norm(mu_o, trO, p)
+    mu_v = M.sqrt_up(ma['v1'] ** 2 + ma['v2'] ** 2)
+    v = lambda p: M.frob_norm(mu_v, il.C[('v1', 'v1')].hi + il.C[('v2', 'v2')].hi, p)
+    mu_nu = M.sqrt_up(ma['nu1'] ** 2 + ma['nu2'] ** 2)
+    nu = lambda p: M.frob_norm(mu_nu, il.C[('nu1', 'nu1')].hi + il.C[('nu2', 'nu2')].hi, p)
+    e = lambda p: R * nu(p) + v(p)
+    kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p)
+    c1 = lambda p: 6 * k1 + R * T(p)
+    c2 = lambda p: 6 * k1 + R * T(p) + R * R * tau(p)
+    norm36 = 36 * k0 * k0 * m3b0
+    # the mean of Omega': ||mu_Omega'||_F <= s_mu r + e_mu (Taylor models; beta_O = r^2 on the diagonal)
+    s_mu2 = e_mu2 = Fr(0)
+    for nm, mult in zip(M.ONAMES, (1, 1, 2)):
+        sa_, ea_ = tm_lin(tml.alpha[nm], R)
+        sb_, eb_ = tm_lin(tml.beta[nm], R)
+        s_mu2 += mult * (b1 * sa_ + k1 * sb_) ** 2
+        e_mu2 += mult * (b1 * ea_ + k1 * eb_) ** 2
+    muO = M.sqrt_up(s_mu2) * R + M.sqrt_up(e_mu2)                      # sup over (0, R] (it multiplies r^2 or more)
+    # the regression of Omega' on B: E[Omega' - mu | B] = beta (B - mu_B) + DY, beta = (r - r^3/4)/2 + ..., DY remainder-only
+    G = tml.G
+    beta_t = G['o11'][0]
+    bc = beta_t.frac_coeffs()
+    require(abs(bc[0]) <= TINY, 'regression of Omega on B at r = 0')
+    for (o, l) in (('o22', 1), ('o12', 2)):
+        assert_flat(G[o][l] - beta_t, 0, 'regression of %s on B' % o)
+    for o, ls in (('o11', (1, 2)), ('o22', (0, 2)), ('o12', (0, 1))):
+        for l in ls:
+            assert_flat(G[o][l], 0, 'off-diagonal regression of %s on B' % o)
+    dG2 = Fr(0)
+    for o, mult in zip(('o11', 'o22', 'o12'), (1, 1, 2)):
+        for l in range(3):
+            t = G[o][l] - beta_t if (o, l) in (('o11', 0), ('o22', 1), ('o12', 2)) else G[o][l]
+            dG2 += mult * Fr(t.bound(), 1 << TM_P) ** 2
+    dGn = M.sqrt_up(dG2)
+    b1c = bc[1]                                                         # = 1/2
+    require(b1c > 0, 'sign of the first-order regression coefficient of Omega on B')
+    sb_beta = sum(abs(x) * R ** (j - 2) for j, x in enumerate(bc) if j >= 2)
+    e_beta = beta_t.frac_e()
+    # E[D tr(A* Omega') 1] = beta (2 E[D^2 1] - E[D tr(A* mu_B) 1]) + R1, tr(A* mu_B) = b tr B + tr(A* dmu), ||dmu||_F <= eps
+    DTlo = cm(1, 1, bA).lo - M.sqrt_up(cm(2, 2, b1).hi) * M.sqrt_up(pR0)
+    m3up = cm(2, 0, b1).hi
+    rho_lo = exp_minus(M, Dl * L3dn).lo - tail_c / m3b0                 # m3_r / m3(b)
+    rho_hi = exp_minus(M, -(Dl * L3up)).hi + out_D2 / m3b0
+    tau_lo = b0 * max(DTlo, Fr(0)) / m3up                               # b E[D tr B 1] / m3(b)
+    tau_hi = b1 * DT / m3b0
+    X3lo = 2 * rho_lo - tau_hi - eps * DT / m3b0
+    X3hi = 2 * rho_hi - tau_lo + eps * DT / m3b0
+    X3m = max(abs(X3lo), abs(X3hi))
+    # |R1|/m3(b) <= (s_mu r + e_mu) DT/m3 + ||DG|| sqrt(E[D^2 tr^2 1]) sqrt(tr Cb)/m3, so r |R1|/m3 <= r^2 s_mu DT/m3 + R (...)
+    R1a = M.sqrt_up(s_mu2) * DT / m3b0
+    R1e = (M.sqrt_up(e_mu2) * DT + dGn * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0
+    # -r E[D tr(A* Omega') 1]/m3(b) = -(b1c r^2 + (r beta - b1c r^2)) X3 - r R1/m3(b), |r beta - b1c r^2| <= sb_beta R r^2 + e_beta r
+    U3_up = (max(Fr(0), -b1c * X3lo) + sb_beta * R * X3m + R1a, R * (e_beta * X3m + R1e))
+    U3_dn = (max(Fr(0), b1c * X3hi) + sb_beta * R * X3m + R1a, R * (e_beta * X3m + R1e))
+    if MUT == 'no-om-term':
+        U3_up = U3_dn = (Fr(0), Fr(0))
+    # explained parts of Omega' by B (for U4) and by zeta = (B, v, nu) (for U6, U7): traces of C_Oz C_zz^-1 C_zO
+    zB = ('a11', 'a22', 'a12')
+    zA = zB + tuple(M.VNAMES) + ('nu1', 'nu2')
+    def cov_Oz(o, z):
+        if z in zB:
+            s_, e_ = tm_lin(tml.Cu[(o, z)], R)
+            return s_ * R + e_
+        return M.absup(il.C[(o, z)])
+    def explained(zs):
+        rows = []
+        for o, mult in zip(M.ONAMES, (1, 1, 2)):
+            rows.append([cov_Oz(o, z) for z in zs])
+        D = [il.C[(z, z)].lo for z in zs]
+        Ef2 = Fr(0)
+        for i in range(len(zs)):
+            for j in range(len(zs)):
+                if i != j:
+                    c = M.absup(il.C[(zs[i], zs[j])])
+                    Ef2 += c * c / (D[i] * D[j])
+        Ef = M.sqrt_up(Ef2)
+        require(Ef < Fr(1, 2), 'diagonal scaling (explained part of Omega)')
+        return sum(mult * sum(x * x / Dz for x, Dz in zip(row, D)) for row, mult in zip(rows, (1, 1, 2))) / (1 - Ef)
+    tB, tK = explained(zB), explained(zA)
+    YB4, YK4 = M.frob_norm(0, tB, 4), M.frob_norm(0, tK, 4)
+    detX = M.absup(il.Cg[('o11', 'o22')] - il.Cg[('o12', 'o12')])     # |E[det(Omega' - E[Omega' | B])]|
+    E1 = up_moment(1, 0)
+    # second order (explicit r^2)
+    # signed kappa term: 6k mu_tau E[D^2 1] + 6k E[(tau - mu_tau) D^2 1] - E[T^2 D^2 1] - r E[T tau D^2 1]
+    ED2lo = max(Fr(0), cm(2, 0, bA).lo - tail_c)
+    mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(ED2lo, ED2)
+    side = 6 * k1 * il.bnorm['tau'] * M.sqrt_up(trCb) * M.sqrt_up(ED4) + R * T(4) * tau(4) * M.sqrt_up(ED4)
+    U2_up = max(Fr(0), mt.hi + side) / norm36
+    U2_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * M.sqrt_up(ED4)) / norm36
+    if MUT == 'no-om-term':
+        U2_up = U2_dn = Fr(0)
+    U2 = Fr(0)
+    U3r = R * kap(4) * M.sqrt_up(ED2T2) * Om(4)
+    # U4 = r^2 E[c1 c2 D det Omega' 1]: E[det Omega' | B] = det(mu + Y_B) + E[det X'] (36 k^2 cancels; relative)
+    U4rel = (detX * E1 + M.sqrt_up(ED2) * (muO + YB4) ** 2 / 2) / m3b0
+    U4 = R * R * kap(4) * M.sqrt_up(ED2) * Om(8) ** 2 / 2
+    U5 = M.sqrt_up(ED2T2) * ((2 * T(4) + R * tau(4)) * v(8) ** 2 + 2 * c1(4) * nu(8) * v(8) + R * c1(4) * nu(8) ** 2)
+    # U6, U7: the 6k parts through E[Omega' | B, v, nu] = mu + Y_K (Omega' - E[Omega' | zeta] has conditional mean 0)
+    U67rel = (M.sqrt_up(ED2) * (muO * e(4) ** 2 + e(8) ** 2 * YK4) + M.sqrt_up(ET4) * (muO * v(4) ** 2 + v(8) ** 2 * YK4)) / (6 * k0 * m3b0)
+    U6 = R * T(8) * M.sqrt_up(ED2) * Om(8) * e(8) ** 2
+    U7 = R * (T(8) + R * tau(8)) * M.sqrt_up(ET4) * Om(8) * v(8) ** 2
+    U8 = R * c2(8) * Om(16) ** 2 / 2 * M.iroot(ET4, 4) * v(16) ** 2
+    U9 = M.sqrt_up(ET4) * v(8) ** 2 * e(8) ** 2
+    U10 = R * M.iroot(ET4, 4) * v(8) ** 2 * Om(4) * e(8) ** 2
+    # the thin set {0 < lambda_1 <= r h} on G_0: conditional law of B given Y = (v, Omega', nu), eigenvalue coordinates
+    Yn = tuple(M.VNAMES) + tuple(M.ONAMES) + ('nu1', 'nu2')
+    lmB = gersh_min(Cb)
+    rows = []
+    for nm in Yn:
+        if nm in M.ONAMES:
+            cov = [(tm_lin(tml.Cu[(nm, an)], R)[0] * R + tm_lin(tml.Cu[(nm, an)], R)[1]) * f for an, f in zip(names, (1, 1, 2))]
+            rows.append([M.sqrt_up(sum(x * x for x in cov)) / lmB])
+        else:
+            rows.append([M.absup(x) for x in il.bt[nm]])
+    Wt = wq_bound(M, il.Cg, Yn, rows)
+    c_hi = sig * sig
+    cl = 1 / (1 / lmB + Wt)
+    KBp = M.rup_rel((M.PI * M.SQRT2 / (2 * M.PI * M.SQRT2PI)).hi / (cl * M.sqrt_down(cl)))
+    m1 = muB + c_hi * M.sqrt_up(Wt) * M.chi_norm(7, 8)
+    Hs = half_moments(M, c_hi)
+    Ja = m1 ** 4 / 4 + sum(M.binom(3, i) * m1 ** (3 - i) * Hs[i] for i in range(4))
+    Jb = m1 ** 3 / 3 + sum(M.binom(2, i) * m1 ** (2 - i) * Hs[i] for i in range(3))
+    h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + Om(p)
+    nhg = h(4) * (Fr(15, 2) * k1 * h(8) + v(16) ** 2) * (9 * k1 * (h(8) + Om(8)) + e(16) ** 2)
+    nhgO = h(8) * (Fr(15, 2) * k1 * h(16) + v(32) ** 2) * (9 * k1 * (h(16) + Om(16)) + e(32) ** 2) * Om(8)
+    bad = R * KBp * (Ja * nhg + R * Jb * nhgO)
+    if MUT == 'no-bad-set':
+        bad = Fr(0)
+    q = M.gauss_tail2(ma['T'], il.sdQ['T'], 3 * k0 / (2 * R)) + M.gauss_tail2(ma['tau'], il.sdQ['tau'], 3 * k0 / (2 * R * R))
+    P1n = c1(8) * BF(16) ** 2 + R * BF(8) * v(16) ** 2
+    P2n = c2(8) * (BF(16) + R * Om(16)) ** 2 + R * (BF(8) + R * Om(8)) * e(16) ** 2
+    g0c = P1n * P2n * M.sqrt_up(q) if q > 0 else Fr(0)
+    X = ((U2 + U3r + U4 + U5 + U6 + U7 + U8 + U9 + U10 + bad) / norm36 + U4rel + U67rel, g0c / norm36)
+    z_up = ae_add(exp_up(M, (Fr(0), Dl * L3up), R), (Fr(0), out_D2 / m3b0), U3_up, X, (U2_up, Fr(0)))
+    z_dn = ae_add((Fr(0), Dl * L3dn), (Fr(0), tail_c / m3b0), U3_dn, X, (U2_dn, Fr(0)))
+    p_up, p_dn = pi_ratio(M, tml, 8, Q, R)
+    return combine(M, p_up, p_dn, z_up, z_dn, R)
+
+
+def rate_sweep(d, tml):
+    """For every R in R_STARS and every sub-box: (a_up, eps_up, a_dn, eps_dn) as decimal strings (rounded up)."""
+    M = ENG[d]
+    fn = bounds_d2 if d == 2 else bounds_d3
+    out = {}
+    for R in R_STARS:
+        il = M.band_law(Fr(0), R)
+        rows = []
+        for bb, kb in SUBBOXES:
+            up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, tml, il)
+            rows.append([dec_up(up[0], 6), dec_up(up[1], 20), dec_up(dn[0], 6), dec_up(dn[1], 20)])
+        out[str(R)] = {'boxes': rows,
+                       'a_up': max((r[0] for r in rows), key=Fr), 'eps_up': max((r[1] for r in rows), key=Fr),
+                       'a_dn': max((r[2] for r in rows), key=Fr), 'eps_dn': max((r[3] for r in rows), key=Fr)}
+    return out
+
+
+# ============================================================================= 4. certified reference integrals
+PREC = 240
+
+
+def iroot_int(N, p):
+    """Largest y >= 0 with y^p <= N."""
+    if N < 2:
+        return N
+    y = 1 << ((N.bit_length() + p - 1) // p)
+    while True:
+        z = ((p - 1) * y + N // y ** (p - 1)) // p
+        if z >= y:
+            break
+        y = z
+    while y ** p > N:
+        y -= 1
+    while (y + 1) ** p <= N:
+        y += 1
+    return y
+
+
+def pow_frac(M, x, num, den):
+    """Enclosure of x^(num/den), rational x > 0."""
+    x = Fr(x) ** num
+    S = 1 << (den * PREC)
+    lo_n = (x.numerator * S) // x.denominator
+    hi_n = -((-x.numerator * S) // x.denominator)
+    ylo, yhi = iroot_int(lo_n, den), iroot_int(hi_n, den)
+    if yhi ** den < hi_n:
+        yhi += 1
+    return M.IV(Fr(ylo, 1 << PREC), Fr(yhi, 1 << PREC))
+
+
+def lower_gamma(M, an, ad, x):
+    """gamma(a, x) = x^a e^-x sum_n x^n / (a (a+1) ... (a+n)), a = an/ad > 0, rational x > 0 (series with a geometric tail)."""
+    a, x = Fr(an, ad), Fr(x)
+    term = s = 1 / a
+    n = 0
+    while True:
+        n += 1
+        term = term * x / (a + n)
+        s += term
+        if a + n + 1 > 2 * x and term < Fr(1, 10 ** 70) * s:
+            break
+    rho = x / (a + n + 1)
+    return pow_frac(M, x, an, ad) * M.exp_neg_point(x) * M.IV(s, s + term * rho / (1 - rho))
+
+
+def k_integral(M, qn, qd, k0, k1):
+    """int_{k0}^{k1} k^q exp(-12 k^2) dk = (1/2) 12^(-a) [gamma(a, 12 k1^2) - gamma(a, 12 k0^2)], a = (q + 1)/2."""
+    an, ad = qn + qd, 2 * qd
+    g = math.gcd(an, ad)
+    an, ad = an // g, ad // g
+    pre = pow_frac(M, Fr(1, 12), an, ad) * Fr(1, 2)
+    return pre * (lower_gamma(M, an, ad, 12 * Fr(k1) ** 2) - lower_gamma(M, an, ad, 12 * Fr(k0) ** 2))
+
+
+def m_ref(d, b):
+    """The reference contact moment: F(b, sqrt2) = E[(b + sqrt2 z)_+^2] (d = 2), m_3(b) = E[det(B)^2 1{B > 0}] (d = 3)."""
+    if d == 2:
+        s = E2.isqrt_iv(E2.IV(2))
+        return E2.IV(E2.elam2_pos(Fr(b), s.lo).lo, E2.elam2_pos(Fr(b), s.hi).hi)
+    return E3.contact_moment(2, 0, Fr(b))
+
+
+def m_ref_derivs(d, b):
+    """Upper bounds of m, m', m'' at b (each increasing in b)."""
+    if d == 2:
+        s = E2.isqrt_iv(E2.IV(2)).hi
+        return E2.elam2_pos(Fr(b), s).hi, 2 * E2.elam_pos(Fr(b), s).hi, Fr(2)
+    cm = E3.contact_moment
+    return cm(2, 0, Fr(b)).hi, 2 * cm(1, 1, Fr(b)).hi, 2 * cm(0, 2, Fr(b)).hi + 4 * cm(1, 0, Fr(b)).hi
+
+
+def b_cells(d):
+    """Enclosures of int over [j h, (j+1) h] of exp(-3b^2/4) m(b) db, h = 1/NH (midpoint rule, |g''| bounded per block)."""
+    M = ENG[d]
+    h = Fr(1, NH)
+    out = []
+    for blk in range(NH // BLOCK):
+        x0, x1 = blk * BLOCK * h, (blk + 1) * BLOCK * h
+        m, m1, m2 = m_ref_derivs(d, x1)
+        g2 = M.exp_neg_point(3 * x0 * x0 / 4).hi * ((Fr(9, 4) * x1 * x1 + Fr(3, 2)) * m + 3 * x1 * m1 + m2)
+        err = h ** 3 / 24 * g2
+        for j in range(BLOCK):
+            c = x0 + (j + Fr(1, 2)) * h
+            out.append(M.exp_neg_point(3 * c * c / 4) * m_ref(d, c) * h + M.IV(-err, err))
+    return out
+
+
+def b_integral(M, cells, b0, b1):
+    i0, i1 = Fr(b0) * NH, Fr(b1) * NH
+    require(i0.denominator == 1 and i1.denominator == 1, 'b-box off the cell grid')
+    return sum((cells[j] for j in range(int(i0), int(i1))), M.IV(0))
+
+
+def K_const(d):
+    """|S^(d-1)| 144 (2 pi)^(-(d+1)) / sqrt12: A_0^ref/(3 k^p) = K k^(2-p) e^(-12 k^2) e^(-3 b^2/4) m(b)."""
+    M = ENG[d]
+    tp = 2 * M.PI
+    pw = M.IV(1)
+    for _ in range(d + 1):
+        pw = pw * tp
+    return (tp if d == 2 else 2 * tp) * 144 / pw / M.isqrt_iv(M.IV(12))
+
+
+def reference(d):
+    M = ENG[d]
+    cells = b_cells(d)
+    K = K_const(d)
+    Ib = b_integral(M, cells, 0, 1)
+    c_ref = K * Ib * k_integral(M, 4, 3, Fr(1, 2), Fr(2))
+    J23, J43, J13 = {}, {}, {}
+    for kb in SUB_K:
+        J23[kb] = k_integral(M, 2, 3, kb[0], kb[1])
+        J43[kb] = k_integral(M, 4, 3, kb[0], kb[1])
+    for kb in CAP_K:
+        J13[kb] = k_integral(M, 1, 3, kb[0], kb[1])
+    Ibs = {bb: b_integral(M, cells, bb[0], bb[1]) for bb in set(SUB_B) | set(CAP_B)}
+    return {'c_ref': [dec_down(c_ref.lo, 24), dec_up(c_ref.hi, 24)],
+            'w_c': [dec_up((K * Ibs[bb] * J43[kb]).hi, 24) for bb, kb in SUBBOXES],          # int A_0/(3 k^(2/3))
+            'w_2': [dec_up((K * Ibs[bb] * J23[kb]).hi, 24) for bb, kb in SUBBOXES],          # int A_0/(3 k^(4/3))
+            'w_cap': [dec_up((K * Ibs[bb] * J13[kb]).hi, 24) for bb, kb in CAP_BOXES]}       # int A_0/(3 k^(5/3))
+
+
+# ============================================================================= 5. the refined cap sweep (engines of #206, #212)
+CAPM = {}
+
+
+def cap_init(d):
+    CAPM['M'] = ENG[d]
+    CAPM['d'] = d
+    if MUT == 'cap-half':
+        CAPM['M'].MUT = 'cap-half'
+
+
+def cap_band(args):
+    """Certified bound of sup over the band of Q^W(G_r^c)/r^3 on every refined box (rounded up, 4 places)."""
+    d, r0, r1 = args
+    M = ENG[d]
+    if MUT == 'cap-half':
+        M.MUT = 'cap-half'
+    law = M.band_law(r0, r1)
+    x0, th = (M.COMBOS[0] if d == 2 else M.COMBO)
+    return {'r': [str(r0), str(r1)],
+            'bound': [M.dec_up(M.box_bound(law, bb, kb, r0, r1, last_band=(r0 == 0), x0=x0, theta=th)['total'])
+                      for bb, kb in CAP_BOXES]}
+
+
+def cap_sweep(d, procs, which=None):
+    bl = ENG[d].bands()
+    idx = range(len(bl)) if which is None else which
+    jobs = [(d, bl[i][0], bl[i][1]) for i in idx]
+    if procs <= 1:
+        return [cap_band(j) for j in jobs]
+    with multiprocessing.Pool(procs, initializer=cap_init, initargs=(d,)) as pool:
+        return pool.map(cap_band, jobs, chunksize=1)
+
+
+def cap_sample(d):
+    """Band indices replayed by --check: the band ending at each r_*, the band ending at 2^-20 and the last band."""
+    bl = ENG[d].bands()
+    ends = set(R_STARS) | {Fr(1, 2 ** 20)}
+    return sorted({i for i, (r0, r1) in enumerate(bl) if r1 in ends} | {0})
+
+
+# ============================================================================= 6. assembly
+def assemble(d, rate, ref, cap):
+    """The C_{B,K}(r_*) table, the second-order density constants, and the (12.1), (12.3) constants."""
+    c_lo, c_hi = Fr(ref['c_ref'][0]), Fr(ref['c_ref'][1])
+    wc = [Fr(x) for x in ref['w_c']]
+    w2 = [Fr(x) for x in ref['w_2']]
+    wcap = [Fr(x) for x in ref['w_cap']]
+    out = {}
+    for R in R_STARS:
+        rr = rate[str(R)]
+        a_up, e_up, a_dn, e_dn = Fr(rr['a_up']), Fr(rr['eps_up']), Fr(rr['a_dn']), Fr(rr['eps_dn'])
+        c2_up = sum(Fr(row[0]) * w for row, w in zip(rr['boxes'], w2))
+        c2_dn = sum(Fr(row[2]) * w for row, w in zip(rr['boxes'], w2))
+        e_c_up = sum(Fr(row[1]) * w for row, w in zip(rr['boxes'], wc))
+        e_c_dn = sum(Fr(row[3]) * w for row, w in zip(rr['boxes'], wc))
+        Cbox = [Fr(0)] * len(CAP_BOXES)
+        for rec in cap:
+            if Fr(rec['r'][1]) <= R:
+                for i, x in enumerate(rec['bound']):
+                    Cbox[i] = max(Cbox[i], Fr(x))
+        S = sum(c * w for c, w in zip(Cbox, wcap))
+        CBK = (1 + a_up * R * R + e_up) * S
+        out[str(R)] = {'C_BK': dec_up(CBK, 4), 'C_121': dec_up(Fr(3, 5) * CBK, 4),
+                       'c2_up': dec_up(c2_up, 12), 'c2_dn': dec_up(c2_dn, 12),
+                       'epsc_up': dec_up(e_c_up, 30), 'epsc_dn': dec_up(e_c_dn, 30),
+                       'rel2_up': dec_up(c2_up / c_lo, 6), 'rel2_dn': dec_up(c2_dn / c_lo, 6),
+                       'ell_max': '%s' % (R ** 3 / 2)}
+    return out
+
+
+def params():
+    return {'R_STARS': [str(x) for x in R_STARS], 'TM': {'s': TM_S, 'N': TM_N, 'P': TM_P}, 'R0_COUPLING': str(R0C),
+            'SUB_B': [[str(a), str(b)] for a, b in SUB_B], 'SUB_K': [[str(a), str(b)] for a, b in SUB_K],
+            'NH': NH, 'BLOCK': BLOCK, 'CAP_B': [[str(a), str(b)] for a, b in CAP_B], 'CAP_K': [[str(a), str(b)] for a, b in CAP_K],
+            'band': {'b': ['0', '1'], 'k': ['1/2', '2']}, 'L_min': 10}
+
+
+def compute_core(d):
+    tml = tm_law(d)
+    return tm_summary(tml), rate_sweep(d, tml), reference(d)
+
+
+def full_run(procs):
+    res = {'schema': 1, 'object': 'CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1', 'scientific_effect': 'NONE',
+           'certified': True, 'mutant': None, 'parameters': params(), 'dims': {}}
+    for d in (2, 3):
+        summ, rate, ref = compute_core(d)
+        cap = cap_sweep(d, procs)
+        res['dims'][str(d)] = {'tm_law': summ, 'rate': rate, 'reference': ref, 'table': assemble(d, rate, ref, cap),
+                               'cap_boxes': [[str(bb[0]), str(bb[1]), str(kb[0]), str(kb[1])] for bb, kb in CAP_BOXES],
+                               'cap_bands': cap}
+    return res
+
+
+def results_text(res):
+    """Canonical layout: everything indented except the cap-band records, one line each."""
+    caps = {d: res['dims'][d].pop('cap_bands') for d in res['dims']}
+    head = json.dumps(res, indent=1, sort_keys=True)
+    for d in res['dims']:
+        res['dims'][d]['cap_bands'] = caps[d]
+    tail = ',\n "cap_bands": {\n' + ',\n'.join(
+        '  "%s": [\n' % d + ',\n'.join('   ' + json.dumps(rec, sort_keys=True) for rec in caps[d]) + '\n  ]'
+        for d in sorted(caps)) + '\n }\n}\n'
+    return head[:-2] + tail
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        res = json.load(fh)
+    for d, recs in res.pop('cap_bands').items():
+        res['dims'][d]['cap_bands'] = recs
+    return res
+
+
+def check_run(procs, full=False):
+    res = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(res), 'RESULTS.json is not in canonical layout')
+    require(res['parameters'] == params(), 'parameters differ')
+    require(res['mutant'] is None and res['certified'] is True, 'header')
+    for d in (2, 3):
+        rd = res['dims'][str(d)]
+        tml = tm_law(d)
+        require(tm_summary(tml) == rd['tm_law'], 'Taylor-model law differs (d = %d)' % d)
+        rate = rate_sweep(d, tml)
+        require(rate == rd['rate'], 'rate sweep differs (d = %d)' % d)
+        ref = reference(d)
+        require(ref == rd['reference'], 'reference integrals differ (d = %d)' % d)
+        c_lo, c_hi = Fr(ref['c_ref'][0]), Fr(ref['c_ref'][1])
+        cx = C_REF_197 if d == 2 else C_REF_205
+        require(c_lo <= cx <= c_hi, 'reference c_{B,K} does not contain the value of Math-#197/#205 (d = %d)' % d)
+        require(rd['cap_boxes'] == [[str(bb[0]), str(bb[1]), str(kb[0]), str(kb[1])] for bb, kb in CAP_BOXES], 'cap boxes')
+        bl = ENG[d].bands()
+        require([rec['r'] for rec in rd['cap_bands']] == [[str(a), str(b)] for a, b in bl], 'cap band layout (d = %d)' % d)
+        which = None if full else cap_sample(d)
+        recs = cap_sweep(d, procs, which)
+        for i, rec in zip(range(len(bl)) if which is None else which, recs):
+            require(rec == rd['cap_bands'][i], 'cap band %d differs (d = %d)' % (i, d))
+        require(assemble(d, rate, ref, rd['cap_bands']) == rd['table'], 'assembled table differs (d = %d)' % d)
+    print('CHECK PASSED (%s)' % ('full' if full else 'sample'))
+
+
+# ============================================================================= 7. floating controls (not part of the certificate)
+def controls():
+    """Floating estimates of the reference-kernel cap-route benchmarks Gamma_{B,K} = |S| int A_0^ref c_G / (3 k^(5/3)) for comparison with C_{B,K}:
+    planar c_G = R(b) J(k) of Math-#203 with J(k) = 2359296 k^3 (1 + delta(k)), delta(1/2) = 0.0232, delta(3/4) = 1.3e-4;
+    d = 3 from the floating R_3(b) R(b) and J^(3)/J ratios of Math-#208."""
+    from math import exp, pi, sqrt, erf
+    out = {}
+    K2 = 2 * pi * 144 / (2 * pi) ** 3 / sqrt(12)
+    Ib2 = sqrt(pi) / 2 * erf(1) / (2 * sqrt(pi))                 # int_0^1 e^{-b^2} db / (2 sqrt(pi)) (m R = p_w(0))
+    def J2(k):
+        dl = 0.0232 + (k - 0.5) / 0.25 * (1.3e-4 - 0.0232) if k < 0.75 else 1.3e-4 * max(0.0, (1 - k) / 0.25)
+        return 2359296 * k ** 3 * (1 + dl)
+    n = 200000
+    Ik2 = sum(((0.5 + (i + 0.5) * 1.5 / n) ** (1 / 3)) * exp(-12 * (0.5 + (i + 0.5) * 1.5 / n) ** 2) * J2(0.5 + (i + 0.5) * 1.5 / n)
+              * 1.5 / n for i in range(n))
+    out['Gamma_2'] = K2 * Ib2 * Ik2
+    bs = [0, 0.25, 0.5, 0.75, 1.0]
+    m3 = [0.6715728752625627, 1.195532362807149, 2.05794812257308, 3.43079921823, 5.54806027367761]
+    r3r = [1.1880833639847637, 0.9085531055205767, 0.6853903187865128, 0.5096151204445851, 0.37314989157732215]
+    f = [exp(-0.75 * b * b) * m * r for b, m, r in zip(bs, m3, r3r)]
+    Ib3 = 0.25 / 3 * (f[0] + 4 * f[1] + 2 * f[2] + 4 * f[3] + f[4])
+    K3 = 4 * pi * 144 / (2 * pi) ** 4 / sqrt(12)
+    def J3(k):
+        dl = 0.1617 + (k - 0.5) / 0.25 * (0.0012 - 0.1617) if k < 0.75 else 0.0012 * max(0.0, (1 - k) / 0.25)
+        return 2359296 * k ** 3 * (1 + dl)
+    Ik3 = sum(((0.5 + (i + 0.5) * 1.5 / n) ** (1 / 3)) * exp(-12 * (0.5 + (i + 0.5) * 1.5 / n) ** 2) * J3(0.5 + (i + 0.5) * 1.5 / n)
+              * 1.5 / n for i in range(n))
+    out['Gamma_3'] = K3 * Ib3 * Ik3
+    return out
+
+
+def mc_ratio(d, n, seed=20261001):
+    """Floating Monte Carlo of (A_r/A_0^ref - 1)/r^2 for the reference kernel at r = 1/8, 1/16: the exact law at r from the
+    series (all r, not only r <= 1/64), Z_r/r^2 by sampling W_r/r^2 = P1 P2 1{typed}, and the exact pin ratio."""
+    import random
+    M = ENG[d]
+    names = ('w', 'v', 'om', 'T', 'tau', 'nu') if d == 2 else \
+        ('a11', 'a22', 'a12', 'v1', 'v2', 'o11', 'o22', 'o12', 'T', 'tau', 'nu1', 'nu2')
+    pins, targets = M.functionals()
+    P = [pins[x] for x in M.PIN_NAMES]
+    idx = [M.TARGET_NAMES.index(x) for x in names]
+    Tf = [targets[x] for x in names]
+    tml = tm_law(d)
+    def cov(F, G, key, r):
+        ser, w, _ = M.pair_series(F, G, key)
+        return sum(c * r ** p for p, c in ser.items())
+    def law(r, b, k):
+        n_ = len(P)
+        S = [[None] * n_ for _ in range(n_)]
+        for i in range(n_):
+            for j in range(i, n_):
+                S[i][j] = S[j][i] = cov(P[i], P[j], ('p', i, j), r)
+        Stp = [[cov(Tf[a], P[j], ('tp', idx[a], j), r) for j in range(n_)] for a in range(len(Tf))]
+        Mx = [list(S[i]) + [Stp[a][i] for a in range(len(Tf))] for i in range(n_)]
+        for col in range(n_):
+            for i in range(col + 1, n_):
+                f = Mx[i][col] / Mx[col][col]
+                for j in range(col, len(Mx[i])):
+                    Mx[i][j] -= f * Mx[col][j]
+        W = []
+        for a in range(len(Tf)):
+            x = [None] * n_
+            for i in reversed(range(n_)):
+                x[i] = (Mx[i][n_ + a] - sum(Mx[i][j] * x[j] for j in range(i + 1, n_))) / Mx[i][i]
+            W.append(x)
+        kv = [-r ** 3 / 2, -r ** 2, Fr(12)] + [Fr(0)] * (n_ - 3)
+        mu = [float(b * W[a][0] + k * sum(W[a][j] * kv[j] for j in range(n_))) for a in range(len(Tf))]
+        def stt(a, c):
+            i, j = idx[a], idx[c]
+            return cov(Tf[a], Tf[c], ('tt', i, j), r) if i <= j else cov(Tf[c], Tf[a], ('tt', j, i), r)
+        C = [[float(stt(a, c) - sum(W[a][l] * Stp[c][l] for l in range(n_))) for c in range(len(Tf))] for a in range(len(Tf))]
+        L = [[0.0] * len(C) for _ in C]
+        for i in range(len(C)):
+            for j in range(i + 1):
+                s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j))
+                L[i][j] = math.sqrt(max(s, 0.0)) if i == j else (s / L[j][j] if L[j][j] > 0 else 0.0)
+        return mu, L
+    rng = random.Random(seed)
+    out = []
+    for r in (Fr(1, 8), Fr(1, 16)):
+        rf = float(r)
+        for b, k in ((Fr(0), Fr(1, 2)), (Fr(0), Fr(2)), (Fr(1), Fr(1, 2)), (Fr(1), Fr(2))):
+            mu, L = law(r, b, k)
+            kf = float(k)
+            tot = tot2 = 0.0
+            for _ in range(n):
+                z = [rng.gauss(0.0, 1.0) for _ in mu]
+                x = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(len(mu))]
+                if d == 2:
+                    w, v, om, T, tau, nu = x
+                    lam = -w
+                    c1, c2 = 6 * kf - rf * T, 6 * kf + rf * T + rf * rf * tau
+                    P1 = c1 * lam - rf * v * v
+                    P2 = c2 * (lam - rf * om) + rf * (rf * nu - v) ** 2
+                    Wv = P1 * P2 if (c1 > 0 and P1 > 0 and P2 > 0) else 0.0
+                else:
+                    a11, a22, a12, v1, v2, o11, o22, o12, T, tau, nu1, nu2 = x
+                    B = (-a11, -a22, -a12)
+                    S_ = (B[0] - rf * o11, B[1] - rf * o22, B[2] - rf * o12)
+                    detB, detS = B[0] * B[1] - B[2] ** 2, S_[0] * S_[1] - S_[2] ** 2
+                    c1, c2 = 6 * kf - rf * T, 6 * kf + rf * T + rf * rf * tau
+                    e1, e2 = rf * nu1 - v1, rf * nu2 - v2
+                    P1 = c1 * detB - rf * (B[1] * v1 * v1 - 2 * B[2] * v1 * v2 + B[0] * v2 * v2)
+                    P2 = c2 * detS + rf * (S_[1] * e1 * e1 - 2 * S_[2] * e1 * e2 + S_[0] * e2 * e2)
+                    Wv = 0.0
+                    if B[0] > 0 and detB > 0 and P1 > 0 and P2 > 0:
+                        # index(H_S) = 2: sign changes of the leading minors (-S_11, det S, r P2) of H_S reordered (y1, y2, x)
+                        m1, m2, m3 = -S_[0], detS, rf * P2
+                        if m1 != 0 and m2 != 0:
+                            neg = (m1 < 0) + (m2 / m1 < 0) + (m3 / m2 < 0)
+                            Wv = P1 * P2 if neg == 2 else 0.0
+                tot += Wv
+                tot2 += Wv * Wv
+            mean = tot / n
+            se = math.sqrt(max(tot2 / n - mean * mean, 0.0) / n)
+            mref = float(m_ref(d, b).mid())
+            zr, zse = mean / (36 * kf * kf * mref), se / (36 * kf * kf * mref)
+            ev = lambda t: sum(float(c) * rf ** j for j, c in enumerate(t.frac_coeffs()))
+            E = ev(tml.e_bb) * float(b) ** 2 + 2 * ev(tml.e_bk) * float(b) * kf + ev(tml.e_kk) * kf * kf
+            pr = math.sqrt(12 / ev(tml.det)) * math.exp(-(E - (1.5 * float(b) ** 2 + 24 * kf * kf)) / 2)
+            out.append({'d': d, 'r': str(r), 'b': str(b), 'k': str(k), 'n': n, 'Z_ratio': round(zr, 5), 'Z_se': round(zse, 5),
+                        'pin_ratio': round(pr, 6), 'coef': round((pr * zr - 1) / rf ** 2, 3), 'coef_se': round(pr * zse / rf ** 2, 3)})
+    return out
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--controls', action='store_true')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.controls:
+        out = controls()
+        out['mc'] = mc_ratio(2, 200000) + mc_ratio(3, 100000)
+        print(json.dumps(out, indent=1))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    require(MUT is None, 'a mutant cannot regenerate RESULTS.json')
+    res = full_run(args.procs)
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+    print(json.dumps({d: {R: row['C_BK'] for R, row in res['dims'][d]['table'].items()} for d in res['dims']}))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_theorem_a_constants_d3_20261001/NOTE.md b/frontiers/c8_theorem_a_constants_d3_20261001/NOTE.md
new file mode 100644
index 000000000..718dd17eb
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_d3_20261001/NOTE.md
@@ -0,0 +1,543 @@
+# Explicit cap-route constants for Theorem A in dimension three: `Q^W(G_r^c) <= C3 r^3` with certified `C3`
+
+**Object:** `CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1`. **Author lane:** Anthropic / Claude.
+**Kind:** author-side theorem (Theorem E3), proved with explicit constants and certified by exact rational interval
+arithmetic (standard library only). **Scientific effect:** NONE. Catalog entry C8 (`reviews/candidates_pending_20260928/CANDIDATES.md`:
+"explicit finite `C`, `r_*`, `z_*`, `c_{B,K}`, `c_{d,L}` on a declared compact-mark / torus band; OPEN") stays OPEN, and no
+catalog edit is made. What this packet supplies, in `d = 3`:
+- on the band `B = [0, 1]`, `K = [1/2, 2]`, uniformly over frames and torus sides `L >= 10`, explicit cap-route pairs
+  `(C3, r_*)` for [LP] (7.8), and therefore for [LP] Theorem A (1.1);
+- a normalizer floor `z3` on `(0, r_*]`.
+
+It is the three-dimensional counterpart of Math-#206 (the planar Theorem E). No STATUS, PROOF_INDEX, GRAPH or catalog edit.
+Same GitHub account as every lane; zero organizational-independence credit. The author will not merge. Scope claim: Math-#208
+comment 5922961739.
+
+**Sources** (pins in `SOURCE_MAP.json`):
+- [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, sections 1, 3, 7 and 8 in `d = 3`:
+  - the model, the law `Q`, `W_r`, `Z_r`;
+  - the preconditioned pins (3.1) and their values (3.3), with `m = 2` transverse directions;
+  - the good event `G_r`, Theorem A (1.1) and the cap bound (7.8);
+  - the §8 implication "on `G_r` the global elder partner of `M` is `S`".
+- [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md`, section 1: the cylinder `D = [-2r, 2r] x ball(0, 2r)`, the
+  partial-block operator norms `M_3`, `M_4` (suprema over independently chosen transverse unit vectors) and condition (1).
+- [G3] Math-#208 at `da48373`, read-only: Theorem G_d, the `d = 3` contact law (its Lemma 1) and the certified
+  `c_G^(3)(0, 2) >= 22424320.655069`, used for the lower-bound sanity check and the comparisons.
+- [E] Math-#206 at `0cb8ad3`, read-only: the planar Theorem E, whose proof this packet follows section by section.
+
+Like [E], this packet does **not** consume [LP] (4.1), (4.2), (5.4), (5.5) or (7.5)–(7.7). Every estimate below is proved here
+with its constant. From [LP] it uses only the definitions, the pins, and (for the consequence on `1 - p_r` alone) the §8
+implication.
+
+## 0. Statement
+
+**Setting** ([LP] section 1 with `d = 3`, `m = 2`; as in [G3] section 0):
+- The torus `X = (R / L Z)^3` and the kernel `K_L(z) = sum_n exp(-|z + Ln|^2/2) / sum_n exp(-|Ln|^2/2)`.
+- An orthonormal frame `R` with axial unit vector `u`, and the pins `M = -(r/2) u`, `S = (r/2) u`, `f(M) = b`,
+  `f(S) = b - k r^3`, `grad f(M) = grad f(S) = 0`.
+- The Gaussian regression law `Q = Q_{r,b,k,R}`, the typed weight `W_r = |det H_M det H_S| 1{H_M < 0, index H_S = 2}`,
+  `Z_r = E_Q W_r`, and `dQ^W = (W_r / Z_r) dQ`.
+- In frame coordinates `(x, y) in R x R^2`: the cylinder `D = [-2r, 2r] x ball(0, 2r)`, the partial-block norms
+  `M_j = max_{a + c = j} sup_D ||partial_x^a D_y^c f||_op` ([CAP] section 1), and the good event
+  `G_r = {lambda_1 > (4/(3k)) r M_3^2, r M_4 <= 3k/10}` with `lambda_1 = lambda_min(-D_y^2 f(M))`.
+- Theorem A of [LP] is `0 <= 1 - p_r <= C r^3` ([LP] (1.1)). Its proof bounds the larger probability `Q^W(G_r^c)` in (7.8) and
+  uses `1 - p_r <= Q^W(G_r^c)` (§8).
+- A **cap-route constant** is any `C` with `Q^W(G_r^c) <= C r^3` for `0 < r <= r_*`. Every cap-route constant is a constant
+  for [LP] (1.1); the converse fails (Math-#203 v1.2, section 5, item 1).
+
+**Theorem E3.** Let `d = 3`, `B = [0, 1]` and `K = [1/2, 2]`. For every torus side `L >= 10`, every orthonormal frame `R`, every
+`b in B`, `k in K`, every `r_*` in the table, and every `0 < r <= r_*`,
+
+    Q^W(G_r^c) <= C3(r_*) r^3        and        Z_r / r^2 >= z3(r_*).
+
+The same holds for the reference kernel `exp(-|z|^2/2)` on `R^3`. Every enclosure below contains the case `M_1 = 0`,
+`Theta = 1` of section 2. The table lists, for each `r_*`:
+- `C3(r_*)`, an exact rational, with its decimal;
+- upper bounds for `C3 / c_G^(3)(0, 2)` and `C3 r_*^3`, and the floor `z3(r_*)`;
+- the cell (`r`-band; `b`; `k`) at which the maximum is attained.
+
+| `r_*` | `C3(r_*)` | `C3 / c_G^(3)(0,2) <=` | `C3 r_*^3 <=` | `z3(r_*)` | maximizing cell |
+|---|---|---|---|---|---|
+| `1/4096` | `56887004491/2500 = 22754801.7964` | `1.014738` | `0.0003311260` | `6.021091` | `[15/65536, 1/4096]`; `[0, 1/512]`; `[63/32, 2]` |
+| `1/2048` | `57060167561/2500 = 22824067.0244` | `1.017827` | `0.0026570712` | `5.998005` | `[15/32768, 1/2048]`; `[0, 1/512]`; `[63/32, 2]` |
+| `1/1024` | `5743533573/250 = 22974134.2920` | `1.024519` | `0.0213963299` | `5.951763` | `[15/16384, 1/1024]`; `[0, 1/512]`; `[63/32, 2]` |
+| `1/512` | `14583076547/625 = 23332922.4752` | `1.040519` | `0.1738438195` | `5.859005` | `[15/8192, 1/512]`; `[439/1024, 421/512]`; `[31/16, 63/32]` |
+| `1/256` | `3714819319587/625 = 5943710911.3392` | `265.06` | `354.3` | `5.672384` | `[7/2048, 15/4096]`; `[0, 1/2]`; `[1/2, 3/4]` |
+| `1/128` | `69971360017.7629` | `3120.3` | `33365` | `5.294648` | `[15/2048, 1/128]`; `[1/2, 1]`; `[1/2, 3/4]` |
+| `1/64` | `17254355056897.0472` | `769448` | `6.6 x 10^7` | `4.520628` | `[15/1024, 1/64]`; `[1/2, 1]`; `[1/2, 3/4]` |
+
+`c_G^(3)(0, 2)` is replaced by its certified lower end `22424320.655069` ([G3]).
+
+**Corollary (explicit Theorem A in `d = 3` on the band).**
+- **Explicit pairs.** Since `0 <= 1 - p_r <= Q^W(G_r^c)` ([LP] §8), [LP] (1.1) holds in `d = 3` on `B x K`, uniformly over frames
+  and `L >= 10`, with every pair `(C3(r_*), r_*)` of the table. The pairs that are informative at `r = r_*` (`C3 r_*^3 < 1`)
+  are exactly those with `r_* <= 1/512`. Examples:
+  - `(C, r_*) = (23332922.4752, 1/512)`, where the bound at `r = r_*` is below `0.174`;
+  - `(22974134.2920, 1/1024)`, where it is below `0.0214`.
+- **What was missing.** [LP] §16 lists "a numerical `r_*` or `C` on a prescribed band" among the items it does not supply.
+  [CAP]'s probabilistic corollary is planar only ("No 3D eight-pin matrix-boundary estimate ... is proved here"). No
+  explicit constant for (1.1) or (7.8) in `d >= 3` is recorded in [LP], [CAP] or the open packets of this repository.
+- **Near-optimal for the cap route.**
+  - Every `d = 3` cap-route constant valid for the reference kernel at `(b, k) = (0, 2)` is at least
+    `c_G^(3)(0, 2) > 22424320.65` ([G3] Corollary 1).
+  - This certificate's scope includes the reference kernel.
+  - The constants are within `1.5%`–`4.1%` of that limit for `r_* <= 1/512`.
+  - The cap route in `d = 3` can be informative only below `r = c_G^(3)(0, 2)^(-1/3) = 3.546 x 10^-3`, so `r_* = 1/256`
+    (`C3 r_*^3 >= c_G^(3) r_*^3 = 1.34`) is uninformative for every cap-route constant. At `1/256` and above, the certified
+    constants are dominated by the rare branches at `k near 1/2`.
+- **Against the plane.**
+  - `C3(r_*)/C(r_*)` is `4.249` at `r_* = 1/4096` and `4.308` at `1/512` ([E] Theorem E).
+  - The sharp ratio is `c_G^(3)(0, 2)/c_G(0, 2) = R_3(0) = (32 + 28 sqrt2)/17 = 4.2116` ([G3]).
+- **Normalizer floor.** `z3(r_*)` is an explicit `z_*` for [LP] (5.5) in `d = 3` on `(0, r_*]`.
+  - The contact value at the band's weakest corner `(b, k) = (0, 1/2)` is `36 k^2 m_(3,0) = 9 (7 - 4 sqrt2)/2 = 6.0442`.
+  - `z3(1/4096) = 6.0211` lies within `0.4%` of it; the last band `[0, 2^-29]` gives `6.044155`.
+  - `Z_r / r^2 >= 4.5206` on all of `(0, 1/64]`.
+
+## 1. Coordinates and exact identities
+
+In frame coordinates `(x, y) = (x, y_1, y_2)`, `M = (-h, 0, 0)` and `S = (h, 0, 0)` with `h = r/2`; the midpoint is `0`.
+
+**Pins.** The eight preconditioned pins are those of [LP] (3.1):
+- `U0 = (f(M) + f(S))/2`, `U1 = (f(S) - f(M))/r`, `U2 = (f_x(S) - f_x(M))/r`, `U3 = (6/r^2)[f_x(M) + f_x(S) - 2(f(S) - f(M))/r]`;
+- for `j = 1, 2`: `V_j0 = (f_yj(M) + f_yj(S))/2` and `V_j1 = (f_yj(S) - f_yj(M))/r`.
+
+Under `Q` their values are exactly `(b - k r^3/2, -k r^2, 0, 12k, 0, 0, 0, 0)` ([LP] (3.3)).
+
+**Targets.** The 31 targets are:
+
+    A_ij  = f_yiyj(M)                                         (3; B = -A is the transverse Hessian of [LP] section 7),
+    v_j   = f_xyj(M)/r + (f_yj(M) - f_yj(S))/r^2               (2),
+    O_ij  = (f_yiyj(S) - f_yiyj(M))/r                         (3; the matrix Omega'),
+    Y_ijk = f_yiyjyk(0)                                       (4),
+    T, tau as in [E] section 1                                (2),
+    nu_j  = (f_xyj(M) + f_xyj(S))/r^2 + 2 (f_yj(M) - f_yj(S))/r^3   (2),
+    X_alpha = partial^alpha f(0),   |alpha| = 4                (15).
+
+Each is a sum of terms `c r^p partial_x^a partial_y1^e1 partial_y2^e2 f(x_0 r, 0, 0)` with a common **weight** `a - p`. Since the
+pins hold exactly under `Q`, almost surely
+
+    f_xx(M) = r^2 T - 6kr,  f_xx(S) = r^2 (T + r tau) + 6kr,  grad_y f_x(M) = r v,  grad_y f_x(S) = r (r nu - v),
+    D_y^2 f(S) = -B + r Omega'.                                                                              (I1)
+
+Write `c1 = 6k - rT`, `c2 = 6k + rT + r^2 tau`, `e = r nu - v` (a vector) and `B_S = B - r Omega'`. The Hessians are
+`H_M = [[-r c1, r v^T], [r v, -B]]` and `H_S = [[r c2, r e^T], [r e, -B_S]]`, and Schur complements give
+
+    det H_M = -r P1,   P1 = c1 det B - r v^T adj(B) v,
+    det H_S =  r P2,   P2 = c2 det B_S + r e^T adj(B_S) e.                                                  (I2)
+
+(For `2 x 2` matrices `det(-B) = det B` and `det(B) B^-1 = adj B`.)
+
+**The typed support.** `H_M < 0` holds exactly when `B > 0` and `c1 > r v^T B^-1 v`; then `c1 > 0` and `P1 > 0`. Index two
+for `H_S` forces `det H_S > 0`, so `P2 > 0`. Hence, in every case,
+
+    W_r <= r^2 P1 P2 1{B > 0, P1 > 0, P2 > 0}.                                                                (I3)
+
+Unlike the plane, (I3) is only an inequality: `det H_S > 0` also allows index zero. Conversely, `B > 0`, `B_S > 0`,
+`c1 > r v^T B^-1 v` and `c2 > 0` imply the typed event with `W_r = r^2 P1 P2` (section 6).
+
+## 2. The Gaussian law on an `r`-band (Lemma 1)
+
+**(a) Exact series.** Let `F`, `F'` be functionals of weights `w_F`, `w_F'`, with `w = w_F + w_F'`. For the reference kernel
+`G(z) = exp(-|z|^2/2)`,
+
+    Cov(F, F') = sum_gamma h(g_x) h(g_1) h(g_2) pi_gamma r^(g_x - w),        gamma = (g_x, g_1, g_2).
+
+- `h(n) = (d/dt)^n exp(-t^2/2)` at `t = 0`, which is `(-1)^(n/2) (n-1)!!` for even `n` and `0` for odd `n`.
+- `pi_gamma` is the exact rational `sum c c' (-1)^(a' + e1' + e2') (x_0 - x_0')^m / m!` over pairs of terms with
+  `g_1 = e1 + e1'`, `g_2 = e2 + e2'` and `m = g_x - a - a' >= 0`.
+- **Regularity:** `pi_gamma = 0` for every `g_x < w`. This is checked exactly for all 780 unordered pairs of the 39 functionals,
+  so no negative power of `r` occurs.
+
+**(b) Truncation at `g_x <= NX = 60`.** As in [E] section 2(b), with `|h(g_y)|` replaced by `|h(g_1) h(g_2)|`: for `r <= 1/64`
+the terms with `g_x > 60` of one pair sum to at most `2 |c c'| |h(g_1) h(g_2)| 61^A r^(61 - w)`, `A = a + a'`.
+
+**(c) Torus remainder, uniform in `L >= 10`, the frame and `r`.**
+- **Decomposition.** `K_L = (G + rho)/Theta` with `rho(z) = sum_{n != 0} G(z + p_n)`, `p_n` in the rotated lattice `L R^T Z^3`,
+  and `Theta = 1 + sum_{n != 0} exp(-|p_n|^2/2)` in `[1, 1 + theta_1(L)]`.
+- **Bound on `rho`.** `rho` is entire. On the complex polydisc `{|z_1|, |z_2|, |z_3| <= 1}`:
+  - `|exp(-(z + p).(z + p)/2)| <= e^(3/2) exp(-|Re z + p|^2/2)`, since `|Im z|^2 <= 3`;
+  - exactly `24 j^2 + 2` lattice points have sup-norm index `j`, each at distance `>= jL`;
+  - `|Re z| <= sqrt 3 < 7/4`.
+
+  Hence `|rho| <= M_1(L) := e^(3/2) sum_{j >= 1} (24 j^2 + 2) exp(-(jL - 7/4)^2/2)`, independently of the frame.
+- **Cauchy estimates** give `|partial^gamma rho(0)| <= g_x! g_1! g_2! M_1`, and the `rho`-part of a covariance is bounded by
+  `sum |c c'| g_1! g_2! max(w, 1)^A M_1 / (1 - e r)` exactly as in [E] section 2(c).
+- **Values.** `M_1(10) < 1.94 x 10^-13` and `theta_1(10) < 5.1 x 10^-21`. Both decrease in `L`, so the enclosures computed at
+  `L = 10` hold for every `L >= 10`, every frame, and the reference kernel (`rho = 0`, `Theta = 1`).
+
+**(d) Regression on the pins.** Interval Gaussian elimination on the eight pins gives, uniformly over `r in [r_0, r_1]`:
+- the `Q`-means `mu_F = b alpha_F + k beta_F` of the 31 targets;
+- their `Q`-covariance, scaled by `Theta^-1 in [1/(1 + theta_1), 1]`;
+- the pin energy `E = e_bb b^2 + 2 e_bk b k + e_kk k^2`, scaled by `Theta`.
+
+At contact (reference kernel) these reduce to the law of [G3] Lemma 1:
+- `B ~ b I + GOE_2`;
+- `v ~ N(0, I/2)`, `Omega' ~ GOE_2`;
+- `Y` with independent entries of variances `6, 2, 2, 6`;
+- all of them independent.
+
+The certificate's band laws reproduce this to `10^-9` on the last band.
+
+**(e) Regression on the transverse Hessian.** Use the Frobenius-isometric coordinates `b~ = (B_11, B_22, sqrt2 B_12)`, so that
+`||b~ - b~'|| = ||B - B'||_F`, with `Q`-mean `mu~` and covariance `C~` (at contact `2 I`).
+- For every other target, `beta~_F = Cov(F, b~) C~^-1` and `F_g = F - beta~_F . (b~ - mu~)`. Then `F_g` is independent of
+  `B` under `Q`, with the same mean and the covariance `C - Cov(., b~) C~^-1 Cov(b~, .)`.
+- Gershgorin bounds give `lambda_min(C~) >= sigma_B^2` and `lambda_max(C~) <= sigma^2`, with `sigma` rational.
+- By the `y_1`- and `y_2`-parities of the reference kernel, `beta~_v = beta~_Y = beta~_nu = 0` there; the certificate
+  carries their torus-sized enclosures.
+
+**(f) The free blocks.** In the coordinates `v`, `omega = (O_11, O_22, sqrt2 O_12)` and
+`Y~ = (Y_111, sqrt3 Y_112, sqrt3 Y_122, Y_222)`, Euclidean norms are `||v||`, `||Omega'||_F` and `||Y||_F` (the Frobenius norm of
+the symmetric 3-tensor).
+- An interval Cholesky factor `L` of the `g`-covariance of these nine coordinates has blocks `L_vv`, `L_ov`, `L_oo`, `L_Yv`,
+  `L_Yo` and `L_YY`.
+- Their operator norms are bounded by `max |diagonal|` plus the Frobenius norm of the strictly lower part (square blocks),
+  and by the Frobenius norm otherwise.
+- At contact `L_vv = I/sqrt2`, `L_oo = sqrt2 I`, `L_YY = sqrt6 I`, and the off-diagonal blocks vanish.
+
+## 3. Third and fourth derivatives on `D` (Lemmas 2 and 3)
+
+Write `A_(a,c) = sup_D ||partial_x^a D_y^c f||_op` for `a + c = 4`, so that `M_4 = max A_(a,c)`.
+
+**Lemma 2 (averaging identities, blockwise).** For every `C^4` function satisfying the pins, and every `z in D`:
+
+    |f_xxx(z)|            <= 12k        + r Delta_1,    Delta_1 = (5/2) A_(4,0) + 2 A_(3,1),
+    ||grad_y f_xx(z)||    <= 2 ||v||     + r Delta_2,    Delta_2 = (5/2) A_(3,1) + 2 A_(2,2),
+    ||D_y^2 f_x(z)||_op   <= ||Omega'||  + r Delta_3,    Delta_3 = (5/2) A_(2,2) + 2 A_(1,3),
+    ||D_y^3 f(z)||_op     <= ||Y||_op    + r Delta_4,    Delta_4 = 2 A_(1,3) + 2 A_(0,4),
+
+so `M_3 <= max_i (B_i + r Delta_i)` with `B = (12k, 2||v||, ||Omega'||_op, ||Y||_op) <= (12k, 2||v||, ||omega||, ||Y~||)`.
+
+*Proof.* Let `g(t) = f(t, 0, 0)` and `q(t) = grad_y f(t, 0, 0)`. The four averages are:
+- `U3 = integral K(t) g'''(t) dt` with the Peano kernel `K(t) = 3(h^2 - t^2)/(4h^3) >= 0`, `integral K = 1`, and `U3 = 12k`;
+- `-2v = integral (2(h - t)/r^2) q''(t) dt` (each component; `q(+-h) = 0` by the pins), weights `>= 0` with total `1`;
+- `Omega' = (1/r) integral D_y^2 f_x(t, 0, 0) dt`;
+- `Y = D_y^3 f(0)`.
+
+The norm of an average is at most `sup_t` of the norm at `(t, 0, 0)`. Along the segment from `(t, 0, 0)` to `z`, the
+derivative of `partial_x^a D_y^c f` is `(z_x - t) partial_x^(a+1) D_y^c f + D_y^(c+1) partial_x^a f [z_y, ...]`. Its operator
+norm is at most `|z_x - t| A_(a+1,c) + ||z_y|| A_(a,c+1)`, for multilinear operator norms with independently chosen unit
+vectors. Use `|z_x - t| <= 5r/2` and `||z_y|| <= 2r` (and `|z_x| <= 2r` for `Y`). The operator norm of a symmetric tensor is at
+most its Frobenius norm (Cauchy-Schwarz).
+
+**Lemma 3 (fourth derivatives).**
+- **Taylor bound.** The field is almost surely entire under `Q` ([E] Lemma 3). Hence, with `X_(a,c) = partial_x^a D_y^c f(0)`,
+
+      A_(a,c) <= ||X_(a,c)||_F + T_(a,c),      T_(a,c) = sum_{gamma != 0} ||partial_x^a D_y^c partial^gamma f(0)||_F (2r)^|gamma| / gamma!,
+
+  where `||X||_F^2 = sum_beta (c!/beta!) X_beta^2` for a symmetric `c`-tensor in two variables.
+- **Gaussian bounds.** For every `delta`, `s_delta^2 = prod_i (2 delta_i - 1)!! + (2 delta)! M_1` bounds the unconditioned
+  variance of `partial^delta f(0)`, and `|mean| <= s_delta sqrt(E)`.
+- **Norm bound.** For a Gaussian vector, `|| ||X - EX|| ||_p <= sqrt(tr Cov) max(1, ||N(0,1)||_p)` (Minkowski in `L^(p/2)` on
+  the eigen-decomposition). Hence `||T_(a,c)||_p <= S_(a,c) (sqrt(E) + max(1, ||N||_p))` with
+
+      S_(a,c)(r_1) = sum_{1 <= |gamma| <= 24} sigma_(a,c),gamma (2 r_1)^|gamma| / gamma!  +  (tail),
+      sigma_(a,c),gamma^2 = sum_beta (c!/beta!) s^2_(a + gamma_x, beta + gamma_y).
+
+  The tail uses `s_delta <= 2^|delta| delta! sqrt(1 + M_1)`, `(alpha + gamma)!/gamma! <= (|gamma| + 4)^4` and
+  `(n + 1)(n + 2)/2` multi-indices of order `n`. It requires `4 r_1 <= 1/16`.
+- **Dependence on `B`.** The regression coefficient of `partial^delta f(0)` on `b~` has norm at most `s_delta / sigma_B`. So
+  `A_(a,c) <= A^g_(a,c) + B_(a,c) ||B - mu_B||_F`, where
+  - `B_(a,c) = (sum_beta (c!/beta!) ||beta~_X(a,beta)||^2)^(1/2) + S_(a,c)/sigma_B`;
+  - `A^g_(a,c)` is the same bound for the `g`-field: same means, smaller variances.
+
+## 4. Regression on `B` and the failure set (Lemma 4)
+
+On the typed support `B > 0`. With `lambda_1 <= lambda_2` the eigenvalues of `B`,
+
+    ||B - mu_B||_F <= ||B||_F + ||mu_B||_F <= lambda_1 + lambda_2 + ||mu_B||_F.
+
+**The `g`-measurable bound on `M_3`.** By Lemmas 2 and 3 and the regression, `M_3 <= max_i (y_i + bar-beta_i ||B - mu_B||_F)`, with
+
+    y_1 = 12k + r Delta_1^g,   y_2 = 2 ||v_g|| + r Delta_2^g,   y_3 = ||omega_g|| + r Delta_3^g,   y_4 = ||Y~_g|| + r Delta_4^g,
+    bar-beta_1 = r ((5/2) B_(4,0) + 2 B_(3,1)),
+    bar-beta_2 = 2 ||beta_v|| + r ((5/2) B_(3,1) + 2 B_(2,2)),
+    bar-beta_3 = ||beta_omega|| + r ((5/2) B_(2,2) + 2 B_(1,3)),
+    bar-beta_4 = ||beta_Y~|| + r (2 B_(1,3) + 2 B_(0,4)),
+
+where `||beta_v||^2 = sum_j ||beta~_vj||^2`, and similarly for `omega` and `Y~` with the Frobenius weights.
+
+**Lemma 4.** Put `y_i'' = y_i + bar-beta_i (||mu_B||_F + lambda_2)`, which is measurable in the `g`-variables and `lambda_2`. Put
+`a = 4/(3k)` and `x_i = a r bar-beta_i y_i''`. If `lambda_1 <= a r M_3^2`, then for some `i`,
+`lambda_1 <= a r (y_i'' + bar-beta_i lambda_1)^2`. For `x_i <= 1/4` the roots of this quadratic satisfy the bounds of [E] Lemma 4:
+
+    lambda_-^(i) <= a r y_i''^2 / (1 - 3 x_i),        lambda_+^(i) >= 1 / (4 a r bar-beta_i^2).
+
+**The cover.** With `X_0 = 1/64` and `kappa = 3/(1 - 3 X_0)`, on the typed support
+
+    G_r^c  subset  N  u  F  u  X  u  A,
+    N = {lambda_1 <= r U,  x_i <= X_0 for all i},    U = max_i a y_i''^2 (1 + kappa x_i),
+    F = {lambda_1 >= 1/(4 a r bar-beta^2)},  bar-beta = max_i bar-beta_i,
+    X = {x_i > X_0 for some i},    A = {r M_4 > 3k/10}.
+
+The threshold `U` is a function of the `g`-variables and of `lambda_2`. The `g`-variables are independent of `B`. The failure
+window in `lambda_1` is `[0, r U]`.
+
+## 5. The main term
+
+**Eigenvalue coordinates.** Write `B = R_phi diag(lambda_1, lambda_2) R_phi^T` with `phi in [0, pi)`. Then
+`dB_11 dB_22 dB_12 = (lambda_2 - lambda_1) d lambda_1 d lambda_2 d phi` (the Jacobian determinant is `-(lambda_2 - lambda_1)`).
+
+**An angle-free density majorant.**
+- The density of `(B_11, B_22, B_12)` is `sqrt2` times that of `b~`, which is at most
+  `(2 pi)^(-3/2) det(C~)^(-1/2) exp(-||b~ - mu~||^2 / (2 sigma^2))`.
+- Let `epsilon` be the supremum over the box of `||mu_B - b I||_F`. Since `||B - mu_B||_F >= ||B - b I||_F - epsilon`, we have
+  `||B - bI||_F^2 = (lambda_1 - b)^2 + (lambda_2 - b)^2` and `||B - bI||_F <= |lambda_1 - b| + |lambda_2 - b|`.
+- Integrating over `phi` therefore bounds the joint density of `(lambda_1, lambda_2)` by
+
+      p(lambda_1, lambda_2) <= K_B (lambda_2 - lambda_1) g(lambda_1) g(lambda_2),
+      K_B = pi sqrt2 (2 pi)^(-3/2) det(C~)^(-1/2),   g(lambda) = exp(-(lambda - b)^2/(2 sigma^2) + epsilon |lambda - b| / sigma^2).
+
+- At contact (`epsilon = 0`, `C~ = 2I`) this is the exact `GOE_2` eigenvalue density
+  `(lambda_2 - lambda_1) exp(-((lambda_1 - b)^2 + (lambda_2 - b)^2)/4) / (4 sqrt(2 pi))` of [G3] section 1.
+
+**The weight on `N`.** Write `Omega = ||Omega'||_op`. On the typed support:
+- `P1 <= |c1| lambda_1 lambda_2`, since `adj B >= 0`;
+- `|det B_S| <= (lambda_1 + r Omega)(lambda_2 + r Omega)`, by Weyl's inequalities;
+- `|e^T adj(B_S) e| <= ||e||^2 (lambda_2 + r Omega)`, since `adj B_S` has the eigenvalues of `B_S`.
+
+Hence
+
+    W_r <= r^2 |c1| lambda_1 lambda_2 (lambda_2 + r Omega) [|c2| (lambda_1 + r Omega) + r ||e||^2],
+
+and with `lambda_1 = r u`, `W_r <= r^4 |c1| u lambda_2 (lambda_2 + r Omega) [|c2| (u + Omega) + ||e||^2]`.
+
+**Majorants.** On `N`, since `lambda_1 <= lambda_2`, every target satisfies
+`|F| <= bar-F = |F_g| + ||beta~_F|| (||mu_B||_F + 2 lambda_2)`. In particular
+`Omega <= bar-Omega = ||omega_g|| + ||beta_omega|| (||mu_B||_F + 2 lambda_2)` and `||e||^2 <= bar-e = (r ||bar-nu|| + ||bar-v||)^2`.
+
+**Integration.** Integrate `lambda_1 = r u` over `[0, r U]`, extend `lambda_2` to `(0, infinity)`, and use
+`lambda_2 - lambda_1 <= lambda_2`:
+
+    E_Q[W_r 1_N] <= K_B r^5 E_g integral_0^inf lambda_2^2 (lambda_2 + r bar-Omega) g(lambda_2)
+                                     integral_0^U g(r u) |bar-c1| u [|bar-c2| (u + bar-Omega) + bar-e] du d lambda_2.
+
+Write `lambda_2^2 (lambda_2 + r bar-Omega) g(lambda_2) d lambda_2 = I_3 nu(d lambda_2) + r bar-Omega I_2 nu'(d lambda_2)` with
+`I_j = integral_0^inf lambda^j g(lambda) d lambda`, and the probability measures `nu ∝ lambda^3 g` and `nu' ∝ lambda^2 g`.
+- Under `g x nu` and `g x nu'`, `lambda_2` is independent of the `g`-variables.
+- `E_nu[lambda_2^p] = I_(3+p)/I_3`, and it dominates `E_nu'[lambda_2^p]` (Chebyshev's association inequality for the increasing
+  functions `lambda^p` and `lambda` under `nu'`).
+
+**Bounds on `I_j`.** For `lambda >= 0`, `epsilon |lambda - b| <= epsilon (lambda + b)`, so
+
+    g(lambda) <= exp((4 b epsilon + epsilon^2)/(2 sigma^2)) exp(-(lambda - b - epsilon)^2/(2 sigma^2)),
+
+and `I_j <= exp((4 b_hi epsilon + epsilon^2)/(2 sigma^2)) J_j(b_hi + epsilon, sigma)`.
+- Here `J_j(m, sigma) = integral_0^inf lambda^j exp(-(lambda - m)^2/(2 sigma^2)) d lambda` increases in `m`.
+- It satisfies `J_0 = sigma sqrt(2 pi) Phi(m/sigma)`, `J_1 = m J_0 + sigma^2 exp(-m^2/(2 sigma^2))` and
+  `J_(j+1) = m J_j + j sigma^2 J_(j-1)`, by integration by parts.
+- Also `I_3 >= J_3(b_lo, sigma)`.
+
+**Density in `lambda_1`.** Two bounds hold on the failure window:
+- `g(lambda_1) <= bar-g = exp(epsilon^2/(2 sigma^2))`, the supremum of `g`;
+- `g(lambda_1) <= g_0 + lambda_1 L`, where
+  - `g_0` is the supremum of `g(0)` over the box;
+  - `L = min(exp(-1/2) bar-g / sigma, g_0 c (x - 1)/log x)`, with `c = (b_hi + epsilon)/sigma^2` and `log x = log(bar-g/g_0)`.
+
+  The second slope is the chord of `min(bar-g, g_0 exp(c lambda))`, valid because `g(lambda) <= g(0) exp((b + epsilon) lambda / sigma^2)`
+  for `lambda >= 0`. It matters at larger `b`, where `g` rises on the failure window. When `g_0 >= bar-g`, only the first
+  bound is used.
+
+**Normalization by `K = 12k`.** As in [E] section 5, `U = a K^2 tilde-U` with `a K^2 = 192k`, and
+`|c1 c2|/(36k^2) <= 1 + eps_T` with `eps_T <= (2r|bar-T| + r^2|bar-tau|)/(6k) + r|bar-T|(r|bar-T| + r^2|bar-tau|)/(36k^2)`. Then
+
+    E_Q[W_r 1_N] / (36 k^2 r^5)  <=  K_B min( bar-g Lambda,  g_0 Lambda + r L Lambda' ),
+    Lambda  = I_3 (A + B + C) + r I_2 D,       Lambda' = I_3 (A' + B' + C') + r I_2 D'.
+
+- `A, B, C, A', B', C'` are the terms of [E] section 5, with the factor `bar-om` replaced by `bar-Omega` and the factor
+  `bar-e` by its vector form.
+- `D` and `D'` bound `E[bar-Omega (integrand of A + B + C)]` and the same with one more power of `tilde-U`, under `g x nu'`.
+  - The part of `bar-Omega` sharing the variable `chi_3` of `u_3` is integrated exactly against `tilde-U^3` (resp.
+    `tilde-U^4`).
+  - The rest goes through Hölder's inequality.
+
+**Moments of `tilde-U`.** These follow [E] section 5, with three changes.
+- **Chi variables.** The free variables are `u_2 = (2 ||L_vv||/K) chi_2`, `u_3 = (||L_oo||/K) chi_3` and
+  `u_4 = (||L_YY||/K) chi_4`, independent. Their positive-part moments are in closed form:
+  - `E[u^e (u^n - c^n)_+] = s^(n+e) E[chi^(n+e) 1{chi > c/s}] - c^n s^e E[chi^e 1{chi > c/s}]`;
+  - `E[chi_nu^j 1{chi_nu > x}] = C_nu sqrt(2 pi) M_(j+nu-1)(x)`, with `C_nu = 1/(2^(nu/2 - 1) Gamma(nu/2))` and the
+    incomplete Gaussian moments `M_j` of [E].
+- **Cross terms.** The cross-Cholesky terms `||L_ov|| chi_2` and `||L_Yv|| chi_2 + ||L_Yo|| chi_3` enter `delta_3` and
+  `delta_4`.
+- **`lambda_2`.** It enters `tilde-y_1` through `(bar-beta_1/K) lambda_2` and each `delta_i` through `(bar-beta_i/K) lambda_2`,
+  with `L^p` norms under `nu`. Since `bar-beta_1 = O(r^2)`, this is invisible in the leading term.
+
+## 6. The normalizer floor (Lemma 5')
+
+**Pointwise bound.** Let `G_0 = {r|T| <= 3k/2, r^2|tau| <= 3k/2}`. On `G_0`:
+- `c1 >= 9k/2`, `3k <= c2 <= 9k`, and `c1 c2 >= 36k^2 (1 - eps_T)` with `eps_T = (2r|T| + r^2|tau|)/(6k)`, as in [E].
+- Write `ell = (lambda_1)_+` and `Omega = ||Omega'||_op`. Pointwise,
+
+      W_r / r^2  >=  1_{G_0} { 36k^2 [ (1 - eps_T) ell^2 lambda_2^2 - r Omega ell lambda_2 (ell + lambda_2) ] - c2 r ||v||^2 ell lambda_2^2 }.
+
+*Proof.* Suppose `B > 0`, `lambda_1 > r Omega`, `c1 lambda_1 > r ||v||^2` and `c2 > 0`. Then the event is typed and
+`W_r = r^2 P1 P2`:
+- `lambda_1 > r Omega` gives `B_S > 0` by Weyl;
+- `c1 lambda_1 > r ||v||^2` gives `c1 > r v^T B^-1 v`;
+- then `P1 >= lambda_2 (c1 lambda_1 - r ||v||^2)`, since `v^T adj(B) v <= lambda_2 ||v||^2`;
+- and `P2 >= c2 (lambda_1 - r Omega)(lambda_2 - r Omega)`, since `adj(B_S) >= 0`.
+
+So `W_r / r^2 >= 1_{G_0} c2 lambda_2 (lambda_2 - r Omega)_+ (c1 lambda_1 - r ||v||^2)_+ (lambda_1 - r Omega)_+` in every case.
+Then use:
+- `(x - y)_+ z >= xz - yz`;
+- `lambda_1 (lambda_1 - r Omega)_+ lambda_2 (lambda_2 - r Omega)_+ >= ell^2 lambda_2^2 - r Omega ell lambda_2 (ell + lambda_2)`;
+- `c2 <= 9k`.
+
+**Taking expectations.** With `F = (det B)^2 1{B > 0} = ell^2 lambda_2^2` and `DT = det B tr B 1{B > 0}` (note
+`ell lambda_2 (ell + lambda_2) = DT` and `ell lambda_2^2 <= DT`):
+
+    Z_r / (36 k^2 r^2)  >=  E[F] - ||F||_2 P(G_0^c)^(1/2) - ||eps_T||_2 ||F||_2 - r E[Omega DT] - (r/(4k)) E[||v||^2 DT].
+
+The regression splits the last two terms:
+- `E[Omega DT] <= E||omega_g|| E[DT] + ||beta_omega|| E[(tr B + ||mu_B||_F) DT]`;
+- `E[||v||^2 DT] <= (17/16) E||v_g||^2 E[DT] + 17 ||beta_v||^2 E[(tr B + ||mu_B||_F)^2 DT]`.
+
+**Contact moments in closed form.** For `B = m I + GOE_2`:
+- The eigenvalues are `s -+ rho`, where `s = m + (G_11 + G_22)/2 ~ N(m, 1)` and
+  `rho = (((G_11 - G_22)/2)^2 + G_12^2)^(1/2)` is Rayleigh and independent of `s`.
+- Hence `det B = s^2 - rho^2`, `tr B = 2s`, and `B > 0` exactly when `s > rho`.
+- With `u = rho^2/2`, `integral_0^s (s^2 - rho^2)^a rho e^(-rho^2/2) d rho = P_a(s) - e^(-s^2/2) Q_a(s)` for explicit
+  polynomials `P_a`, `Q_a`.
+- `E[s^n e^(-s^2/2) 1{s > 0}] = (e^(-m^2/4)/sqrt2) E[(m/2 + U/sqrt2)^n 1{U > -m/sqrt2}]` with `U ~ N(0, 1)`.
+
+So every `E[det(B)^a tr(B)^c 1{B > 0}]` is a finite combination of incomplete Gaussian moments. In particular
+
+    m_(3,m) = E[(det B)^2 1{B > 0}] = E[(s^4 - 4 s^2 + 8) 1{s > 0}] - 4 sqrt2 e^(-m^2/4) Phi(m/sqrt2),
+
+which gives `m_(3,0) = (7 - 4 sqrt2)/2`, the value of [G3] section 1. The certificate checks this value.
+
+**Loewner coupling.**
+- Write `b~ = mu~ + L~ z`, with `L~` the Cholesky factor of `C~` and `z` standard in `R^3`, and put `G = mat(sqrt2 z) ~ GOE_2`.
+- Then `B - [(b - epsilon) I + G] >= -eta ||z|| I`, where `eta = ||L~ - sqrt2 I||`, because `||mat(y)||_op <= ||y||`.
+- Every `h(B) = (det B)^a (tr B)^c 1{B > 0}` is monotone in the Loewner order.
+- So on `{||z|| <= R_0 = 12}`, `h((b_lo - epsilon - eta R_0) I + G) <= h(B) <= h((b_hi + epsilon + eta R_0) I + G)`.
+
+Hence:
+- `E[F] >= m_(3, b'_lo) - E[F((b'_lo) I + G)^2]^(1/2) P(chi_3 > 12)^(1/2)`;
+- the correction moments are bounded at `b'' = b_hi + epsilon + eta R_0`, plus outside-ball terms;
+- `m_(3,b)` and the correction moments increase in `b`.
+
+## 7. Rare branches
+
+**Weight bound.** On the typed support, with `D = det B 1{B > 0}`,
+
+    W_r / r^2 <= |c1| D [ |c2| (D + r Omega tr B + r^2 Omega^2) + r ||e||^2 (tr B + r Omega) ].
+
+Hölder gives `||W_r 1_typed||_q <= r^2 g_1(q) g_2(q)`.
+- `||D||_p` (for `p = 4q <= 64`) comes from the contact moments at `b''` (Loewner coupling).
+- The outside-ball part is bounded by `2^-p E[(||mu_B||_F + sigma chi_3)^(2p) 1{chi_3 > 12}]`, using `D <= ||B||_F^2/2`.
+
+Then `E_Q[W_r 1_E] <= ||W_r 1_typed||_q P(E)^(1 - 1/q)` for `E = F, X, A` and `q in {2, 4, 8, 16}`, the smallest bound being
+used.
+
+**`P(A)`.** For each block `(a, c)`, the threshold `t = 3k/(10r)` is split as `f t + (1 - f) t`, `f in {7/8, 15/16}`, and the
+smaller bound is used:
+- `c = 0`: a Gaussian tail of `X_(4,0,0)`.
+- `c = 1`: `||X_(3,.)|| <= ||mean|| + lambda_max^(1/2) chi_2`, a chi tail.
+- `c >= 2`: `||X_(a,c)||_op = max_theta |Q(theta)|`, where `Q(theta) = partial_x^a partial_(u_theta)^c f(0)` is a trigonometric
+  polynomial of degree `c`.
+  - Bernstein's inequality gives `|Q'| <= c max|Q|`.
+  - A net of 258 rational unit vectors (`((1 - t^2), 2t)/(1 + t^2)`, `t = j/128`, and their rotations by `pi/2`) has angular
+    gaps at most `2/128`.
+  - Together these give `max|Q| <= max_j |Q(theta_j)| / (1 - c/128)`.
+  - The certificate uses the union bound with the largest net variance of the `Q`-law and `|mean| <= ||mean||_F`.
+- In every case, plus Markov's inequality at `p = 32` on `T_(a,c)`.
+
+The Frobenius norm would be far too generous here. The bi-Laplacian direction of the quartic block carries Frobenius variance
+`~ 144` against `~ 96` for any single direction.
+
+**`P(F)`.** `P(F) <= P(B_11 >= lambda_far)`, since `lambda_1 <= B_11`.
+
+**`P(X)`.** `P(X) <= sum_i P(y_i' + bar-beta_i lambda_2 > X_0/(a r bar-beta_i))`, split into:
+- Markov's inequality on `r Delta_1` (`i = 1`), and chi tails of `u_i` plus Markov's inequality on `delta_i` (`i >= 2`);
+- `P(lambda_2 > s) <= P(sigma chi_3 > s - ||mu_B||_F)`.
+
+## 8. Bands, boxes, and the last band
+
+**Bands.** `(0, 1/64]` is covered by `[0, 2^-29]` and 120 bands: the octaves `[2^-(j+1), 2^-j]` are split into 8 equal
+sub-bands for `j = 6, ..., 12` and into 4 for `j = 13, ..., 28`. On each band the law is enclosed uniformly (Lemma 1), and every
+`r`-monotone quantity is taken at its worst endpoint.
+
+**Boxes.** There are 39 boxes `(b, k)`, listed as `GRID` in `theorem_a3.py` and in `RESULTS.json`:
+- the `k`-boxes are `[1/2, 3/4], [3/4, 1], [1, 5/4], [5/4, 3/2], [3/2, 13/8], [13/8, 7/4], [7/4, 15/8], [15/8, 31/16], [31/16, 63/32]` and
+  `[63/32, 2]`;
+- each `k`-box has its own `b`-grid, refined toward the corner `(b, k) = (0, 2)`; the top `k`-box has 11 `b`-boxes, starting
+  with `[0, 1/512]`.
+
+The grids were chosen so that every box stays below the corner box at `r <= 1/4096` and `r <= 1/512`. This matters because
+the bound grows with `b_hi` through `I_3` while `c_G^(3)` decreases in `b`. A box only affects how close `C3` comes to
+`c_G^(3)`, never validity.
+
+`C3(r_*)` is the maximum of the box bounds over the bands with `r_1 <= r_*`. `z3(r_*)` is the minimum over the same boxes of
+`36 k_lo^2` times the floor of Lemma 5'.
+
+**The last band `[0, R]`, `R = 2^-29`.** The argument of [E] section 8 carries over:
+- every Gaussian or chi tail argument is `>= 3` at `R` (checked), and it scales at least like `1/r`;
+- no tail probability reaches `1` at `R` (checked);
+- for a `chi_nu` tail with `nu <= 5`, `P(chi_nu > 2x) <= 2^nu e^(-3x^2/2) P(chi_nu > x) < P(chi_nu > x)/64` when `x >= 3`;
+- every Markov piece scales like `r^32` or faster.
+
+So the tail term over `r^3` attains its supremum over `(0, R]` on `[R/2, R]`.
+
+## 9. Arithmetic
+
+As in [E] section 9:
+- exact rationals;
+- `exp`, `Phi`, `pi` and `sqrt` enclosed with outward rounding to `2^-160`;
+- upper bounds rounded upward to 200 significant bits;
+- interval Gaussian elimination and Cholesky decompositions with positive pivots.
+
+In addition:
+- The `chi` constants `C_nu sqrt(2 pi)` come from `Gamma` at integers and half-integers.
+- The contact moments use exact polynomial coefficients.
+- The `J_j` come from the stable three-term recursion. All its terms are positive for `m >= 0`.
+
+## 10. Controls (floating point; not part of the certificate)
+
+`theorem_a3.py --mc R B K N SEED` is an end-to-end control computed from the definitions. It works as follows:
+- It samples the degree-7 Taylor jet at the midpoint under `Q`, from the exact pin series evaluated in floating point.
+- It evaluates `W_r` from the Hessians at `M` and `S`.
+- It evaluates `M_3`, `M_4` on a 23-point grid of `D`, maximizing the binary forms over 24 angles.
+
+Pooled over 16 seeds:
+
+| `(r, b, k)` | samples | `Q^W(G_r^c)/r^3` | `Z_r/r^2` | certified `C3(r)` |
+|---|---|---|---|---|
+| `(1/512, 0, 2)` | `320000` | `1.572e7 +- 0.026e7` | `95.7 +- 1.1` | `23332922.48` |
+| `(1/1024, 0, 2)` | `640000` | `1.906e7 +- 0.028e7` | `97.5 +- 0.9` | `22974134.29` |
+
+- **The normalizer.** `Z_r/r^2` agrees with the contact value `36 k^2 m_(3,0) = 96.68`.
+- **The failure probability.**
+  - `Q^W(G_r^c)/r^3` lies below `c_G^(3)(0, 2) = 2.2424e7` at these radii.
+  - It approaches the limit linearly: `2 x 1.903e7 - 1.569e7 = 2.237e7`, within `0.3%` of `c_G^(3)` ([G3]).
+  - The first-order correction is negative and about three times the planar one (the planar control of [E] at
+    `(1/512, 0, 2)` was `9%` below `c_G`). It comes from the eigenvalue Jacobian `lambda_2 - lambda_1`, which the certificate
+    bounds by `lambda_2`.
+- **Caveats.** The grid suprema lie slightly below the true ones, so the Monte Carlo failure probability is, if anything,
+  underestimated. These are consistency checks, not part of the proof.
+
+## 11. What this does not do
+
+- **`d = 3` only.** The method extends to every `d` with `m = d - 1` transverse directions: `GOE_m` eigenvalue coordinates, and
+  `m`-variable blocks for `M_3` and `M_4`. This packet certifies `d = 3` only.
+- **The band only.** No `k < 1/2`. In particular there is no three-dimensional SIDE24 point: the [CAP] SIDE24 law is planar.
+- **No lower bound for the constant of (1.1).** Every bound here is for the cap route (7.8). As in [G3] Corollary 2, the
+  cap criterion overstates the true failure rate by the planar factor of Math-#203 (about `2.4 x 10^6` at `(0, 2)`).
+- **Other C8 constants.** `c_{B,K}` and `c_{d,L}` are not addressed (see Math-#205 and Math-#209 for `c_{B,K}` in `d = 3, 4`).
+- **Not a review.** This is not a review of [LP]'s proof or of [CAP]. The [LP] §8 implication and [CAP]'s deterministic
+  theorem are used as statements, only for the consequence on `1 - p_r`.
+- **Precision.** The cap-route constants for `r_* >= 1/256` are valid but far from sharp, because the rare branches at small
+  `k` dominate there. A finer sweep or sign-aware weight bounds could improve them, but they cannot become informative,
+  since `c_G^(3) (1/256)^3 > 1`.
+- **Status.** C8 stays OPEN.
+- **Independence.** Same GitHub account as every lane; zero organizational-independence credit. The author will not merge.
+
+## 12. Revisions
+
+- **v1 (`c4b8ec3`):** Theorem E3 on the band.
+- **v1.1:** documentation only. It changes two docstrings in `theorem_a3.py`, byte-identical copies of which are carried by
+  Math-#215 and Math-#227. The AST with docstrings removed is unchanged, and so are `RESULTS.json` and every constant.
+  - `sqrt_up` promised a relative error below `2^-200`; at `x = 49/100` the excess is `(8/7) 2^-200` (C59-DOC-01). The
+    stated bound is now `2^-198`: the root step adds less than `2^-209` and the rounding to 200 bits less than `2^-199`.
+  - `box_bound` said "every choice gives a valid bound". It now states the proved domain: `0 < x0 < 2/9` and
+    `0 < theta < 1` (C61). `COMBO = (1/64, 1/32)` lies inside it.
diff --git a/frontiers/c8_theorem_a_constants_d3_20261001/README.md b/frontiers/c8_theorem_a_constants_d3_20261001/README.md
new file mode 100644
index 000000000..bfd92f786
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_d3_20261001/README.md
@@ -0,0 +1,47 @@
+# Explicit cap-route constants for Theorem A in dimension three (C8: certified `C`, `r_*`, `z_*` on a declared band, `d = 3`)
+
+**Object:** `CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1`. **Author lane:** Anthropic / Claude. **Kind:** author-side theorem
+with explicit constants, certified by exact rational interval arithmetic (standard library only). **Scientific effect:**
+NONE: no register, catalog, GRAPH or STATUS change, and catalog entry C8 stays OPEN.
+
+**Setting.** The eight-pin law of [LP] in `d = 3` with the good event `G_r = {lambda_1 > (4/(3k)) r M_3^2, r M_4 <= 3k/10}` of
+[LP] section 7 / [CAP] (1) (partial-block operator norms on `D = [-2r, 2r] x ball(0, 2r)`). Theorem E3 (`NOTE.md`) holds on the
+band `b in [0, 1]`, `k in [1/2, 2]`, uniformly over frames and torus sides `L >= 10`, and for the reference kernel:
+
+    Q^W(G_r^c) <= C3(r_*) r^3   and   Z_r / r^2 >= z3(r_*)      for 0 < r <= r_*,
+
+| `r_*` | `C3(r_*)` | `C3 / c_G^(3)(0,2)` | `C3 r_*^3` | `z3(r_*)` |
+|---|---|---|---|---|
+| `1/4096` | `22754801.7964` | `1.0148` | `3.4e-4` | `6.0210` |
+| `1/2048` | `22824067.0244` | `1.0179` | `2.7e-3` | `5.9980` |
+| `1/1024` | `22974134.2920` | `1.0246` | `0.0214` | `5.9517` |
+| `1/512` | `23332922.4752` | `1.0406` | `0.174` | `5.8590` |
+
+(`1/256`, `1/128` and `1/64` are also certified, with `C3 r_*^3 > 1`; every cap-route constant has `C r^3 > 1` at
+`r = 1/256` in `d = 3`.)
+
+**What follows.**
+- **Explicit Theorem A in `d = 3`.** Since `1 - p_r <= Q^W(G_r^c)` ([LP] §8), [LP] Theorem A (1.1) holds in `d = 3` on this band
+  with explicit pairs, for example `(C, r_*) = (23332922.4752, 1/512)` (the bound at `r = r_*` is below `0.174`) or `(22974134.2920, 1/1024)`. [LP] §16 lists such a pair among the items it does not supply.
+- **Near-optimal for the cap route.** The constants are within `1.5%`–`4.1%` of the smallest possible `d = 3` cap-route constant for `r_* <= 1/512`. The sharp cap coefficient `c_G^(3)(0, 2) = 22424320.655` of Math-#208 is a
+  lower bound for every `d = 3` cap-route constant.
+- **Normalizer floor.** `z3(1/4096) = 6.0210`, against the contact value `9 m_(3,0) = 6.0442` at `(b, k) = (0, 1/2)`.
+
+**Files.**
+- `NOTE.md`: statement, the exact identities (I1)–(I3), Lemmas 1–5' (exact covariance series with a frame- and `L`-uniform torus
+  remainder on the polydisc in `C^3`; regression on the transverse Hessian; blockwise averaging identities for `M_3`; Frobenius
+  Taylor bounds for `M_4`; the root bounds; the normalizer floor through closed-form contact moments and a Loewner coupling),
+  the main term in eigenvalue coordinates, the rare branches (angular nets for the `M_4` blocks), bands, values, Monte Carlo
+  controls, and non-claims.
+- `theorem_a3.py`: the certificate and its replay.
+  - `--check --procs N` replays the parameters, 11 sampled bands and the `C` table exactly against `RESULTS.json`.
+  - `--check-full` replays all 121 bands.
+  - `--mutant {no-delta, cap-half, no-zfloor-correction, no-om-term, no-lambda2}` must exit 1.
+  - With no flags it regenerates `RESULTS.json`.
+  - `--mc R B K N SEED` is the floating Monte Carlo control.
+- `RESULTS.json` (one line per band: 39 box bounds, 39 floors, the largest tail part), `SOURCE_MAP.json` (pins [LP] and [CAP]
+  on `main`; read-only references to Math-#208 and Math-#206), `SOURCE_FILES.json`.
+- Workflow `.github/workflows/c8-theorem-a-constants-d3.yml` replays the manifest, the pins, `--check-full` and `--check` in
+  the two interpreter modes, the five mutants, and a clean tree.
+
+Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge.
diff --git a/frontiers/c8_theorem_a_constants_d3_20261001/RESULTS.json b/frontiers/c8_theorem_a_constants_d3_20261001/RESULTS.json
new file mode 100644
index 000000000..e227c1aca
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_d3_20261001/RESULTS.json
@@ -0,0 +1,445 @@
+{
+ "C_table": {
+  "1/1024": {
+   "C": "5743533573/250",
+   "C_over_cG3_02": "1.024519",
+   "C_rstar3": "0.0213963299",
+   "argmax": {
+    "b": [
+     "0",
+     "1/512"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "15/16384",
+     "1/1024"
+    ]
+   },
+   "z_star": "5951763/1000000"
+  },
+  "1/128": {
+   "C": "699713600177629/10000",
+   "C_over_cG3_02": "3120.333548",
+   "C_rstar3": "33364.9444664779",
+   "argmax": {
+    "b": [
+     "1/2",
+     "1"
+    ],
+    "k": [
+     "1/2",
+     "3/4"
+    ],
+    "r": [
+     "15/2048",
+     "1/128"
+    ]
+   },
+   "z_star": "661831/125000"
+  },
+  "1/2048": {
+   "C": "57060167561/2500",
+   "C_over_cG3_02": "1.017827",
+   "C_rstar3": "0.0026570712",
+   "argmax": {
+    "b": [
+     "0",
+     "1/512"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "15/32768",
+     "1/2048"
+    ]
+   },
+   "z_star": "1199601/200000"
+  },
+  "1/256": {
+   "C": "3714819319587/625",
+   "C_over_cG3_02": "265.056454",
+   "C_rstar3": "354.2727775180",
+   "argmax": {
+    "b": [
+     "0",
+     "1/2"
+    ],
+    "k": [
+     "1/2",
+     "3/4"
+    ],
+    "r": [
+     "7/2048",
+     "15/4096"
+    ]
+   },
+   "z_star": "88631/15625"
+  },
+  "1/4096": {
+   "C": "56887004491/2500",
+   "C_over_cG3_02": "1.014738",
+   "C_rstar3": "0.0003311260",
+   "argmax": {
+    "b": [
+     "0",
+     "1/512"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "15/65536",
+     "1/4096"
+    ]
+   },
+   "z_star": "6021091/1000000"
+  },
+  "1/512": {
+   "C": "14583076547/625",
+   "C_over_cG3_02": "1.040519",
+   "C_rstar3": "0.1738438195",
+   "argmax": {
+    "b": [
+     "439/1024",
+     "421/512"
+    ],
+    "k": [
+     "31/16",
+     "63/32"
+    ],
+    "r": [
+     "15/8192",
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+  {"bound": ["8665137068393.1518", "9956019454279.8119", "2191371516008.7021", "2449753272085.5618", "951336463998.4963", "1034848412477.0997", "540879232497.8232", "573470648590.7270", "235823333311.7894", "288280513026.0400", "116674561868.1804", "283242329700.6185", "88335645844.6949", "62551397819.6754", "94936655705.1922", "199340068680.9127", "71497644630.9826", "38459729418.3000", "47132932677.2517", "67959296947.7603", "129804930941.9260", "58560640216.3836", "30443142904.3316", "33684447707.9762", "40406762750.6002", "55938974468.6970", "99343133556.3185", "59678966758.8825", "26811416987.6454", "26912366466.5422", "27106981044.4760", "27491475014.8567", "28177598400.6187", "29484295263.9806", "31928726932.7159", "36764501939.2241", "47257613493.3256", "73601349136.8585", "84880582065.7067"], "r": ["7/512", "15/1024"], "tail_max": "167014041.672379346462", "zfloor": ["4.614793", "15.063644", "10.832262", "34.970195", "19.624164", "63.040533", "30.966404", "99.210408", "45.312956", "135.029132", "55.934077", "105.998869", "388.714792", "66.960462", "92.301516", "160.661687", "482.144852", "77.757647", "91.632524", "121.848151", "199.635673", "522.750121", "83.449892", "90.661659", "104.829687", "135.182920", "210.415897", "492.850389", "86.528905", "86.925851", "87.519015", "88.508442", "90.316616", "93.605641", "99.844941", "111.907492", "136.609963", "193.378685", "366.284867"]},
+  {"bound": ["15174797427265.5022", "17254355056897.0472", "3862733277574.1783", "4271623345254.2016", "1691123367404.9855", "1818572004394.0915", "969831263734.9964", "1015917103460.9613", "424374873249.9529", "512776099953.9125", "211039720200.5779", "507901064449.7291", "157045094001.9905", "113936980702.0093", "171875790846.9824", "357898787286.4434", "127600200070.4190", "70296070608.1778", "85844141641.3811", "123136037326.5608", "233369855330.7983", "104715238863.2923", "55741998830.1991", "61570244463.0367", "73626531501.0033", "101452141041.3803", "178872391108.5589", "106841875214.1672", "49184656220.5629", "49365727240.1784", "49715894810.8599", "50408780433.3970", "51644562252.3530", "53997766756.0053", "58395007342.3763", "67070667835.8411", "85889949258.7039", "132965416794.4147", "152275033950.0566"], "r": ["15/1024", "1/64"], "tail_max": "139486768.697320494657", "zfloor": ["4.520628", "14.836201", "10.652791", "34.536776", "19.333699", "62.339195", "30.539097", "98.178873", "44.751449", "133.607988", "55.469004", "104.662309", "387.059061", "66.562952", "91.643120", "159.043016", "480.249447", "77.366782", "91.137049", "121.058371", "197.885776", "520.604413", "83.063781", "90.223497", "104.287583", "134.361138", "208.739790", "490.576917", "86.157532", "86.552310", "87.141832", "88.124618", "89.920727", "93.187013", "99.382608", "111.362959", "135.856227", "192.058685", "363.952845"]}
+ ]
+}
diff --git a/frontiers/c8_theorem_a_constants_d3_20261001/SOURCE_FILES.json b/frontiers/c8_theorem_a_constants_d3_20261001/SOURCE_FILES.json
new file mode 100644
index 000000000..764ab9897
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_d3_20261001/SOURCE_FILES.json
@@ -0,0 +1,40 @@
+{
+ "schema": 1,
+ "scientific_effect": "NONE",
+ "certified": true,
+ "review_status": "AUTHOR_SIDE_THEOREM_WITH_CERTIFIED_CONSTANTS",
+ "object": "CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1",
+ "files": [
+  {
+   "path": "NOTE.md",
+   "bytes": 36029,
+   "sha256": "a93366be08211820492613b611c8c0e205158e520617716485707afb7ea0fc19",
+   "git_blob": "718dd17eb6f1e94391ea8f3ad4d7c1bfd9d48d2d"
+  },
+  {
+   "path": "README.md",
+   "bytes": 3624,
+   "sha256": "b096e42b1f31b30f025c813b64eda44b03a725496ea1a298de0f51edc3249b94",
+   "git_blob": "bfd92f786151b154b86fdde34f94675995045df9"
+  },
+  {
+   "path": "RESULTS.json",
+   "bytes": 159177,
+   "sha256": "9d9e889018b197ef4d997744d05bb1e7b84881abd29ae14874d452343d9b67c0",
+   "git_blob": "e227c1aca350afed080923fa0bb6b3e976f967ee"
+  },
+  {
+   "path": "SOURCE_MAP.json",
+   "bytes": 6153,
+   "sha256": "d2dfd680219ea576231725dfb48b55aae97067bc74a73cf1729131bf1fe84e14",
+   "git_blob": "9c08564351fbeadb2edff27a953999e47346414b"
+  },
+  {
+   "path": "theorem_a3.py",
+   "bytes": 76209,
+   "sha256": "310d4ab3d3a38a377d753753f3b814591a9090a947f87d6843ab7389389572d3",
+   "git_blob": "267890232119b35206be7a8cc74d730de9a68644"
+  }
+ ],
+ "nonauthor_reads": []
+}
diff --git a/frontiers/c8_theorem_a_constants_d3_20261001/SOURCE_MAP.json b/frontiers/c8_theorem_a_constants_d3_20261001/SOURCE_MAP.json
new file mode 100644
index 000000000..9c0856435
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_d3_20261001/SOURCE_MAP.json
@@ -0,0 +1,85 @@
+{
+  "schema": 1,
+  "object": "CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1",
+  "scientific_effect": "NONE",
+  "certified": true,
+  "review_status": "AUTHOR_SIDE_THEOREM_WITH_CERTIFIED_CONSTANTS",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "main_at_creation": "3e0a91b69892b9717564c592e7a966492a81566a",
+  "note": "Theorem E3: for the d = 3 eight-pin law of [LP] and the good event G_r of [LP] section 7 / [CAP] (1) (partial-block operator norms on D = [-2r, 2r] x ball(0, 2r)), on the band b in [0, 1], k in [1/2, 2], uniformly over frames and torus sides L >= 10 (and for the reference kernel), Q^W(G_r^c) <= C3(r_*) r^3 and Z_r / r^2 >= z3(r_*) for 0 < r <= r_*, with C3(1/4096) = 22754801.7964 (1.0148 c_G^(3)(0, 2)), C3(1/1024) = 22974134.2920, C3(1/512) = 23332922.4752 (C3 r_*^3 <= 0.174), z3(1/512) = 5.859005 and three further certified (uninformative) pairs. The proof is self-contained apart from the definitions and the pins of [LP] (3.1)-(3.3): it does not consume [LP] (4.1), (4.2), (5.4), (5.5) or (7.5)-(7.7). With [LP] section 8 (1 - p_r <= Q^W(G_r^c)) this gives explicit pairs (C, r_*) for [LP] Theorem A (1.1) in d = 3. Every C3 here is a cap-route constant (a constant for [LP] (7.8)), hence >= c_G^(3)(0, 2) = 2.2424e7 (Math-#208). C8 stays OPEN: no catalog edit; c_{B,K}, c_{d,L} and d >= 4 are not addressed.",
+  "sources": [
+    {
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "id": "LP",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "section 1: the model in every dimension (torus kernel K_L, pins, Q, W_r with index(H_S) = d - 1, Z_r, Q^W) and Theorem A (1.1); section 3: the preconditioned pins (3.1) with m = d - 1 transverse rows and their values (3.3); section 7: the good event G_r and the cap bound (7.8); section 8: on G_r the global elder partner of M is S, so 1 - p_r <= Q^W(G_r^c) (used only for the consequence on 1 - p_r); section 16: a numerical r_*, C on a prescribed band is listed as not supplied",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
+      "bytes": 15160,
+      "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc",
+      "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1",
+      "id": "CAP",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "section 1: the cylinder D = [-2r, 2r] x ball(0, 2r), the partial-block operator norms M_3, M_4 (suprema over independently chosen transverse unit vectors) and condition (1) defining G_r; the deterministic cap theorem behind [LP] section 8 (consumed as a statement)",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c8_cap_failure_coefficient_dims_20261001/NOTE.md",
+      "bytes": 21719,
+      "sha256": "d6b4992c41b781101152e888bf601b5371e7d018b9a556563094acd700aeeec6",
+      "git_blob": "17e6866b8dd5b2d14845032b3aac3de68dec03b0",
+      "id": "G3",
+      "commit": "da483732664d128728950dcad6d4fdbe5181f7cd",
+      "pull_request": "Math-#208",
+      "role": "Theorem G_d, the d = 3 contact law (Lemma 1: B ~ b I + GOE_2, v ~ N(0, I/2), Omega' ~ GOE_2, Y with variances 6, 2, 2, 6) and m_(3,0) = (7 - 4 sqrt2)/2, reproduced independently here; the certified lower end c_G^(3)(0, 2) >= 22424320.655069 (its RESULTS.json, below) is used only for the --check sanity rule C3 > c_G^(3)(0, 2) and for the comparisons in NOTE section 0; not consumed by the proof; Math-#208 merged at b55b8f3, where NOTE.md has since changed by the C208-C-01 wording fix only (blob b42be5b8), so this NOTE pin stays historical",
+      "current_required": false,
+      "on_main": false
+    },
+    {
+      "path": "frontiers/c8_cap_failure_coefficient_dims_20261001/RESULTS.json",
+      "bytes": 6837,
+      "sha256": "1406d5138ac3d33bcb0358f6a3005a67e3a4a2dbfc2c43bc3ac2593667550608",
+      "git_blob": "e9fd6f377d1fd9dca9d09cc884d2bad2422e8b17",
+      "id": "G3-RESULTS",
+      "commit": "da483732664d128728950dcad6d4fdbe5181f7cd",
+      "pull_request": "Math-#208",
+      "role": "the certified enclosure c_G^(3)(0, 2) in [22424320.655069, 22424320.655072]; comparison and sanity check only; Math-#208 merged at b55b8f3 with this RESULTS.json byte-identical, so the pin is checked on the tree",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c8_theorem_a_constants_planar_20260930/NOTE.md",
+      "bytes": 31497,
+      "sha256": "d943f39d0cce065397a3ecc0d698a42e4a97c6fe0d8d1491f91a5daa910dbe94",
+      "git_blob": "0b0f54e247d68c45e9560a362738f7ce614110e0",
+      "id": "E",
+      "commit": "0cb8ad3ca1fe1733e737570ab682f469b66d76e4",
+      "pull_request": "Math-#206",
+      "role": "the planar Theorem E: the proof structure (sections 1-9) followed here and the planar constants quoted for comparison; every step used is restated or generalized in this NOTE",
+      "current_required": false,
+      "on_main": false
+    }
+  ],
+  "catalog_reference": {
+    "path": "reviews/candidates_pending_20260928/CANDIDATES.md",
+    "entry": "C8. Numerical constants (still open as of 2026-09-29): explicit finite C, r_*, z_*, c_{B,K}, c_{d,L} on a declared compact-mark / torus band; disposition OPEN",
+    "effect_here": "none; this packet supplies certified cap-route pairs (C3, r_*) and z3 on (0, r_*] for d = 3 on the band b in [0, 1], k in [1/2, 2], L >= 10; no catalog edit; C8 stays OPEN pending nonauthor reads and its other entries"
+  },
+  "related_records": [
+    "Math-#206 (the planar Theorem E: explicit cap-route constants in d = 2)",
+    "Math-#208 (sharp cap coefficient c_G^(d); lower bound for every d = 3 cap-route constant)",
+    "Math-#203 (planar sharp cap coefficient c_G)",
+    "Math-#205 (c_{B,K} in d = 3)"
+  ],
+  "nonauthor_reads": [],
+  "nonauthor_acceptance": false,
+  "governing_status_changed": false,
+  "main_at_update": "3e0a91b69892b9717564c592e7a966492a81566a"
+}
diff --git a/frontiers/c8_theorem_a_constants_d3_20261001/theorem_a3.py b/frontiers/c8_theorem_a_constants_d3_20261001/theorem_a3.py
new file mode 100644
index 000000000..267890232
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_d3_20261001/theorem_a3.py
@@ -0,0 +1,1714 @@
+#!/usr/bin/env python3
+"""Explicit cap-route constants for Theorem A of [LP] in dimension d = 3, for the torus kernel K_L with L >= 10 in every
+frame: for 0 < r <= r_*, b in [0, 1], k in [1/2, 2],
+
+    1 - p_r(b, k, R)  <=  Q^W(G_r^c)  <=  C3(r_*) r^3,        Z_r / r^2 >= z3(r_*),
+
+with G_r the good event of [LP] section 7 / [CAP] (1) (partial-block operator norms on the cylinder
+D = [-2r, 2r] x ball(0, 2r)) and C3(r_*), z3(r_*) certified here (NOTE.md Theorem E3).  The proof bounds
+Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0:
+
+  * covariances of the eight preconditioned pins and the 31 targets are exact power series in r, obtained from the Taylor
+    coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded
+    uniformly in r and in the frame by Cauchy estimates on a polydisc in C^3;
+  * the failure event is reduced to {lambda_1 <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded blockwise through exact
+    averaging identities of the pinned jets and M_4 through the Taylor series at the midpoint (Frobenius norms);
+  * the main term is integrated over the transverse Hessian B = -D_y^2 f(M) in eigenvalue coordinates after an exact
+    regression on its three entries, against an angle-free Gaussian majorant of its density; the remaining expectations are
+    closed-form incomplete Gaussian and chi moments plus Hoelder-controlled perturbations;
+  * the normalizer Z_r is bounded below through a Loewner-order coupling with the contact law, whose moments
+    E[det(B)^a tr(B)^c 1{B > 0}] are in closed form; the rare branches are Gaussian and chi tails.
+All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only.
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term', 'no-lambda2')
+MUT = None
+PBITS = 160
+L_MIN = 10
+NX = 60                    # Taylor order in x of the covariance series (exact); the rest is a bounded tail
+NB = 24                    # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded)
+X0 = Fr(1, 64)             # default threshold for the quadratic near-branch analysis (records use COMBOS)
+THETA = Fr(1, 32)          # default splitting parameter in the max-moment lemma (records use COMBOS)
+THETA_E = Fr(1, 16)        # splitting parameter for the ||r nu - v||^2 factor
+QS = (2, 4, 8, 16)         # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used
+R0_COUPLING = 12           # radius of the Loewner coupling of the transverse Hessian with the contact law (floor, section 6)
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+BAND_B = (Fr(0), Fr(1))
+BAND_K = (Fr(1, 2), Fr(2))
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203 / #206)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+def phi_pt(x):
+    """phi(x) for rational x (interval)."""
+    x = Fr(x)
+    return exp_neg_point(x * x / 2) / SQRT2PI
+
+
+def Phibar_pt(x):
+    """1 - Phi(x) for rational x (interval)."""
+    x = Fr(x)
+    if abs(x) <= 12:
+        p = phi_point(x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 0:
+        return IV(0, (phi_pt(x) / x).hi)            # Mills: Phibar(x) <= phi(x)/x
+    return IV(1 - (phi_pt(-x) / (-x)).hi, 1)
+
+
+def dfact(n):
+    """(n)!! for n >= -1."""
+    return math.prod(range(n, 0, -2)) if n > 0 else 1
+
+
+def gauss_moment(j):
+    """E[Z^j] for Z ~ N(0, 1)."""
+    return 0 if j % 2 else dfact(j - 1)
+
+
+def upper_moments(x, J):
+    """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x (intervals).  For x < 0,
+    M_j(x) = E[Z^j] - (-1)^j M_j(-x)."""
+    x = Fr(x)
+    if x < 0:
+        M = upper_moments(-x, J)
+        return [IV(gauss_moment(j)) - M[j] * (-1) ** j for j in range(J + 1)]
+    ph = phi_pt(x)
+    M = [Phibar_pt(x), ph]
+    for j in range(2, J + 1):
+        M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1))
+    return M[:J + 1]
+
+
+def rup_rel(x, bits=200):
+    """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds)."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh)
+    return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh)
+
+
+def rdown_rel(x, bits=200):
+    """Round a rational x >= 0 down to a dyadic with `bits` significant bits."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr((x.numerator << sh) // x.denominator, 1 << sh)
+    return Fr((x.numerator // (x.denominator << -sh)) << -sh)
+
+
+def sqrt_up(x):
+    """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root
+    step adds less than 2^-209 and rup_rel(., 200) less than 2^-199.  (The earlier '2^-200' was too sharp: at x = 49/100 the
+    excess is (8/7) 2^-200, review C59-DOC-01.)"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m))
+
+
+def sqrt_down(x):
+    """Rational lower bound of sqrt(x), x >= 0 rational."""
+    x = Fr(x)
+    if x <= 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rdown_rel(Fr(math.isqrt(nd << (2 * m)), d << m))
+
+
+def iroot(x, p):
+    """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two."""
+    v = Fr(x)
+    while p > 1:
+        v = sqrt_up(v)
+        p //= 2
+    return v
+
+
+def pow2(p):
+    q = 1
+    while q < p:
+        q *= 2
+    return q
+
+
+ZN = {}
+
+
+def znorm(p):
+    """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5)."""
+    if p == 1:
+        return Fr(4, 5)
+    if p not in ZN:
+        ZN[p] = iroot(Fr(dfact(p - 1)), p)
+    return ZN[p]
+
+
+def znorm1(p):
+    """max(1, ||N(0,1)||_p): the factor in ||(|mu| + ||Sigma^(1/2) z||)||_p <= |mu| + sqrt(tr Sigma) znorm1(p)."""
+    return Fr(1) if p <= 2 else znorm(p)
+
+
+def frob_norm(mu_abs, tr_hi, p):
+    """Upper bound of || ||X|| ||_p for a Gaussian vector X with ||E X|| <= mu_abs and tr Cov X <= tr_hi (Minkowski in
+    L^(p/2) on the eigen-decomposition for p >= 2; Lyapunov for p <= 2)."""
+    return rup_rel(mu_abs + sqrt_up(tr_hi) * znorm1(pow2(p)))
+
+
+def gamma_half(n2):
+    """Gamma(n2 / 2) for an integer n2 >= 1, as an interval (sqrt(pi) for odd n2)."""
+    if n2 % 2 == 0:
+        return IV(math.factorial(n2 // 2 - 1))
+    n = (n2 - 1) // 2                                 # Gamma(n + 1/2) = (2n - 1)!! sqrt(pi) / 2^n
+    return isqrt_iv(PI) * Fr(dfact(2 * n - 1), 2 ** n)
+
+
+def chi_moment(nu, j):
+    """E[chi_nu^j] = 2^(j/2) Gamma((nu + j)/2) / Gamma(nu/2) (interval), integer j >= 0."""
+    if j % 2 == 0:
+        return IV(math.prod(nu + 2 * i for i in range(j // 2)))
+    out = gamma_half(nu + j) / gamma_half(nu)
+    for _ in range(j):
+        out = out * SQRT2
+    return out
+
+
+CN = {}
+
+
+def chi_norm(nu, p):
+    """Upper bound of ||chi_nu||_p for p a power of two."""
+    if p == 1:
+        return sqrt_up(Fr(nu))
+    key = (nu, p)
+    if key not in CN:
+        CN[key] = iroot(Fr(math.prod(nu + 2 * i for i in range(p // 2))), p)
+    return CN[key]
+
+
+CC = {}
+
+
+def chi_const(nu):
+    """C_nu sqrt(2 pi) with C_nu = 1 / (2^(nu/2 - 1) Gamma(nu/2)): E[chi_nu^j 1{chi > x}] = C_nu sqrt(2 pi) M_(j+nu-1)(x)."""
+    if nu not in CC:
+        p2 = IV(Fr(2) ** ((nu - 2) // 2)) if nu % 2 == 0 else IV(Fr(2) ** ((nu - 3) // 2)) * SQRT2
+        CC[nu] = SQRT2PI / (p2 * gamma_half(nu))
+    return CC[nu]
+
+
+def chi_upper_moments(nu, x, J):
+    """[E[chi_nu^j 1{chi_nu > x}] for j = 0..J] (intervals), rational x >= 0."""
+    M = upper_moments(x, J + nu - 1)
+    c = chi_const(nu)
+    return [M[j + nu - 1] * c for j in range(J + 1)]
+
+
+def chi_tail(nu, x):
+    """Rational upper bound of P(chi_nu > x) for rational x >= 0 (Mills-type bound beyond the series range)."""
+    x = Fr(x)
+    if x <= 0:
+        return Fr(1)
+    if x <= 12:
+        return min(Fr(1), chi_upper_moments(nu, x, 0)[0].hi)
+    # x > 12: P(chi_nu > x) = c_nu sqrt(2pi) M_(nu-1)(x), M_j(x) <= phi(x) x^(j-1) (1 + 2 (j-1)/x^2) for x^2 >= 2 j, and
+    # phi(x) <= m!/(x^2/2)^m / sqrt(2 pi) (m = 40)
+    y = x * x / 2
+    ph = Fr(math.factorial(40)) / y ** 40 * Fr(2, 5)          # 1/sqrt(2 pi) < 2/5
+    j = nu - 1
+    Mj = ph * (x ** (j - 1) if j >= 1 else 1 / x) * (1 + Fr(2 * max(j - 1, 1)) / (x * x))
+    return min(Fr(1), Mj * Fr(51, 20))                          # c_nu sqrt(2 pi) <= sqrt(2 pi) < 51/20
+
+
+def phibar_upper(x):
+    """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound
+    phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor."""
+    x = Fr(x)
+    if x <= 12:
+        return Phibar_pt(x).hi
+    y = x * x / 2
+    m = 40
+    return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2))      # sqrt(2 pi) > 5/2
+
+
+def gauss_tail2(mu_abs, sd, t):
+    """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd."""
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_tail1(mu_abs, sd, t):
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_norm(mu_abs, sd, p):
+    """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd."""
+    return rup_rel(mu_abs + sd * znorm(pow2(p)))
+
+
+def absup(x):
+    return max(abs(x.lo), abs(x.hi))
+
+
+def pow_tail(x, q):
+    """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two)."""
+    if x <= 0:
+        return Fr(0)
+    return iroot(rup_rel(x, 64) ** (q - 1), q)
+
+
+# ----------------------------------------------------------------------------- functionals and exact covariance series (d = 3)
+HALF = Fr(1, 2)
+XM, XS, X0PT = -HALF, HALF, Fr(0)
+JET4_IDX = {'X%d%d%d' % (a, b1, 4 - a - b1): (a, b1, 4 - a - b1) for a in range(5) for b1 in range(5 - a)}
+JET4 = tuple(sorted(JET4_IDX, key=lambda n: (-JET4_IDX[n][0], -JET4_IDX[n][1])))
+BLOCKS = ((4, 0), (3, 1), (2, 2), (1, 3), (0, 4))           # partial blocks (a, c) of the fourth derivatives, a + c = 4
+
+
+def block_members(a, c):
+    """[(name, Frobenius multiplicity c!/beta!)] of the 4-jet block d_x^a D_y^c f(0)."""
+    out = []
+    for b1 in range(c, -1, -1):
+        b2 = c - b1
+        out.append(('X%d%d%d' % (a, b1, b2), math.factorial(c) // (math.factorial(b1) * math.factorial(b2))))
+    return out
+
+
+def term(c, p, a, b1, b2, x0):
+    """c r^p d_x^a d_y1^b1 d_y2^b2 f(x0 r, 0, 0)."""
+    return (Fr(c), p, a, b1, b2, Fr(x0))
+
+
+def functionals():
+    pins = {
+        'p1': [term(HALF, 0, 0, 0, 0, XM), term(HALF, 0, 0, 0, 0, XS)],
+        'p2': [term(-1, -1, 0, 0, 0, XM), term(1, -1, 0, 0, 0, XS)],
+        'p3p': [term(6, -2, 1, 0, 0, XM), term(6, -2, 1, 0, 0, XS), term(12, -3, 0, 0, 0, XM), term(-12, -3, 0, 0, 0, XS)],
+        'p4': [term(-1, -1, 1, 0, 0, XM), term(1, -1, 1, 0, 0, XS)],
+    }
+    for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))):
+        pins['q1%d' % j] = [term(HALF, 0, 0, e1, e2, XM), term(HALF, 0, 0, e1, e2, XS)]
+        pins['q2%d' % j] = [term(-1, -1, 0, e1, e2, XM), term(1, -1, 0, e1, e2, XS)]
+    targets = {
+        'a11': [term(1, 0, 0, 2, 0, XM)], 'a22': [term(1, 0, 0, 0, 2, XM)], 'a12': [term(1, 0, 0, 1, 1, XM)],
+        'o11': [term(1, -1, 0, 2, 0, XS), term(-1, -1, 0, 2, 0, XM)],
+        'o22': [term(1, -1, 0, 0, 2, XS), term(-1, -1, 0, 0, 2, XM)],
+        'o12': [term(1, -1, 0, 1, 1, XS), term(-1, -1, 0, 1, 1, XM)],
+        'T': [term(1, -2, 2, 0, 0, XM), term(4, -3, 1, 0, 0, XM), term(2, -3, 1, 0, 0, XS), term(6, -4, 0, 0, 0, XM),
+              term(-6, -4, 0, 0, 0, XS)],
+        'tau': [term(1, -3, 2, 0, 0, XS), term(-1, -3, 2, 0, 0, XM), term(-6, -4, 1, 0, 0, XM), term(-6, -4, 1, 0, 0, XS),
+                term(-12, -5, 0, 0, 0, XM), term(12, -5, 0, 0, 0, XS)],
+    }
+    for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))):
+        targets['v%d' % j] = [term(1, -1, 1, e1, e2, XM), term(1, -2, 0, e1, e2, XM), term(-1, -2, 0, e1, e2, XS)]
+        targets['nu%d' % j] = [term(1, -2, 1, e1, e2, XM), term(1, -2, 1, e1, e2, XS), term(2, -3, 0, e1, e2, XM),
+                               term(-2, -3, 0, e1, e2, XS)]
+    for b1 in range(3, -1, -1):
+        targets['y%s' % ('1' * b1 + '2' * (3 - b1))] = [term(1, 0, 0, b1, 3 - b1, X0PT)]
+    for n, (a, b1, b2) in JET4_IDX.items():
+        targets[n] = [term(1, 0, a, b1, b2, X0PT)]
+    return pins, targets
+
+
+PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q11', 'q21', 'q12', 'q22')
+BNAMES = ('a11', 'a22', 'a12')                              # the transverse Hessian A = D_y^2 f(M) = -B
+VNAMES = ('v1', 'v2')
+ONAMES = ('o11', 'o22', 'o12')
+YNAMES = ('y111', 'y112', 'y122', 'y222')
+FREE = VNAMES + ONAMES + YNAMES                              # the free third-derivative blocks, in Cholesky order
+FREE_W = (1, 1, 1, 1, 2, 1, 3, 3, 1)                         # Frobenius multiplicities (Euclidean norms in these weights)
+TARGET_NAMES = BNAMES + FREE + ('T', 'tau', 'nu1', 'nu2') + JET4
+NP = len(PIN_NAMES)
+
+
+def weight(F):
+    """Scaling weight a - (power of r), common to all terms of a preconditioned functional."""
+    ws = {t[2] - t[1] for t in F}
+    require(len(ws) == 1, 'functional not homogeneous')
+    return ws.pop()
+
+
+def hcoef(n):
+    """d^n/dx^n exp(-x^2/2) at 0."""
+    if n % 2:
+        return 0
+    return (-1) ** (n // 2) * dfact(n - 1)
+
+
+PAIR_CACHE = {}
+
+
+def pair_series(F, G, key):
+    """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG), gamma = (gx, gy1, gy2):
+    returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', gy1, gy2) for the remainder bounds)."""
+    if key in PAIR_CACHE:
+        return PAIR_CACHE[key]
+    wF, wG = weight(F), weight(G)
+    w = wF + wG
+    pi = {}
+    absterms = []
+    for (cf, pf, af, bf1, bf2, xf) in F:
+        for (cg, pg, ag, bg1, bg2, xg) in G:
+            g1, g2 = bf1 + bg1, bf2 + bg2
+            sgn = (-1) ** (ag + bg1 + bg2)
+            d = xf - xg
+            absterms.append((abs(cf * cg), af + ag, g1, g2))
+            for gx in range(af + ag, NX + 1):
+                m = gx - af - ag
+                val = cf * cg * sgn * d ** m / math.factorial(m)
+                if val != 0:
+                    pi[(gx, g1, g2)] = pi.get((gx, g1, g2), Fr(0)) + val
+    for (gx, g1, g2), v in pi.items():
+        require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)')
+    ser = {}
+    for (gx, g1, g2), v in pi.items():
+        kap = hcoef(gx) * hcoef(g1) * hcoef(g2)
+        if kap and v:
+            ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v
+    out = ({p: c for p, c in ser.items() if c != 0}, w, absterms)
+    PAIR_CACHE[key] = out
+    return out
+
+
+E_UP = Fr(27183, 10000)                                      # > e
+SQRT3_UP = Fr(7, 4)                                          # > sqrt(3)
+
+
+def torus_M1(L):
+    """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2|, |z_3| <= 1 (every frame, |p_n| = |n| L):
+    exp(3/2) sum_{j >= 1} (24 j^2 + 2) exp(-(j L - sqrt 3)^2 / 2); 24 j^2 + 2 lattice points have sup-norm j."""
+    L = Fr(L)
+    e32 = exp_neg_point(Fr(3, 2)).inv().hi
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += (24 * j * j + 2) * exp_neg_point((j * L - SQRT3_UP) ** 2 / 2).hi
+    # j >= 6: (24 j^2 + 2) exp(-(jL - 7/4)^2/2) decreases by more than 1/2 per step for L >= 10
+    tail = 2 * (24 * 36 + 2) * exp_neg_point((6 * L - SQRT3_UP) ** 2 / 2).hi
+    return e32 * (tot + tail)
+
+
+def theta_minus_one(L):
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += (24 * j * j + 2) * exp_neg_point((j * L) ** 2 / 2).hi
+    return tot + 2 * (24 * 36 + 2) * exp_neg_point((6 * L) ** 2 / 2).hi
+
+
+def cov_band(F, G, key, r0, r1, M1):
+    """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1]."""
+    ser, w, absterms = pair_series(F, G, key)
+    val = IV(0)
+    for p, c in ser.items():
+        val = val + IV(r0 ** p, r1 ** p) * c
+    A = max((t[1] for t in absterms), default=0)
+    # Taylor truncation tail of the reference kernel (orders gx > NX)
+    tail = Fr(0)
+    for (acc, aa, g1, g2) in absterms:
+        tail += acc * abs(hcoef(g1) * hcoef(g2))
+    tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w)
+    # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy1! gy2! M1 on the unit polydisc
+    tor = Fr(0)
+    for (acc, aa, g1, g2) in absterms:
+        tor += acc * math.factorial(g1) * math.factorial(g2) * Fr(max(w, 1)) ** aa
+    tor = tor * M1 / (1 - E_UP * r1)
+    err = tail + tor
+    return val + IV(-err, err)
+
+
+# ----------------------------------------------------------------------------- interval linear algebra
+def solve_iv(A, B):
+    """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c]."""
+    n = len(A)
+    M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    for col in range(n):
+        p = max(range(col, n), key=lambda i: abs(M[i][col].mid()))
+        M[col], M[p] = M[p], M[col]
+        piv = M[col][col]
+        for i in range(col + 1, n):
+            f = M[i][col] / piv
+            for j in range(col, len(M[i])):
+                M[i][j] = M[i][j] - f * M[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            s = M[i][n + c]
+            for j in range(i + 1, n):
+                s = s - M[i][j] * x[j]
+            x[i] = s / M[i][i]
+        X.append(x)
+    return X
+
+
+def cholesky(C):
+    """Interval Cholesky of an n x n SPD interval matrix (pivots must be positive)."""
+    n = len(C)
+    L = [[IV(0)] * n for _ in range(n)]
+    for i in range(n):
+        for j in range(i + 1):
+            s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0))
+            if i == j:
+                require(s.lo > 0, 'Cholesky pivot not positive')
+                L[i][j] = isqrt_iv(s)
+            else:
+                L[i][j] = s / L[j][j]
+    return L
+
+
+def opnorm_lower_tri(L, rows, cols):
+    """Upper bound of the operator norm of the block L[rows][cols] of an interval lower-triangular matrix:
+    max |diagonal| + Frobenius norm of the rest (square diagonal blocks), Frobenius norm otherwise."""
+    if rows == cols:
+        dg = max(absup(L[i][i]) for i in rows)
+        off = sum(absup(L[i][j]) ** 2 for i in rows for j in cols if j < i)
+        return rup_rel(dg + sqrt_up(off))
+    return sqrt_up(sum(absup(L[i][j]) ** 2 for i in rows for j in cols))
+
+
+def sym3_eig_bounds(C):
+    """(lower bound of lambda_min, upper bound of lambda_max) of a symmetric 3 x 3 interval matrix (Gershgorin)."""
+    lo = min(C[i][i].lo - sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3))
+    hi = max(C[i][i].hi + sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3))
+    return lo, hi
+
+
+def det3(C):
+    a, b, c = C[0]
+    d, e, f = C[1]
+    g, h, i = C[2]
+    return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
+
+
+# ----------------------------------------------------------------------------- the conditional law on an r-band
+class Law:
+    pass
+
+
+def band_law(r0, r1, L=L_MIN):
+    """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the
+    targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients; then the
+    regression on the transverse Hessian (three entries) and the g-law of the remaining targets."""
+    pins, targets = functionals()
+    P = [pins[n] for n in PIN_NAMES]
+    T = [targets[n] for n in TARGET_NAMES]
+    M1 = torus_M1(L)
+    th1 = theta_minus_one(L)
+    nt = len(T)
+    Spp = [[None] * NP for _ in range(NP)]
+    for i in range(NP):
+        for j in range(i, NP):
+            Spp[i][j] = Spp[j][i] = cov_band(P[i], P[j], ('p', i, j), r0, r1, M1)
+    Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(NP)] for i in range(nt)]
+    Stt = [[None] * nt for _ in range(nt)]
+    for i in range(nt):
+        for j in range(i, nt):
+            Stt[i][j] = Stt[j][i] = cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1)
+    r = IV(r0, r1)
+    kvec = [-(r * r * r) * HALF, -(r * r), IV(12)] + [IV(0)] * (NP - 3)
+    e1 = [IV(1)] + [IV(0)] * (NP - 1)
+    W = solve_iv(Spp, Stp)
+    xs = solve_iv(Spp, [e1, kvec])
+    law = Law()
+    law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1
+    law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)}
+    law.beta = {n: sum((W[i][j] * kvec[j] for j in range(NP)), IV(0)) for i, n in enumerate(TARGET_NAMES)}
+    scale = IV(1 / (1 + th1), 1)          # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1]
+    C = {}
+    for i, a in enumerate(TARGET_NAMES):
+        for j in range(i, nt):
+            c = TARGET_NAMES[j]
+            v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(NP)), IV(0))
+            C[(a, c)] = C[(c, a)] = v * scale
+    law.C = C
+    escale = IV(1, 1 + th1)
+    law.e_bb = xs[0][0] * escale
+    law.e_bk = sum((kvec[j] * xs[0][j] for j in range(NP)), IV(0)) * escale
+    law.e_kk = sum((kvec[j] * xs[1][j] for j in range(NP)), IV(0)) * escale
+    regress_on_B(law)
+    law.S = {blk: taylor_tail_coefficient(blk, r1, M1) for blk in BLOCKS}
+    law.net_var = {blk: net_max_variance(law.C, blk) for blk in BLOCKS if blk[1] >= 2}
+    return law
+
+
+NET_M = 128                                # rational unit vectors ((1 - t^2), 2 t)/(1 + t^2), t = j/NET_M, and their rotations
+
+
+def circle_net():
+    """Rational unit vectors u_j covering angles [0, pi) with consecutive angular gaps <= 2/NET_M (theta = 2 arctan t has
+    d theta/dt <= 2), together with the rotated copies (-u2, u1)."""
+    pts = []
+    for j in range(NET_M + 1):
+        t = Fr(j, NET_M)
+        pts.append(((1 - t * t) / (1 + t * t), 2 * t / (1 + t * t)))
+    return pts + [(-u2, u1) for (u1, u2) in pts]
+
+
+NET = circle_net()
+
+
+def block_form_weights(blk, u):
+    """w_beta = (c!/beta!) u1^beta1 u2^beta2: d_x^a d_u^c f(0) = sum_beta w_beta X_(a,beta)."""
+    a, c = blk
+    return [(n, m * u[0] ** int(n[2]) * u[1] ** int(n[3])) for n, m in block_members(a, c)]
+
+
+def net_max_variance(C, blk):
+    """Upper bound of max_j Var_Q(d_x^a d_(u_j)^c f(0)) over the net."""
+    best = Fr(0)
+    for u in NET:
+        w = block_form_weights(blk, u)
+        v = sum(abs(wi) * abs(wj) * absup(C[(ni, nj)]) if ni != nj else wi * wi * C[(ni, ni)].hi
+                for ni, wi in w for nj, wj in w)
+        best = max(best, v)
+    return best
+
+
+FROB_B = (IV(1), IV(1), SQRT2)               # (B11, B22, B12) -> Frobenius-isometric coordinates (B11, B22, sqrt2 B12)
+
+
+def regress_on_B(law):
+    """Regression of every target on the three entries of the transverse Hessian, in the Frobenius-isometric coordinates
+    b~ = (B11, B22, sqrt2 B12) of B = -A: F = F_g + beta~_F . (b~ - mu~), with F_g independent of B."""
+    C = law.C
+    # covariance of b~ (B = -A: signs cancel in the covariance)
+    Cb = [[C[(BNAMES[i], BNAMES[j])] * FROB_B[i] * FROB_B[j] for j in range(3)] for i in range(3)]
+    law.Cb = Cb
+    law.detCb = det3(Cb)
+    require(law.detCb.lo > 0, 'transverse Hessian covariance not positive')
+    lo, hi = sym3_eig_bounds(Cb)
+    require(lo > 0, 'transverse Hessian covariance eigenvalue bound')
+    law.sB_lo2, law.sB_hi2 = lo, hi
+    law.sB_lo = sqrt_down(lo)
+    # regression coefficients of F on b~: beta~_F = Cov(F, b~) Cb^-1, Cov(F, b~_i) = -Cov(F, A_i) FROB_B[i]
+    others = [n for n in TARGET_NAMES if n not in BNAMES]
+    rhs = [[-C[(n, BNAMES[i])] * FROB_B[i] for i in range(3)] for n in others]
+    sol = solve_iv(Cb, rhs)
+    law.bt = {n: sol[i] for i, n in enumerate(others)}
+    law.bnorm = {n: sqrt_up(sum(absup(x) ** 2 for x in law.bt[n])) for n in others}
+    Cg = {}
+    for a in others:
+        for c in others:
+            Cg[(a, c)] = C[(a, c)] - sum((law.bt[a][i] * (-C[(c, BNAMES[i])] * FROB_B[i]) for i in range(3)), IV(0))
+    law.Cg = Cg
+    # the free blocks in Frobenius coordinates: v, omega = (o11, o22, sqrt2 o12), Y~ = (y111, sqrt3 y112, sqrt3 y122, y222)
+    fw = [isqrt_iv(IV(w)) if w != 1 else IV(1) for w in FREE_W]
+    law.Lfree = cholesky([[Cg[(FREE[i], FREE[j])] * fw[i] * fw[j] for j in range(9)] for i in range(9)])
+    law.fw = fw
+    law.sdg = {n: sqrt_up(max(Cg[(n, n)].hi, Fr(0))) for n in others}
+    law.sdQ = {n: sqrt_up(C[(n, n)].hi) for n in TARGET_NAMES}
+
+
+def sd0_sq(g, M1):
+    """Upper bound of Var d^g f(0) = |d^(2g) K_L(0)| (normalized kernel, every frame, L >= L_MIN), g = (gx, gy1, gy2):
+    the reference value prod (2 g_i - 1)!! plus the Cauchy bound (2g)! M1 of the torus remainder."""
+    return math.prod(dfact(2 * x - 1) for x in g) + math.prod(math.factorial(2 * x) for x in g) * M1
+
+
+def taylor_tail_coefficient(blk, r1, M1):
+    """S_(a,c)(r1) = sum_{gamma != 0} sigma_(a,c),gamma (2 r1)^|gamma| / gamma!, sigma^2 = sum_beta (c!/beta!) s^2_(a+gx, beta+gy):
+    the Frobenius norm of the Taylor tail of the block d_x^a D_y^c f at the midpoint over D."""
+    a, c = blk
+    tot = Fr(0)
+    for n in range(1, NB + 1):
+        for gx in range(n + 1):
+            for g1 in range(n - gx + 1):
+                g2 = n - gx - g1
+                s2 = Fr(0)
+                for b1 in range(c + 1):
+                    b2 = c - b1
+                    mult = math.factorial(c) // (math.factorial(b1) * math.factorial(b2))
+                    s2 += mult * sd0_sq((a + gx, b1 + g1, b2 + g2), M1)
+                tot += sqrt_up(s2) * (2 * r1) ** n / (math.factorial(gx) * math.factorial(g1) * math.factorial(g2))
+    # tail |gamma| = n > NB: s_delta <= 2^|delta| delta! sqrt(1 + M1) and (alpha + gamma)!/gamma! <= (n + 4)^4, so
+    # sigma (2 r1)^n / gamma! <= 2^(c/2) 2^(4 + n) (n + 4)^4 (2 r1)^n sqrt(1 + M1); (n+1)(n+2)/2 multi-indices of order n;
+    # the ratio of consecutive bounds is below 2 q for these n (q = 4 r1 <= 1/16)
+    q = 4 * r1
+    require(q <= Fr(1, 16), 'Taylor tail ratio too large')
+    n = NB + 1
+    base = 2 * 2 ** 4 * sqrt_up(Fr(2) ** c * (1 + M1))
+    tail = base * Fr((n + 1) * (n + 2), 2) * Fr(n + 4) ** 4 * q ** n / (1 - 2 * q)
+    return rup_rel(tot + tail)
+
+
+# ----------------------------------------------------------------------------- closed forms for the contact transverse Hessian
+# B = m I + GOE_2 (diagonal N(0, 2), off-diagonal N(0, 1)): its eigenvalues are s -+ rho with s = m + (G11 + G22)/2 ~ N(m, 1)
+# and rho = sqrt(((G11 - G22)/2)^2 + G12^2) Rayleigh, independent; det B = s^2 - rho^2, tr B = 2 s, B > 0 iff s > rho.
+def upper_moments_iv(x, J):
+    """[M_0(x), ..., M_J(x)] over an interval x whose interior does not contain 0 (each M_j is monotone there)."""
+    if x.lo == x.hi:
+        return upper_moments(x.lo, J)
+    require(not (x.lo < 0 < x.hi), 'incomplete moment over an interval containing 0')
+    A, B = upper_moments(x.lo, J), upper_moments(x.hi, J)
+    return [IV(min(a.lo, b.lo), max(a.hi, b.hi)) for a, b in zip(A, B)]
+
+
+def es_pos(m, n):
+    """[E[s^j 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m (intervals)."""
+    M = upper_moments(-Fr(m), n)
+    return [sum((M[i] * (math.comb(j, i) * Fr(m) ** (j - i)) for i in range(j + 1)), IV(0)) for j in range(n + 1)]
+
+
+def isqrt2_pow(i):
+    """2^(-i/2) (interval)."""
+    return IV(Fr(1, 2 ** (i // 2))) if i % 2 == 0 else IV(Fr(1, 2 ** (i // 2))) / SQRT2
+
+
+def es_pos_exp(m, n):
+    """[E[s^j exp(-s^2/2) 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m:
+    = exp(-m^2/4)/sqrt2 . E[(m/2 + U/sqrt2)^j 1{U > -m/sqrt2}], U ~ N(0, 1)."""
+    m = Fr(m)
+    M = upper_moments_iv(-(IV(m) / SQRT2), n)
+    pre = exp_neg_point(m * m / 4) / SQRT2
+    out = []
+    for j in range(n + 1):
+        tot = IV(0)
+        for i in range(j + 1):
+            tot = tot + M[i] * (math.comb(j, i) * (m / 2) ** (j - i)) * isqrt2_pow(i)
+        out.append(pre * tot)
+    return out
+
+
+def inner_rho_polys(a):
+    """integral_0^s (s^2 - rho^2)^a rho exp(-rho^2/2) d rho = P_a(s) - exp(-s^2/2) Q_a(s) (exact polynomial coefficient
+    dicts {power: Fraction}); with u = rho^2/2 it is integral_0^(s^2/2) (s^2 - 2u)^a e^(-u) du."""
+    P, Q = {}, {}
+    for i in range(a + 1):
+        c = Fr(math.comb(a, i) * (-2) ** i * math.factorial(i))
+        P[2 * (a - i)] = P.get(2 * (a - i), Fr(0)) + c
+        for l in range(i + 1):
+            pw = 2 * (a - i) + 2 * l
+            Q[pw] = Q.get(pw, Fr(0)) + c / (2 ** l * math.factorial(l))
+    return P, Q
+
+
+def contact_moment(a, c, m):
+    """E[det(B)^a tr(B)^c 1{B > 0}] for B = m I + GOE_2, rational m (interval)."""
+    P, Q = inner_rho_polys(a)
+    n = 2 * a + c
+    E1 = es_pos(m, n)
+    E2 = es_pos_exp(m, n)
+    tot = IV(0)
+    for pw, cf in P.items():
+        tot = tot + E1[pw + c] * (cf * 2 ** c)
+    for pw, cf in Q.items():
+        tot = tot - E2[pw + c] * (cf * 2 ** c)
+    return tot
+
+
+def m3(m):
+    """m_(3,m) = E[det(B)^2 1{B > 0}] = E[(s^4 - 4 s^2 + 8) 1{s > 0}] - 4 sqrt2 exp(-m^2/4) Phi(m/sqrt2)."""
+    return contact_moment(2, 0, m)
+
+
+def gauss_half_moment(j, m, sig):
+    """J_j(m, sig) = integral_0^inf lambda^j exp(-(lambda - m)^2/(2 sig^2)) d lambda for rational m, sig > 0 (interval):
+    sig sqrt(2 pi) E[(m + sig Z)^j 1{m + sig Z > 0}]."""
+    m, sig = Fr(m), Fr(sig)
+    M = upper_moments(-m / sig, j)
+    tot = sum((M[i] * (math.comb(j, i) * m ** (j - i) * sig ** i) for i in range(j + 1)), IV(0))
+    return tot * SQRT2PI * sig
+
+
+# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k)
+def binom(n, k):
+    return math.comb(n, k)
+
+
+def pos_part_moments_chi(s, c, nu, nmax, extra_max):
+    """P[(n, e)] = upper bound of E[u^e (u^n - c^n)_+], u = s chi_nu, for 1 <= n <= nmax, 0 <= e <= extra_max."""
+    x = c / s
+    M = chi_upper_moments(nu, x, nmax + extra_max)
+    out = {}
+    for n in range(1, nmax + 1):
+        for e in range(extra_max + 1):
+            out[(n, e)] = rup_rel((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e).hi)
+    return out
+
+
+NU = {2: 2, 3: 3, 4: 4}                      # chi degrees of freedom of the free blocks v (2), omega (3), Y~ (4)
+
+
+def normalized_moments(q, nmax, theta):
+    """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax, under the product of the g-law and nu:
+      E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmt1 + (r1/Klo) D1;
+      Tn[n], To[n], Tv[n]   : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2),
+                              y_i <= u_i + delta_i (u_i = s_i chi_nu_i independent), threshold c = q['c'].
+    o0 = Klo u_3 and v0^2 = Klo^2 u_2^2 / 4 are unnormalized."""
+    th = theta
+    cp = q['c'] * (1 - th)                  # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli)
+    PP = {i: pos_part_moments_chi(q['s'][i], cp, NU[i], nmax, 2) for i in (2, 3, 4)}
+    Eo0 = (chi_moment(3, 1) * q['s3']).hi
+    Ev0sq = (chi_moment(2, 2) * q['s2'] * q['s2']).hi / 4
+    n_o0_2 = q['s3'] * chi_norm(3, 2)
+    n_v0sq_2 = q['s2'] * q['s2'] * chi_norm(2, 4) ** 2 / 4
+    Klo = q['Klo']
+    E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {}
+    base = rup_rel(1 + q['cmt1'])
+    bp = [Fr(1)]
+    for _ in range(nmax):
+        bp.append(rup_rel(bp[-1] * base))
+    # (r/K) D1 terms: ||(rK D1)^j||_1 <= (rK ||D1||_pow2(j))^j and ||(rK D1)^j||_2 <= (rK ||D1||_pow2(2j))^j
+    d1 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(j))) ** j) for j in range(1, nmax + 1)]
+    d2 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(2 * j))) ** j) for j in range(1, nmax + 1)]
+    for n in range(0, nmax + 1):
+        ey = bp[n]
+        eo = bp[n] * Eo0
+        ev = bp[n] * Ev0sq
+        for j in range(1, n + 1):
+            coef = binom(n, j) * bp[n - j]
+            ey += coef * d1[j]
+            eo += coef * n_o0_2 * d2[j]
+            ev += coef * n_v0sq_2 * d2[j]
+        E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev)
+        if n == 0:
+            continue
+        f1 = (1 + th) ** (n - 1)
+        f2 = (1 + 1 / th) ** (n - 1)
+        tn = to = tv = Fr(0)
+        for i in (2, 3, 4):
+            P = PP[i]
+            dn = rup_rel(q['nd'](i, pow2(n)) ** n)
+            dn2 = rup_rel(q['nd'](i, pow2(2 * n)) ** n)
+            tn += f1 * P[(n, 0)] + f2 * dn
+            to += (f1 * Klo * P[(n, 1)] if i == 3 else f1 * Eo0 * P[(n, 0)]) + f2 * n_o0_2 * dn2
+            tv += (f1 * Klo * Klo / 4 * P[(n, 2)] if i == 2 else f1 * Ev0sq * P[(n, 0)]) + f2 * n_v0sq_2 * dn2
+        Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv)
+    return E1, E1o, E1v, Tn, To, Tv
+
+
+def half_gauss_moments(m, sig, J):
+    """[J_j(m, sig) for j = 0..J], J_j = integral_0^inf l^j exp(-(l - m)^2/(2 sig^2)) dl, rational m >= 0, sig > 0 (intervals):
+    J_0 = sig sqrt(2 pi) Phi(m/sig), J_1 = m J_0 + sig^2 exp(-m^2/(2 sig^2)), J_(j+1) = m J_j + j sig^2 J_(j-1)
+    (integration by parts; every term is positive for m >= 0)."""
+    m, sig = Fr(m), Fr(sig)
+    require(m >= 0 and sig > 0, 'half-line Gaussian moments')
+    s2 = sig * sig
+    out = [SQRT2PI * sig * (1 - Phibar_pt(m / sig))]
+    out.append(out[0] * m + exp_neg_point(m * m / (2 * s2)) * s2)
+    for j in range(1, J):
+        nxt = out[j] * m + out[j - 1] * (j * s2)
+        out.append(IV(rdown_rel(nxt.lo), rup_rel(nxt.hi)))
+    return out
+
+
+def lam2_moments(b_lo, b_hi, eps, sig):
+    """Bounds on the lambda_2-integrals of the density majorant g(l) = exp(-(l - b)^2/(2 sig^2) + eps |l - b|/sig^2),
+    uniformly over b in [b_lo, b_hi] (b >= 0): I_hi[j] >= integral_0^inf l^j g(l) dl (j = 0..JM), I3_lo <= integral l^3 g.
+    For l >= 0, eps |l - b| <= eps (l + b), so g(l) <= exp((4 b eps + eps^2)/(2 sig^2)) exp(-(l - b - eps)^2/(2 sig^2));
+    each J_j(m, sig) increases with m; and g(l) >= exp(-(l - b)^2/(2 sig^2))."""
+    JM = 3 + 64
+    shift = (4 * b_hi * eps + eps * eps) / (2 * sig * sig)
+    fac = exp_neg_point(shift).inv().hi
+    Jh = half_gauss_moments(b_hi + eps, sig, JM)
+    I_hi = [rup_rel(x.hi * fac) for x in Jh]
+    I3_lo = rdown_rel(half_gauss_moments(b_lo, sig, 3)[3].lo)
+    return I_hi, I3_lo
+
+
+DETN = {}
+
+
+def det_norm(p, bpp, muB, sig):
+    """Upper bound of ||det(B) 1{B > 0}||_p (p a power of two) for B = mu_B + Gaussian with Loewner coupling
+    B <= (b'' + eta ||z||) I + GOE (b'' rounded up to a 1/1024 grid): the contact moment E[det^p 1{B > 0}] at b'' on
+    {||z|| <= R0}, plus E[(||B||_F^2/2)^p 1{||z|| > R0}] <= 2^-p E[(||mu_B|| + sig chi_3)^(2p) 1{chi_3 > R0}]."""
+    bg = Fr(math.ceil(bpp * 1024), 1024)
+    key = (p, bg)
+    if key not in DETN:
+        DETN[key] = contact_moment(p, 0, bg).hi
+    cm = DETN[key]
+    R0 = Fr(R0_COUPLING)
+    U = chi_upper_moments(3, R0, 2 * p)
+    out = sum((U[i] * (binom(2 * p, i) * muB ** (2 * p - i) * sig ** i) for i in range(2 * p + 1)), IV(0)).hi / 2 ** p
+    return iroot(rup_rel(cm + out), p)
+
+
+def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None):
+    """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch
+    threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5).  The bound is proved for 0 < x0 < 2/9 and
+    0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and
+    (1 + kappa x0) c^2 <= 1.  No claim is made outside that domain.  COMBO = (1/64, 1/32), also used by Math-#215's caller,
+    lies inside it (review C61)."""
+    x0 = X0 if x0 is None else Fr(x0)
+    theta = THETA if theta is None else Fr(theta)
+    b = IV(bb[0], bb[1])
+    k = IV(kb[0], kb[1])
+    blo, bhi = bb
+    klo, khi = kb[0], kb[1]
+    require(blo >= 0, 'b >= 0 assumed by the density majorant')
+    Klo = 12 * klo
+    mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES}
+    mabs = {n: absup(mu[n]) for n in TARGET_NAMES}
+    E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk
+    require(E.hi > 0, 'energy')
+    sqE = sqrt_up(E.hi)
+    nl2 = MUT != 'no-lambda2'
+    # ---- the transverse Hessian B = -A: mean in Frobenius coordinates, its distance to b I, the density majorant
+    dmu = [b * (-law.alpha['a11'] - 1) + k * (-law.beta['a11']), b * (-law.alpha['a22'] - 1) + k * (-law.beta['a22']),
+           (b * law.alpha['a12'] + k * law.beta['a12']) * SQRT2]
+    eps = sqrt_up(sum(absup(x) ** 2 for x in dmu))                 # sup over the box of ||mu_B - b I||_F
+    muB = rup_rel(bhi * sqrt_up(Fr(2)) + eps)                       # sup ||mu_B||_F
+    sig = sqrt_up(law.sB_hi2)                                       # rational, >= sqrt(lambda_max(Cov b~))
+    sB_lo = law.sB_lo                                               # <= sqrt(lambda_min(Cov b~))
+    KB = rup_rel((PI * SQRT2 / (2 * PI * SQRT2PI) / isqrt_iv(law.detCb)).hi)   # pi sqrt2 (2 pi)^(-3/2) det^(-1/2)
+    s2g = sig * sig
+    ex0 = blo * blo / (2 * s2g) - eps * bhi / s2g          # g(0; b) <= exp(-ex0) over the box
+    g0 = exp_neg_point(ex0).hi if ex0 >= 0 else exp_neg_point(-ex0).inv().hi
+    gbar = exp_neg_point(eps * eps / (2 * s2g)).inv().hi
+    # slope of the linear density bound g(l) <= g0 + l Lg on l >= 0: the global |g'| <= exp(-1/2) gbar / sig, or the chord
+    # of min(gbar, g0 exp(c l)) with c = (b_hi + eps)/sig^2 (g(l; b) <= g(0; b) exp((b + eps) l / sig^2) for l >= 0):
+    # g0 c (x - 1)/log x with log x = log(gbar/g0) = y (rational), (x - 1)/log x increasing in x
+    Lg = rup_rel(exp_neg_point(Fr(1, 2)).hi * gbar / sig)
+    y = (eps * eps + blo * blo - 2 * eps * bhi) / (2 * s2g)
+    if y > 0:
+        cc = (bhi + eps) / s2g
+        Lg = min(Lg, rup_rel(g0 * cc * (exp_neg_point(y).inv().hi - 1) / y))
+    else:
+        Lg = Fr(0)                                    # gbar <= g0: the bound g <= gbar alone is the smaller one
+    I_hi, I3_lo = lam2_moments(blo, bhi, eps, sig)
+    require(I3_lo > 0, 'lambda_2 moment')
+
+    def l2norm(p):                                  # ||lambda_2||_p under nu (and under nu'), p a power of two
+        if not nl2:
+            return Fr(0)
+        return iroot(rup_rel(I_hi[3 + p] / I3_lo), p)
+    # ---- regression on B and the coefficients bar-beta_i (NOTE section 4)
+    bn = law.bnorm
+    bv = sqrt_up(bn['v1'] ** 2 + bn['v2'] ** 2)
+    bo = sqrt_up(bn['o11'] ** 2 + bn['o22'] ** 2 + 2 * bn['o12'] ** 2)
+    by = sqrt_up(bn['y111'] ** 2 + 3 * bn['y112'] ** 2 + 3 * bn['y122'] ** 2 + bn['y222'] ** 2)
+    bnu = sqrt_up(bn['nu1'] ** 2 + bn['nu2'] ** 2)
+    a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1)      # the cap constant 4/(3k) (mutant: halved)
+
+    def Bblk(blk):
+        return rup_rel(sqrt_up(sum(m * bn[n] ** 2 for n, m in block_members(*blk))) + law.S[blk] / sB_lo)
+    Bb = {blk: Bblk(blk) for blk in BLOCKS}
+    bb1 = rup_rel(Fr(5, 2) * Bb[(4, 0)] + 2 * Bb[(3, 1)])           # bar-beta_1 / r
+    bbars = {1: rup_rel(r1 * bb1),
+             2: rup_rel(2 * bv + r1 * (Fr(5, 2) * Bb[(3, 1)] + 2 * Bb[(2, 2)])),
+             3: rup_rel(bo + r1 * (Fr(5, 2) * Bb[(2, 2)] + 2 * Bb[(1, 3)])),
+             4: rup_rel(by + r1 * (2 * Bb[(1, 3)] + 2 * Bb[(0, 4)]))}
+    bbar = max(bbars.values())
+    # ---- free blocks: Cholesky factors in Frobenius coordinates (v | omega | Y~)
+    Lf = law.Lfree
+    Iv, Io, Iy = (0, 1), (2, 3, 4), (5, 6, 7, 8)
+    Lvv, Low, Loo = opnorm_lower_tri(Lf, Iv, Iv), opnorm_lower_tri(Lf, Io, Iv), opnorm_lower_tri(Lf, Io, Io)
+    Lyv, Lyo, Lyy = opnorm_lower_tri(Lf, Iy, Iv), opnorm_lower_tri(Lf, Iy, Io), opnorm_lower_tri(Lf, Iy, Iy)
+    mu_v = sqrt_up(mabs['v1'] ** 2 + mabs['v2'] ** 2)
+    mu_o = sqrt_up(mabs['o11'] ** 2 + mabs['o22'] ** 2 + 2 * mabs['o12'] ** 2)
+    mu_y = sqrt_up(mabs['y111'] ** 2 + 3 * mabs['y112'] ** 2 + 3 * mabs['y122'] ** 2 + mabs['y222'] ** 2)
+    mu_nu = sqrt_up(mabs['nu1'] ** 2 + mabs['nu2'] ** 2)
+    s2, s3, s4 = 2 * Lvv, Loo, Lyy
+
+    memo = {}
+
+    def blk_g_norm(blk, p):                         # || ||X_blk,g||_F ||_p
+        mem = block_members(*blk)
+        m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem))
+        tr = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in mem)
+        return frob_norm(m_, tr, p)
+
+    def nA(blk, p):                                 # ||A^g_blk||_p, A^g = ||X_blk,g||_F + T^g_blk
+        key = ('A', blk, p)
+        if key not in memo:
+            memo[key] = rup_rel(blk_g_norm(blk, p) + law.S[blk] * (sqE + znorm1(pow2(p))))
+        return memo[key]
+
+    def nDelta(i, p):
+        key = ('D', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nDelta0(i, p))
+        return memo[key]
+
+    def nDelta0(i, p):
+        if MUT == 'no-delta':
+            return Fr(0)
+        if i == 1:
+            return Fr(5, 2) * nA((4, 0), p) + 2 * nA((3, 1), p)
+        if i == 2:
+            return Fr(5, 2) * nA((3, 1), p) + 2 * nA((2, 2), p)
+        if i == 3:
+            return Fr(5, 2) * nA((2, 2), p) + 2 * nA((1, 3), p)
+        return 2 * nA((1, 3), p) + 2 * nA((0, 4), p)
+
+    def nd(i, p, lam=True):
+        key = ('d', i, p, lam)
+        if key not in memo:
+            memo[key] = rup_rel(nd0(i, p, lam))
+        return memo[key]
+
+    def nd0(i, p, lam):     # ||delta_i||_p / Klo, delta_i including bar-beta_i (||mu_B|| + lambda_2)
+        cm = bbars[i] * (muB + (l2norm(p) if lam else 0))
+        if i == 2:
+            v = 2 * mu_v + cm + r1 * nDelta(2, p)
+        elif i == 3:
+            v = mu_o + Low * chi_norm(2, pow2(p)) + cm + r1 * nDelta(3, p)
+        else:
+            v = mu_y + Lyv * chi_norm(2, pow2(p)) + Lyo * chi_norm(3, pow2(p)) + cm + r1 * nDelta(4, p)
+        return v / Klo
+
+    kappa = 3 / (1 - 3 * x0)
+    Fk = 1 + kappa * x0                      # (1 + kappa x_i) <= Fk on the event {x_i <= x0}
+    cth = 1 - kappa * x0 / 2                 # <= Fk^{-1/2}
+    NMAX = 24
+    q = {'cmt1': bbars[1] * muB / Klo, 'rK': r1 / Klo,
+         'nD1': lambda p: rup_rel(nDelta(1, p) + bb1 * l2norm(pow2(p))),
+         's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth, 'Klo': Klo}
+    E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta)
+    et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac)     # eta_1 K, k-free
+    aK2_hi = 192 * khi * a_fac                    # a K^2 = 192 k
+
+    def EUt(p):                                   # upper bound of E[Ut^p], Ut = U/(a K^2) <= max(Z1, Fk y_i^2)
+        key = ('U', p)
+        if key not in memo:
+            memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p])
+        return memo[key]
+
+    def EoUt(p):
+        return sum(binom(p, l) * et1 ** l * E1o[2 * p + l] for l in range(p + 1)) + Fk ** p * To[2 * p]
+
+    def EvUt(p):
+        return sum(binom(p, l) * et1 ** l * E1v[2 * p + l] for l in range(p + 1)) + Fk ** p * Tv[2 * p]
+
+    def sq(x):
+        return sqrt_up(x)
+
+    def nUt(m, p):                                # ||Ut^m||_p <= Fk^m ||max_i y_i||_2mp^2m (on the event x_i <= X0)
+        require(2 * m * p <= NMAX and pow2(p) == p, 'norm order')
+        key = ('nUt', m, p)
+        if key not in memo:
+            memo[key] = rup_rel(Fk ** m * iroot(E1[2 * m * p] + Tn[2 * m * p], p))
+        return memo[key]
+
+    def nU(p):
+        return rup_rel(aK2_hi * nUt(1, p))
+
+    def ndel(p):                                   # || ||B - mu_B||_F ||_p <= ||mu_B|| + lambda_1 + lambda_2 <= ||mu_B|| + 2 lambda_2
+        return rup_rel(muB + 2 * l2norm(p))
+
+    def nbar(n, p):                                # ||F_bar||_p, F_bar = |F_g| + ||beta_F|| ||B - mu_B||_F
+        return rup_rel(gauss_norm(mabs[n], law.sdg[n], p) + bn[n] * ndel(p))
+
+    six_klo = 6 * klo
+    nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4)
+    epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2
+    nT8, nt8 = nbar('T', 8), nbar('tau', 8)
+    epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2
+    U2_2, U3_2, U4_2 = sq(EUt(4)), sq(EUt(6)), sq(EUt(8))
+    U2_4, U3_4 = nUt(2, 4), nUt(3, 4)
+    # the Omega majorant: Omega_bar = o0 + d_o, o0 = s3 chi_3 (shares u_3), d_o = rest
+    ndo = lambda p: rup_rel(mu_o + Low * chi_norm(2, pow2(p)) + bo * ndel(p))
+    no4 = rup_rel(s3 * chi_norm(3, 4) + ndo(4))
+    # the ||r nu - v||^2 majorant (Term C)
+    th = THETA_E
+    dv = lambda p: rup_rel(mu_v + bv * ndel(p))
+    nnu = lambda p: rup_rel(frob_norm(mu_nu, law.Cg[('nu1', 'nu1')].hi + law.Cg[('nu2', 'nu2')].hi, p)
+                            + bnu * ndel(p))
+    ebar4 = rup_rel((r1 * nnu(8) + Lvv * chi_norm(2, 8) + dv(8)) ** 2)        # ||e_bar||_4
+    ebar8 = rup_rel((r1 * nnu(16) + Lvv * chi_norm(2, 16) + dv(16)) ** 2)     # ||e_bar||_8
+    cC = (1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nnu(4) ** 2
+    # ---- Term A, B, C (one power of U per lambda_1 integration), normalized by 36 k^2 r^5 and by the lambda_2 weight
+    termA = aK2_hi ** 3 / 3 * (EUt(3) + epsc2 * U3_2)
+    termB = aK2_hi ** 2 / 2 * (EoUt(2) + (ndo(2) + epsc4 * no4) * U2_2)
+    termC = aK2_hi ** 2 / (12 * klo) * ((1 + th) ** 2 * EvUt(2) + cC * U2_2 + r1 * nT4 / six_klo * ebar4 * U2_2)
+    termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * U4_2)
+    termBp = aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2)
+    termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt(3) + cC * U3_2 + r1 * nT4 / six_klo * ebar4 * U3_2)
+    if MUT == 'no-om-term':
+        termB = termBp = Fr(0)
+    # ---- the Jacobian-correction terms: the part r Omega_bar of (lambda_2 + r Omega_bar), integrated against nu'
+    # D  = E[Omega_bar (A + B + C integrands)],  D' = E[Omega_bar (A' + B' + C' integrands)]
+    nT16 = nbar('T', 16)
+    termD = (aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2)
+             + aK2_hi ** 2 / 2 * no4 * no4 * (U2_2 + epsc4 * U2_4)
+             + aK2_hi ** 2 / (12 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U2_4)
+    termDp = (aK2_hi ** 4 / 4 * (EoUt(4) + (ndo(2) + epsc4 * no4) * U4_2)
+              + aK2_hi ** 3 / 3 * no4 * no4 * (U3_2 + epsc4 * U3_4)
+              + aK2_hi ** 3 / (18 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U3_4)
+    if MUT == 'no-om-term':
+        termD = termDp = Fr(0)
+    I2, I3 = I_hi[2], I_hi[3]
+    lead = I3 * (termA + termB + termC) + r1 * I2 * termD
+    leadp = I3 * (termAp + termBp + termCp) + r1 * I2 * termDp
+    # ---- normalizer floor Z_r / (36 k^2 r^2) >= m3(b'_lo) - corrections (NOTE section 6)
+    Lb = cholesky([[IV(x.lo, x.hi) for x in row] for row in law.Cb])
+    eta = sqrt_up(sum(absup(Lb[i][j] - (SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1)))
+    R0 = Fr(R0_COUPLING)
+    bp_lo = blo - eps - eta * R0
+    bp_hi = bhi + eps + eta * R0
+    bp_lo = Fr(math.floor(bp_lo * 2 ** 40), 2 ** 40)
+    bp_hi = Fr(math.ceil(bp_hi * 2 ** 40), 2 ** 40)
+    pR0 = chi_tail(3, R0)
+    m3lo = m3(bp_lo).lo
+    tail_c = sqrt_up(contact_moment(4, 0, bp_lo).hi) * sqrt_up(pR0)
+    # upper bounds of E[h(B)] for Loewner-monotone h through b'' = bp_hi, plus the outside-ball part
+    nBF16 = rup_rel(muB + sig * chi_norm(3, 16))                 # || ||B||_F ||_16
+    out_ball = sqrt_up(pR0)
+
+    def up_moment(a, c):                    # E[det^a tr^c 1{B > 0}] <= contact value at b'' + outside-ball part
+        crude = rup_rel(nBF16 ** (2 * a + c))      # det <= ||B||^2, tr <= sqrt2 ||B|| <= 2 ||B||: crude, times P^(1/2)
+        return rup_rel(contact_moment(a, c, bp_hi).hi + 2 ** c * crude * out_ball)
+    F2 = up_moment(4, 0)                    # E[(det B)^4 1{B > 0}]
+    DT, TDT, TTDT = up_moment(1, 1), up_moment(1, 2), up_moment(1, 3)
+    sdT, sdtau = law.sdQ['T'], law.sdQ['tau']
+    q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1))
+    nF = sqrt_up(F2)
+    corr_G0 = nF * sqrt_up(q_bad) if q_bad > 0 else Fr(0)
+    corr_eps = (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) / six_klo * nF
+    trO = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in zip(ONAMES, (1, 1, 2)))
+    trV = law.Cg[('v1', 'v1')].hi + law.Cg[('v2', 'v2')].hi
+    Eog = rup_rel(mu_o + sqrt_up(trO))
+    corr_om = Eog * DT + bo * (TDT + muB * DT)
+    Evg2 = mu_v ** 2 + trV
+    corr_v = (Fr(17, 16) * Evg2 * DT + 17 * bv ** 2 * (TTDT + 2 * muB * TDT + muB * muB * DT)) / (4 * klo)
+    corr = tail_c + corr_G0 + corr_eps + r1 * (corr_om + corr_v)
+    if MUT == 'no-zfloor-correction':
+        corr = Fr(0)
+    zt = m3lo - corr
+    require(zt > 0, 'normalizer floor not positive')
+    main = KB * min(gbar * lead, g0 * lead + r1 * Lg * leadp) / zt
+    # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q), ||W 1_typed||_q / r^2 <= g1 g2
+    six_khi = 6 * khi
+    trB = sum(x.hi for x in (law.Cb[0][0], law.Cb[1][1], law.Cb[2][2]))
+    trOQ = sum(m * law.C[(n, n)].hi for n, m in zip(ONAMES, (1, 1, 2)))
+    trVQ = law.C[('v1', 'v1')].hi + law.C[('v2', 'v2')].hi
+    trNQ = law.C[('nu1', 'nu1')].hi + law.C[('nu2', 'nu2')].hi
+
+    # W_r / r^2 <= |c1| D [|c2| (D + r Om tr + r^2 Om^2) + r ||e||^2 (tr + r Om)] on the typed support, D = det B 1{B > 0},
+    # tr = tr B, Om = ||Omega'||_F (NOTE section 7)
+    sd_tr = sqrt_up(law.C[('a11', 'a11')].hi + law.C[('a22', 'a22')].hi + 2 * absup(law.C[('a11', 'a22')]))
+    m_tr = absup(mu['a11'] + mu['a22'])
+
+    def wnorm(qq):
+        p4, p8 = 4 * qq, 8 * qq
+        nD = lambda p: det_norm(p, bp_hi, muB, sig)
+        ntr = lambda p: gauss_norm(m_tr, sd_tr, p)
+        no = lambda p: frob_norm(mu_o, trOQ, p)
+        nT = lambda p: gauss_norm(mabs['T'], sdT, p)
+        nc2 = lambda p: six_khi + r1 * nT(p) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, p)
+        ne = lambda p: r1 * frob_norm(mu_nu, trNQ, p) + frob_norm(mu_v, trVQ, p)
+        g1 = (six_khi + r1 * nT(p4)) * nD(p4)
+        g2 = (nc2(p4) * (nD(p4) + r1 * no(p8) * ntr(p8) + r1 * r1 * no(p8) ** 2)
+              + r1 * ne(p8) ** 2 * (ntr(p4) + r1 * no(p4)))
+        return g1 * g2
+    thr = 3 * klo / (10 * r1)
+    qA2 = Fr(0)
+    args = []
+    for blk in BLOCKS:
+        a_, c_ = blk
+        mem = block_members(*blk)
+        m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem))       # |mean of d_x^a d_u^c f(0)| <= ||mean||_F for unit u
+        best = None
+        for frac in (Fr(7, 8), Fr(15, 16)):                        # split t = frac t + (1 - frac) t between jet and Taylor tail
+            if c_ == 0:                                            # scalar block
+                x_ = (thr * frac - m_) / law.sdQ[mem[0][0]]
+                pj = gauss_tail2(m_, law.sdQ[mem[0][0]], thr * frac)
+            elif c_ == 1:                                          # vector block: ||X - mean|| <= sqrt(lambda_max) chi_2
+                lmax = max(sum(absup(law.C[(ni, nj)]) for nj, mj in mem) for ni, mi in mem)
+                x_ = (thr * frac - m_) / sqrt_up(lmax)
+                pj = chi_tail(2, x_) if x_ > 0 else Fr(1)
+            else:                                                  # ||X||_op = max_theta |Q(theta)|, Q a trigonometric
+                rho = 1 - Fr(c_, NET_M)                            # polynomial of degree c: Bernstein |Q'| <= c max|Q| and
+                sdn = sqrt_up(law.net_var[blk])                    # net gaps <= 2/NET_M give max|Q| <= max_j |Q_j| / rho
+                x_ = (thr * frac * rho - m_) / sdn
+                pj = min(Fr(1), len(NET) * gauss_tail2(m_, sdn, thr * frac * rho))
+            pj += (law.S[blk] * (sqE + znorm(32)) / (thr * (1 - frac))) ** 32
+            if best is None or pj < best[0]:
+                best = (pj, x_)
+        args.append(best[1])
+        qA2 += best[0]
+    a_hi = Fr(4, 3) / klo * a_fac
+    qfar = Fr(0)
+    qx = Fr(0)
+    sdB11 = sqrt_up(law.C[('a11', 'a11')].hi)
+    if bbar > 0:
+        lam_far = 1 / (4 * a_hi * r1 * bbar * bbar)
+        args.append((lam_far - absup(mu['a11'])) / sdB11)
+        qfar = gauss_tail1(absup(mu['a11']), sdB11, lam_far)
+
+    def lam2_tail(s):                       # P(lambda_2 > s) <= P(sig chi_3 > s - ||mu_B||)
+        x_ = (s - muB) / sig
+        args.append(x_)
+        return chi_tail(3, x_) if x_ > 0 else Fr(1)
+    for i in (1, 2, 3, 4):                         # x_i > X0  <=>  y_i' + bar-beta_i lambda_2 > X0 / (a r1 bar-beta_i)
+        if bbars[i] == 0:
+            continue
+        tx = x0 / (a_hi * r1 * bbars[i])
+        if i == 1:
+            cb = 12 * khi + bbars[1] * muB
+            if tx <= cb:
+                qx += Fr(1)
+            else:
+                qx += (r1 * nDelta(1, 32) / ((tx - cb) / 2)) ** 32
+                qx += lam2_tail((tx - cb) / (2 * bbars[1]))
+        else:
+            s_i = {2: s2, 3: s3, 4: s4}[i]
+            x_ = tx / (3 * s_i)
+            args.append(x_)
+            qx += chi_tail(NU[i], x_) + (nd(i, 32, lam=False) * Klo / (tx / 3)) ** 32 + lam2_tail(tx / (3 * bbars[i]))
+    qs = [min(Fr(1), x) for x in (qA2, qfar, qx)]
+    tail_num = min(wnorm(qq) * sum(pow_tail(x, qq) for x in qs) for qq in QS)
+    if last_band:
+        # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian or chi tail argument scales at least like 1/r and is
+        # >= 3 at R, so each such piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32;
+        # after the power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of
+        # tail / r^3 over (0, R] is attained on [R/2, R]
+        require(min(args) >= 3, 'last-band tail arguments')
+        require(max(qA2, qfar, qx) < 1, 'last-band tail pieces')
+        rdiv = r1 / 2
+    else:
+        rdiv = r0
+    tail = tail_num / (36 * klo * klo * zt * rdiv ** 3)
+    total = main + tail
+    return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'tail_num': tail_num, 'KB': KB, 'g0': g0, 'gbar': gbar, 'I3': I3,
+            'termA': termA, 'termB': termB, 'termC': termC, 'termD': termD, 'qA2': qA2, 'qfar': qfar, 'qx': qx,
+            'm3lo': m3lo, 'corr': corr, 'eps': eps, 'eta': eta, 'min_arg': min(args)}
+# ----------------------------------------------------------------------------- the (k, b) boxes of the sweep
+F_ = Fr
+GRID = (
+    ((F_(1, 2), F_(3, 4)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(3, 4), F_(1)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(1), F_(5, 4)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(5, 4), F_(3, 2)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(3, 2), F_(13, 8)), (F_(0), F_(243, 512), F_(1))),
+    ((F_(13, 8), F_(7, 4)), (F_(0), F_(309, 1024), F_(899, 1024), F_(1))),
+    ((F_(7, 4), F_(15, 8)), (F_(0), F_(37, 256), F_(105, 256), F_(933, 1024), F_(1))),
+    ((F_(15, 8), F_(31, 16)), (F_(0), F_(37, 512), F_(207, 1024), F_(225, 512), F_(113, 128), F_(1))),
+    ((F_(31, 16), F_(63, 32)), (F_(0), F_(37, 1024), F_(103, 1024), F_(111, 512), F_(439, 1024), F_(421, 512), F_(1))),
+    ((F_(63, 32), F_(2)), (F_(0), F_(1, 512), F_(5, 1024), F_(5, 512), F_(19, 1024), F_(35, 1024), F_(1, 16), F_(29, 256),
+                           F_(209, 1024), F_(377, 1024), F_(685, 1024), F_(1))),
+)
+
+
+# ----------------------------------------------------------------------------- bands, boxes, runs
+TOP_OCTAVES = 7            # octaves [2^-(j+1), 2^-j], j = 6..12 (r up to 1/64), split into 8 sub-bands
+LOW_OCTAVES = 16           # octaves j = 13..28, split into 4 sub-bands
+R_LAST = Fr(1, 2 ** 29)    # last band [0, 2^-29]
+COMBO = (Fr(1, 64), Fr(1, 32))     # (X_0, theta) of NOTE sections 4-5
+
+
+def bands():
+    """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST]."""
+    out = []
+    for j in range(6, 6 + TOP_OCTAVES + LOW_OCTAVES):
+        hi = Fr(1, 2 ** j)
+        lo = hi / 2
+        m = 8 if j < 6 + TOP_OCTAVES else 4
+        for i in range(m):
+            out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m))
+    out.append((Fr(0), R_LAST))
+    require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover')
+    out = sorted(out)
+    require(all(out[i][1] == out[i + 1][0] for i in range(len(out) - 1)), 'bands contiguous')
+    return out
+
+
+def dec_down(x, places=4):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_up(x, places=4):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def grid_boxes():
+    out = []
+    for kb, bg in GRID:
+        require(bg[0] == 0 and bg[-1] == 1 and all(bg[i] < bg[i + 1] for i in range(len(bg) - 1)), 'b grid')
+        for i in range(len(bg) - 1):
+            out.append(((Fr(bg[i]), Fr(bg[i + 1])), (Fr(kb[0]), Fr(kb[1]))))
+    ks = [kb for kb, _ in GRID]
+    require(ks[0][0] == BAND_K[0] and ks[-1][1] == BAND_K[1] and all(ks[i][1] == ks[i + 1][0] for i in range(len(ks) - 1)),
+            'k boxes cover K')
+    return tuple(out)
+
+
+CG3_02_LO = Fr('22424320.655069')     # certified lower end of c_G^(3)(0, 2) (Math-#208)
+
+
+def band_record(args):
+    """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down,
+    6 places) for each box in BOXES order, and the largest tail part (rounded up, 12 places)."""
+    r0, r1 = args
+    law = band_law(r0, r1)
+    last = (r0 == 0)
+    bound, zfloor, tails = [], [], []
+    for bb, kb in BOXES:
+        res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=COMBO[0], theta=COMBO[1])
+        bound.append(dec_up(res['total']))
+        zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6))
+        tails.append(res['tail'])
+    return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12)}
+
+
+def full_run(procs):
+    bl = bands()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, bl)
+    return assemble(recs)
+
+
+def assemble(recs):
+    table = {}
+    for rs in R_STARS:
+        mx, arg, zmin = Fr(0), None, None
+        for rec in recs:
+            if Fr(rec['r'][1]) <= rs:
+                for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']):
+                    v = Fr(bs)
+                    if v > mx:
+                        mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]}
+                    z = Fr(zs)
+                    zmin = z if zmin is None or z < zmin else zmin
+        table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG3_02': dec_up(mx / CG3_02_LO, 6),
+                          'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1', 'scientific_effect': 'NONE',
+            'certified': True, 'mutant': MUT, 'L_min': L_MIN, 'dimension': 3,
+            'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]},
+            'parameters': params(), 'C_table': table, 'bands': recs}
+
+
+def params():
+    return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBO_X0_THETA': [str(COMBO[0]), str(COMBO[1])], 'THETA_E': str(THETA_E),
+            'QS': list(QS), 'R0_COUPLING': R0_COUPLING, 'NET_M': NET_M, 'R_STARS': [str(x) for x in R_STARS],
+            'TOP_OCTAVES': TOP_OCTAVES, 'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST),
+            'GRID': [[[str(kb[0]), str(kb[1])], [str(x) for x in bg]] for kb, bg in GRID]}
+
+
+def results_text(res):
+    """RESULTS.json layout: the summary indented, then one line per band record."""
+    head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True)
+    lines = ',\n'.join('  ' + json.dumps(rec, sort_keys=True) for rec in res['bands'])
+    return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n'
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def replay_sample():
+    """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band."""
+    bl = bands()
+    idx = []
+    for rs in R_STARS:
+        j = max(i for i, b in enumerate(bl) if b[1] == rs)
+        idx.append(j)
+    idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))]
+    idx.append(0)
+    return sorted(set(idx))
+
+
+def check_run(procs, full=False):
+    ref = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout')
+    require(ref['parameters'] == params(), 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json produced under a mutant')
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure')
+    require(abs(m3(0).mid() - (Fr(7, 2) - 2 * SQRT2.mid())) < Fr(1, 10 ** 40), 'm_(3,0) closed form')
+    bl = bands()
+    require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs')
+    idx = list(range(len(bl))) if full else replay_sample()
+    with multiprocessing.Pool(procs) as pool:          # in order; the first differing band exits (and stops the pool)
+        for i, rec in zip(idx, pool.imap(band_record, [bl[i] for i in idx])):
+            require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r']))
+    # the table is the maximum over the stored records
+    asm = assemble(ref['bands'])
+    require(asm['C_table'] == ref['C_table'], 'C table is not the maximum of the stored bounds')
+    # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G^(3)(0, 2) of Math-#208 (a lower bound for every
+    # d = 3 cap-route constant valid at (0, 2) for the reference kernel, which this certificate covers)
+    for rs, row in ref['C_table'].items():
+        require(Fr(row['C']) > CG3_02_LO, 'C below the sharp cap coefficient')
+    print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()},
+                      'z_star': {rs: row['z_star'] for rs, row in ref['C_table'].items()}, 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.mc:
+        rr, bb, kk, nn, sd = args.mc
+        print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd))))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({'C': {rs: row['C'] for rs, row in res['C_table'].items()},
+                      'z': {rs: row['z_star'] for rs, row in res['C_table'].items()}}))
+BOXES = grid_boxes()
+
+
+# ----------------------------------------------------------------------------- floating control (not part of the certificate)
+def mc_event(r, b, k, n, seed, deg=7, nang=24):
+    """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel in d = 3 (floating point): the
+    Taylor jet of order deg at the midpoint is sampled under Q (exact pin series evaluated in floating point), W_r is
+    evaluated from the Hessians at M and S, and M_3, M_4 are the partial-block operator norms ([CAP] (1)) maximized over a
+    grid of the cylinder D (binary forms maximized over nang angles; floating, so slightly below the true suprema)."""
+    import random
+    pins, _ = functionals()
+    P = [pins[nm] for nm in PIN_NAMES]
+    jets = [(a, b1, m - a - b1) for m in range(deg + 1) for a in range(m, -1, -1) for b1 in range(m - a, -1, -1)]
+    J = [[term(1, 0, a, b1, b2, X0PT)] for (a, b1, b2) in jets]
+    nj = len(J)
+
+    def ev(F, G, key):
+        ser, w, _ = pair_series(F, G, key)
+        return sum(float(c) * r ** p for p, c in ser.items())
+    Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(NP)] for i in range(NP)]
+    Sjp = [[ev(J[i], P[j], ('jp3', deg, i, j)) for j in range(NP)] for i in range(nj)]
+    Sjj = [[ev(J[i], J[j], ('jj3', deg, i, j)) for j in range(nj)] for i in range(nj)]
+
+    def solve(A, bv):
+        m = len(A)
+        M = [A[i][:] + [bv[i]] for i in range(m)]
+        for c in range(m):
+            p = max(range(c, m), key=lambda i: abs(M[i][c]))
+            M[c], M[p] = M[p], M[c]
+            for i in range(c + 1, m):
+                f = M[i][c] / M[c][c]
+                for j in range(c, m + 1):
+                    M[i][j] -= f * M[c][j]
+        x = [0.0] * m
+        for i in reversed(range(m)):
+            x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i]
+        return x
+    vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k] + [0.0] * (NP - 3)
+    Wt = [solve(Spp, Sjp[i]) for i in range(nj)]
+    mu = [sum(Wt[i][j] * vr[j] for j in range(NP)) for i in range(nj)]
+    C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(NP)) for j in range(nj)] for i in range(nj)]
+    # Cholesky with clamped pivots (the conditional covariance is positive semidefinite up to rounding)
+    L = [[0.0] * nj for _ in range(nj)]
+    for i in range(nj):
+        for j in range(i + 1):
+            s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j))
+            if i == j:
+                L[i][i] = math.sqrt(s) if s > 1e-300 else 0.0
+            else:
+                L[i][j] = s / L[j][j] if L[j][j] > 0 else 0.0
+    index = {al: i for i, al in enumerate(jets)}
+    fact = [math.factorial(i) for i in range(deg + 2)]
+
+    def deriv(cf, be, p):
+        """d^be f(p) from the jet (Taylor at 0)."""
+        tot = 0.0
+        for al, i in index.items():
+            if al[0] >= be[0] and al[1] >= be[1] and al[2] >= be[2]:
+                e = (al[0] - be[0], al[1] - be[1], al[2] - be[2])
+                tot += cf[i] * (p[0] ** e[0] * p[1] ** e[1] * p[2] ** e[2]) / (fact[e[0]] * fact[e[1]] * fact[e[2]])
+        return tot
+    angs = [(math.cos(math.pi * t / nang), math.sin(math.pi * t / nang)) for t in range(nang)]
+    pts = [(x, 0.0, 0.0) for x in (-2 * r, -r, 0.0, r, 2 * r)]
+    pts += [(x, 2 * r * math.cos(2 * math.pi * t / 6), 2 * r * math.sin(2 * math.pi * t / 6)) for x in (-2 * r, 0.0, 2 * r)
+            for t in range(6)]
+
+    def block_norm(vals, c):
+        """Operator norm of the symmetric c-form with components vals[(b1, b2)] (b1 + b2 = c) over unit vectors."""
+        if c == 0:
+            return abs(vals[(0, 0)])
+        best = 0.0
+        for (u1, u2) in angs:
+            q = sum(math.comb(c, b1) * vals[(b1, c - b1)] * u1 ** b1 * u2 ** (c - b1) for b1 in range(c + 1))
+            best = max(best, abs(q))
+        return best
+
+    def Mj(cf, j):
+        best = 0.0
+        for p in pts:
+            for a in range(j + 1):
+                c = j - a
+                vals = {(b1, c - b1): deriv(cf, (a, b1, c - b1), p) for b1 in range(c + 1)}
+                best = max(best, block_norm(vals, c))
+        return best
+
+    def hess(cf, p):
+        return [[deriv(cf, tuple(int(x == i) + int(x == j) for x in range(3)), p) for j in range(3)] for i in range(3)]
+
+    def det3f(H):
+        return (H[0][0] * (H[1][1] * H[2][2] - H[1][2] * H[2][1]) - H[0][1] * (H[1][0] * H[2][2] - H[1][2] * H[2][0])
+                + H[0][2] * (H[1][0] * H[2][1] - H[1][1] * H[2][0]))
+
+    def neg_def(H):
+        return H[0][0] < 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and det3f(H) < 0
+
+    def index2(H):
+        """Number of negative eigenvalues of a symmetric 3 x 3 matrix is 2: det > 0 and not positive definite."""
+        d = det3f(H)
+        pd = H[0][0] > 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and d > 0
+        return d > 0 and not pd
+    rng = random.Random(seed)
+    num = den = 0.0
+    acap = 4 / (3 * k)
+    for _ in range(n):
+        z = [rng.gauss(0, 1) for _ in range(nj)]
+        cf = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(nj)]
+        HM, HS = hess(cf, (-r / 2, 0.0, 0.0)), hess(cf, (r / 2, 0.0, 0.0))
+        if not (neg_def(HM) and index2(HS)):
+            continue
+        Wv = abs(det3f(HM) * det3f(HS))
+        den += Wv
+        a11, a22, a12 = -HM[1][1], -HM[2][2], -HM[1][2]
+        lam1 = (a11 + a22) / 2 - math.sqrt(((a11 - a22) / 2) ** 2 + a12 * a12)
+        M3 = Mj(cf, 3)
+        if lam1 <= acap * r * M3 * M3:
+            num += Wv
+            continue
+        if r * Mj(cf, 4) > 3 * k / 10:
+            num += Wv
+    return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3 if den else None,
+            'Z_over_r2': den / n / r ** 2}
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_theorem_a_constants_planar_20260930/NOTE.md b/frontiers/c8_theorem_a_constants_planar_20260930/NOTE.md
new file mode 100644
index 000000000..49ebecf43
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_planar_20260930/NOTE.md
@@ -0,0 +1,512 @@
+# Explicit cap-route constants for Theorem A in the plane: `Q^W(G_r^c) <= C r^3` with certified `C`
+
+**Object:** `CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1`. **Author lane:** Anthropic / Claude.
+**Kind:** author-side theorem (Theorem E), proved with explicit constants and certified by exact rational interval
+arithmetic (standard library only). **Scientific effect:** NONE. Catalog entry C8 (`reviews/candidates_pending_20260928/CANDIDATES.md`:
+"explicit finite `C`, `r_*`, `z_*`, `c_{B,K}`, `c_{d,L}` on a declared compact-mark / torus band; OPEN") stays OPEN, and no
+catalog edit is made. What this packet supplies, in `d = 2`:
+- on the band `B = [0, 1]`, `K = [1/2, 2]`, uniformly over frames and torus sides `L >= 10`, explicit cap-route pairs
+  `(C, r_*)` for [LP] (7.8), and therefore for [LP] Theorem A (1.1);
+- a normalizer floor `z_*` on `(0, r_*]`;
+- (v1.1) the same at the single fixed-axis SIDE24 point `(b, k) = (6/5, 1/6)` of [CAP] section 1 (Theorem E'). There the only
+  constant on record, `2.4 x 10^23`, is replaced by `39322.8` for `r <= 1/4096`.
+
+No STATUS, PROOF_INDEX, GRAPH or catalog edit. Same GitHub account as every lane; zero organizational-independence credit.
+The author will not merge.
+
+**Sources** (pins in `SOURCE_MAP.json`):
+- [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md`, sections 1, 3, 7 and 8:
+  - the model and the law `Q`, `W_r`, `Z_r`;
+  - the contact frame (3.1)–(3.3);
+  - the good event `G_r`, Theorem A (1.1) and the cap bound (7.8);
+  - the §8 implication "on `G_r` the global elder partner of `M` is `S`".
+- [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md`, section 1: the cylinder `D`, the norms `M_3`, `M_4`
+  and condition (1).
+- [G] Math-#203 at `a388580`, read-only: Theorem G and the sharp cap coefficient `c_G(0, 2)`, used for the lower-bound sanity
+  check and the comparison.
+- [Z] Math-#195 at `6e7746b`, read-only: the floor on `[1/64, 1/2]`, comparison only.
+
+Unlike [G], this packet does **not** consume [LP] (4.1), (4.2), (5.4), (5.5) or (7.5)–(7.7). Every estimate below is proved
+here with its constant. From [LP] it uses only the definitions, the pins, and (for the consequence on `1 - p_r` alone) the §8
+implication.
+
+## 0. Statement
+
+**Setting** ([LP] section 1, `d = 2`; as in [G] section 0):
+- The torus `X = (R / L Z)^2` and the kernel
+  `K_L(z) = sum_n exp(-|z + Ln|^2/2) / sum_n exp(-|Ln|^2/2)`.
+- An orthonormal frame `R` with axial unit vector `u`, and the pins `M = -(r/2) u`, `S = (r/2) u`, `f(M) = b`,
+  `f(S) = b - k r^3`, `grad f(M) = grad f(S) = 0`.
+- The Gaussian regression law `Q = Q_{r,b,k,R}`, the typed weight `W_r = |det H_M det H_S| 1{H_M < 0, det H_S < 0}`,
+  `Z_r = E_Q W_r`, and `dQ^W = (W_r / Z_r) dQ`.
+- In frame coordinates `(x, y)`: `D = [-2r, 2r]^2`, `M_j = max_{a + c = j} sup_D |partial_x^a partial_y^c f|`, and the good
+  event `G_r = {-f_yy(M) > (4/(3k)) r M_3^2, r M_4 <= 3k/10}` ([LP] section 7, [CAP] (1)).
+- Theorem A of [LP] is `0 <= 1 - p_r <= C r^3` ([LP] (1.1)). Its proof bounds the larger probability `Q^W(G_r^c)` in (7.8) and
+  uses `1 - p_r <= Q^W(G_r^c)` (§8).
+- A **cap-route constant** is any `C` with `Q^W(G_r^c) <= C r^3` for `0 < r <= r_*`. Every cap-route constant is a constant
+  for [LP] (1.1); the converse fails (see [G] v1.2, section 5, item 1).
+
+**Theorem E.** Let `B = [0, 1]` and `K = [1/2, 2]`. For every torus side `L >= 10`, every orthonormal frame `R`, every
+`b in B`, `k in K`, every `r_*` in the table, and every `0 < r <= r_*`,
+
+    Q^W(G_r^c) <= C(r_*) r^3        and        Z_r / r^2 >= z(r_*).
+
+The same holds for the reference kernel `exp(-|z|^2/2)` on `R^2`, since every enclosure below contains the case
+`M_1 = 0`, `Theta = 1` of section 2. The table lists, for each `r_*`:
+- `C(r_*)`, an exact rational, with its decimal;
+- upper bounds for `C / c_G(0, 2)` and `C r_*^3`, and the floor `z(r_*)`;
+- the cell (`r`-band; `b`; `k`) at which the maximum is attained.
+
+| `r_*` | `C(r_*)` | `C / c_G(0,2) <=` | `C r_*^3 <=` | `z(r_*)` | maximizing cell |
+|---|---|---|---|---|---|
+| `1/4096` | `26776575319/5000 = 5355315.0638` | `1.005814` | `0.0000779301` | `8.995379` | `[31/131072, 1/4096]`; `[0, 1/64]`; `[63/32, 2]` |
+| `1/2048` | `53623024479/10000 = 5362302.4479` | `1.007127` | `0.0006242542` | `8.990755` | `[31/65536, 1/2048]`; `[0, 1/64]`; `[63/32, 2]` |
+| `1/1024` | `2151134031/400 = 5377835.0775` | `1.010044` | `0.0050084993` | `8.981500` | `[31/32768, 1/1024]`; `[0, 1/64]`; `[63/32, 2]` |
+| `1/512` | `54156592391/10000 = 5415659.2391` | `1.017148` | `0.0403498057` | `8.962953` | `[31/16384, 1/512]`; `[0, 1/64]`; `[63/32, 2]` |
+| `1/256` | `93707999311/10000 = 9370799.9311` | `1.759987` | `0.5585432012` | `8.921798` | `[15/4096, 1/256]`; `[1/4, 1]`; `[1/2, 1]` |
+| `1/128` | `87390113247/250 = 349560452.9880` | `65.653041` | `166.6834130230` | `8.841791` | `[5/1024, 11/2048]`; `[1/4, 1]`; `[1/2, 1]` |
+| `1/64` | `14397396179923/10000 = 1439739617.9923` | `270.406115` | `5492.1707839673` | `8.676320` | `[15/1024, 1/64]`; `[1/4, 1]`; `[1/2, 1]` |
+
+`c_G(0, 2)` is replaced by its certified lower end `5324360.4426` ([G]). v1 had `C(1/256) = 34071677.7914`; the Hölder exponents of
+section 7 bring it below `16777216 = 256^3`. All other entries are equal to v1's or below them by less than `10^-7`
+relative.
+
+**Theorem E' (the SIDE24 point).** The same holds at the single point `(b, k) = (6/5, 1/6)`, for every `L >= 10`, every frame
+and the reference kernel. That point lies outside the band (`k < 1/2`). It is the parameter point of the fixed-x-axis
+SIDE24 law of [CAP] section 1 (torus side 24, `0 < r <= 1/20`).
+
+| `r_*` | `C'(r_*)` | `C' / c_G(6/5, 1/6) <=` | `C' r_*^3 <=` | `z'(r_*)` |
+|---|---|---|---|---|
+| `1/4096` | `19661409/500 = 39322.8180` | `1.100348` | `5.8 x 10^-7` | `3.226690` |
+| `1/2048` | `215461191/5000 = 43092.2382` | `1.205825` | `5.1 x 10^-6` | `3.222401` |
+| `1/1024` | `1604824613/1250 = 1283859.6904` | `35.93` | `0.0012` | `3.213821` |
+| `1/512` | `1173169654911/10000 = 117316965.4911` | `3283` | `0.8741` | `3.196649` |
+| `1/256`, `1/128` | `815443379659/5000 = 163088675.9318` | `4564` | `> 1` | `3.162091`, `3.092963` |
+| `1/64` | `5553365927589/5000 = 1110673185.5178` | `31080` | `> 1` | `2.953979` |
+
+`c_G(6/5, 1/6)` is replaced by its certified lower end `35736.7352` ([G]). The contact value of the floor at this point is
+`z_0 = 36 k^2 E[w^2 1{w < 0}] = 3.2310`.
+
+**Corollary (explicit Theorem A on the band).**
+- **Explicit pairs.** Since `0 <= 1 - p_r <= Q^W(G_r^c)` ([LP] §8), [LP] (1.1) holds on `B x K`, uniformly over frames and
+  `L >= 10`, with every pair `(C(r_*), r_*)` of the table. The pairs that are informative at `r = r_*` (`C r_*^3 < 1`) are those
+  with `r_* <= 1/256`. Examples: `(C, r_*) = (5415659.2391, 1/512)`, where the bound at `r = r_*` is below `0.0404`, and
+  `(9370799.9311, 1/256)`, where it is below `0.559`.
+- **What was missing.** [LP] §16 lists "a numerical `r_*` or `C` on a prescribed band" among the items it does not supply.
+  The only explicit constant on record before this packet is [CAP] (2)–(3): `1 - p_elder(r) <= min(1, 2.4 x 10^23 r^3)` for
+  the fixed-axis SIDE24 law, nontrivial only below `r = 1.6 x 10^-8`.
+- **The SIDE24 law (Theorem E').** For that law, `1 - p_elder(r) <= 39322.8180 r^3` for `r <= 1/4096` and
+  `<= 43092.2382 r^3` for `r <= 1/2048`, within `10.1%` and `21%` of the sharp cap coefficient `c_G(6/5, 1/6) = 35736.8`. The
+  bound stays below `1` up to `r = 1/512` (`C' r_*^3 <= 0.874`).
+- **Near-optimal for the cap route.** The constants are within `0.6%`–`1.8%` of the smallest possible cap-route constant on
+  the band for `r_* <= 1/512`. Every cap-route constant valid for the reference kernel at `(b, k) = (0, 2)` is at least
+  `c_G(0, 2) > 5324360.44` ([G] Theorem G, v1.2 section 5 item 1), and this certificate's scope includes the reference kernel.
+  This statement concerns the cap route only. The true elder-failure coefficient at `(0, 2)` is
+  `alpha_1 + alpha_2 = 2.18` ([G] section 5 from [S]/[NUM]), so the cap criterion overstates the failure rate by about
+  `2.4 x 10^6`; this certificate cannot remove that slack.
+- **Normalizer floor.** `z(r_*)` is an explicit `z_*` for [LP] (5.5) on `(0, r_*]`. The contact value at the band's
+  weakest corner, `(b, k) = (0, 1/2)`, is `36 k^2 E[w^2 1{w < 0}] = 9`, and `z(1/4096) = 8.9954` lies within `0.06%` of it.
+  [Z] gives `Z_r / r^2 >= 3.5044` on `[1/64, 1/2]` for `L >= 12`. Together: `Z_r / r^2 >= 3.5044` on all of `(0, 1/2]` for
+  `L >= 12`, and `Z_r / r^2 >= 8.6763` on `(0, 1/64]` already for `L >= 10`.
+
+## 1. Coordinates and exact identities
+
+In frame coordinates `M = (-h, 0)`, `S = (h, 0)` with `h = r/2`, and the midpoint is `0`. The six pins are the
+preconditioned functionals of [LP] (3.1): `U0 = (f(M) + f(S))/2`, `U1 = (f(S) - f(M))/r`, `U2 = (f_x(S) - f_x(M))/r`,
+`U3 = (6/r^2)[f_x(M) + f_x(S) - 2(f(S) - f(M))/r]`, `V0 = (f_y(M) + f_y(S))/2`, `V1 = (f_y(S) - f_y(M))/r`. Under `Q` their
+values are exactly `(b - k r^3/2, -k r^2, 0, 12k, 0, 0)` ([LP] (3.3)). The twelve targets are:
+
+    w   = f_yy(M),
+    v   = f_xy(M)/r + (f_y(M) - f_y(S))/r^2,
+    om  = (f_yy(S) - f_yy(M))/r,
+    Y   = f_yyy(0),
+    T   = f_xx(M)/r^2 + (4 f_x(M) + 2 f_x(S))/r^3 + 6 (f(M) - f(S))/r^4,
+    tau = (f_xx(S) - f_xx(M))/r^3 - 6 (f_x(M) + f_x(S))/r^4 - 12 (f(M) - f(S))/r^5,
+    nu  = (f_xy(M) + f_xy(S))/r^2 + 2 (f_y(M) - f_y(S))/r^3,
+    X_alpha = partial^alpha f(0),   |alpha| = 4.
+
+Each functional is a sum of terms `c r^p partial_x^a partial_y^e f(x_0 r, 0)` with a common **weight** `a - p`. Since the pins
+hold exactly under `Q`, almost surely:
+
+    f_xx(M) = r^2 T - 6kr,   f_xx(S) = r^2 (T + r tau) + 6kr,   f_xy(M) = r v,   f_xy(S) = r (r nu - v),   f_yy(S) = w + r om.   (I1)
+
+Write `lambda = -w`, `c1 = 6k - rT`, `c2 = 6k + rT + r^2 tau` and `e = (r nu - v)^2`. Then `det H_M = r P1` and
+`det H_S = -r P2`, with
+
+    P1 = c1 lambda - r v^2,     P2 = c2 (lambda - r om) + r e,                                    (I2)
+
+so `H_M < 0, det H_S < 0` holds exactly when `c1 > 0, P1 > 0, P2 > 0`, and then `W_r = r^2 P1 P2`. (`f_xx(M) = -r c1`.)
+
+## 2. The Gaussian law on an `r`-band (Lemma 1)
+
+**(a) Exact series.** Let `F`, `F'` be functionals of weights `w_F`, `w_F'`, with `w = w_F + w_F'`. For the reference kernel
+`G(z) = exp(-|z|^2/2)`,
+
+    Cov(F, F') = sum_gamma h(g_x) h(g_y) pi_gamma r^(g_x - w).
+
+- `h(n) = (d/dt)^n exp(-t^2/2)` at `t = 0`, which is `(-1)^(n/2) (n-1)!!` for even `n` and `0` for odd `n`.
+- `pi_gamma` is the exact rational `sum c c' (-1)^(a'+e') (x_0 - x_0')^m / m!`. The sum runs over pairs of terms with
+  `g_y = e + e'` and `m = g_x - a - a' >= 0`.
+- **Regularity:** `pi_gamma = 0` for every `g_x < w`. This is checked exactly for all 252 ordered pairs of the 18 functionals,
+  so no negative power of `r` occurs.
+
+**(b) Truncation at `g_x <= NX = 60`.** For one pair with `A = a + a'`, the `g_x`-th term is at most
+`|c c'| |h(g_y)| (g_x^A / g_x!!) r^(g_x - w)`, because `|h(n)|/(n - A)! <= n^A / n!!`. For `r <= 1/64` the terms with
+`g_x > 60` therefore sum to at most `2 |c c'| |h(g_y)| 61^A r^(61 - w)`.
+
+**(c) Torus remainder, uniform in `L >= 10`, the frame and `r`.**
+- **Decomposition.** `K_L = (G + rho) / Theta` with `rho(z) = sum_{n != 0} G(z + p_n)`, where `p_n` runs over the rotated
+  lattice `L R^T Z^2`, and `Theta = 1 + sum_{n != 0} exp(-|p_n|^2/2)`, which lies in `[1, 1 + theta_1(L)]`.
+- **Bound on `rho`.** `rho` is entire. On the complex polydisc `{|z_1|, |z_2| <= 1}`:
+  - `|exp(-(z + p)^2/2)| <= e exp(-|Re z + p|^2/2)`;
+  - at most `8j` lattice points have sup-norm index `j`, each at distance `>= jL`;
+  - `|Re z| <= 3/2`.
+
+  Hence `|rho| <= M_1(L) := e sum_{j >= 1} 8j exp(-(jL - 3/2)^2/2)`, independently of the frame.
+- **Cauchy estimates** give `|partial^gamma rho(0)| <= g_x! g_y! M_1`.
+- **Effect on the covariance.** The regularity in (a) is an identity in the Taylor coefficients of any kernel, so the
+  `rho`-part of the covariance is `sum_{g_x >= w} partial^gamma rho(0) pi_gamma r^(g_x - w)`. Using
+  `g_x!/(g_x - A)! <= w^A e^(g_x - w)` (valid because `A <= w`: every power `p` is `<= 0`), it is bounded by
+  `sum |c c'| g_y! max(w, 1)^A M_1 / (1 - e r)`.
+- **Values.** `M_1(10) < 4.46 x 10^-15` and `theta_1(10) < 1.55 x 10^-21`. Both decrease in `L`, so the enclosures computed
+  at `L = 10` hold for every `L >= 10`, for every frame, and for the reference kernel (`rho = 0`, `Theta = 1`).
+- **Why no `r^-10`.** The remainder is applied to the regular series, not to the raw pin functionals. This removes the
+  `r^-10` amplification that forces `L >= 12` in [Z].
+
+**(d) Regression.** Interval Gaussian elimination on the six pins gives, uniformly over `r in [r_0, r_1]` (each power `r^p`
+is enclosed by `[r_0^p, r_1^p]`):
+- the `Q`-means `mu_F = b alpha_F + k beta_F` of the twelve targets;
+- their `Q`-covariance `C`, scaled by `Theta^-1 in [1/(1 + theta_1), 1]`;
+- the pin energy `E = v_r^T Sigma_r^-1 v_r = e_bb b^2 + 2 e_bk b k + e_kk k^2`, scaled by `Theta in [1, 1 + theta_1]`.
+
+At contact (reference kernel) these reduce to `w ~ N(-b, 2)`, `v ~ N(0, 1/2)`, `om ~ N(0, 2)`, `Y ~ N(0, 6)` (as in [G]
+section 2.2) and `E -> 3b^2/2 + 24 k^2`.
+
+The **regression on `w`**: `beta_F = C(F, w)/C(w, w)` and `F_g = F - beta_F (w - mu_w)`. Then `F_g` is independent of `w`
+under `Q`, with mean `mu_F` and covariance `C - C(., w) C(w, .)/C(w, w)`. The `g`-block of `(v, om, Y)` is factored by an
+interval Cholesky decomposition with lower-triangular entries `l_ij`.
+
+## 3. Third and fourth derivatives on `D` (Lemmas 2 and 3)
+
+Write `A_alpha = sup_D |partial^alpha f|` for `|alpha| = 4`.
+
+**Lemma 2 (averaging identities).** For every `C^4` function satisfying the pins, and every `z in D`:
+
+    |f_xxx(z)| <= 12k   + r Delta_1,    Delta_1 = (5/2) A_40 + 2 A_31,
+    |f_xxy(z)| <= 2|v|  + r Delta_2,    Delta_2 = (5/2) A_31 + 2 A_22,
+    |f_xyy(z)| <= |om|  + r Delta_3,    Delta_3 = (5/2) A_22 + 2 A_13,
+    |f_yyy(z)| <= |Y|   + r Delta_4,    Delta_4 = 2 A_13 + 2 A_04,
+
+so `M_3 <= max_i (B_i + r Delta_i)` with `B = (12k, 2|v|, |om|, |Y|)`.
+
+*Proof.* Let `g(t) = f(t, 0)` and `q(t) = f_y(t, 0)`.
+- **`f_xxx`.** `U3` annihilates quadratics and equals `g'''` on cubics. Its Peano kernel is
+  `U3 = integral_{-h}^{h} K(t) g'''(t) dt` with `K(t) = 3(h^2 - t^2)/(4h^3) >= 0` and `integral K = 1`; under `Q`,
+  `U3 = 12k`.
+- **`f_xxy`.** The pins give `q(+-h) = 0`, so Taylor's formula at `-h` yields
+  `-2v = integral_{-h}^{h} (2(h - t)/r^2) q''(t) dt`, an average of `f_xxy(t, 0)` (weights `>= 0` with total `1`).
+- **`f_xyy`.** `om = (1/r) integral_{-h}^{h} f_xyy(t, 0) dt`.
+- **`f_yyy`.** `Y = f_yyy(0)`.
+
+In each case `|partial^beta f(z) - partial^beta f(t, 0)| <= |z_x - t| A_{beta + e_x} + |z_y| A_{beta + e_y}` along the segment,
+which stays in the convex set `D`. Use `|z_x - t| <= 5r/2` and `|z_y| <= 2r` (and `|z_x| <= 2r` for `Y`).
+
+**Lemma 3 (fourth derivatives).**
+- **Entire field.** Under `Q` the field is almost surely entire. The torus field is
+  `sum_m sqrt(lambda_m)(xi_m cos + eta_m sin)(2 pi m.z/L)` with `lambda_m` decaying like `exp(-2 pi^2 |m|^2/L^2)`, which
+  converges on compacts of `C^2`, and the regression adds finitely many entire functions.
+- **Taylor bound.** Hence `A_alpha <= |X_alpha| + T_alpha`, where
+  `T_alpha = sum_{beta != 0} |partial^(alpha+beta) f(0)| (2r)^|beta| / beta!`.
+- **Gaussian bounds.** For every `gamma`, the `Q`-law of `partial^gamma f(0)` is Gaussian with standard deviation at most
+  `s_gamma` and `|mean| <= s_gamma sqrt(E)` (Cauchy–Schwarz in the `Sigma_r^-1` inner product), where
+  `s_gamma^2 = (2g_x - 1)!!(2g_y - 1)!! + (2g_x)!(2g_y)! M_1` bounds the unconditioned variance.
+- **Norm bound.** Therefore `||T_alpha||_p <= S_alpha (sqrt(E) + ||N(0,1)||_p)`, with
+
+      S_alpha(r_1) = sum_{1 <= |beta| <= 24} s_{alpha+beta} (2r_1)^|beta| / beta!  +  2^(|alpha|+1) 26 . 29^8 (4r_1)^25 / (1 - 8r_1).
+
+  The tail term uses `s_gamma <= 2^|gamma| gamma! sqrt(1 + M_1)` and `(alpha+beta)!/beta! <= (n + 4)^4` at `|beta| = n`, and
+  requires `4 r_1 <= 1/16`.
+- **Dependence on `w`.** The regression coefficient of `partial^gamma f(0)` on `w` is at most `s_gamma / sigma_w`. So
+  `A_alpha <= A_alpha^g + B_alpha |w - mu_w|`, with `B_alpha = |beta_{X_alpha}| + S_alpha / sigma_w`, where `A_alpha^g` is the
+  same bound for the `w`-independent field. The `g`-part of each jet has the same mean and a smaller variance.
+
+## 4. Regression on `w` and the failure set (Lemma 4)
+
+**The `g`-measurable bound on `M_3`.** By Lemmas 2–3 and the regression,
+
+    M_3 <= max_i (y_i + bar-beta_i |w - mu_w|),
+
+with the `g`-measurable parts
+
+    y_1 = 12k + r Delta_1^g,   y_2 = 2|v_g| + r Delta_2^g,   y_3 = |om_g| + r Delta_3^g,   y_4 = |Y_g| + r Delta_4^g,
+
+and the coefficients
+
+    bar-beta_1 = r((5/2) B_40 + 2 B_31),
+    bar-beta_2 = 2|beta_v| + r((5/2) B_31 + 2 B_22),
+    bar-beta_3 = |beta_om| + r((5/2) B_22 + 2 B_13),
+    bar-beta_4 = |beta_Y| + r(2 B_13 + 2 B_04).
+
+On `lambda >= 0`, `|w - mu_w| <= lambda + |mu_w|`. Put `y_i' = y_i + bar-beta_i |mu_w|`, `a = 4/(3k)` and
+`x_i = a r bar-beta_i y_i'`.
+
+**Lemma 4.** If `lambda <= a r M_3^2`, then for some `i`, `lambda <= a r (y_i' + bar-beta_i lambda)^2`. For `x_i <= 1/4` the two
+roots of this quadratic satisfy
+
+    lambda_-^(i) <= a r y_i'^2 / (1 - 3 x_i),        lambda_+^(i) >= (1 - 3x_i) / (a r bar-beta_i^2) >= 1 / (4 a r bar-beta_i^2).
+
+*Proof.* `lambda_- = 2 a r y'^2 / (1 - 2x + sqrt(1 - 4x))` and `lambda_+ = (1 - 2x + sqrt(1 - 4x)) / (2 a r bar-beta^2)`.
+Then use `sqrt(1 - 4x) >= 1 - 4x`.
+
+**The cover.** Put `X_0 = 1/64` and `kappa = 3/(1 - 3X_0)`, so that `1/(1 - 3x) <= 1 + kappa x` for `x <= X_0`. On the typed
+support,
+
+    G_r^c  subset  N  u  F  u  X  u  A,
+    N = {lambda <= r U,  x_i <= X_0 for all i},    U = max_i a y_i'^2 (1 + kappa x_i),
+    F = {lambda >= lambda_far},  lambda_far = 1/(4 a r bar-beta^2),  bar-beta = max_i bar-beta_i,
+    X = {x_i > X_0 for some i},    A = {r M_4 > 3k/10}.
+
+`U` is `g`-measurable, so on `N` the failure is a `lambda`-interval `[0, rU]` that depends only on the `g`-parts.
+
+## 5. The main term
+
+On `N`, every target `F` is bounded by its `g`-measurable majorant `bar-F = |F_g| + |beta_F| (|mu_w| + rU)`.
+
+**Density of `lambda` (v1.1).** Two bounds hold for the density `p` of `lambda` on the failure window `[0, rU]`:
+- `p <= bar-p = phi(max(0, mu_w)/sigma_w)/sigma_w`, its supremum on `[0, infinity)`;
+- `p(lambda) <= p_0 + lambda L_p`, where:
+  - `p_0 = phi(m_0/sigma_hi)/sigma_lo` bounds `p(0)` over the box, with `m_0` the smallest `|mu_w|` in the box (`0` if the
+    enclosure of `mu_w` contains `0`);
+  - `L_p = 0.24198/sigma_lo^2 >= phi(1)/sigma_w^2 = sup |p'|`.
+
+The second bound matters when `-mu_w = b > 0` is far from `0`: at `b = 6/5` the first gives `0.2821` while `p(0) = 0.1967`. The
+term `lambda L_p` costs one more power of `U`. From (I2),
+
+    W_r <= r^2 |c1| lambda (|c2| (lambda + r |om|) + r e)
+
+on the typed support (positive parts; this covers both signs of `c1`, `c2`). Integrating over `lambda = r u in [0, rU]`
+gives
+
+    E_Q[W_r 1_N] <= r^5 bar-p E_g[ |bar-c1 bar-c2| (U^3/3 + bar-om U^2/2) + |bar-c1| bar-e U^2/2 ],
+
+and, with the linear density bound, the same integral with one more factor `lambda <= rU`:
+
+    E_Q[W_r 1_N] <= r^5 p_0 E_g[ ... as above ... ] + r^6 L_p E_g[ |bar-c1 bar-c2| (U^4/4 + bar-om U^3/3) + |bar-c1| bar-e U^3/3 ].
+
+The certificate uses the smaller of the two.
+
+**Normalization by `K = 12k`.** Write `U = a K^2 tilde-U` with `a K^2 = 192k`, and
+`|c1 c2|/(36k^2) <= 1 + eps`, where `eps <= (2r|T| + r^2|tau|)/(6k) + r|T|(r|T| + r^2|tau|)/(36k^2)`. Then
+
+    E_Q[W_r 1_N] / (36 k^2 r^5)  <=  min( bar-p (A + B + C),  p_0 (A + B + C) + r L_p (A' + B' + C') ),
+    A  = (aK^2)^3/3 . E[tilde-U^3 (1 + eps)],   B  = (aK^2)^2/2 . E[bar-om tilde-U^2 (1 + eps)],
+    C  = (aK^2)^2/(12k) . E[(1 + r|bar-T|/(6k)) bar-e tilde-U^2],
+    A' = (aK^2)^4/4 . E[tilde-U^4 (1 + eps)],   B' = (aK^2)^3/3 . E[bar-om tilde-U^3 (1 + eps)],
+    C' = (aK^2)^3/(18k) . E[(1 + r|bar-T|/(6k)) bar-e tilde-U^3].
+
+**Moments of `tilde-U`.** With `tilde-y_i = y_i'/K`, `F_k = 1 + kappa X_0`, `eta_1 = 16 kappa r bar-beta_1` and
+`Z_1 = tilde-y_1^2 (1 + eta_1 tilde-y_1)`:
+- `tilde-U <= max(Z_1, F_k tilde-y_i^2)` for `i = 2, 3, 4`.
+- Since `tilde-y_1 >= 1`,
+  `max(Z_1, F_k tilde-y_i^2)^p <= Z_1^p + F_k^p sum_i (tilde-y_i^(2p) - c^(2p))_+` with `c = 1 - kappa X_0/2 <= F_k^(-1/2)`.
+- Each `tilde-y_i <= u_i + delta_i` (`i >= 2`), where `u_i` is the half-normal of the `i`-th Cholesky variable and
+  `delta_i` collects the means, the off-diagonal Cholesky terms, `bar-beta_i |mu_w|` and `r Delta_i^g`.
+- With a parameter `theta > 0`:
+  `(u + delta)^n - c^n <= (1 + theta)^(n-1) (u^n - (c(1 - theta))^n)_+ + (1 + 1/theta)^(n-1) delta^n`.
+- `E[u^q (u^n - c'^n)_+]` is in closed form through the incomplete Gaussian moments
+  `M_j(x) = integral_x^infinity t^j phi(t) dt = x^(j-1) phi(x) + (j - 1) M_(j-2)(x)`.
+- `tilde-y_1 = 1 + bar-beta_1 |mu_w|/K + (r/K) Delta_1^g` is expanded binomially.
+- The `delta`-terms and every cross term are bounded by Hölder's inequality, with Gaussian `L^p` norms
+  (`||N||_p <= ((p - 1)!!)^(1/p)` for `p` a power of two, `||N||_1 <= 4/5`) and the bounds of Lemma 3.
+
+**Parameter choices (v1.1).** Every pair in the proved domain `0 < X_0 < 2/9`, `0 < theta < 1` gives a valid bound; both
+pairs used below lie in that domain. Each box uses the smaller of the bounds for
+`(X_0, theta) = (1/64, 1/32)` and `(1/1024, 1/256)`. The second pair matters at small `k`, where the free third derivatives
+`2|v|, |om|, |Y|` often exceed `12k`, so the factor `F_k^3 = (1 + kappa X_0)^3` and the split loss `(1 + theta)^5` multiply
+the leading term. In v1 these were `1.155` and `1.166`; with the second pair they are `1.009` and `1.020`.
+
+**The factors `bar-om` and `bar-e`.**
+- In `bar-om`, the part that shares its Gaussian variable with `u_3` is integrated exactly; the rest goes through Hölder.
+- `bar-e = (r |bar-nu| + |bar-v|)^2`. Write `|bar-v| <= v_0 + d_v`, where `v_0 = l_11 |z_1|` is the Cholesky part, the
+  variable of `u_2`, and `d_v = |mu_v| + |beta_v| (|mu_w| + rU)`. With `theta_E = 1/16`:
+
+      bar-e <= (1 + theta_E)^2 v_0^2 + (1 + theta_E)(1 + 1/theta_E) d_v^2 + (1 + 1/theta_E) r^2 bar-nu^2.
+
+  The `v_0^2` term is integrated exactly against `u_2`; the rest goes through Hölder.
+
+## 6. The normalizer floor (Lemma 5)
+
+Let `G_0 = {r|T| <= 3k/2, r^2|tau| <= 3k/2}`. On `G_0`:
+
+    c1 >= 9k/2,   3k <= c2 <= 9k,   c1 c2 >= 36k^2 (1 - (2r|T| + r^2|tau|)/(6k)),
+
+and, pointwise,
+
+    W_r / r^2  >=  1_{G_0} { 36k^2 [ lambda_+^2 - r om_+ lambda_+ - ((2r|T| + r^2|tau|)/(6k)) lambda_+^2 ] - r c2 v^2 lambda_+ }.
+
+*Proof.* By (I2), `W_r/r^2 >= 1_{G_0} c2 (c1 lambda - r v^2)_+ (lambda - r om_+)_+`. If `lambda <= r om_+`, the right side
+above is at most `0`. Otherwise use `(x - y)_+ z >= xz - yz` and `c1 c2 lambda (lambda - r om_+) >= 36k^2 (1 - ...) lambda (lambda - r om_+)`.
+
+Taking expectations, with Cauchy–Schwarz on `G_0^c` and Hölder on the `T`, `tau` terms:
+
+    Z_r / (36 k^2 r^2)  >=  E[lambda_+^2] - ||lambda||_4^2 P(G_0^c)^(1/2) - (2r||T||_2 + r^2||tau||_2) ||lambda||_4^2/(6k)
+                            - r E[|om| lambda_+] - (r/(3k)) E[v^2 lambda_+]
+
+(the certificate uses `1/(3k)` where `c2 <= 9k` would allow `1/(4k)`).
+- `E[lambda_+^2] = E[(sigma z + m)_+^2]` is increasing in `m` and `sigma`, so it is evaluated at the lower endpoints.
+- `E[|om| lambda_+]` and `E[v^2 lambda_+]` are split by the regression; the `g`-parts are independent of `lambda`.
+- `P(G_0^c)` is bounded by Gaussian tails of `T` and `tau`.
+
+The bound of the main term is divided by this floor.
+
+## 7. Rare branches
+
+**Weight bound.** For `q in {2, 4, 8, 16}`, by Hölder,
+
+    ||W_r 1_typed||_q <= r^2 g_1(q) g_2(q),
+    g_1(q) = (6k + r||T||_{4q}) ||lambda||_{4q},
+    g_2(q) = (6k + r||T||_{4q} + r^2||tau||_{4q})(||lambda||_{4q} + r||om||_{4q}) + r (r||nu||_{4q} + ||v||_{4q})^2.
+
+Then `E_Q[W_r 1_E] <= ||W_r 1_typed||_q P(E)^(1 - 1/q)` for `E = F, X, A`, and each box uses the smallest of the four bounds.
+v1 used `q = 2` only. A larger `q` gains when the probabilities are small and loses when they are not, because `||W||_q`
+grows with `q`.
+- `P(A) <= sum_alpha [P(|X_alpha| > (7/8) t) + P(T_alpha > t/8)]` with `t = 3k/(10r)`: a Gaussian tail plus Markov's
+  inequality at `p = 32` with Lemma 3.
+- `P(F) <= bar-Phi((lambda_far - |mu_w|)/sigma_w)`.
+- `P(X) <= sum_i P(y_i' > X_0/(a r bar-beta_i))`: Markov at `p = 32` on `r Delta_1` for `i = 1`; a Gaussian tail of `u_i`
+  plus Markov on `delta_i` for `i >= 2`.
+
+`bar-Phi(x)` is computed from its series for `x <= 12`, and beyond that from Mills' bound `phi(x)/x` with
+`e^(-y) <= 40!/y^40`.
+
+## 8. Bands, boxes, and the last band
+
+**Bands.** `(0, 1/64]` is covered by `[0, 2^-29]` and 152 bands. Each octave `[2^-(j+1), 2^-j]` is split into equal
+sub-bands:
+- 8 sub-bands for `j = 6, 7, 8`;
+- 16 for `j = 9, ..., 12`;
+- 4 for `j = 13, ..., 28`.
+
+**Per band.** On each band `[r_0, r_1]`:
+- the law is enclosed uniformly (Lemma 1);
+- every `r`-monotone quantity is taken at its worst endpoint: thresholds and `S_alpha` at `r_1`, the tail divided by `r_0^3`.
+
+**Boxes.** 28 boxes per band:
+
+    b in {[0, 1/64], [1/64, 1/16], [1/16, 1/4], [1/4, 1]},
+    k in {[1/2, 1], [1, 3/2], [3/2, 7/4], [7/4, 15/8], [15/8, 31/16], [31/16, 63/32], [63/32, 2]}.
+
+That makes 4284 box bounds. `C(r_*)` is their maximum over the bands with `r_1 <= r_*`, and `z(r_*)` is the minimum over
+the same boxes of `36 k_lo^2` times the floor of Lemma 5. The SIDE24 point is evaluated on every band as a degenerate box
+`b = 6/5`, `k = 1/6`, which gives `C'(r_*)` and `z'(r_*)` in the same way.
+
+**The last band `[0, R]`, `R = 2^-29`.** The main term does not depend on `r`, and the tail is divided by `(R/2)^3`. This is
+justified as follows:
+- every Gaussian-tail argument is `>= 3` at `R` (checked), and it at least doubles when `r` halves;
+- no tail probability reaches `1` at `R` (checked);
+- every Markov piece scales like `r^32` or faster.
+
+So each halving of `r` divides each probability by more than `64`. After the power `1 - 1/q` with `q >= 2` it still divides
+each piece by more than `8`, for the `q` chosen at `R`. The tail term over `r^3` therefore attains its supremum over `(0, R]`
+on `[R/2, R]`.
+
+## 9. Arithmetic
+
+- **Representation:** exact rationals.
+- **Transcendentals:** `exp`, `Phi`, `pi` and `sqrt` are enclosed with outward rounding to `2^-160`, by series with explicit
+  remainders (`exp` as a fixed-point series; `pi` from Machin's formula).
+- **Upper bounds:** rounded upward to 200 significant bits (dyadic).
+- **Square roots of upper bounds:** taken without an absolute rounding floor.
+- **Matrices:** interval Gaussian elimination with midpoint pivoting, and an interval Cholesky decomposition whose pivots
+  must be positive.
+
+Each failed precondition exits through `require`. Standard library only.
+
+## 10. Controls (floating point; not part of the certificate)
+
+`theorem_a.py --mc R B K N SEED` evaluates `Q^W(G_r^c)/r^3` and `Z_r/r^2` end to end from the definitions:
+- the degree-8 jet at `0` is sampled under `Q` for the reference kernel;
+- the field near `D` is rebuilt from it;
+- `W_r`, `M_3` and `M_4` are evaluated directly (maxima over a `9 x 9` grid of `D`).
+
+Means over seeds, with the standard error of the mean:
+
+| `(r, b, k)` | seeds x samples | `Q^W(G_r^c)/r^3` | `Z_r/r^2` | certified (cell containing the point) |
+|---|---|---|---|---|
+| `(1/512, 0, 2)` | `4 x 20000` | `4.840e6 +- 0.035e6` | `144.2 +- 0.8` | `C <= 5415659.24`; `Z/r^2 >= 139.21` |
+| `(1/1024, 0, 2)` | `6 x 20000` | `5.192e6 +- 0.053e6` | `144.7 +- 0.6` | `C <= 5377835.08`; `Z/r^2 >= 139.37` |
+| `(1/512, 0, 1/2)` | `6 x 20000` | `8.74e4 +- 0.19e4` | `9.04 +- 0.04` | cell bound `<= 724033.37` (the cell reaches `k = 1`); `Z/r^2 >= 8.9629` |
+
+The limits as `r -> 0` are `c_G(0, 2) = 5324360.44`, `c_G(0, 1/2) = 85120.5` and `z_0 = 144`, `9`.
+
+Two biases point the same way:
+- the grid maxima slightly underestimate `M_3` and `M_4`;
+- at finite `r` the density of `lambda` decreases across the failure window (by about `8%` at `r = 1/512`, `(0, 2)`).
+
+Both explain why the finite-`r` estimates at `k = 2` sit below `c_G`, and why the estimate approaches `c_G` as `r` halves.
+The certified `C` must exceed `c_G` (it holds as `r -> 0`), and it does by the margins in the table. The controls only check
+that no gross error separates the certificate from the definitions.
+
+## 11. What this does not do
+
+- **Proofs and reviews of sources.** Not a proof or a review of [LP]'s proof of Theorem A or of [CAP]. The consequence for
+  `1 - p_r` uses [LP] §8, which rests on the deterministic cap theorem of [CAP], as a statement.
+- **Scope.** Planar only (`d = 2`); `d >= 3` would need the eigenvalue boundary layer of [LP] section 7. The band
+  `b in [0, 1]`, `k in [1/2, 2]`, and the single SIDE24 point `(6/5, 1/6)`. No other `k < 1/2`: a smaller `k` lowers every
+  threshold `3k/(10r)` and `12k`, and at the SIDE24 point the tail branches already dominate for `r > 1/2048`.
+- **Lower bounds.** No lower bound for the constant of [LP] (1.1) itself (see [G] v1.2).
+- **Large `r_*`.** For `r_* >= 1/128` (band) and `r_* >= 1/256` (SIDE24 point) the table values are certified but exceed `1`
+  at `r = r_*`. There the tail branches `A` and `X` dominate, and the method stops being informative.
+- **Other C8 constants.** `c_{B,K}` and `c_{d,L}` are not addressed (Math-#197 and the SIDE24 coefficient packets).
+- **Controls.** The Monte Carlo controls are floating point and are not part of the certificate. There is none at the SIDE24
+  point: there the failure probability (`c_G r^3 < 5 x 10^-6` for `r <= 1/2048`) is beyond reach of sampling.
+- **C8.** C8 stays OPEN; this packet changes no catalog entry. Nonauthor reads are requested on the PR.
+
+## 12. Verification and provenance
+
+- `python3 -B -S theorem_a.py --check --procs N` (also with `-B -O -S`) replays exactly, and compares with `RESULTS.json`:
+  - the parameters, the band list, and 11 sampled bands (the last band and the band ending at each `r_*` among them);
+  - both `C` tables as the maxima of the stored records;
+  - `C > c_G(0, 2)` and `C' > c_G(6/5, 1/6)` for every `r_*`;
+  - the canonical layout of `RESULTS.json` (one line per band record).
+
+  It takes about one minute on four cores.
+- `--check-full` replays all 153 bands (about ten minutes on four cores).
+- Mutants `no-delta` (drop the fourth-derivative terms of Lemma 2), `cap-half` (halve the cap constant `4/(3k)`),
+  `no-zfloor-correction` (drop the corrections of Lemma 5) and `no-om-term` (drop the `om` terms `B`, `B'`) exit 1.
+- Without flags, the script regenerates `RESULTS.json` (about ten minutes on four cores).
+- Workflow `.github/workflows/c8-theorem-a-constants-planar.yml` checks:
+  - the manifest and the main-resident pins ([LP], [CAP]);
+  - `--check-full` and `--check` in the two interpreter modes;
+  - the four mutants;
+  - a clean tree.
+
+**Revisions.** v1 (`f3d41b5`): Theorem E on the band.
+
+v1.1 adds:
+- Theorem E' at the fixed-axis SIDE24 point;
+- the linear density bound for `lambda` (section 5);
+- the per-box choice between two `(X_0, theta)` pairs (section 5);
+- the Hölder exponents `q in {2, 4, 8, 16}` for the rare branches (section 7);
+- `RESULTS.json` regenerated.
+
+The band constants change only in the last digits, except `C(1/256)`, which drops from `3.41 x 10^7` to `9.37 x 10^6`.
+
+v1.2 is documentation only: two docstrings in `theorem_a.py`, byte-identical copies of which are carried by Math-#215 and
+Math-#227. The AST with docstrings removed is unchanged, and so are `RESULTS.json` and every constant.
+- `sqrt_up` promised a relative error below `2^-200`; at `x = 49/100` the excess is `(8/7) 2^-200` (C59-DOC-01). The
+  stated bound is now `2^-198`: the root step adds less than `2^-209` and the rounding to 200 bits less than `2^-199`.
+- `box_bound` said "every choice gives a valid bound". It now states the proved domain: `0 < x0 < 2/9` and
+  `0 < theta < 1` (C61). Both `COMBOS` pairs, `(1/64, 1/32)` and `(1/1024, 1/256)`, lie inside it.
+
+v1.2.1 is one wording fix in section 0, from the Slice C review 5397446329 (C206-C-01). The SIDE24 constant
+`39322.8180` exceeds the certified lower end `35736.7352` of `c_G(6/5, 1/6)` by the factor `1.100348` (`RESULTS.json`),
+so "within `10%`" now reads "within `10.1%`". The `21%` statement (factor `1.205825`) was already correct. No code,
+`RESULTS.json` or constant changed.
+
+v1.2.2 is one wording fix in section 5, from the Slice A review 5397500398 (C206-A-01). "Every pair `(X_0, theta)`
+gives a valid bound" now names the proved domain `0 < X_0 < 2/9`, `0 < theta < 1`, matching the `box_bound` docstring
+since v1.2; both `COMBOS` pairs, `(1/64, 1/32)` and `(1/1024, 1/256)`, lie inside it. No code, `RESULTS.json` or constant
+changed.
+
+Author lane Anthropic / Claude, 30 September – 2 October 2026. Scientific effect NONE. The author will not merge.
diff --git a/frontiers/c8_theorem_a_constants_planar_20260930/README.md b/frontiers/c8_theorem_a_constants_planar_20260930/README.md
new file mode 100644
index 000000000..bb03e4d15
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_planar_20260930/README.md
@@ -0,0 +1,57 @@
+# Explicit cap-route constants for Theorem A in the plane (C8: certified `C`, `r_*`, `z_*` on a declared band)
+
+**Object:** `CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** author-side theorem
+with explicit constants, certified by exact rational interval arithmetic (standard library only). **Scientific effect:**
+NONE: no register, catalog, GRAPH or STATUS change, and catalog entry C8 stays OPEN.
+
+**Setting.** The planar six-pin law of [LP] with the good event
+`G_r = {lambda_min(-A_M) > (4/(3k)) r M_3^2, r M_4 <= 3k/10}` of [LP] section 7 / [CAP] (1). Theorem E (`NOTE.md`) holds on
+the band `b in [0, 1]`, `k in [1/2, 2]`, uniformly over frames and torus sides `L >= 10`, and for the reference kernel:
+
+    Q^W(G_r^c) <= C(r_*) r^3   and   Z_r / r^2 >= z(r_*)      for 0 < r <= r_*,
+
+| `r_*` | `C(r_*)` | `C r_*^3` | `z(r_*)` |
+|---|---|---|---|
+| `1/4096` | `5355315.0638` | `7.8e-5` | `8.9953` |
+| `1/2048` | `5362302.4479` | `6.3e-4` | `8.9907` |
+| `1/1024` | `5377835.0775` | `5.1e-3` | `8.9815` |
+| `1/512` | `5415659.2391` | `0.0404` | `8.9629` |
+| `1/256` | `9370799.9311` | `0.559` | `8.9217` |
+
+(`1/128` and `1/64` are also certified, with `C r_*^3 > 1`.)
+
+**Theorem E' (v1.1).** The same holds at the fixed-axis SIDE24 point `(b, k) = (6/5, 1/6)` of [CAP] §1:
+
+| `r_*` | `C'(r_*)` | `C' / c_G(6/5, 1/6)` | `C' r_*^3` |
+|---|---|---|---|
+| `1/4096` | `39322.818` | `1.100` | `5.8e-7` |
+| `1/2048` | `43092.238` | `1.206` | `5.1e-6` |
+| `1/1024` | `1283859.690` | `35.9` | `0.0012` |
+| `1/512` | `117316965.49` | `3283` | `0.874` |
+
+This replaces [CAP]'s `2.4 x 10^23`, which is nontrivial only below `r = 1.6 x 10^-8`.
+
+**What follows.**
+- **Explicit Theorem A.** Since `1 - p_r <= Q^W(G_r^c)` ([LP] §8), [LP] Theorem A (1.1) holds on this band with explicit
+  pairs, for example `(C, r_*) = (5415659.2391, 1/512)` or `(9370799.9311, 1/256)`. [LP] §16 lists such a pair among the items
+  it does not supply. For the SIDE24 law, `1 - p_elder(r) <= 39322.818 r^3` for `r <= 1/4096`.
+- **Near-optimal for the cap route.** The constants are within `0.6%`–`1.8%` of the sharp cap coefficient
+  `c_G(0, 2) = 5324360.44` of Math-#203, below which no cap-route constant can go.
+- **Normalizer floor.** Together with Math-#195, `Z_r / r^2 >= 3.5044` on all of `(0, 1/2]` for `L >= 12`.
+
+**Files.**
+- `NOTE.md`: statement, the exact identities, Lemmas 1–5 (exact covariance series with a frame- and `L`-uniform torus
+  remainder; averaging identities for `M_3`; the Taylor bound for `M_4`; the root bounds after regression on `f_yy(M)`; the
+  normalizer floor), the main term, the rare branches, bands, values, Monte Carlo controls, and non-claims.
+- `theorem_a.py`: the certificate and its replay.
+  - `--check --procs N` replays the parameters, 11 sampled bands and both `C` tables exactly against `RESULTS.json`.
+  - `--check-full` replays all 153 bands, in about ten minutes on four cores.
+  - `--mutant {no-delta, cap-half, no-zfloor-correction, no-om-term}` must exit 1.
+  - With no flags it regenerates `RESULTS.json`.
+  - `--mc R B K N SEED` is the floating Monte Carlo control.
+- `RESULTS.json` (one line per band: 28 box bounds, 28 floors, the largest tail part, the SIDE24 bound and floor),
+  `SOURCE_MAP.json` (pins [LP] and [CAP] on `main`; read-only references to Math-#203 and Math-#195), `SOURCE_FILES.json`.
+- Workflow `.github/workflows/c8-theorem-a-constants-planar.yml` replays the manifest, the pins, `--check-full` and `--check`
+  in the two interpreter modes, the four mutants, and a clean tree.
+
+Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge.
diff --git a/frontiers/c8_theorem_a_constants_planar_20260930/RESULTS.json b/frontiers/c8_theorem_a_constants_planar_20260930/RESULTS.json
new file mode 100644
index 000000000..ed35fc61a
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_planar_20260930/RESULTS.json
@@ -0,0 +1,474 @@
+{
+ "C_table": {
+  "1/1024": {
+   "C": "2151134031/400",
+   "C_over_cG02": "1.010044",
+   "C_rstar3": "0.0050084993",
+   "argmax": {
+    "b": [
+     "0",
+     "1/64"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "31/32768",
+     "1/1024"
+    ]
+   },
+   "z_star": "17963/2000"
+  },
+  "1/128": {
+   "C": "87390113247/250",
+   "C_over_cG02": "65.653041",
+   "C_rstar3": "166.6834130230",
+   "argmax": {
+    "b": [
+     "1/4",
+     "1"
+    ],
+    "k": [
+     "1/2",
+     "1"
+    ],
+    "r": [
+     "5/1024",
+     "11/2048"
+    ]
+   },
+   "z_star": "8841791/1000000"
+  },
+  "1/2048": {
+   "C": "53623024479/10000",
+   "C_over_cG02": "1.007127",
+   "C_rstar3": "0.0006242542",
+   "argmax": {
+    "b": [
+     "0",
+     "1/64"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "31/65536",
+     "1/2048"
+    ]
+   },
+   "z_star": "1798151/200000"
+  },
+  "1/256": {
+   "C": "93707999311/10000",
+   "C_over_cG02": "1.759987",
+   "C_rstar3": "0.5585432012",
+   "argmax": {
+    "b": [
+     "1/4",
+     "1"
+    ],
+    "k": [
+     "1/2",
+     "1"
+    ],
+    "r": [
+     "15/4096",
+     "1/256"
+    ]
+   },
+   "z_star": "4460899/500000"
+  },
+  "1/4096": {
+   "C": "26776575319/5000",
+   "C_over_cG02": "1.005814",
+   "C_rstar3": "0.0000779301",
+   "argmax": {
+    "b": [
+     "0",
+     "1/64"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "31/131072",
+     "1/4096"
+    ]
+   },
+   "z_star": "8995379/1000000"
+  },
+  "1/512": {
+   "C": "54156592391/10000",
+   "C_over_cG02": "1.017148",
+   "C_rstar3": "0.0403498057",
+   "argmax": {
+    "b": [
+     "0",
+     "1/64"
+    ],
+    "k": [
+     "63/32",
+     "2"
+    ],
+    "r": [
+     "31/16384",
+     "1/512"
+    ]
+   },
+   "z_star": "8962953/1000000"
+  },
+  "1/64": {
+   "C": "14397396179923/10000",
+   "C_over_cG02": "270.406115",
+   "C_rstar3": "5492.1707839673",
+   "argmax": {
+    "b": [
+     "1/4",
+     "1"
+    ],
+    "k": [
+     "1/2",
+     "1"
+    ],
+    "r": [
+     "15/1024",
+     "1/64"
+    ]
+   },
+   "z_star": "54227/6250"
+  }
+ },
+ "C_table_side24": {
+  "1/1024": {
+   "C": "1604824613/1250",
+   "C_over_cG": "35.925490",
+   "C_rstar3": "0.0011956876",
+   "argmax_r": [
+    "31/32768",
+    "1/1024"
+   ],
+   "z_star": "3213821/1000000"
+  },
+  "1/128": {
+   "C": "815443379659/5000",
+   "C_over_cG": "4563.614304",
+   "C_rstar3": "77.7667407665",
+   "argmax_r": [
+    "1/512",
+    "9/4096"
+   ],
+   "z_star": "3092963/1000000"
+  },
+  "1/2048": {
+   "C": "215461191/5000",
+   "C_over_cG": "1.205825",
+   "C_rstar3": "0.0000050166",
+   "argmax_r": [
+    "31/65536",
+    "1/2048"
+   ],
+   "z_star": "3222401/1000000"
+  },
+  "1/256": {
+   "C": "815443379659/5000",
+   "C_over_cG": "4563.614304",
+   "C_rstar3": "9.7208425959",
+   "argmax_r": [
+    "1/512",
+    "9/4096"
+   ],
+   "z_star": "3162091/1000000"
+  },
+  "1/4096": {
+   "C": "19661409/500",
+   "C_over_cG": "1.100348",
+   "C_rstar3": "0.0000005723",
+   "argmax_r": [
+    "31/131072",
+    "1/4096"
+   ],
+   "z_star": "322669/100000"
+  },
+  "1/512": {
+   "C": "1173169654911/10000",
+   "C_over_cG": "3282.811506",
+   "C_rstar3": "0.8740795068",
+   "argmax_r": [
+    "31/16384",
+    "1/512"
+   ],
+   "z_star": "3196649/1000000"
+  },
+  "1/64": {
+   "C": "5553365927589/5000",
+   "C_over_cG": "31079.313186",
+   "C_rstar3": "4236.8819637978",
+   "argmax_r": [
+    "15/1024",
+    "1/64"
+   ],
+   "z_star": "2953979/1000000"
+  }
+ },
+ "L_min": 10,
+ "band": {
+  "b": [
+   "0",
+   "1"
+  ],
+  "k": [
+   "1/2",
+   "2"
+  ]
+ },
+ "certified": true,
+ "mutant": null,
+ "object": "CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1",
+ "parameters": {
+  "B_BOXES": [
+   [
+    "0",
+    "1/64"
+   ],
+   [
+    "1/64",
+    "1/16"
+   ],
+   [
+    "1/16",
+    "1/4"
+   ],
+   [
+    "1/4",
+    "1"
+   ]
+  ],
+  "COMBOS_X0_THETA": [
+   [
+    "1/64",
+    "1/32"
+   ],
+   [
+    "1/1024",
+    "1/256"
+   ]
+  ],
+  "HIGH_OCTAVES": 3,
+  "K_BOXES": [
+   [
+    "1/2",
+    "1"
+   ],
+   [
+    "1",
+    "3/2"
+   ],
+   [
+    "3/2",
+    "7/4"
+   ],
+   [
+    "7/4",
+    "15/8"
+   ],
+   [
+    "15/8",
+    "31/16"
+   ],
+   [
+    "31/16",
+    "63/32"
+   ],
+   [
+    "63/32",
+    "2"
+   ]
+  ],
+  "LOW_OCTAVES": 16,
+  "NB": 24,
+  "NX": 60,
+  "PBITS": 160,
+  "QS": [
+   2,
+   4,
+   8,
+   16
+  ],
+  "R_LAST": "1/536870912",
+  "R_STARS": [
+   "1/4096",
+   "1/2048",
+   "1/1024",
+   "1/512",
+   "1/256",
+   "1/128",
+   "1/64"
+  ],
+  "THETA_E": "1/16",
+  "TOP_OCTAVES": 4
+ },
+ "schema": 1,
+ "scientific_effect": "NONE",
+ "side24_point": {
+  "b": "6/5",
+  "k": "1/6"
+ },
+ "bands": [
+  {"bound": ["688669.0013", "2260517.0655", "3585872.5655", "4408747.4482", "4863642.2266", "5102380.2144", "5348823.2086", "676614.8681", "2220950.1759", "3523107.4584", "4331579.2618", "4778511.8571", "5013071.1318", "5255200.5507", "641489.0702", "2105651.7204", "3340208.7798", "4106709.5595", "4530440.0859", "4752822.4450", "4982381.9518", "515657.2978", "1692616.0096", "2685007.5875", "3301154.8228", "3641768.1650", "3820528.9432", "4005059.0303"], "r": ["0", "1/536870912"], "side24": ["37767.8416", "3.230978"], "tail_max": "0.000000000001", "zfloor": ["8.999999", "35.999999", "80.999998", "110.249998", "126.562498", "135.140623", "139.535154", "9.159780", "36.639120", "82.438020", "112.207305", "128.809406", "137.539822", "142.012371", "9.652497", "38.609991", "86.872480", "118.243098", "135.738251", "144.938288", "149.651421", "11.833305", "47.333222", "106.499749", "144.957992", "166.405858", "177.684478", "183.462459"]},
+  {"bound": ["688669.0031", "2260517.0583", "3585872.5455", "4408747.4196", "4863642.1928", "5102380.1779", "5348823.1694", "676614.8899", "2220950.2648", "3523107.6162", "4331579.4672", "4778512.0903", "5013071.3798", "5255200.8145", "641489.0914", "2105651.8055", "3340208.9303", "4106709.7552", "4530440.3079", "4752822.6812", "4982382.2029", "515657.3168", "1692616.0818", "2685007.7131", "3301154.9853", "3641768.3490", "3820529.1387", "4005059.2381"], "r": ["1/536870912", "5/2147483648"], "side24": ["37767.8445", "3.230978"], "tail_max": "0.000000000001", "zfloor": ["8.999999", "35.999999", "80.999999", "110.249999", "126.562499", "135.140624", "139.535155", "9.159780", "36.639120", "82.438021", "112.207306", "128.809408", "137.539823", "142.012372", "9.652497", "38.609991", "86.872481", "118.243099", "135.738252", "144.938289", "149.651423", "11.833305", "47.333222", "106.499750", "144.957993", "166.405860", "177.684479", "183.462460"]},
+  {"bound": ["688669.0082", "2260517.0671", "3585872.5552", "4408747.4300", "4863642.2036", "5102380.1889", "5348823.1808", "676614.9149", "2220950.3694", "3523107.8032", "4331579.7112", "4778512.3673", "5013071.6746", "5255201.1281", "641489.1156", "2105651.9057", "3340209.1087", "4106709.9878", "4530440.5720", "4752822.9621", "4982382.5018", "515657.3384", "1692616.1665", "2685007.8619", "3301155.1786", "3641768.5681", "3820529.3716", "4005059.4858"], "r": ["5/2147483648", "3/1073741824"], "side24": ["37767.8474", "3.230978"], "tail_max": "0.000000000001", "zfloor": ["8.999999", "35.999999", "80.999999", "110.249999", "126.562499", "135.140624", "139.535155", "9.159780", "36.639120", "82.438021", "112.207306", "128.809408", "137.539823", "142.012372", "9.652497", "38.609991", "86.872481", "118.243099", "135.738252", "144.938289", "149.651423", "11.833305", "47.333222", "106.499750", "144.957993", "166.405860", "177.684479", "183.462460"]},
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+{
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+  "object": "CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1",
+  "scientific_effect": "NONE",
+  "certified": true,
+  "review_status": "AUTHOR_SIDE_THEOREM_WITH_CERTIFIED_CONSTANTS",
+  "source_repository": "d6g8k5htny-coder/Math-",
+  "main_at_creation": "3e0a91b69892b9717564c592e7a966492a81566a",
+  "note": "Theorem E: for the planar six-pin law of [LP] and the good event G_r of [LP] section 7 / [CAP] (1), on the band b in [0, 1], k in [1/2, 2], uniformly over frames and torus sides L >= 10 (and for the reference kernel), Q^W(G_r^c) <= C(r_*) r^3 and Z_r / r^2 >= z(r_*) for 0 < r <= r_*, with C(1/512) = 5415659.2391, C(1/256) = 9370799.9311 (C r_*^3 <= 0.559), z(1/512) = 8.962953 and five further certified pairs. Theorem E' (v1.1): the same at the fixed-axis SIDE24 point (b, k) = (6/5, 1/6) of [CAP] section 1, C'(1/4096) = 39322.818 (1.100 c_G), C'(1/2048) = 43092.2382, C'(1/512) r^3 <= 0.874, replacing [CAP]'s recorded 2.4e23. The proof is self-contained apart from the definitions and the pins of [LP] (3.1)-(3.3): it does not consume [LP] (4.1), (4.2), (5.4), (5.5) or (7.5)-(7.7). With [LP] section 8 (1 - p_r <= Q^W(G_r^c)) this gives explicit pairs (C, r_*) for [LP] Theorem A (1.1). Every C here is a cap-route constant (a constant for [LP] (7.8)). C8 stays OPEN: no catalog edit; c_{B,K}, c_{d,L} and d >= 3 are not addressed.",
+  "sources": [
+    {
+      "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+      "bytes": 40261,
+      "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+      "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+      "id": "LP",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "section 1: the model (torus kernel K_L, pins, Q, W_r, Z_r, Q^W) and Theorem A (1.1); section 3: the preconditioned pins (3.1) and their values (3.3); section 7: the good event G_r and the cap bound (7.8); section 8: on G_r the global elder partner of M is S, so 1 - p_r <= Q^W(G_r^c) (used only for the consequence on 1 - p_r); section 16: a numerical r_*, C on a prescribed band is listed as not supplied",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
+      "bytes": 15160,
+      "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc",
+      "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1",
+      "id": "CAP",
+      "commit": "3e0a91b69892b9717564c592e7a966492a81566a",
+      "role": "section 1: the cylinder D, the partial-block norms M_3, M_4 (coordinate partials in d = 2) and condition (1) defining G_r; the deterministic cap theorem behind [LP] section 8 (consumed as a statement); the probabilistic corollary (2)-(3): the fixed-axis SIDE24 law (b = 6/5, kappa = 1/6, 0 < r <= 1/20), whose parameter point is Theorem E', and its recorded constant 2.4e23, quoted for comparison",
+      "current_required": true,
+      "on_main": true
+    },
+    {
+      "path": "frontiers/c8_cap_failure_coefficient_planar_20260930/NOTE.md",
+      "bytes": 28432,
+      "sha256": "0cb75475d9ee387a4045ff01ab9edb43a73c8c4edb6fc78d30518ced5bb02c5a",
+      "git_blob": "0b64f1c2828dfae7e6c104bac044004fc72c221d",
+      "id": "G",
+      "commit": "a388580eee356a88ce31aeed3999ab3fdc302312",
+      "pull_request": "Math-#203",
+      "role": "Theorem G and the certified lower ends c_G(0, 2) >= 5324360.4426 and c_G(6/5, 1/6) >= 35736.7352; v1.2 section 5 item 1: every cap-route constant is >= c_G. Used only for the --check sanity rules C > c_G (necessary conditions) and for the comparisons in NOTE section 0; not consumed by the proof",
+      "current_required": false,
+      "on_main": false
+    },
+    {
+      "path": "frontiers/c8_normalizer_floor_planar_20260930/NOTE.md",
+      "bytes": 17915,
+      "sha256": "3dffe22dc58bf4262ab580197b3fd5ef0f1ba71e0a2e3f7030748c4574fd9eab",
+      "git_blob": "d321d52e83e9fa6cd2467f82310b6267c8be4e29",
+      "id": "Z",
+      "commit": "6e7746bdacd04f45810638f7c612fc8cfba39d1e",
+      "pull_request": "Math-#195",
+      "role": "the certified floor Z_r / r^2 >= 3.5044 on [1/64, 1/2] at L >= 12, combined with z(1/64) in NOTE section 0; comparison only",
+      "current_required": false,
+      "on_main": false
+    }
+  ],
+  "catalog_reference": {
+    "path": "reviews/candidates_pending_20260928/CANDIDATES.md",
+    "entry": "C8. Numerical constants (still open as of 2026-09-29): explicit finite C, r_*, z_*, c_{B,K}, c_{d,L} on a declared compact-mark / torus band; disposition OPEN",
+    "effect_here": "none; this packet supplies certified cap-route pairs (C, r_*) and z_* on (0, r_*] for d = 2 on the band b in [0, 1], k in [1/2, 2], L >= 10; no catalog edit; C8 stays OPEN pending nonauthor reads and its other entries"
+  },
+  "related_records": [
+    "Math-#203 (sharp cap coefficient c_G; lower bound for every cap-route constant)",
+    "Math-#195 (z_* on [1/64, 1/2])",
+    "Math-#197 (c_{B,K})",
+    "Math-#199, Math-#200, Math-#204 (c_{d,L} coefficients)"
+  ],
+  "nonauthor_reads": [],
+  "nonauthor_acceptance": false,
+  "governing_status_changed": false,
+  "main_at_update": "3e0a91b69892b9717564c592e7a966492a81566a"
+}
diff --git a/frontiers/c8_theorem_a_constants_planar_20260930/theorem_a.py b/frontiers/c8_theorem_a_constants_planar_20260930/theorem_a.py
new file mode 100644
index 000000000..ae279e901
--- /dev/null
+++ b/frontiers/c8_theorem_a_constants_planar_20260930/theorem_a.py
@@ -0,0 +1,1249 @@
+#!/usr/bin/env python3
+"""Explicit constants for Theorem A of [LP] in the plane (d = 2), for the torus kernel K_L with L >= 10 in every frame:
+for 0 < r <= r_*, b in [0, 1], k in [1/2, 2],
+
+    1 - p_r(b, k, R)  <=  Q^W(G_r^c)  <=  C(r_*) r^3,
+
+with G_r the good event of [LP] section 7 / [CAP] (1) and C(r_*) certified here (NOTE.md Theorem E); the same at the single
+fixed-axis SIDE24 point (b, k) = (6/5, 1/6) of [CAP] section 1 (Theorem E').  The proof bounds
+Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0:
+
+  * covariances of the preconditioned pin and target functionals are exact power series in r, obtained from the Taylor
+    coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded
+    uniformly in r and in the frame by Cauchy estimates on a polydisc;
+  * the failure event is reduced to {lambda <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded through exact averaging
+    identities of the pinned jets and M_4 through the Taylor series at the midpoint;
+  * the main term is integrated in lambda = -f_yy(M) against its Gaussian density after an exact regression on w = f_yy(M);
+    the remaining expectations are closed-form incomplete Gaussian moments plus Hoelder-controlled perturbations;
+  * the normalizer Z_r is bounded below in closed form; the rare branches are Gaussian tails.
+All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only.
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term')
+MUT = None
+PBITS = 160
+L_MIN = 10
+NX = 60                    # Taylor order in x of the covariance series (exact); the rest is a bounded tail
+NB = 24                    # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded)
+X0 = Fr(1, 64)             # default threshold for the quadratic near-branch analysis (records use COMBOS)
+THETA = Fr(1, 32)          # default splitting parameter in the max-moment lemma (records use COMBOS)
+THETA_E = Fr(1, 16)        # splitting parameter for the (r nu + v)^2 factor
+QS = (2, 4, 8, 16)         # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+BAND_B = (Fr(0), Fr(1))
+BAND_K = (Fr(1, 2), Fr(2))
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+
+
+def phi_pt(x):
+    """phi(x) for rational x (interval)."""
+    x = Fr(x)
+    return exp_neg_point(x * x / 2) / SQRT2PI
+
+
+def Phibar_pt(x):
+    """1 - Phi(x) for rational x (interval)."""
+    p = phi_point(Fr(x)) if abs(Fr(x)) <= 12 else None
+    if p is None:
+        if x > 0:
+            # Mills: Phibar(x) <= phi(x)/x
+            return IV(0, (phi_pt(x) / x).hi)
+        return IV(1 - (phi_pt(-x) / (-x)).hi, 1)
+    return IV(1 - p.hi, 1 - p.lo)
+
+
+def upper_moments(x, J):
+    """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x >= 0 (intervals)."""
+    x = Fr(x)
+    require(x >= 0, 'upper moment at negative point')
+    ph = phi_pt(x)
+    M = [Phibar_pt(x), ph]
+    for j in range(2, J + 1):
+        M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1))
+    return M
+
+
+def dfact(n):
+    """(n)!! for n >= -1."""
+    return math.prod(range(n, 0, -2)) if n > 0 else 1
+
+
+def rup_rel(x, bits=200):
+    """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds)."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh)
+    return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh)
+
+
+def sqrt_up(x):
+    """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root
+    step adds less than 2^-209 and rup_rel(., 200) less than 2^-199.  (The earlier '2^-200' was too sharp: at x = 49/100 the
+    excess is (8/7) 2^-200, review C59-DOC-01.)"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m))
+
+
+def iroot(x, p):
+    """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two."""
+    v = Fr(x)
+    while p > 1:
+        v = sqrt_up(v)
+        p //= 2
+    return v
+
+
+def pow2(p):
+    q = 1
+    while q < p:
+        q *= 2
+    return q
+
+
+ZN = {}
+
+
+def znorm(p):
+    """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5)."""
+    if p == 1:
+        return Fr(4, 5)
+    if p not in ZN:
+        ZN[p] = iroot(Fr(dfact(p - 1)), p)
+    return ZN[p]
+
+
+def habs_moment(j):
+    """E|zeta|^j (interval)."""
+    if j % 2 == 0:
+        return IV(dfact(j - 1))
+    return SQRT2 / isqrt_iv(PI) * (2 ** ((j - 1) // 2) * math.factorial((j - 1) // 2))
+
+
+# ----------------------------------------------------------------------------- functionals and exact covariance series
+HALF = Fr(1, 2)
+XM, XS, X0PT = -HALF, HALF, Fr(0)
+
+
+def term(c, p, a, b, x0):
+    """c r^p d_x^a d_y^b f(x0 r, 0)."""
+    return (Fr(c), p, a, b, Fr(x0))
+
+
+def functionals():
+    pins = {
+        'p1': [term(HALF, 0, 0, 0, XM), term(HALF, 0, 0, 0, XS)],
+        'p2': [term(-1, -1, 0, 0, XM), term(1, -1, 0, 0, XS)],
+        'p3p': [term(6, -2, 1, 0, XM), term(6, -2, 1, 0, XS), term(12, -3, 0, 0, XM), term(-12, -3, 0, 0, XS)],
+        'p4': [term(-1, -1, 1, 0, XM), term(1, -1, 1, 0, XS)],
+        'q1': [term(HALF, 0, 0, 1, XM), term(HALF, 0, 0, 1, XS)],
+        'q2': [term(-1, -1, 0, 1, XM), term(1, -1, 0, 1, XS)],
+    }
+    targets = {
+        'w': [term(1, 0, 0, 2, XM)],
+        'v': [term(1, -1, 1, 1, XM), term(1, -2, 0, 1, XM), term(-1, -2, 0, 1, XS)],
+        'om': [term(1, -1, 0, 2, XS), term(-1, -1, 0, 2, XM)],
+        'Y': [term(1, 0, 0, 3, X0PT)],
+        'T': [term(1, -2, 2, 0, XM), term(4, -3, 1, 0, XM), term(2, -3, 1, 0, XS), term(6, -4, 0, 0, XM), term(-6, -4, 0, 0, XS)],
+        'tau': [term(1, -3, 2, 0, XS), term(-1, -3, 2, 0, XM), term(-6, -4, 1, 0, XM), term(-6, -4, 1, 0, XS),
+                term(-12, -5, 0, 0, XM), term(12, -5, 0, 0, XS)],
+        'nu': [term(1, -2, 1, 1, XM), term(1, -2, 1, 1, XS), term(2, -3, 0, 1, XM), term(-2, -3, 0, 1, XS)],
+        'X40': [term(1, 0, 4, 0, X0PT)], 'X31': [term(1, 0, 3, 1, X0PT)], 'X22': [term(1, 0, 2, 2, X0PT)],
+        'X13': [term(1, 0, 1, 3, X0PT)], 'X04': [term(1, 0, 0, 4, X0PT)],
+    }
+    return pins, targets
+
+
+PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q1', 'q2')
+TARGET_NAMES = ('w', 'v', 'om', 'Y', 'T', 'tau', 'nu', 'X40', 'X31', 'X22', 'X13', 'X04')
+JET4 = ('X40', 'X31', 'X22', 'X13', 'X04')
+JET4_IDX = {'X40': (4, 0), 'X31': (3, 1), 'X22': (2, 2), 'X13': (1, 3), 'X04': (0, 4)}
+
+
+def weight(F):
+    """Scaling weight a - (power of r), common to all terms of a preconditioned functional."""
+    ws = {t[2] - t[1] for t in F}
+    require(len(ws) == 1, 'functional not homogeneous')
+    return ws.pop()
+
+
+def hcoef(n):
+    """d^n/dx^n exp(-x^2/2) at 0."""
+    if n % 2:
+        return 0
+    return (-1) ** (n // 2) * dfact(n - 1)
+
+
+PAIR_CACHE = {}
+
+
+def pair_series(F, G, key):
+    """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG):
+    returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', b+b') for the remainder bounds)."""
+    if key in PAIR_CACHE:
+        return PAIR_CACHE[key]
+    wF, wG = weight(F), weight(G)
+    w = wF + wG
+    pi = {}
+    absterms = []
+    for (cf, pf, af, bf, xf) in F:
+        for (cg, pg, ag, bg, xg) in G:
+            gy = bf + bg
+            sgn = (-1) ** (ag + bg)
+            d = xf - xg
+            absterms.append((abs(cf * cg), af + ag, gy))
+            for gx in range(af + ag, NX + 1):
+                m = gx - af - ag
+                val = cf * cg * sgn * d ** m / math.factorial(m)
+                if val != 0:
+                    pi[(gx, gy)] = pi.get((gx, gy), Fr(0)) + val
+    for (gx, gy), v in pi.items():
+        require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)')
+    ser = {}
+    for (gx, gy), v in pi.items():
+        kap = hcoef(gx) * hcoef(gy)
+        if kap and v:
+            ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v
+    out = ({p: c for p, c in ser.items() if c != 0}, w, absterms)
+    PAIR_CACHE[key] = out
+    return out
+
+
+def torus_M1(L):
+    """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2| <= 1 (every frame, |p_n| = |n| L):
+    exp(1) * sum_{j >= 1} 8 j exp(-(j L - 3/2)^2 / 2)."""
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += 8 * j * exp_neg_point((j * L - Fr(3, 2)) ** 2 / 2).hi
+    tail = 8 * 7 * exp_neg_point((6 * L - Fr(3, 2)) ** 2 / 2).hi * 4      # j >= 6, very crude
+    return Fr(27183, 10000) * (tot + tail)
+
+
+def theta_minus_one(L):
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += 8 * j * exp_neg_point((j * L) ** 2 / 2).hi
+    return tot + 8 * 7 * exp_neg_point((6 * L) ** 2 / 2).hi * 4
+
+
+def cov_band(F, G, key, r0, r1, M1):
+    """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1]."""
+    ser, w, absterms = pair_series(F, G, key)
+    val = IV(0)
+    for p, c in ser.items():
+        val = val + IV(r0 ** p, r1 ** p) * c
+    A = max((t[1] for t in absterms), default=0)
+    # Taylor truncation tail of the reference kernel (orders gx > NX)
+    tail = Fr(0)
+    for (acc, aa, gy) in absterms:
+        tail += acc * abs(hcoef(gy))
+    tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w)
+    # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy! M1 on the unit polydisc
+    tor = Fr(0)
+    if True:
+        for (acc, aa, gy) in absterms:
+            tor += acc * math.factorial(gy) * Fr(max(w, 1)) ** aa
+        tor = tor * M1 / (1 - Fr(27183, 10000) * r1)
+    err = tail + tor
+    return val + IV(-err, err)
+
+
+# ----------------------------------------------------------------------------- interval linear algebra
+def solve_iv(A, B):
+    """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c]."""
+    n = len(A)
+    M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    for col in range(n):
+        p = max(range(col, n), key=lambda i: abs(M[i][col].mid()))
+        M[col], M[p] = M[p], M[col]
+        piv = M[col][col]
+        for i in range(col + 1, n):
+            f = M[i][col] / piv
+            for j in range(col, len(M[i])):
+                M[i][j] = M[i][j] - f * M[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            s = M[i][n + c]
+            for j in range(i + 1, n):
+                s = s - M[i][j] * x[j]
+            x[i] = s / M[i][i]
+        X.append(x)
+    return X
+
+
+def cholesky3(C):
+    """Interval Cholesky of a 3 x 3 SPD interval matrix."""
+    L = [[IV(0)] * 3 for _ in range(3)]
+    for i in range(3):
+        for j in range(i + 1):
+            s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0))
+            if i == j:
+                require(s.lo > 0, 'Cholesky pivot not positive')
+                L[i][j] = isqrt_iv(s)
+            else:
+                L[i][j] = s / L[j][j]
+    return L
+
+
+def absup(x):
+    return max(abs(x.lo), abs(x.hi))
+
+
+# ----------------------------------------------------------------------------- the conditional law on an r-band
+class Law:
+    pass
+
+
+def band_law(r0, r1, L=L_MIN):
+    """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the
+    targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients."""
+    pins, targets = functionals()
+    P = [pins[n] for n in PIN_NAMES]
+    T = [targets[n] for n in TARGET_NAMES]
+    M1 = torus_M1(L)
+    th1 = theta_minus_one(L)
+    nt = len(T)
+    Spp = [[cov_band(P[i], P[j], ('p', i, j), r0, r1, M1) for j in range(6)] for i in range(6)]
+    Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(6)] for i in range(nt)]
+    Stt = [[cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1) for j in range(nt)] for i in range(nt)]
+    r = IV(r0, r1)
+    kvec = [-(r * r * r) * HALF, -(r * r), IV(12), IV(0), IV(0), IV(0)]
+    e1 = [IV(1), IV(0), IV(0), IV(0), IV(0), IV(0)]
+    W = solve_iv(Spp, Stp)
+    xs = solve_iv(Spp, [e1, kvec])
+    law = Law()
+    law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1
+    law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)}
+    law.beta = {n: sum((W[i][j] * kvec[j] for j in range(6)), IV(0)) for i, n in enumerate(TARGET_NAMES)}
+    scale = IV(1 / (1 + th1), 1)          # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1]
+    C = {}
+    for i, a in enumerate(TARGET_NAMES):
+        for j, c in enumerate(TARGET_NAMES):
+            if j < i:
+                continue
+            v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(6)), IV(0))
+            if i != j:
+                v2 = Stt[j][i] - sum((W[j][l] * Stp[i][l] for l in range(6)), IV(0))
+                v = v.meet(v2)
+            C[(a, c)] = C[(c, a)] = v * scale
+    law.C = C
+    escale = IV(1, 1 + th1)
+    law.e_bb = xs[0][0] * escale
+    law.e_bk = sum((kvec[j] * xs[0][j] for j in range(6)), IV(0)) * escale
+    law.e_kk = sum((kvec[j] * xs[1][j] for j in range(6)), IV(0)) * escale
+    # regression on w and the g-parts
+    sw2 = C[('w', 'w')]
+    require(sw2.lo > 0, 'Var w not positive')
+    law.sw2 = sw2
+    law.sw = isqrt_iv(sw2)
+    law.bw = {n: C[(n, 'w')] / sw2 for n in TARGET_NAMES if n != 'w'}
+    Cg = {}
+    for a in TARGET_NAMES:
+        for c in TARGET_NAMES:
+            if a != 'w' and c != 'w':
+                Cg[(a, c)] = C[(a, c)] - C[(a, 'w')] * C[(c, 'w')] / sw2
+    law.Cg = Cg
+    law.L3 = cholesky3([[Cg[(a, c)] for c in ('v', 'om', 'Y')] for a in ('v', 'om', 'Y')])
+    law.sdg = {n: isqrt_iv(IV(max(Cg[(n, n)].lo, Fr(0)), Cg[(n, n)].hi)) if Cg[(n, n)].lo > 0 else IV(0, isqrt_iv(IV(Cg[(n, n)].hi)).hi)
+               for n in TARGET_NAMES if n != 'w'}
+    law.sdQ = {n: isqrt_iv(C[(n, n)]) for n in TARGET_NAMES}
+    # crude higher jets: Taylor-tail coefficients S_alpha(r1) = sum_{beta != 0} sd0_{alpha+beta} (2 r1)^{|beta|} / beta!
+    law.S = {n: taylor_tail_coefficient(JET4_IDX[n], r1, M1) for n in JET4}
+    return law
+
+
+def sd0_bound(gx, gy, M1):
+    """Upper bound of sqrt(Var d^gamma f(0)) = sqrt(|d^{2 gamma} K_L(0)|) (normalized, every frame, L >= L_MIN)."""
+    v = dfact(2 * gx - 1) * dfact(2 * gy - 1) + math.factorial(2 * gx) * math.factorial(2 * gy) * M1
+    return isqrt_iv(IV(v)).hi
+
+
+def taylor_tail_coefficient(alpha, r1, M1):
+    ax, ay = alpha
+    tot = Fr(0)
+    for n in range(1, NB + 1):
+        for bx in range(n + 1):
+            by = n - bx
+            tot += sd0_bound(ax + bx, ay + by, M1) * (2 * r1) ** n / (math.factorial(bx) * math.factorial(by))
+    # tail |beta| > NB: sd0_gamma <= 2^|gamma| gx! gy! sqrt(1 + M1) (binom(2m, m) <= 4^m), so a term is at most
+    # 2^|alpha| (4 + n)^4 (4 r1)^n 2 per multi-index; n + 1 multi-indices of order n
+    q = 4 * r1
+    require(q <= Fr(1, 16), 'Taylor tail ratio too large')
+    tail = Fr(0)
+    base = 2 ** (ax + ay) * 2
+    n = NB + 1
+    # sum_{n > NB} (n + 1)(n + 4)^8 q^n <= (NB + 2)(NB + 5)^8 q^(NB+1) / (1 - 2 q)   (consecutive ratio <= 2q for these n)
+    tail = base * (NB + 2) * Fr(NB + 5) ** 8 * q ** (NB + 1) / (1 - 2 * q)
+    return tot + tail
+
+
+# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k)
+def binom(n, k):
+    return math.comb(n, k)
+
+
+def gauss_norm(mu_abs, sd, p):
+    """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd."""
+    return rup_rel(mu_abs + sd * znorm(pow2(p)))
+
+
+def pos_part_moments(s, c, nmax, extra_max):
+    """P[n][e] = upper bound of E[u^e (u^n - c^n)_+], u = s |zeta|, for 1 <= n <= nmax, 0 <= e <= extra_max."""
+    x = c / s
+    M = upper_moments(x, nmax + extra_max)
+    out = {}
+    for n in range(1, nmax + 1):
+        for e in range(extra_max + 1):
+            out[(n, e)] = rup_rel(((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e) * 2).hi)
+    return out
+
+
+def normalized_moments(q, nmax, theta):
+    """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax:
+      E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmu1/K + (r1/K) Delta1;
+      Tn[n], To[n], Tv[n]   : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2),
+                              y_i <= u_i + delta_i (u_i = s_i |zeta| independent half-normals), threshold c = q['c'].
+    o0 = s3 |z2| and v0^2 = s2^2 z1^2 / 4 are unnormalized."""
+    th = theta
+    cp = q['c'] * (1 - th)                  # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli)
+    PP = {i: pos_part_moments(q['s'][i], cp, nmax, 2) for i in (2, 3, 4)}
+    Eu3 = (habs_moment(1) * q['s3']).hi
+    Eu2sq = (habs_moment(2) * q['s2'] * q['s2']).hi
+    n_o0_2 = q['s3'] * znorm(2)
+    n_u2sq_2 = q['s2'] * q['s2'] * znorm(4) ** 2
+    r3 = q['s3'] / q['s'][3]
+    r2sq = (q['s2'] / q['s'][2]) ** 2
+    E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {}
+    base = 1 + q['cmt1']
+    for n in range(0, nmax + 1):
+        ey = base ** n
+        eo = base ** n * Eu3
+        ev = base ** n * Eu2sq
+        for j in range(1, n + 1):
+            coef = binom(n, j) * base ** (n - j) * q['rK'] ** j
+            ey += coef * q['nD1'](pow2(j)) ** j
+            eo += coef * n_o0_2 * q['nD1'](pow2(2 * j)) ** j
+            ev += coef * n_u2sq_2 * q['nD1'](pow2(2 * j)) ** j
+        E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev / 4)
+        if n == 0:
+            continue
+        f1 = (1 + th) ** (n - 1)
+        f2 = (1 + 1 / th) ** (n - 1)
+        tn = to = tv = Fr(0)
+        for i in (2, 3, 4):
+            P = PP[i]
+            dn = q['nd'](i, pow2(n)) ** n
+            dn2 = q['nd'](i, pow2(2 * n)) ** n
+            tn += f1 * P[(n, 0)] + f2 * dn
+            to += (f1 * r3 * P[(n, 1)] if i == 3 else f1 * Eu3 * P[(n, 0)]) + f2 * n_o0_2 * dn2
+            tv += (f1 * r2sq * P[(n, 2)] if i == 2 else f1 * Eu2sq * P[(n, 0)]) + f2 * n_u2sq_2 * dn2
+        Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv / 4)
+    return E1, E1o, E1v, Tn, To, Tv
+
+
+def elam_pos(m, s):
+    """E[(s z + m)_+] at a point (rational m, s > 0)."""
+    x = Fr(m) / s
+    return m * Phi(IV(x)) + phi_pt(x) * s
+
+
+def elam2_pos(m, s):
+    """E[(s z + m)_+^2] at a point."""
+    x = Fr(m) / s
+    return (m * m + s * s) * Phi(IV(x)) + phi_pt(x) * (m * s)
+
+
+def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None):
+    """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch
+    threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5).  The bound is proved for 0 < x0 < 2/9 and
+    0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and
+    (1 + kappa x0) c^2 <= 1.  No claim is made outside that domain.  Both COMBOS pairs, and the (1/64, 1/32) of Math-#215's
+    caller, lie inside it (review C61)."""
+    x0 = X0 if x0 is None else Fr(x0)
+    theta = THETA if theta is None else Fr(theta)
+    b = IV(bb[0], bb[1])
+    k = IV(kb[0], kb[1])
+    klo, khi = kb[0], kb[1]
+    Klo = 12 * klo
+    mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES}
+    mabs = {n: absup(mu[n]) for n in TARGET_NAMES}
+    E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk
+    require(E.hi > 0, 'energy')
+    sqE = isqrt_iv(IV(E.hi)).hi
+    sw_lo, sw_hi = law.sw.lo, law.sw.hi
+    muw_abs = mabs['w']
+    bw = {n: absup(law.bw[n]) for n in law.bw}
+    t = {n: law.S[n] / sw_lo for n in JET4}
+    a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1)      # the cap constant 4/(3k) (mutant: halved)
+    B = lambda n: bw[n] + t[n]
+    bbars = {1: r1 * (Fr(5, 2) * B('X40') + 2 * B('X31')),
+             2: 2 * bw['v'] + r1 * (Fr(5, 2) * B('X31') + 2 * B('X22')),
+             3: bw['om'] + r1 * (Fr(5, 2) * B('X22') + 2 * B('X13')),
+             4: bw['Y'] + r1 * (2 * B('X13') + 2 * B('X04'))}
+    bbars = {i: rup_rel(v) for i, v in bbars.items()}
+    bbar = max(bbars.values())
+    L3 = law.L3
+    s2, s3, s4 = 2 * L3[0][0].hi, L3[1][1].hi, L3[2][2].hi
+    l21, l31, l32 = absup(L3[1][0]), absup(L3[2][0]), absup(L3[2][1])
+
+    memo = {}
+
+    def nA(n, p):     # ||A_alpha^g||_p, A_alpha^g = |X_alpha,g| + T_alpha^g
+        key = ('A', n, p)
+        if key not in memo:
+            memo[key] = rup_rel(gauss_norm(mabs[n], law.sdg[n].hi, p) + law.S[n] * (sqE + znorm(pow2(p))))
+        return memo[key]
+
+    def nDelta(i, p):
+        key = ('D', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nDelta0(i, p))
+        return memo[key]
+
+    def nDelta0(i, p):
+        if MUT == 'no-delta':
+            return Fr(0)
+        if i == 1:
+            return Fr(5, 2) * nA('X40', p) + 2 * nA('X31', p)
+        if i == 2:
+            return Fr(5, 2) * nA('X31', p) + 2 * nA('X22', p)
+        if i == 3:
+            return Fr(5, 2) * nA('X22', p) + 2 * nA('X13', p)
+        return 2 * nA('X13', p) + 2 * nA('X04', p)
+
+    def nd(i, p):
+        key = ('d', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nd0(i, p))
+        return memo[key]
+
+    def nd0(i, p):     # ||delta_i||_p / Klo, delta_i including beta_i |mu_w|
+        zp = znorm(pow2(p))
+        cm = bbars[i] * muw_abs
+        if i == 2:
+            v = 2 * mabs['v'] + cm + r1 * nDelta(2, p)
+        elif i == 3:
+            v = mabs['om'] + l21 * zp + cm + r1 * nDelta(3, p)
+        else:
+            v = mabs['Y'] + (l31 + l32) * zp + cm + r1 * nDelta(4, p)
+        return v / Klo
+
+    kappa = 3 / (1 - 3 * x0)
+    Fk = 1 + kappa * x0                      # (1 + kappa x_i) <= Fk on the event {x_i <= x0}
+    cth = 1 - kappa * x0 / 2                 # <= Fk^{-1/2}
+    NMAX = 24
+    q = {'cmt1': bbars[1] * muw_abs / Klo, 'rK': r1 / Klo, 'nD1': lambda p: nDelta(1, p),
+         's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth}
+    E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta)
+    et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac)     # eta_1 K, k-free
+    aK2_hi = 192 * khi * a_fac                    # a K^2 = 192 k
+
+    # moments of Ut = U / (a K^2) <= max(Z1, Fk y_i^2), Z1 = y1^2 (1 + et1 y1)
+    def EUt(p):                                   # upper bound of E[Ut^p]
+        key = ('U', p)
+        if key not in memo:
+            memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p])
+        return memo[key]
+
+    def EoUt2():
+        return sum(binom(2, l) * et1 ** l * E1o[4 + l] for l in range(3)) + Fk ** 2 * To[4]
+
+    def EvUt2():
+        return sum(binom(2, l) * et1 ** l * E1v[4 + l] for l in range(3)) + Fk ** 2 * Tv[4]
+
+    def sq(x):
+        return isqrt_iv(IV(x)).hi
+
+    U3_2 = sq(EUt(6))                              # ||Ut^3||_2
+    U2_2 = sq(EUt(4))                              # ||Ut^2||_2
+
+    def nU(p):                                     # ||U||_p <= a K^2 Fk ||max_i y_i||_2p^2 (on the event x_i <= X0)
+        require(2 * p <= NMAX and pow2(p) == p, 'norm order')
+        key = ('nU', p)
+        if key not in memo:
+            memo[key] = rup_rel(aK2_hi * Fk * iroot(E1[2 * p] + Tn[2 * p], p))
+        return memo[key]
+
+    def nbar(n, p):                                # ||F_bar||_p, F_bar = |F_g| + |beta_F| (|mu_w| + r1 U)
+        return gauss_norm(mabs[n], law.sdg[n].hi, p) + bw[n] * (muw_abs + r1 * nU(p))
+
+    six_klo = 6 * klo
+    nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4)
+    epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2
+    nT8, nt8 = nbar('T', 8), nbar('tau', 8)
+    epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2
+    # Term A / (36 k^2)
+    A_n = EUt(3) + epsc2 * U3_2
+    termA = aK2_hi ** 3 / 3 * A_n
+    # Term B / (36 k^2)
+    ndo = lambda p: l21 * znorm(pow2(p)) + mabs['om'] + bw['om'] * (muw_abs + r1 * nU(p))
+    no4 = s3 * znorm(4) + ndo(4)
+    B_n = EoUt2() + (ndo(2) + epsc4 * no4) * U2_2
+    termB = aK2_hi ** 2 / 2 * B_n
+    if MUT == 'no-om-term':
+        termB = Fr(0)
+    # Term C / (36 k^2) = (a K^2)^2 / (12 k) * E[(1 + r T/(6k)) e Ut^2]
+    th = THETA_E
+    dv = lambda p: mabs['v'] + bw['v'] * (muw_abs + r1 * nU(p))
+    ebar4 = (r1 * nbar('nu', 8) + L3[0][0].hi * znorm(8) + dv(8)) ** 2
+    C_n = ((1 + th) ** 2 * EvUt2()
+           + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U2_2
+           + r1 * nT4 / six_klo * ebar4 * U2_2)
+    termC = aK2_hi ** 2 / (12 * klo) * C_n
+    # one more power of U for the density refinement: the same three terms with lambda inserted, divided by r
+    # (int_0^{rU} lambda W dlambda <= r^6 |c1| [|c2| (U^4/4 + om U^3/3) + e U^3/3])
+    U3_2_ = U3_2
+    termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * sq(EUt(8)))
+    EoUt3 = sum(binom(3, l) * et1 ** l * E1o[6 + l] for l in range(4)) + Fk ** 3 * To[6]
+    EvUt3 = sum(binom(3, l) * et1 ** l * E1v[6 + l] for l in range(4)) + Fk ** 3 * Tv[6]
+    termBp = aK2_hi ** 3 / 3 * (EoUt3 + (ndo(2) + epsc4 * no4) * U3_2_)
+    if MUT == 'no-om-term':
+        termBp = Fr(0)
+    termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt3
+                                         + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U3_2_
+                                         + r1 * nT4 / six_klo * ebar4 * U3_2_)
+    # density of lambda = -w: pbar = its supremum on [0, inf); on the failure window [0, rU] also p(lambda) <= p0 + lambda Lp
+    # with p0 = sup over the box of the density at 0 and Lp = sup |p'| <= phi(1)/sigma^2, phi(1) < 0.24198
+    mlo = max(Fr(0), mu['w'].lo)
+    pbar = (phi_pt(mlo / sw_hi) / sw_lo).hi
+    m0 = Fr(0) if mu['w'].lo <= 0 <= mu['w'].hi else min(abs(mu['w'].lo), abs(mu['w'].hi))
+    p0 = (phi_pt(m0 / sw_hi) / sw_lo).hi
+    Lp = Fr(24198, 100000) / (sw_lo * sw_lo)
+    # ---- normalizer floor Z_r / (36 k^2 r^2) >= El2 - corr
+    m_lam = -mu['w']
+    El2 = elam2_pos(m_lam.lo, sw_lo).lo
+    Elp = elam_pos(m_lam.hi, sw_hi).hi
+    lam_n = lambda p: muw_abs + sw_hi * znorm(pow2(p))
+    sdT, sdtau = law.sdQ['T'].hi, law.sdQ['tau'].hi
+    q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1))
+    Eom_g = mabs['om'] + law.sdg['om'].hi * Fr(4, 5)
+    Evg2 = mabs['v'] ** 2 + law.sdg['v'].hi ** 2
+    E_om_lam = Eom_g * Elp + bw['om'] * (muw_abs * lam_n(1) + lam_n(2) ** 2)
+    E_v2_lam = 2 * Evg2 * Elp + 2 * bw['v'] ** 2 * (sw_hi * znorm(4)) ** 2 * lam_n(2)
+    corr = (lam_n(4) ** 2 * iroot(q_bad, 2) if q_bad > 0 else Fr(0)) \
+        + (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) * lam_n(4) ** 2 / six_klo \
+        + r1 * (E_v2_lam / (3 * klo) + E_om_lam)
+    if MUT == 'no-zfloor-correction':
+        corr = Fr(0)
+    zt = El2 - corr
+    require(zt > 0, 'normalizer floor not positive')
+    main = min(pbar * (termA + termB + termC), p0 * (termA + termB + termC) + r1 * Lp * (termAp + termBp + termCp)) / zt
+    # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q) (Hoelder), ||W 1_typed||_q / r^2 <= g1 g2,
+    # for each q in QS; the smallest resulting bound is used
+    six_khi = 6 * khi
+
+    def wnorm(q):
+        p4 = 4 * q
+        nTQ = gauss_norm(mabs['T'], sdT, p4)
+        ntauQ = gauss_norm(mabs['tau'], sdtau, p4)
+        g1 = (six_khi + r1 * nTQ) * lam_n(p4)
+        g2 = (six_khi + r1 * nTQ + r1 * r1 * ntauQ) * (lam_n(p4) + r1 * gauss_norm(mabs['om'], law.sdQ['om'].hi, p4)) \
+            + r1 * (r1 * gauss_norm(mabs['nu'], law.sdQ['nu'].hi, p4) + gauss_norm(mabs['v'], law.sdQ['v'].hi, p4)) ** 2
+        return g1 * g2
+    thr = 3 * klo / (10 * r1)
+    qA2 = Fr(0)
+    args = []
+    for n in JET4:
+        args.append((thr * Fr(7, 8) - mabs[n]) / law.sdQ[n].hi)
+        qA2 += gauss_tail2(mabs[n], law.sdQ[n].hi, thr * Fr(7, 8))
+        qA2 += (law.S[n] * (sqE + znorm(32)) / (thr / 8)) ** 32
+    a_hi = Fr(4, 3) / klo * a_fac
+    qfar = Fr(0)
+    qx = Fr(0)
+    if bbar > 0:
+        lam_far = 1 / (4 * a_hi * r1 * bbar * bbar)
+        args.append((lam_far - muw_abs) / sw_hi)
+        qfar = gauss_tail1(muw_abs, sw_hi, lam_far)
+    for i in (1, 2, 3, 4):                         # x_i > X0  <=>  y_i' > X0 / (a r1 beta_i)
+        if bbars[i] == 0:
+            continue
+        tx = x0 / (a_hi * r1 * bbars[i])
+        if i == 1:
+            cb = 12 * khi + bbars[1] * muw_abs
+            qx += Fr(1) if tx <= cb else (r1 * nDelta(1, 32) / (tx - cb)) ** 32
+        else:
+            s_i = {2: s2, 3: s3, 4: s4}[i]
+            args.append(tx / (2 * s_i))
+            qx += gauss_tail2(0, s_i, tx / 2) + (nd(i, 32) * Klo / (tx / 2)) ** 32
+    qs = [min(Fr(1), x) for x in (qA2, qfar, qx)]
+    tail_num = min(wnorm(q) * sum(pow_tail(x, q) for x in qs) for q in QS)
+    if last_band:
+        # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian tail argument scales at least like 1/r and is >= 3 at R,
+        # so each Gaussian piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32; after the
+        # power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of tail / r^3
+        # over (0, R] is attained on [R/2, R]
+        require(min(args) >= 3, 'last-band tail arguments')
+        # no piece is capped at 1, so the factor-64 decrease of every piece per halving carries over to each sum
+        require(max(qA2, qfar, qx) < 1, 'last-band tail pieces')
+        rdiv = r1 / 2
+    else:
+        rdiv = r0
+    tail = tail_num / (36 * klo * klo * zt * rdiv ** 3)
+    total = main + tail
+    return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'pbar': pbar, 'termA': termA, 'termB': termB,
+            'termC': termC, 'qA2': qA2, 'qfar': qfar, 'qx': qx, 'El2': El2, 'corr': corr, 'min_arg': min(args)}
+
+
+def pow_tail(x, q):
+    """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two)."""
+    if x <= 0:
+        return Fr(0)
+    return iroot(rup_rel(x, 64) ** (q - 1), q)
+
+
+def phibar_upper(x):
+    """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound
+    phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor."""
+    x = Fr(x)
+    if x <= 12:
+        return Phibar_pt(x).hi
+    y = x * x / 2
+    m = 40
+    return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2))      # sqrt(2 pi) > 5/2
+
+
+def gauss_tail2(mu_abs, sd, t):
+    """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd."""
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_tail1(mu_abs, sd, t):
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), phibar_upper((t - mu_abs) / sd))
+
+
+# ----------------------------------------------------------------------------- bands, boxes, runs
+B_BOXES = ((Fr(0), Fr(1, 64)), (Fr(1, 64), Fr(1, 16)), (Fr(1, 16), Fr(1, 4)), (Fr(1, 4), Fr(1)))
+K_BOXES = ((Fr(1, 2), Fr(1)), (Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(7, 4)), (Fr(7, 4), Fr(15, 8)), (Fr(15, 8), Fr(31, 16)),
+           (Fr(31, 16), Fr(63, 32)), (Fr(63, 32), Fr(2)))
+HIGH_OCTAVES = 3           # octaves [2^-(j+1), 2^-j], j = 6..8 (r up to 1/64), split into 8 sub-bands
+TOP_OCTAVES = 4            # octaves j = 9..12, split into 16 sub-bands
+LOW_OCTAVES = 16           # octaves j = 13..28, split into 4 sub-bands
+R_LAST = Fr(1, 2 ** 29)    # last band [0, 2^-29]
+
+
+def bands():
+    """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST]."""
+    out = []
+    for j in range(6, 9 + TOP_OCTAVES + LOW_OCTAVES):
+        hi = Fr(1, 2 ** j)
+        lo = hi / 2
+        m = 8 if j < 9 else (16 if j < 9 + TOP_OCTAVES else 4)
+        for i in range(m):
+            out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m))
+    out.append((Fr(0), R_LAST))
+    require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover')
+    return sorted(out)
+
+
+def up(x, digits=6):
+    """Decimal string of a rational rounded UP to `digits` significant digits (as a string float)."""
+    x = Fr(x)
+    if x <= 0:
+        return '0'
+    e = len(str(x.numerator // x.denominator)) if x >= 1 else -len(str(x.denominator // x.numerator))
+    scale = Fr(10) ** (digits - e)
+    n = -((-x.numerator * scale.numerator) // (x.denominator * scale.denominator))
+    return str(Fr(n) / scale) if scale.denominator == 1 else '%s' % (Fr(n) / scale)
+
+
+def dec_down(x, places=4):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_up(x, places=4):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+BOXES = tuple((bb, kb) for bb in B_BOXES for kb in K_BOXES)      # the order of the per-box lists in a band record
+# (x0, theta) choices; each gives a valid bound and the record keeps the smaller one
+COMBOS = ((Fr(1, 64), Fr(1, 32)), (Fr(1, 1024), Fr(1, 256)))
+# the fixed-axis SIDE24 point of [CAP] section 1 (b = 6/5, kappa = 1/6), a single (b, k) point outside the band
+SIDE24_B, SIDE24_K = (Fr(6, 5), Fr(6, 5)), (Fr(1, 6), Fr(1, 6))
+CG_02_LO, CG_SIDE24_LO = Fr('5324360.4426'), Fr('35736.7352')      # certified lower ends of c_G at (0, 2), (6/5, 1/6) ([G])
+
+
+def best_box(law, bb, kb, r0, r1, last):
+    best = None
+    for x0, theta in COMBOS:
+        res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=x0, theta=theta)
+        if best is None or res['total'] < best['total']:
+            best = res
+    return best
+
+
+def band_record(args):
+    """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down,
+    6 places) for each of the 28 boxes in BOXES order, the largest tail part (rounded up, 12 places), and the same two
+    numbers at the SIDE24 point."""
+    r0, r1 = args
+    law = band_law(r0, r1)
+    last = (r0 == 0)
+    bound, zfloor, tails = [], [], []
+    for bb, kb in BOXES:
+        res = best_box(law, bb, kb, r0, r1, last)
+        bound.append(dec_up(res['total']))
+        zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6))
+        tails.append(res['tail'])
+    s24 = best_box(law, SIDE24_B, SIDE24_K, r0, r1, last)
+    return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12),
+            'side24': [dec_up(s24['total']), dec_down(36 * SIDE24_K[0] ** 2 * s24['zt'], 6)]}
+
+
+def full_run(procs):
+    bl = bands()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, bl)
+    return assemble(recs)
+
+
+def assemble(recs):
+    table = {}
+    for rs in R_STARS:
+        mx, arg, zmin = Fr(0), None, None
+        for rec in recs:
+            if Fr(rec['r'][1]) <= rs:
+                for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']):
+                    v = Fr(bs)
+                    if v > mx:
+                        mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]}
+                    z = Fr(zs)
+                    zmin = z if zmin is None or z < zmin else zmin
+        table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG02': dec_up(mx / CG_02_LO, 6),
+                          'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    side = {}
+    for rs in R_STARS:
+        rows = [rec for rec in recs if Fr(rec['r'][1]) <= rs]
+        mx = max(Fr(rec['side24'][0]) for rec in rows)
+        arg = [rec['r'] for rec in rows if Fr(rec['side24'][0]) == mx][0]
+        zmin = min(Fr(rec['side24'][1]) for rec in rows)
+        side[str(rs)] = {'C': str(mx), 'argmax_r': arg, 'C_over_cG': dec_up(mx / CG_SIDE24_LO, 6),
+                         'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1', 'scientific_effect': 'NONE',
+            'certified': True, 'mutant': MUT, 'L_min': L_MIN,
+            'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]},
+            'side24_point': {'b': str(SIDE24_B[0]), 'k': str(SIDE24_K[0])},
+            'parameters': params(), 'C_table': table, 'C_table_side24': side, 'bands': recs}
+
+
+def params():
+    return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBOS_X0_THETA': [[str(a), str(b)] for a, b in COMBOS], 'THETA_E': str(THETA_E), 'QS': list(QS),
+            'R_STARS': [str(x) for x in R_STARS], 'HIGH_OCTAVES': HIGH_OCTAVES, 'TOP_OCTAVES': TOP_OCTAVES,
+            'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST),
+            'B_BOXES': [[str(a), str(b)] for a, b in B_BOXES], 'K_BOXES': [[str(a), str(b)] for a, b in K_BOXES]}
+
+
+def results_text(res):
+    """RESULTS.json layout: the summary indented, then one line per band record."""
+    head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True)
+    lines = ',\n'.join('  ' + json.dumps(rec, sort_keys=True) for rec in res['bands'])
+    return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n'
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def replay_sample():
+    """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band."""
+    bl = bands()
+    idx = []
+    for rs in R_STARS:
+        j = max(i for i, b in enumerate(bl) if b[1] == rs)
+        idx.append(j)
+    idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))]
+    idx.append(0)
+    return sorted(set(idx))
+
+
+def check_run(procs, full=False):
+    ref = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout')
+    require(ref['parameters'] == params(), 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json produced under a mutant')
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure')
+    bl = bands()
+    require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs')
+    idx = list(range(len(bl))) if full else replay_sample()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, [bl[i] for i in idx])
+    for i, rec in zip(idx, recs):
+        require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r']))
+    # the table is the maximum over the stored records
+    asm = assemble(ref['bands'])
+    require(asm['C_table'] == ref['C_table'] and asm['C_table_side24'] == ref['C_table_side24'],
+            'C tables are not the maxima of the stored bounds')
+    # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G(0, 2) of Math-#203 (a lower bound for every
+    # cap-route constant valid for all L >= 10, since c_G^{L,R} -> c_G^{ref} as L -> infinity)
+    for rs, row in ref['C_table'].items():
+        require(Fr(row['C']) > CG_02_LO, 'C below the sharp cap coefficient')
+    for rs, row in ref['C_table_side24'].items():
+        require(Fr(row['C']) > CG_SIDE24_LO, 'SIDE24 C below the sharp cap coefficient')
+    print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()},
+                      'C_table_side24': {rs: row['C'] for rs, row in ref['C_table_side24'].items()}, 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.mc:
+        rr, bb, kk, nn, sd = args.mc
+        print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd))))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({'band': {rs: row['C'] for rs, row in res['C_table'].items()},
+                      'side24': {rs: row['C'] for rs, row in res['C_table_side24'].items()}}))
+
+
+
+# ----------------------------------------------------------------------------- floating control (not part of the certificate)
+def mc_event(r, b, k, n, seed, deg=8, grid=9):
+    """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel (floating point): the Taylor jet of
+    order deg at the midpoint is sampled under Q, the field near D is rebuilt from it, W_r and G_r (M_3, M_4 on a grid of D)
+    are evaluated directly from the definitions."""
+    import random
+    pins, _ = functionals()
+    P = [pins[nm] for nm in PIN_NAMES]
+    jets = [(i, m - i) for m in range(deg + 1) for i in range(m, -1, -1)]
+    J = [[term(1, 0, a, bb, X0PT)] for (a, bb) in jets]
+
+    def ev(F, G, key):
+        ser, w, _ = pair_series(F, G, key)
+        return sum(float(c) * r ** p for p, c in ser.items())
+    Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(6)] for i in range(6)]
+    Sjp = [[ev(J[i], P[j], ('jp', deg, i, j)) for j in range(6)] for i in range(len(J))]
+    Sjj = [[ev(J[i], J[j], ('jj', deg, i, j)) for j in range(len(J))] for i in range(len(J))]
+
+    def solve(A, bv):
+        m = len(A)
+        M = [A[i][:] + [bv[i]] for i in range(m)]
+        for c in range(m):
+            p = max(range(c, m), key=lambda i: abs(M[i][c]))
+            M[c], M[p] = M[p], M[c]
+            for i in range(c + 1, m):
+                f = M[i][c] / M[c][c]
+                for j in range(c, m + 1):
+                    M[i][j] -= f * M[c][j]
+        x = [0.0] * m
+        for i in reversed(range(m)):
+            x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i]
+        return x
+    vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k, 0, 0, 0]
+    Wt = [solve(Spp, Sjp[i]) for i in range(len(J))]
+    mu = [sum(Wt[i][j] * vr[j] for j in range(6)) for i in range(len(J))]
+    C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(6)) for j in range(len(J))] for i in range(len(J))]
+    # symmetric square root (Jacobi)
+    m = len(J)
+    A = [row[:] for row in C]
+    V = [[float(i == j) for j in range(m)] for i in range(m)]
+    for _ in range(100):
+        off = sum(A[p][q] ** 2 for p in range(m) for q in range(p + 1, m))
+        if off < 1e-26:
+            break
+        for p in range(m - 1):
+            for q in range(p + 1, m):
+                if abs(A[p][q]) < 1e-300:
+                    continue
+                th = 0.5 * math.atan2(2 * A[p][q], A[q][q] - A[p][p])
+                c, s = math.cos(th), math.sin(th)
+                for kk in range(m):
+                    akp, akq = A[kk][p], A[kk][q]
+                    A[kk][p], A[kk][q] = c * akp - s * akq, s * akp + c * akq
+                for kk in range(m):
+                    apk, aqk = A[p][kk], A[q][kk]
+                    A[p][kk], A[q][kk] = c * apk - s * aqk, s * apk + c * aqk
+                for kk in range(m):
+                    vkp, vkq = V[kk][p], V[kk][q]
+                    V[kk][p], V[kk][q] = c * vkp - s * vkq, s * vkp + c * vkq
+    lam = [max(A[i][i], 0.0) for i in range(m)]
+    S = [[sum(V[i][kk] * math.sqrt(lam[kk]) * V[j][kk] for kk in range(m)) for j in range(m)] for i in range(m)]
+    fact = [math.factorial(i) for i in range(deg + 2)]
+    pts = [(-2 * r + 4 * r * i / (grid - 1), -2 * r + 4 * r * j / (grid - 1)) for i in range(grid) for j in range(grid)]
+
+    def deriv(cf, a0, b0, x, y):
+        tot = 0.0
+        for idx, (a, bb) in enumerate(jets):
+            if a >= a0 and bb >= b0:
+                tot += cf[idx] * x ** (a - a0) * y ** (bb - b0) / (fact[a - a0] * fact[bb - b0])
+        return tot
+    rng = random.Random(seed)
+    num = den = 0.0
+    acap = 4 / (3 * k)
+    for _ in range(n):
+        z = [rng.gauss(0, 1) for _ in jets]
+        cf = [mu[i] + sum(S[i][j] * z[j] for j in range(m)) for i in range(m)]
+        hM = [deriv(cf, 2, 0, -r / 2, 0.0), deriv(cf, 1, 1, -r / 2, 0.0), deriv(cf, 0, 2, -r / 2, 0.0)]
+        hS = [deriv(cf, 2, 0, r / 2, 0.0), deriv(cf, 1, 1, r / 2, 0.0), deriv(cf, 0, 2, r / 2, 0.0)]
+        dM = hM[0] * hM[2] - hM[1] ** 2
+        dS = hS[0] * hS[2] - hS[1] ** 2
+        if not (hM[0] < 0 and dM > 0 and dS < 0):
+            continue
+        Wv = abs(dM * dS)
+        den += Wv
+        lamv = -hM[2]
+        M3 = max(abs(deriv(cf, a, 3 - a, x, y)) for (x, y) in pts for a in range(4))
+        M4 = max(abs(deriv(cf, a, 4 - a, x, y)) for (x, y) in pts for a in range(5))
+        if not (lamv > acap * r * M3 * M3 and r * M4 <= 3 * k / 10):
+            num += Wv
+    return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3, 'Z_over_r2': den / n / r ** 2}
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md
new file mode 100644
index 000000000..8758ddf22
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/NOTE.md
@@ -0,0 +1,364 @@
+# C8: the radial ledger on the torus with a vanishing remainder (Theorem L′)
+
+**Object:** CL-C8-TORUS-VANISHING-REMAINDER-20261001-v1.
+**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 1 October 2026.
+**Disposition:** author-side theorem with certified constants; no review yet. It extends Math-#215 (Theorem L) and closes
+the item that #215 lists as not given ("a vanishing-remainder rate"), which Math-#193 (comment 5932553031) records among the
+open C8 items as "a vanishing torus remainder".
+**Scientific effect:** NONE. No `STATUS`, `PROOF_INDEX`, `GRAPH`, claim, catalog, prize or source-body change. Same GitHub
+account as every lane; zero organizational-independence credit. Nothing in Math-#215, #206 or #212 is touched.
+**Delivered under:** Dylan Roy's explicit instructions in this session, given after the 2026-09-27 owner stop (quoted in
+`SOURCE_MAP.json`, `delivered_under`). `OWNER_STOP.md` addresses Cursor agents and automations and is untouched.
+
+## 0. Statement
+
+Notation as in Math-#215 ([B]) and [LP]: `A_r(b, k, u) = 12 pi_r(R; v_r) Z_r / r^2` is the radial ledger of [LP] (10.2) for
+the field with kernel `K`, the band is `B x K = [0, 1] x [1/2, 2]`, and `A_0 = lim_{r -> 0} A_r` is the contact integrand
+**of the same kernel**. For the Gaussian kernel `A_0 = A_0^ref` (closed form, [B] §1); for the torus kernel `K_L` it is
+its own `A_0^(L)`, which [B] only locates within `eps_d` of `A_0^ref`.
+
+**Theorem L′.** For `d = 2, 3`, every torus side `L >= 10`, every frame, every `b in B`, `k in K`, every `R` in the
+table and every `0 < r <= R`,
+
+    -(a_dn(R) r^2 + beta_dn r)  <=  A_r(b, k, u) / A_0(b, k, u) - 1  <=  a_up(R) r^2 + beta_up r.
+
+The constants are exact rationals in `RESULTS.json` (key `rate_torus`), shown rounded up:
+
+| `R` | `a_up`, `d = 2` | `a_dn`, `d = 2` | `a_up`, `d = 3` | `a_dn`, `d = 3` |
+|---|---|---|---|---|
+| `1/4096` | `13.822900` | `3.637234` | `25.666574` | `25.516405` |
+| `1/2048` | `13.825546` | `3.643346` | `26.256538` | `26.108063` |
+| `1/1024` | `13.830904` | `3.655573` | `27.446809` | `27.301701` |
+| `1/512` | `13.841883` | `3.680046` | `29.869058` | `29.730582` |
+| `1/256` | `13.864900` | `3.729062` | `34.883096` | `34.757367` |
+| `1/128` | `13.915213` | `3.827402` | `45.611669` | `45.508415` |
+| `1/64` | `14.033374` | `4.025510` | `70.060722` | `69.981331` |
+
+with first-order coefficients
+
+    beta_2 <= 1.036 x 10^-9,        beta_3 <= 1.914 x 10^-7        (every R in the table).
+
+**For the Gaussian kernel the first-order coefficients vanish identically** (`rate_reference_kernel`: `beta = 0` exactly),
+and the `a` columns are the same: `|A_r / A_0^ref - 1| <= a(R) r^2`, with no additive remainder.
+
+What changes against [B]:
+- **The remainder vanishes.** [B] proves `|A_r / A_0^ref - 1| <= a r^2 + eps_d`, with `eps_2 <= 3.96 x 10^-10` and
+  `eps_3 <= 1.79 x 10^-8` independent of `r`. Theorem L′ compares `A_r` with the limit of the same kernel, and every
+  remainder carries a factor `r` or `r^2`. In particular `A_r^(L) = A_0^(L) (1 + O(r))` on the torus with explicit
+  constants, and `A_r = A_0^ref (1 + O(r^2))` for the Gaussian kernel with no remainder at all.
+- **The second-order coefficients are [B]'s.**
+  - `a(R)` is never below [B]'s Theorem L value.
+  - It equals [B]'s to the displayed digits for `R <= 1/2048`.
+  - It exceeds [B]'s by at most `1.1 x 10^-5` for `R <= 1/512` and by at most `7.4 x 10^-4` at `R = 1/64`. The excess comes
+    from the cross terms and the `r^9` remainders, §§2, 5.
+- **The first-order coefficient is a torus artefact.** `beta` is a bound on the torus part of the `r^1` coefficients
+  (§2, Lemma V2). It is not sharp: the actual torus corrections are of order `e^-50`, while `beta` comes from Cauchy
+  estimates at `M_1(10) < 4.46 x 10^-15`.
+
+**Corollary L′1 (the densities with a vanishing remainder).** Let `nu_cand`, `nu_eld` be the compact-window densities of [LP]
+Theorem B in the versions (11.2), for the torus kernel or the Gaussian kernel, and
+`c_{B,K} = int_{B x K x S^(d-1)} A_0(b, k, u)/(3 k^(2/3)) db dk dsigma(u)` the leading constant **of the same kernel**. The
+angular integral uses ordinary surface measure `dsigma`, as in [LP] (11.2)–(11.3). The torus is not assumed isotropic. For `0 < ell < min(r_pop, R)^3 / 2`, with `r_pop` [LP]'s population radius:
+
+    -(c2_dn ell^(2/3) + c1_dn ell^(1/3))  <=  ell^(1/3) nu_cand(ell) - c_{B,K}  <=  c2_up ell^(2/3) + c1_up ell^(1/3),
+    ell^(1/3) nu_cand(ell) - C_{B,K}(R) ell  <=  ell^(1/3) nu_eld(ell)  <=  ell^(1/3) nu_cand(ell),
+
+with `C_{B,K}(R)` of [B] Corollary L1, and (`corollary_torus`; ratios to `c_{B,K}`, rounded up):
+
+| `R` | `c2_up/c`, `d = 2` | `c2_dn/c`, `d = 2` | `c2_up/c`, `d = 3` | `c2_dn/c`, `d = 3` |
+|---|---|---|---|---|
+| `1/4096` | `6.282558` | `4.464593` | `23.153357` | `22.427357` |
+| `1/512` | `6.322008` | `4.505526` | `27.377121` | `26.661848` |
+| `1/64` | `6.643378` | `4.836304` | `67.060479` | `66.396981` |
+
+and `c1/c <= 1.531 x 10^-10` (`d = 2`), `<= 3.985 x 10^-8` (`d = 3`). For the Gaussian kernel `c1 = 0`
+(`corollary_reference_kernel`).
+
+So, on the torus, both densities are `c_{B,K} ell^(-1/3) + O(1) + O(ell^(1/3))` about the **torus** constant, with explicit
+constants and a vanishing remainder. [B] gave the same expansion only about `c^ref_{B,K}`, up to `eps'_d ell^(-1/3)`.
+
+**Corollary L′2 (expected selected counts; derived in review C63, 5387311588).** Let `n_sel(t)` be the expected number per
+unit volume of selected elder pairs of the population with lifetime in `(0, t]`. Multiply by `L^d` for the expected total on
+the torus. For `0 < t < ell_* = min(r_pop, R)^3/2`, integrating Corollary L′1 and its elder line over `(0, t]` gives
+
+    max(0, (3/2) c t^(2/3) - c1_dn t - (3/4) c2_dn t^(4/3) - (3/5) C_{B,K}(R) t^(5/3))
+        <=  n_sel(t)  <=  (3/2) c t^(2/3) + c1_up t + (3/4) c2_up t^(4/3),
+
+with `c = c_{B,K}` of the same kernel. Equivalently, `n_sel(t) = (3/2) c t^(2/3) (1 + delta)` with
+`-dn <= delta <= up`. Every term is increasing in `t`, so the values at `t = ell_*` bound the whole window. The ratios
+use `corollary_torus` here and `C_{B,K}(R)` of [B] (`RESULTS.json` blob `894fae2e`), divided by `c >= c_ref,lo (1 - eps_0)`;
+all are rounded up:
+
+| `d` | `R` | `ell_*` (`r_pop >= R`) | `up` | `dn` |
+|---|---|---|---|---|
+| 2 | `1/4096` | `2^-37` | `1.18e-7` | `5.75e-7` |
+| 2 | `1/1024` | `2^-31` | `1.90e-6` | `3.37e-5` |
+| 2 | `1/512` | `2^-28` | `7.60e-6` | `2.73e-4` |
+| 3 | `1/4096` | `2^-37` | `4.35e-7` | `4.03e-6` |
+| 3 | `1/1024` | `2^-31` | `7.50e-6` | `2.51e-4` |
+| 3 | `1/512` | `2^-28` | `3.29e-5` | `3.84e-3` |
+
+Each scope applies:
+- the upper bound has the torus candidate corollary's scope;
+- the lower bound additionally has the elder corollary's scope, through `C_{B,K}`.
+
+The bound is per unit volume, for the compact-window population only. It is not a count of all short bars.
+
+**Four result scopes, and what each consumes.** Every scope also retains the corrected parent [LP] + [CAP] + [E1] + [E2] +
+[REC] (§8) and its Gaussian-field, typed-weight and Kac–Rice interpretation.
+1. *Theorem L′*, the main radial rate, for both kernels: §§2–4 and the band-law, primitive and moment contracts of [E], [E3].
+2. *The Gaussian-kernel candidate corollary*: the rate and the reference integrals; `eps_0 = 0`.
+3. *The torus candidate corollary*: additionally the finite contact comparison `eps_0` (§5).
+4. *The elder corollary, for either kernel*: additionally [B]'s cap-derived `C_{B,K}(R)` (§5).
+
+## 1. Where [B]'s remainder came from
+
+[B]'s `eps_d` collects five things, each `r`-independent:
+1. the Taylor-model remainder `e` (rounding, truncated powers `r^j`, `j > 8`, the covariance tail beyond `g_x = 60`);
+2. the torus part of each covariance, as one constant `M_1`-bound;
+3. the kernel normalization `Theta` in the pin density (`n theta_1 / 2`);
+4. in `d = 3`, the Loewner coupling's truncation at radius 12 (`out_D2`, `tail_c`), present even for the Gaussian kernel;
+5. the far set `G_0^c`, bounded at its worst radius `R`.
+
+Theorem L′ removes all five:
+1. Every rounding goes into a coefficient radius and every truncation into a remainder proportional to `r^9` (§2).
+2. The torus part is split coefficientwise (Lemma V2). Its constant term cancels against `A_0^(L)`, and its higher terms
+   carry powers of `r`.
+3. `Theta` multiplies only increments (Lemma V3).
+4. The main normalizer term is compared at `r` and at `0` directly, without truncation (Lemmas V4, V5).
+5. The far set is bounded by Markov's inequality at order 64, which gives `O(r^2)` (Lemma V6).
+
+## 2. Interval-coefficient Taylor models (Lemmas V1, V2)
+
+**Lemma V1 (the class `TV`).** A `TV` on `[0, R_0]`, `R_0 = 1/64`, is a list of centers `C_j`, radii `D_j` (`j <= 8`) and a
+remainder coefficient `E`. It encloses a function `f` if there are **fixed** numbers `c_j in [C_j - D_j, C_j + D_j]` and a
+function `rho` with `|rho(r)| <= E r^9` such that `f(r) = sum c_j r^j + rho(r)` for every `r in [0, R_0]`. Then `f(0) = c_0`.
+Sums, products, scalar multiples and reciprocals of enclosed functions are enclosed by the operations of `theorem_v.py`:
+- **Products.** The coefficient `sum_{i+j=k} a_i b_j` lies in the interval product. For `k <= 8` its center is rounded
+  with the error added to the radius. For `k > 8` the term is at most `(|S_k| + Rd_k) R_0^(k-9) r^9`. The cross terms
+  `rho_a g_b`, `rho_b g_a`, `rho_a rho_b` are at most `(E_a |g_b| + E_b |g_a| + E_a E_b R_0^9) r^9`, where `|g| <= sum (|C_j| + D_j) R_0^j`.
+- **Reciprocals.** Write `f = c_0 + h`, with `h(0) = 0` and `u = 1/c_0` in an interval. The interval is centered at
+  `1/C_0`, so that centers stay the reference values. Then `g = h u` has `|g(r)| <= G_1 r`, and
+  `1/f = u sum_{n <= 8} (-g)^n + u theta (G_1 r)^9/(1 - G_1 R_0)`. The certificate requires `G_1 R_0 < 1/2` and `g(0) = 0`.
+
+Nothing is assumed about the coefficients beyond their intervals. In particular the same unknown number appears
+consistently wherever an enclosed function is reused, because each operation is sound for every admissible choice.
+
+**Readers.** For `0 < r <= R`, with `H = sum_{j >= 3} (|C_j| + D_j) R^(j-2) + E R^7`:
+
+    f(r) - f(0)  in  [c_1^lo r + (c_2^lo - H) r^2,  c_1^hi r + (c_2^hi + H) r^2].
+
+**Lemma V2 (the torus part, coefficientwise).** For the covariance of two preconditioned functionals with weight `w`:
+- [E] §2(c) writes the torus part as `sum_{g_x >= w} partial^gamma rho(0) pi_gamma r^(g_x - w)`, with
+  `|partial^gamma rho(0)| <= g_x! g_y! M_1` (Cauchy estimates on the unit polydisc).
+- `|pi_gamma| <= sum |c c'| |d|^m/m!`, with `m = g_x - A`, `|d| <= 1`.
+- `g_x!/(g_x - A)! <= w^A e^(g_x - w)`, because `A log x <= w(x - 1)` for `x = g_x/w >= 1`.
+
+So the coefficient of `r^p` in the torus part is at most `T_0 e^p`, with `T_0 = sum |c c'| g_y! max(w, 1)^A M_1`, the
+quantity [E] uses at `p`-sum level. The certificate adds:
+- radii `T_0 e^p` to the coefficients `p = 0, ..., 8`;
+- `T_0 e^9 / (1 - e R_0) r^9` to the remainder.
+
+The reference series gives the centers exactly. The truncation tail beyond `g_x = 60` is at most
+`2 |c c'| |h(g_y)| 61^A r^(61 - w)` for `r <= 1/64` ([E] §2(b)), so it goes into the `r^9` remainder (`61 - w >= 9` is
+checked). `M_1(10) < 4.46 x 10^-15` (`d = 2`) and `< 1.94 x 10^-13` (`d = 3`) decrease in `L`, so the models hold for every
+`L >= 10` and every frame; with `M_1 = 0` they are the Gaussian kernel's.
+
+The pins (six in `d = 2`, eight in `d = 3`), the targets of [B] §2 and Gaussian elimination without pivoting give `TV`s for `det S_r`, the energies `e_bb`,
+`e_bk`, `e_kk`, the regression coefficients `alpha`, `beta` of the transverse Hessian and of `Omega'`, their covariances, and in
+`d = 3` the regression of `Omega'` on `B = -A`. The certificate checks that the centers are [B]'s reference laws. In
+particular the law of the transverse Hessian is flat in `r` ([B] Lemma 1): every center of `alpha_A + 1`, `beta_A` and
+`Cov(A) - Cov(GOE)` vanishes to `10^-60`, so on the torus only radii survive.
+
+## 3. The pin density (Lemma V3)
+
+With `S_r` the unnormalized pin covariance and `Theta in [1, 1 + theta_1]` the normalization, `pi_r ∝ det(S_r/Theta)^(-1/2)
+exp(-Theta E_r/2)`. So
+
+    log(pi_r / pi_0) = -(1/2) log(det S_r / det S_0) - Theta (E_r - E_0)/2.
+
+`Theta^(n/2)` cancels and `Theta` multiplies only the increment `E_r - E_0`. With `x = (det S_r - det S_0)/det S_0` and
+`|x| <= x_m < 1/2`:
+- `-(1/2) log(1 + x) <= max(0, -x)/(2(1 - x_m))`;
+- `(1/2) log(1 + x) <= max(0, x)/2`.
+
+The increments of `det S` and of the energies are read off the `TV`s as above. On a box with `b, k >= 0` the energy weights
+`b^2`, `2bk`, `k^2` are nonnegative and the bounds are taken coefficientwise. There is no constant term:
+`log(pi_r/pi_0) in [-(dn_a r^2 + dn_b r), up_a r^2 + up_b r]`.
+
+## 4. The normalizer (Lemmas V4–V6)
+
+As in [B] §4 (exact expansion of `W_r = r^2 P1 P2` on the typed set, the three first-order cancellations, Hölder for the
+second-order terms):
+
+    Z_r / (36 k^2 r^2) = m_r + (first-order term) + (second-order terms),
+
+where `m_r = E_r[lambda_+^2]` (`d = 2`) or `E_r[det(B)^2 1{B > 0}]` (`d = 3`), under the law at `r`, and `m_0` is the same
+under the law at `0`. Then `A_r / A_0 = (pi_r/pi_0) (Z_r / (36 k^2 r^2 m_0))`, and every term below is divided by `m_0`.
+
+**Lemma V4 (`d = 2`, main term).** `lambda ~ N(mu, sigma^2)` with `mu = -(b alpha_w + k beta_w)`, `sigma^2 = C_ww / Theta`, and
+`F(mu, sigma) = E[(mu + sigma z)_+^2]`. Then `dF/dmu = 2 E[lambda_+]` and `dF/d(sigma^2) = Phi(mu/sigma) <= 1`. Along the
+segment from `(mu_0, sigma_0^2)` to `(mu_r, sigma_r^2)`, which stays in the band's range,
+
+    |m_r - m_0| <= 2 F_1,max |mu_r - mu_0| + |C_ww(r) - C_ww(0)|,      m_0 >= F_min.
+
+Both increments are `TV` increments. For the Gaussian kernel they vanish ([B] Lemma 1).
+
+**Lemma V5 (`d = 3`, main term).** In natural coordinates `(B_11, B_22, sqrt2 B_12)` the Frobenius norm is Euclidean. Couple
+`B_r = mu_r + C_r^(1/2) Z` and `B_0 = mu_0 + C_0^(1/2) Z`.
+- `g(B) = det(B)^2 1{B > 0}` is continuous, since `det B = 0` on the boundary of the cone.
+- `g` is piecewise smooth along segments, with `|grad g| = 2 |det B| ||adj B||_F <= ||B||_F^3` (`2 x 2`:
+  `|det B| <= ||B||_F^2/2`, `||adj B||_F = ||B||_F`).
+
+Hence
+
+    |m_r - m_0| <= E[max(||B_r||, ||B_0||)^3 ||B_r - B_0||] <= sqrt2 ||B||_8^3 (||mu_r - mu_0|| + ||C_r^(1/2) - C_0^(1/2)||_F).
+
+Square roots satisfy `||C_r^(1/2) - C_0^(1/2)||_F <= ||C_r - C_0||_F / (2 sqrt(lambda_min))`. Proof:
+- `X = C_r^(1/2) - C_0^(1/2)` solves `C_r^(1/2) X + X C_0^(1/2) = C_r - C_0`.
+- In the eigenbases, `|X_ij| (s_i + t_j) = |(C_r - C_0)_ij|`.
+- Here `lambda_min` is the Gershgorin floor over the band.
+
+`||B||_8` is the band's Frobenius moment bound, and `m_0 >= m_3(b) rho_lo`, with `rho_lo` from [B]'s Loewner coupling over
+`[0, R]` (which contains `r = 0`). For the Gaussian kernel the increments vanish.
+
+**First-order terms.** For `d = 2`, the term is `-36 k^2 r E[lambda_+ om]`. For `d = 3`, it is `-36 k^2 r E[D tr(A* Omega') 1]`.
+- [B] writes each through the regression, with coefficients `mu_om`, `bw_om` (`d = 2`) and `beta(r)`, `mu_Omega`, `DG`
+  (`d = 3`).
+- Each `TV` coefficient satisfies `|t(r)| <= c_0 + s r`.
+- The explicit factor `r` turns `c_0` (a torus radius; `0` for the Gaussian kernel) into a contribution to `beta`. It turns
+  `s` into a contribution to `a`.
+- In `d = 3`, `r beta(r) - r^2/2 = c_0 r + (c_1 - 1/2) r^2 + ...`, with `c_1` centered at `1/2`.
+
+**Second-order terms.** These are [B]'s, with its interval band laws, divided by `m_0` instead of the reference moment:
+- in `d = 2`: the signed `kappa` term, `T_4`, `T_5`, `T_6` and the thin set;
+- in `d = 3`: `U_2`, ..., `U_10` and the thin set.
+
+**Lemma V6 (the far set).** `G_0^c = {|T| > 3k/(2r)} u {|tau| > 3k/(2r^2)}` enters through
+`E[W_r 1_(G_0^c)]/r^2 <= P1n P2n P(G_0^c)^(1/2)` ([B] §5). By Markov's inequality at order `p = 64`:
+
+    P(G_0^c) <= (2 ||T||_64/(3 k_0))^64 R^60 r^4 + (2 ||tau||_64/(3 k_0))^64 R^124 r^4,
+
+so this term is `O(r^2)`.
+
+## 5. Assembly
+
+`(1 + p)(1 + z) - 1 <= p + z + p z(R)` and `1 - (1 - p)(1 - z) <= p + z`, with `exp(x) - 1 <= x exp(x(R))` for the pin factor.
+This gives Theorem L′ on each of the 96 boxes `[i/8, (i+1)/8] x [1/2 + j/8, 1/2 + (j+1)/8]` (as in [B]) and each `R`.
+
+**Corollary L′1.** As in [B] Corollary L1:
+- `ell^(1/3) nu_cand(ell) = int_{B x K x S^(d-1)} A_r(b, k, u) / (3 k^(2/3)) db dk dsigma(u)`, with `r = (ell/k)^(1/3) <= R`
+  ([LP] (11.2)).
+- Theorem L′ holds uniformly in the direction `u`. Integrate it box by box, over each box and all of `S^(d-1)`:
+  `a r^2` gives the weight `int_{box x S^(d-1)} A_0 / (3 k^(4/3))`, and `beta r` gives `int_{box x S^(d-1)} A_0 / (3 k)`.
+  No isotropy is used. The reference weights of the pinned `theorem_b.py` are these full angular integrals for `A_0^ref`
+  (its `K_const` includes `|S^(d-1)|`), so the program needs no extra sphere factor (OA-227-C-01, review 5384474393).
+- The weights of the torus `A_0^(L)` are at most `(1 + eps_0)` times the reference weights. Here `eps_0` is [B]'s
+  `|A_0/A_0^ref - 1|` bound, recomputed by the pinned `theorem_b.py` at `R = 1/4096`:
+  `eps_0 <= 3.95 x 10^-10` (`d = 2`), `1.78 x 10^-8` (`d = 3`).
+
+  Only the contact (`r -> 0`) value of that bound is used. Review C60 (5386707554) proves it directly at contact:
+  - it uses the C57 contact intervals, Gaussian log-moment derivatives (`d = 2`), a Loewner and chi-tail comparison
+    (`d = 3`), the normalized pin density, and dominated convergence;
+  - it gives `|A_0^(L)/A_0^ref - 1| <= 3.943 x 10^-10` (`d = 2`) and `<= 1.778 x 10^-8` (`d = 3`) for `L >= 10`, every
+    frame and the compact marks.
+
+  So the torus corollary does not rest on the positive-radius parts of [B]'s Theorem L.
+- The elder line uses [B]'s `C_{B,K}(R)`. Review C61 (5386924911, at Math-#215 `b41eb28`) validates all fourteen published
+  `C_{B,K}` and `C_121` upper constants by a cellwise transfer. The transfer uses this packet's reviewed rate and C60's
+  contact comparison, the two cap engines at [B]'s caller, and the corrected parent set. C61's scope is
+  `0 <= nu_cand - nu_eld <= C_{B,K}(R) ell^(2/3)` for `ell < min(r_pop, R)^3/2` on the compact window. C61 does not claim
+  that every short bar is a candidate, nor an unrestricted difference rate.
+
+## 6. Verification
+
+- `python3 -B -S theorem_v.py --check` recomputes everything and compares it with `RESULTS.json` exactly:
+  - the `TV` laws of both kernels;
+  - the 2 x 7 x 96 box bounds per kernel;
+  - the reference integrals and the `r^1` weights;
+  - `eps_0`;
+  - the corollaries.
+
+  It takes about 3.5 minutes on one core and passes under `-B -S` and `-B -O -S`.
+- Five mutants, each of which must exit 1:
+  - `torus-const-only`: the torus part only in the constant coefficient;
+  - `exp-radius`: `e^p` replaced by `1` in the torus radii;
+  - `no-lin`: drop the `r^1` coefficients;
+  - `no-om-term`: drop the first-order terms;
+  - `coupling-half`: halve Lemma V5's constant.
+- The workflow `.github/workflows/c8-torus-vanishing-remainder.yml` checks:
+  - the manifest;
+  - the main-resident pins [LP], [CAP], [E1], [E2], [REC];
+  - the packet copies [B], [E], [E3], byte for byte;
+  - both modes;
+  - the mutants;
+  - a clean tree.
+
+**Consistency with [B].** With `beta` set aside, `a(R)` reproduces [B]'s Theorem L table, as in §0. The `c2/c` ratios
+reproduce [B]'s Corollary L1 to within `5 x 10^-4`. Both come from an independent assembly, using increments rather than
+deviations from `A_0^ref`.
+
+## 7. What this does not do
+
+- **Still the compact window `B x K`, `L >= 10`, `d = 2, 3`, `R <= 1/64`.** These are [B]'s ranges.
+- **`beta` is not the true first-order coefficient.** It bounds the torus part of the `r^1` coefficients through
+  `M_1`. Whether the torus `A_r` has a nonzero `r^1` term at all (of order `e^-50`) is not decided.
+- **Consumed, with their review status (OpenAI / Codex, delegated; same account, zero organizational-independence credit):**
+  - [LP]'s sections 8–12, as in [B], read with [E1], [E2], [REC];
+  - [B]'s §§2–5 algebra (exact expansions, first-order cancellations, Hölder bounds, Loewner coupling): scoped reads
+    5378815703, 5379687608, 5379747967, 5379848224, 5384256400. Its positive-radius Theorem L is not accepted as a whole;
+    this packet uses only these assembled conclusions of Theorem L, namely its contact value (C60) and its `C_{B,K}` (C61),
+    while the scoped [B] algebra and reference routines above remain consumed;
+  - the band laws, interval and transcendental primitives and moment routines of [E], [E3]: reviewed at the contracts used
+    here (C57, C59).
+
+  [B], [E] and [E3] are author-side and unmerged.
+- No `C`, `r_*`, `z_*` beyond [B], #206, #212 and #195. No new count, moment or elder statement beyond Corollary L′1.
+- Same GitHub account as every lane; zero organizational-independence credit. **I will not merge.**
+
+## 8. Provenance
+
+- **Sources** (`SOURCE_MAP.json`):
+  - [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (on `main`, verified by the workflow), read
+    with [CAP] `imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md` and its mandatory amendments, all on
+    `main` and verified:
+    - [E1] `imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md`;
+    - [E2] `reviews/d1_section9_borel_repair_20260925/REPAIR.md`;
+    - [REC] `reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md`;
+  - [B] Math-#215 `theorem_b.py` at `1aa250d` (v1.5; packet copy `theorem_b.py`), and its `NOTE.md` and `RESULTS.json` at
+    `d77cabf` (provenance);
+  - [E] Math-#206 `theorem_a.py` at `13f932c` (v1.2; packet copy `engine_e2.py`);
+  - [E3] Math-#212 `theorem_a3.py` at `250b907` (v1.1; packet copy `engine_e3.py`).
+- **Files:**
+  - `NOTE.md`, `README.md`;
+  - `theorem_v.py` (new);
+  - the copies `theorem_b.py`, `engine_e2.py`, `engine_e3.py`;
+  - `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`.
+
+## 9. Revisions
+
+- **v1.1 (`75b5fce`):** the explicit angular integrals (OA-227-C-01).
+- **v1.2:** documentation only, recording the review cycle C56–C61. In every changed Python file the AST with docstrings
+  removed is unchanged, and so are `RESULTS.json` and every constant.
+  - **NOTE.**
+    - "six pins" becomes "six in `d = 2`, eight in `d = 3`" (C57).
+    - §0 gains the four result scopes.
+    - §5 records C60's direct contact comparison and C61's `C_{B,K}` disposition.
+    - §§7–8 add the corrected parent set [E1], [E2], [REC] (C61) and the current review status, and §8 pins [CAP] as well,
+      so that this consumer verifies the whole parent set it cites (C63, 5943439412).
+  - **v1.2.3.** It adds Corollary L′2 (§0), the expected selected-count bound derived in review C63, with its relative
+    error table. The integration is checked symbolically, and the numbers are computed with exact rationals from the cited
+    tables.
+  - **v1.2.2 (C63, 5943465973).** Three wording changes:
+    - the `TV` docstring adds C56's integer `E`, integer `s, N >= 0` with `N >= 2` for the second-order readers, and
+      exact rational reader inputs;
+    - Corollary L′1's window becomes `ell < min(r_pop, R)^3/2`, as in §5;
+    - §7 says "these assembled conclusions of Theorem L".
+  - **`theorem_v.py`.**
+    - The `TV` docstring states its preconditions (C56).
+    - `combine_v` states the positivity case split of the lower bound (C58).
+    - A comment at the planar `omega` regression says that `Theta` cancels exactly and `cwlo` is conservative (C58).
+    - A comment in `explained` keeps the diagonal surplus (C58).
+  - **The copies follow Math-#215 v1.5, Math-#206 v1.2 and Math-#212 v1.1.** These change docstrings and comments only:
+    `sqrt_up` (`2^-198`, C59-DOC-01), `box_bound`'s parameter domain (C61), `wq_bound`'s diagonal surplus (C58), and the
+    `3/4` multiplier comment (C60).
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/README.md b/frontiers/c8_torus_vanishing_remainder_20261001/README.md
new file mode 100644
index 000000000..c7aed0099
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/README.md
@@ -0,0 +1,19 @@
+# C8: the radial ledger on the torus with a vanishing remainder
+
+Theorem L′ (`NOTE.md`): for the torus kernel (every side `L >= 10`, every frame) and the Gaussian kernel, `d = 2, 3`, on the
+band `b in [0, 1]`, `k in [1/2, 2]`,
+
+    -(a_dn r^2 + beta_dn r)  <=  A_r / A_0 - 1  <=  a_up r^2 + beta_up r        (0 < r <= R <= 1/64),
+
+with `A_0` the contact integrand of the same kernel. The `a(R)` are Math-#215's Theorem L constants (13.82 / 3.64 in the
+plane, 25.67 / 25.52 in `d = 3` at `R = 1/4096`); `beta_2 <= 1.04e-9`, `beta_3 <= 1.92e-7` on the torus and `beta = 0` for
+the Gaussian kernel. This removes the `r`-independent `eps_d` of Math-#215. Corollary L′1 gives `ell^(1/3) nu(ell)` about
+the torus's own `c_{B,K}` with remainder `c1 ell^(1/3) + c2 ell^(2/3)`.
+
+Replay (standard library, about 3.5 minutes):
+
+    python3 -B -S theorem_v.py --check
+    python3 -B -S theorem_v.py --mutant torus-const-only     # must exit 1 (also exp-radius, no-lin, no-om-term, coupling-half)
+
+`theorem_b.py`, `engine_e2.py` and `engine_e3.py` are byte-identical copies of Math-#215's files (and of the engines of
+Math-#206, #212), pinned in `SOURCE_MAP.json`. Scientific effect NONE; author-side; not merged by its author.
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/RESULTS.json b/frontiers/c8_torus_vanishing_remainder_20261001/RESULTS.json
new file mode 100644
index 000000000..70681a5db
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/RESULTS.json
@@ -0,0 +1,18129 @@
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+  {
+   "path": "theorem_b.py",
+   "bytes": 53141,
+   "sha256": "1e510cc385b2ed6502e2864cfdbd60514d92ed4217b035083dbf822d6e1543a1",
+   "git_blob": "36b39dbd33ef2e6d277ce1c3820b3f0d6659d779"
+  },
+  {
+   "path": "theorem_v.py",
+   "bytes": 41036,
+   "sha256": "1a3bc1c82bab6ea2b5a96cd7ad395fc3c6a7d4bbedd2cee12cda5812d4e8699d",
+   "git_blob": "5fd6b75df48940f293f0c112c1005418dcd75bde"
+  }
+ ]
+}
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json
new file mode 100644
index 000000000..ffc6bbafb
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/SOURCE_MAP.json
@@ -0,0 +1,131 @@
+{
+ "schema": 1,
+ "object": "CL-C8-TORUS-VANISHING-REMAINDER-20261001-v1",
+ "scientific_effect": "NONE",
+ "certified": true,
+ "review_status": "AUTHOR_SIDE_THEOREM_WITH_CERTIFIED_CONSTANTS",
+ "source_repository": "d6g8k5htny-coder/Math-",
+ "main_at_creation": "c2f1270d28192b3c06dd31f728ebf04d03d4246e",
+ "delivered_under": "explicit owner instructions of Dylan Roy in Claude Code session session_015wNj8LPTKXsaT68G3DgPPh, given after OWNER_STOP (2026-09-27): 2026-09-28 \"you have my full permisions and approval to make the proper changes and uploads\"; 2026-09-30 \"Work on closing as many mathematical items as possible while working with and supporting the other models in the GitHub. Also advance the mathematics in every place that's justified based on what we currently know. Ultra tokens, ultra depth @GitHub @Superpowers\" (repeated 2026-09-30 as \"Continue to Work on closing ...\"). Pull requests only, never merged by their author; no register, GRAPH, STATUS, lemma_closed, prize or premise edits. OWNER_STOP.md addresses Cursor agents and automations and is untouched.",
+ "note": "current_required sources ([LP], [CAP] and the mandatory parent amendments [E1], [E2], [REC]) are verified on the tree by the workflow; packet_copy sources ([B], [E], [E3]) are verified against the packet copies byte for byte; [B-NOTE] and [B-RESULTS] are read-only provenance pinned by identity",
+ "sources": [
+  {
+   "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+   "bytes": 40261,
+   "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+   "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+   "id": "LP",
+   "commit": "c2f1270d28192b3c06dd31f728ebf04d03d4246e",
+   "current_required": true,
+   "on_main": true,
+   "role": "sections 1, 3, 8-12: the model, the preconditioned pins, the radial ledger A_r = 12 pi_r Z_r / r^2 (10.2) and its limit A_0 (10.3), the lifetime pushforward (11.1)-(11.3); section 11 records that A_r -> A_0 was not assigned a numerical rate. Consumed as statements, as in Math-#215"
+  },
+  {
+   "path": "imports/lifetime_parent_20260925/MARKED_CYLINDER_CAP_PROOF.md",
+   "bytes": 15160,
+   "sha256": "0bf922b9203c29088b12388807aa0e2ecd020485eb0f6e919679841b5b2636fc",
+   "git_blob": "0633aca3c2a2882b0de4399da0a75d64c2e6b2e1",
+   "id": "CAP",
+   "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449",
+   "current_required": true,
+   "on_main": true,
+   "role": "section 1: the marked-cylinder good event G_r and the cap criterion; part of the corrected parent P + CAP + E1 + E2 + REC that every scope of Corollary L'1 retains (review C63), and through [B] and [E], [E3] the cap route behind the elder line's C_{B,K}"
+  },
+  {
+   "path": "imports/lifetime_parent_20260925/ERRATUM_CONGRUENCE.md",
+   "bytes": 1782,
+   "sha256": "bad7ef609c4ad8c41ad6af562c1b6807921e19a9d556ed793ad1a0db6e202028",
+   "git_blob": "213594d6ca6a86fb938110f4d166d9ce275a02d0",
+   "id": "E1",
+   "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449",
+   "current_required": true,
+   "on_main": true,
+   "role": "mandatory erratum to [LP]: the preconditioning congruence is diag(r^(-1/2), I) (corrected parent P + CAP + E1 + E2 + REC; review C61)"
+  },
+  {
+   "path": "reviews/d1_section9_borel_repair_20260925/REPAIR.md",
+   "bytes": 9062,
+   "sha256": "845abf9f9c99d672c2a10a887b5a2e7206a3d2de3d876f35f75ff6e2dc13e62f",
+   "git_blob": "fe9b9ce4999908bb3814b500ee2d0ceb0c6f704a",
+   "id": "E2",
+   "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449",
+   "current_required": true,
+   "on_main": true,
+   "role": "mandatory repair of [LP] section 9: the marked Kac-Rice identity for the Borel elder indicator (replaces LP section 9 wherever sections 9-11 are consumed)"
+  },
+  {
+   "path": "reviews/d1_chain_reconciliation_20260928/RECONCILIATION.md",
+   "bytes": 23312,
+   "sha256": "451b9d7ffee072a73fc904cab891b6693e3f233808b89df10cce1b57036b65da",
+   "git_blob": "75da2597971510f843f8d90c743950cb8c177342",
+   "id": "REC",
+   "commit": "6f74f7a77bae15d6aa07506bab38977e34dcc449",
+   "current_required": true,
+   "on_main": true,
+   "role": "chain reconciliation and reading rule for the parent: consumers cite P + CAP + E1 + E2 + REC; embedding condition r < L/(4 sqrt 2), met by r <= 1/64, L >= 10"
+  },
+  {
+   "path": "frontiers/c8_lifetime_difference_constant_20261001/theorem_b.py",
+   "bytes": 53141,
+   "sha256": "1e510cc385b2ed6502e2864cfdbd60514d92ed4217b035083dbf822d6e1543a1",
+   "git_blob": "36b39dbd33ef2e6d277ce1c3820b3f0d6659d779",
+   "id": "B",
+   "commit": "1aa250df3020dd71db8b091d1d4516bc5054e2b8",
+   "pull_request": 215,
+   "packet_copy": "theorem_b.py",
+   "current_required": false,
+   "on_main": false,
+   "role": "Math-#215 Theorem L certificate: the Taylor-model law (tm_law), the second-order term bounds of bounds_d2/bounds_d3 (reproduced in theorem_v.py with the new increments), the reference integrals (reference, b_cells, k_integral, K_const) and eps_0; carried byte for byte and used as a library"
+  },
+  {
+   "path": "frontiers/c8_lifetime_difference_constant_20261001/NOTE.md",
+   "bytes": 31488,
+   "sha256": "dfb4f7053354a9616239c833c6e7d05799a2e391c42e0ea6e90dde691a4db830",
+   "git_blob": "2e6a364297e19a6d3224dc78a225d75e58828452",
+   "id": "B-NOTE",
+   "commit": "d77cabf4fee5d36de2a85a810359c1c8812fc28c",
+   "pull_request": 215,
+   "current_required": false,
+   "on_main": false,
+   "role": "read-only provenance: Theorem L and Corollary L1 with the r-independent eps_d, sections 2-5 (exact expansions, first-order cancellations, Hoelder bounds, Loewner coupling), whose algebra theorem_v.py reuses"
+  },
+  {
+   "path": "frontiers/c8_lifetime_difference_constant_20261001/RESULTS.json",
+   "bytes": 463344,
+   "sha256": "77703dbdd7969ab62c179aa49391c6cddb545d37e0d59654476a09166a0b938f",
+   "git_blob": "894fae2e48e67dd1f21ab2ffffaf7050260cfbc1",
+   "id": "B-RESULTS",
+   "commit": "d77cabf4fee5d36de2a85a810359c1c8812fc28c",
+   "pull_request": 215,
+   "current_required": false,
+   "on_main": false,
+   "role": "read-only comparison: Theorem L a(R) and eps_d, Corollary L1 ratios, which theorem_v.py reproduces (a) and removes (eps)"
+  },
+  {
+   "path": "frontiers/c8_theorem_a_constants_planar_20260930/theorem_a.py",
+   "bytes": 52864,
+   "sha256": "c5ea8bf87c54761866315f28f6c93c6ce5c1e0f99d47c5a03f14d095d06871cc",
+   "git_blob": "ae279e9015483a666ece8deb2544cbc458a5e270",
+   "id": "E",
+   "commit": "13f932c2f6ec07d8c4c9996bd030ab22eaa32108",
+   "pull_request": 206,
+   "packet_copy": "engine_e2.py",
+   "current_required": false,
+   "on_main": false,
+   "role": "the planar engine of Math-#206 (as carried by Math-#215): exact covariance series pair_series, the truncation tail (NOTE 2(b)) and the torus remainder M_1 with its coefficient majorant (NOTE 2(c)), interval band laws, transcendentals"
+  },
+  {
+   "path": "frontiers/c8_theorem_a_constants_d3_20261001/theorem_a3.py",
+   "bytes": 76209,
+   "sha256": "310d4ab3d3a38a377d753753f3b814591a9090a947f87d6843ab7389389572d3",
+   "git_blob": "267890232119b35206be7a8cc74d730de9a68644",
+   "id": "E3",
+   "commit": "250b90722e8c47adf8f7e7c41a61e8a8085d9741",
+   "pull_request": 212,
+   "packet_copy": "engine_e3.py",
+   "current_required": false,
+   "on_main": false,
+   "role": "the d = 3 engine of Math-#212 (as carried by Math-#215): exact 3D covariance series, band laws, contact moments E[det^a tr^c 1{B > 0}], Frobenius moment bounds"
+  }
+ ]
+}
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py
new file mode 100644
index 000000000..ae279e901
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e2.py
@@ -0,0 +1,1249 @@
+#!/usr/bin/env python3
+"""Explicit constants for Theorem A of [LP] in the plane (d = 2), for the torus kernel K_L with L >= 10 in every frame:
+for 0 < r <= r_*, b in [0, 1], k in [1/2, 2],
+
+    1 - p_r(b, k, R)  <=  Q^W(G_r^c)  <=  C(r_*) r^3,
+
+with G_r the good event of [LP] section 7 / [CAP] (1) and C(r_*) certified here (NOTE.md Theorem E); the same at the single
+fixed-axis SIDE24 point (b, k) = (6/5, 1/6) of [CAP] section 1 (Theorem E').  The proof bounds
+Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0:
+
+  * covariances of the preconditioned pin and target functionals are exact power series in r, obtained from the Taylor
+    coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded
+    uniformly in r and in the frame by Cauchy estimates on a polydisc;
+  * the failure event is reduced to {lambda <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded through exact averaging
+    identities of the pinned jets and M_4 through the Taylor series at the midpoint;
+  * the main term is integrated in lambda = -f_yy(M) against its Gaussian density after an exact regression on w = f_yy(M);
+    the remaining expectations are closed-form incomplete Gaussian moments plus Hoelder-controlled perturbations;
+  * the normalizer Z_r is bounded below in closed form; the rare branches are Gaussian tails.
+All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only.
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term')
+MUT = None
+PBITS = 160
+L_MIN = 10
+NX = 60                    # Taylor order in x of the covariance series (exact); the rest is a bounded tail
+NB = 24                    # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded)
+X0 = Fr(1, 64)             # default threshold for the quadratic near-branch analysis (records use COMBOS)
+THETA = Fr(1, 32)          # default splitting parameter in the max-moment lemma (records use COMBOS)
+THETA_E = Fr(1, 16)        # splitting parameter for the (r nu + v)^2 factor
+QS = (2, 4, 8, 16)         # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+BAND_B = (Fr(0), Fr(1))
+BAND_K = (Fr(1, 2), Fr(2))
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+
+
+def phi_pt(x):
+    """phi(x) for rational x (interval)."""
+    x = Fr(x)
+    return exp_neg_point(x * x / 2) / SQRT2PI
+
+
+def Phibar_pt(x):
+    """1 - Phi(x) for rational x (interval)."""
+    p = phi_point(Fr(x)) if abs(Fr(x)) <= 12 else None
+    if p is None:
+        if x > 0:
+            # Mills: Phibar(x) <= phi(x)/x
+            return IV(0, (phi_pt(x) / x).hi)
+        return IV(1 - (phi_pt(-x) / (-x)).hi, 1)
+    return IV(1 - p.hi, 1 - p.lo)
+
+
+def upper_moments(x, J):
+    """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x >= 0 (intervals)."""
+    x = Fr(x)
+    require(x >= 0, 'upper moment at negative point')
+    ph = phi_pt(x)
+    M = [Phibar_pt(x), ph]
+    for j in range(2, J + 1):
+        M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1))
+    return M
+
+
+def dfact(n):
+    """(n)!! for n >= -1."""
+    return math.prod(range(n, 0, -2)) if n > 0 else 1
+
+
+def rup_rel(x, bits=200):
+    """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds)."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh)
+    return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh)
+
+
+def sqrt_up(x):
+    """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root
+    step adds less than 2^-209 and rup_rel(., 200) less than 2^-199.  (The earlier '2^-200' was too sharp: at x = 49/100 the
+    excess is (8/7) 2^-200, review C59-DOC-01.)"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m))
+
+
+def iroot(x, p):
+    """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two."""
+    v = Fr(x)
+    while p > 1:
+        v = sqrt_up(v)
+        p //= 2
+    return v
+
+
+def pow2(p):
+    q = 1
+    while q < p:
+        q *= 2
+    return q
+
+
+ZN = {}
+
+
+def znorm(p):
+    """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5)."""
+    if p == 1:
+        return Fr(4, 5)
+    if p not in ZN:
+        ZN[p] = iroot(Fr(dfact(p - 1)), p)
+    return ZN[p]
+
+
+def habs_moment(j):
+    """E|zeta|^j (interval)."""
+    if j % 2 == 0:
+        return IV(dfact(j - 1))
+    return SQRT2 / isqrt_iv(PI) * (2 ** ((j - 1) // 2) * math.factorial((j - 1) // 2))
+
+
+# ----------------------------------------------------------------------------- functionals and exact covariance series
+HALF = Fr(1, 2)
+XM, XS, X0PT = -HALF, HALF, Fr(0)
+
+
+def term(c, p, a, b, x0):
+    """c r^p d_x^a d_y^b f(x0 r, 0)."""
+    return (Fr(c), p, a, b, Fr(x0))
+
+
+def functionals():
+    pins = {
+        'p1': [term(HALF, 0, 0, 0, XM), term(HALF, 0, 0, 0, XS)],
+        'p2': [term(-1, -1, 0, 0, XM), term(1, -1, 0, 0, XS)],
+        'p3p': [term(6, -2, 1, 0, XM), term(6, -2, 1, 0, XS), term(12, -3, 0, 0, XM), term(-12, -3, 0, 0, XS)],
+        'p4': [term(-1, -1, 1, 0, XM), term(1, -1, 1, 0, XS)],
+        'q1': [term(HALF, 0, 0, 1, XM), term(HALF, 0, 0, 1, XS)],
+        'q2': [term(-1, -1, 0, 1, XM), term(1, -1, 0, 1, XS)],
+    }
+    targets = {
+        'w': [term(1, 0, 0, 2, XM)],
+        'v': [term(1, -1, 1, 1, XM), term(1, -2, 0, 1, XM), term(-1, -2, 0, 1, XS)],
+        'om': [term(1, -1, 0, 2, XS), term(-1, -1, 0, 2, XM)],
+        'Y': [term(1, 0, 0, 3, X0PT)],
+        'T': [term(1, -2, 2, 0, XM), term(4, -3, 1, 0, XM), term(2, -3, 1, 0, XS), term(6, -4, 0, 0, XM), term(-6, -4, 0, 0, XS)],
+        'tau': [term(1, -3, 2, 0, XS), term(-1, -3, 2, 0, XM), term(-6, -4, 1, 0, XM), term(-6, -4, 1, 0, XS),
+                term(-12, -5, 0, 0, XM), term(12, -5, 0, 0, XS)],
+        'nu': [term(1, -2, 1, 1, XM), term(1, -2, 1, 1, XS), term(2, -3, 0, 1, XM), term(-2, -3, 0, 1, XS)],
+        'X40': [term(1, 0, 4, 0, X0PT)], 'X31': [term(1, 0, 3, 1, X0PT)], 'X22': [term(1, 0, 2, 2, X0PT)],
+        'X13': [term(1, 0, 1, 3, X0PT)], 'X04': [term(1, 0, 0, 4, X0PT)],
+    }
+    return pins, targets
+
+
+PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q1', 'q2')
+TARGET_NAMES = ('w', 'v', 'om', 'Y', 'T', 'tau', 'nu', 'X40', 'X31', 'X22', 'X13', 'X04')
+JET4 = ('X40', 'X31', 'X22', 'X13', 'X04')
+JET4_IDX = {'X40': (4, 0), 'X31': (3, 1), 'X22': (2, 2), 'X13': (1, 3), 'X04': (0, 4)}
+
+
+def weight(F):
+    """Scaling weight a - (power of r), common to all terms of a preconditioned functional."""
+    ws = {t[2] - t[1] for t in F}
+    require(len(ws) == 1, 'functional not homogeneous')
+    return ws.pop()
+
+
+def hcoef(n):
+    """d^n/dx^n exp(-x^2/2) at 0."""
+    if n % 2:
+        return 0
+    return (-1) ** (n // 2) * dfact(n - 1)
+
+
+PAIR_CACHE = {}
+
+
+def pair_series(F, G, key):
+    """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG):
+    returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', b+b') for the remainder bounds)."""
+    if key in PAIR_CACHE:
+        return PAIR_CACHE[key]
+    wF, wG = weight(F), weight(G)
+    w = wF + wG
+    pi = {}
+    absterms = []
+    for (cf, pf, af, bf, xf) in F:
+        for (cg, pg, ag, bg, xg) in G:
+            gy = bf + bg
+            sgn = (-1) ** (ag + bg)
+            d = xf - xg
+            absterms.append((abs(cf * cg), af + ag, gy))
+            for gx in range(af + ag, NX + 1):
+                m = gx - af - ag
+                val = cf * cg * sgn * d ** m / math.factorial(m)
+                if val != 0:
+                    pi[(gx, gy)] = pi.get((gx, gy), Fr(0)) + val
+    for (gx, gy), v in pi.items():
+        require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)')
+    ser = {}
+    for (gx, gy), v in pi.items():
+        kap = hcoef(gx) * hcoef(gy)
+        if kap and v:
+            ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v
+    out = ({p: c for p, c in ser.items() if c != 0}, w, absterms)
+    PAIR_CACHE[key] = out
+    return out
+
+
+def torus_M1(L):
+    """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2| <= 1 (every frame, |p_n| = |n| L):
+    exp(1) * sum_{j >= 1} 8 j exp(-(j L - 3/2)^2 / 2)."""
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += 8 * j * exp_neg_point((j * L - Fr(3, 2)) ** 2 / 2).hi
+    tail = 8 * 7 * exp_neg_point((6 * L - Fr(3, 2)) ** 2 / 2).hi * 4      # j >= 6, very crude
+    return Fr(27183, 10000) * (tot + tail)
+
+
+def theta_minus_one(L):
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += 8 * j * exp_neg_point((j * L) ** 2 / 2).hi
+    return tot + 8 * 7 * exp_neg_point((6 * L) ** 2 / 2).hi * 4
+
+
+def cov_band(F, G, key, r0, r1, M1):
+    """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1]."""
+    ser, w, absterms = pair_series(F, G, key)
+    val = IV(0)
+    for p, c in ser.items():
+        val = val + IV(r0 ** p, r1 ** p) * c
+    A = max((t[1] for t in absterms), default=0)
+    # Taylor truncation tail of the reference kernel (orders gx > NX)
+    tail = Fr(0)
+    for (acc, aa, gy) in absterms:
+        tail += acc * abs(hcoef(gy))
+    tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w)
+    # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy! M1 on the unit polydisc
+    tor = Fr(0)
+    if True:
+        for (acc, aa, gy) in absterms:
+            tor += acc * math.factorial(gy) * Fr(max(w, 1)) ** aa
+        tor = tor * M1 / (1 - Fr(27183, 10000) * r1)
+    err = tail + tor
+    return val + IV(-err, err)
+
+
+# ----------------------------------------------------------------------------- interval linear algebra
+def solve_iv(A, B):
+    """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c]."""
+    n = len(A)
+    M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    for col in range(n):
+        p = max(range(col, n), key=lambda i: abs(M[i][col].mid()))
+        M[col], M[p] = M[p], M[col]
+        piv = M[col][col]
+        for i in range(col + 1, n):
+            f = M[i][col] / piv
+            for j in range(col, len(M[i])):
+                M[i][j] = M[i][j] - f * M[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            s = M[i][n + c]
+            for j in range(i + 1, n):
+                s = s - M[i][j] * x[j]
+            x[i] = s / M[i][i]
+        X.append(x)
+    return X
+
+
+def cholesky3(C):
+    """Interval Cholesky of a 3 x 3 SPD interval matrix."""
+    L = [[IV(0)] * 3 for _ in range(3)]
+    for i in range(3):
+        for j in range(i + 1):
+            s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0))
+            if i == j:
+                require(s.lo > 0, 'Cholesky pivot not positive')
+                L[i][j] = isqrt_iv(s)
+            else:
+                L[i][j] = s / L[j][j]
+    return L
+
+
+def absup(x):
+    return max(abs(x.lo), abs(x.hi))
+
+
+# ----------------------------------------------------------------------------- the conditional law on an r-band
+class Law:
+    pass
+
+
+def band_law(r0, r1, L=L_MIN):
+    """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the
+    targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients."""
+    pins, targets = functionals()
+    P = [pins[n] for n in PIN_NAMES]
+    T = [targets[n] for n in TARGET_NAMES]
+    M1 = torus_M1(L)
+    th1 = theta_minus_one(L)
+    nt = len(T)
+    Spp = [[cov_band(P[i], P[j], ('p', i, j), r0, r1, M1) for j in range(6)] for i in range(6)]
+    Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(6)] for i in range(nt)]
+    Stt = [[cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1) for j in range(nt)] for i in range(nt)]
+    r = IV(r0, r1)
+    kvec = [-(r * r * r) * HALF, -(r * r), IV(12), IV(0), IV(0), IV(0)]
+    e1 = [IV(1), IV(0), IV(0), IV(0), IV(0), IV(0)]
+    W = solve_iv(Spp, Stp)
+    xs = solve_iv(Spp, [e1, kvec])
+    law = Law()
+    law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1
+    law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)}
+    law.beta = {n: sum((W[i][j] * kvec[j] for j in range(6)), IV(0)) for i, n in enumerate(TARGET_NAMES)}
+    scale = IV(1 / (1 + th1), 1)          # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1]
+    C = {}
+    for i, a in enumerate(TARGET_NAMES):
+        for j, c in enumerate(TARGET_NAMES):
+            if j < i:
+                continue
+            v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(6)), IV(0))
+            if i != j:
+                v2 = Stt[j][i] - sum((W[j][l] * Stp[i][l] for l in range(6)), IV(0))
+                v = v.meet(v2)
+            C[(a, c)] = C[(c, a)] = v * scale
+    law.C = C
+    escale = IV(1, 1 + th1)
+    law.e_bb = xs[0][0] * escale
+    law.e_bk = sum((kvec[j] * xs[0][j] for j in range(6)), IV(0)) * escale
+    law.e_kk = sum((kvec[j] * xs[1][j] for j in range(6)), IV(0)) * escale
+    # regression on w and the g-parts
+    sw2 = C[('w', 'w')]
+    require(sw2.lo > 0, 'Var w not positive')
+    law.sw2 = sw2
+    law.sw = isqrt_iv(sw2)
+    law.bw = {n: C[(n, 'w')] / sw2 for n in TARGET_NAMES if n != 'w'}
+    Cg = {}
+    for a in TARGET_NAMES:
+        for c in TARGET_NAMES:
+            if a != 'w' and c != 'w':
+                Cg[(a, c)] = C[(a, c)] - C[(a, 'w')] * C[(c, 'w')] / sw2
+    law.Cg = Cg
+    law.L3 = cholesky3([[Cg[(a, c)] for c in ('v', 'om', 'Y')] for a in ('v', 'om', 'Y')])
+    law.sdg = {n: isqrt_iv(IV(max(Cg[(n, n)].lo, Fr(0)), Cg[(n, n)].hi)) if Cg[(n, n)].lo > 0 else IV(0, isqrt_iv(IV(Cg[(n, n)].hi)).hi)
+               for n in TARGET_NAMES if n != 'w'}
+    law.sdQ = {n: isqrt_iv(C[(n, n)]) for n in TARGET_NAMES}
+    # crude higher jets: Taylor-tail coefficients S_alpha(r1) = sum_{beta != 0} sd0_{alpha+beta} (2 r1)^{|beta|} / beta!
+    law.S = {n: taylor_tail_coefficient(JET4_IDX[n], r1, M1) for n in JET4}
+    return law
+
+
+def sd0_bound(gx, gy, M1):
+    """Upper bound of sqrt(Var d^gamma f(0)) = sqrt(|d^{2 gamma} K_L(0)|) (normalized, every frame, L >= L_MIN)."""
+    v = dfact(2 * gx - 1) * dfact(2 * gy - 1) + math.factorial(2 * gx) * math.factorial(2 * gy) * M1
+    return isqrt_iv(IV(v)).hi
+
+
+def taylor_tail_coefficient(alpha, r1, M1):
+    ax, ay = alpha
+    tot = Fr(0)
+    for n in range(1, NB + 1):
+        for bx in range(n + 1):
+            by = n - bx
+            tot += sd0_bound(ax + bx, ay + by, M1) * (2 * r1) ** n / (math.factorial(bx) * math.factorial(by))
+    # tail |beta| > NB: sd0_gamma <= 2^|gamma| gx! gy! sqrt(1 + M1) (binom(2m, m) <= 4^m), so a term is at most
+    # 2^|alpha| (4 + n)^4 (4 r1)^n 2 per multi-index; n + 1 multi-indices of order n
+    q = 4 * r1
+    require(q <= Fr(1, 16), 'Taylor tail ratio too large')
+    tail = Fr(0)
+    base = 2 ** (ax + ay) * 2
+    n = NB + 1
+    # sum_{n > NB} (n + 1)(n + 4)^8 q^n <= (NB + 2)(NB + 5)^8 q^(NB+1) / (1 - 2 q)   (consecutive ratio <= 2q for these n)
+    tail = base * (NB + 2) * Fr(NB + 5) ** 8 * q ** (NB + 1) / (1 - 2 * q)
+    return tot + tail
+
+
+# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k)
+def binom(n, k):
+    return math.comb(n, k)
+
+
+def gauss_norm(mu_abs, sd, p):
+    """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd."""
+    return rup_rel(mu_abs + sd * znorm(pow2(p)))
+
+
+def pos_part_moments(s, c, nmax, extra_max):
+    """P[n][e] = upper bound of E[u^e (u^n - c^n)_+], u = s |zeta|, for 1 <= n <= nmax, 0 <= e <= extra_max."""
+    x = c / s
+    M = upper_moments(x, nmax + extra_max)
+    out = {}
+    for n in range(1, nmax + 1):
+        for e in range(extra_max + 1):
+            out[(n, e)] = rup_rel(((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e) * 2).hi)
+    return out
+
+
+def normalized_moments(q, nmax, theta):
+    """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax:
+      E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmu1/K + (r1/K) Delta1;
+      Tn[n], To[n], Tv[n]   : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2),
+                              y_i <= u_i + delta_i (u_i = s_i |zeta| independent half-normals), threshold c = q['c'].
+    o0 = s3 |z2| and v0^2 = s2^2 z1^2 / 4 are unnormalized."""
+    th = theta
+    cp = q['c'] * (1 - th)                  # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli)
+    PP = {i: pos_part_moments(q['s'][i], cp, nmax, 2) for i in (2, 3, 4)}
+    Eu3 = (habs_moment(1) * q['s3']).hi
+    Eu2sq = (habs_moment(2) * q['s2'] * q['s2']).hi
+    n_o0_2 = q['s3'] * znorm(2)
+    n_u2sq_2 = q['s2'] * q['s2'] * znorm(4) ** 2
+    r3 = q['s3'] / q['s'][3]
+    r2sq = (q['s2'] / q['s'][2]) ** 2
+    E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {}
+    base = 1 + q['cmt1']
+    for n in range(0, nmax + 1):
+        ey = base ** n
+        eo = base ** n * Eu3
+        ev = base ** n * Eu2sq
+        for j in range(1, n + 1):
+            coef = binom(n, j) * base ** (n - j) * q['rK'] ** j
+            ey += coef * q['nD1'](pow2(j)) ** j
+            eo += coef * n_o0_2 * q['nD1'](pow2(2 * j)) ** j
+            ev += coef * n_u2sq_2 * q['nD1'](pow2(2 * j)) ** j
+        E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev / 4)
+        if n == 0:
+            continue
+        f1 = (1 + th) ** (n - 1)
+        f2 = (1 + 1 / th) ** (n - 1)
+        tn = to = tv = Fr(0)
+        for i in (2, 3, 4):
+            P = PP[i]
+            dn = q['nd'](i, pow2(n)) ** n
+            dn2 = q['nd'](i, pow2(2 * n)) ** n
+            tn += f1 * P[(n, 0)] + f2 * dn
+            to += (f1 * r3 * P[(n, 1)] if i == 3 else f1 * Eu3 * P[(n, 0)]) + f2 * n_o0_2 * dn2
+            tv += (f1 * r2sq * P[(n, 2)] if i == 2 else f1 * Eu2sq * P[(n, 0)]) + f2 * n_u2sq_2 * dn2
+        Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv / 4)
+    return E1, E1o, E1v, Tn, To, Tv
+
+
+def elam_pos(m, s):
+    """E[(s z + m)_+] at a point (rational m, s > 0)."""
+    x = Fr(m) / s
+    return m * Phi(IV(x)) + phi_pt(x) * s
+
+
+def elam2_pos(m, s):
+    """E[(s z + m)_+^2] at a point."""
+    x = Fr(m) / s
+    return (m * m + s * s) * Phi(IV(x)) + phi_pt(x) * (m * s)
+
+
+def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None):
+    """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch
+    threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5).  The bound is proved for 0 < x0 < 2/9 and
+    0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and
+    (1 + kappa x0) c^2 <= 1.  No claim is made outside that domain.  Both COMBOS pairs, and the (1/64, 1/32) of Math-#215's
+    caller, lie inside it (review C61)."""
+    x0 = X0 if x0 is None else Fr(x0)
+    theta = THETA if theta is None else Fr(theta)
+    b = IV(bb[0], bb[1])
+    k = IV(kb[0], kb[1])
+    klo, khi = kb[0], kb[1]
+    Klo = 12 * klo
+    mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES}
+    mabs = {n: absup(mu[n]) for n in TARGET_NAMES}
+    E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk
+    require(E.hi > 0, 'energy')
+    sqE = isqrt_iv(IV(E.hi)).hi
+    sw_lo, sw_hi = law.sw.lo, law.sw.hi
+    muw_abs = mabs['w']
+    bw = {n: absup(law.bw[n]) for n in law.bw}
+    t = {n: law.S[n] / sw_lo for n in JET4}
+    a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1)      # the cap constant 4/(3k) (mutant: halved)
+    B = lambda n: bw[n] + t[n]
+    bbars = {1: r1 * (Fr(5, 2) * B('X40') + 2 * B('X31')),
+             2: 2 * bw['v'] + r1 * (Fr(5, 2) * B('X31') + 2 * B('X22')),
+             3: bw['om'] + r1 * (Fr(5, 2) * B('X22') + 2 * B('X13')),
+             4: bw['Y'] + r1 * (2 * B('X13') + 2 * B('X04'))}
+    bbars = {i: rup_rel(v) for i, v in bbars.items()}
+    bbar = max(bbars.values())
+    L3 = law.L3
+    s2, s3, s4 = 2 * L3[0][0].hi, L3[1][1].hi, L3[2][2].hi
+    l21, l31, l32 = absup(L3[1][0]), absup(L3[2][0]), absup(L3[2][1])
+
+    memo = {}
+
+    def nA(n, p):     # ||A_alpha^g||_p, A_alpha^g = |X_alpha,g| + T_alpha^g
+        key = ('A', n, p)
+        if key not in memo:
+            memo[key] = rup_rel(gauss_norm(mabs[n], law.sdg[n].hi, p) + law.S[n] * (sqE + znorm(pow2(p))))
+        return memo[key]
+
+    def nDelta(i, p):
+        key = ('D', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nDelta0(i, p))
+        return memo[key]
+
+    def nDelta0(i, p):
+        if MUT == 'no-delta':
+            return Fr(0)
+        if i == 1:
+            return Fr(5, 2) * nA('X40', p) + 2 * nA('X31', p)
+        if i == 2:
+            return Fr(5, 2) * nA('X31', p) + 2 * nA('X22', p)
+        if i == 3:
+            return Fr(5, 2) * nA('X22', p) + 2 * nA('X13', p)
+        return 2 * nA('X13', p) + 2 * nA('X04', p)
+
+    def nd(i, p):
+        key = ('d', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nd0(i, p))
+        return memo[key]
+
+    def nd0(i, p):     # ||delta_i||_p / Klo, delta_i including beta_i |mu_w|
+        zp = znorm(pow2(p))
+        cm = bbars[i] * muw_abs
+        if i == 2:
+            v = 2 * mabs['v'] + cm + r1 * nDelta(2, p)
+        elif i == 3:
+            v = mabs['om'] + l21 * zp + cm + r1 * nDelta(3, p)
+        else:
+            v = mabs['Y'] + (l31 + l32) * zp + cm + r1 * nDelta(4, p)
+        return v / Klo
+
+    kappa = 3 / (1 - 3 * x0)
+    Fk = 1 + kappa * x0                      # (1 + kappa x_i) <= Fk on the event {x_i <= x0}
+    cth = 1 - kappa * x0 / 2                 # <= Fk^{-1/2}
+    NMAX = 24
+    q = {'cmt1': bbars[1] * muw_abs / Klo, 'rK': r1 / Klo, 'nD1': lambda p: nDelta(1, p),
+         's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth}
+    E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta)
+    et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac)     # eta_1 K, k-free
+    aK2_hi = 192 * khi * a_fac                    # a K^2 = 192 k
+
+    # moments of Ut = U / (a K^2) <= max(Z1, Fk y_i^2), Z1 = y1^2 (1 + et1 y1)
+    def EUt(p):                                   # upper bound of E[Ut^p]
+        key = ('U', p)
+        if key not in memo:
+            memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p])
+        return memo[key]
+
+    def EoUt2():
+        return sum(binom(2, l) * et1 ** l * E1o[4 + l] for l in range(3)) + Fk ** 2 * To[4]
+
+    def EvUt2():
+        return sum(binom(2, l) * et1 ** l * E1v[4 + l] for l in range(3)) + Fk ** 2 * Tv[4]
+
+    def sq(x):
+        return isqrt_iv(IV(x)).hi
+
+    U3_2 = sq(EUt(6))                              # ||Ut^3||_2
+    U2_2 = sq(EUt(4))                              # ||Ut^2||_2
+
+    def nU(p):                                     # ||U||_p <= a K^2 Fk ||max_i y_i||_2p^2 (on the event x_i <= X0)
+        require(2 * p <= NMAX and pow2(p) == p, 'norm order')
+        key = ('nU', p)
+        if key not in memo:
+            memo[key] = rup_rel(aK2_hi * Fk * iroot(E1[2 * p] + Tn[2 * p], p))
+        return memo[key]
+
+    def nbar(n, p):                                # ||F_bar||_p, F_bar = |F_g| + |beta_F| (|mu_w| + r1 U)
+        return gauss_norm(mabs[n], law.sdg[n].hi, p) + bw[n] * (muw_abs + r1 * nU(p))
+
+    six_klo = 6 * klo
+    nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4)
+    epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2
+    nT8, nt8 = nbar('T', 8), nbar('tau', 8)
+    epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2
+    # Term A / (36 k^2)
+    A_n = EUt(3) + epsc2 * U3_2
+    termA = aK2_hi ** 3 / 3 * A_n
+    # Term B / (36 k^2)
+    ndo = lambda p: l21 * znorm(pow2(p)) + mabs['om'] + bw['om'] * (muw_abs + r1 * nU(p))
+    no4 = s3 * znorm(4) + ndo(4)
+    B_n = EoUt2() + (ndo(2) + epsc4 * no4) * U2_2
+    termB = aK2_hi ** 2 / 2 * B_n
+    if MUT == 'no-om-term':
+        termB = Fr(0)
+    # Term C / (36 k^2) = (a K^2)^2 / (12 k) * E[(1 + r T/(6k)) e Ut^2]
+    th = THETA_E
+    dv = lambda p: mabs['v'] + bw['v'] * (muw_abs + r1 * nU(p))
+    ebar4 = (r1 * nbar('nu', 8) + L3[0][0].hi * znorm(8) + dv(8)) ** 2
+    C_n = ((1 + th) ** 2 * EvUt2()
+           + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U2_2
+           + r1 * nT4 / six_klo * ebar4 * U2_2)
+    termC = aK2_hi ** 2 / (12 * klo) * C_n
+    # one more power of U for the density refinement: the same three terms with lambda inserted, divided by r
+    # (int_0^{rU} lambda W dlambda <= r^6 |c1| [|c2| (U^4/4 + om U^3/3) + e U^3/3])
+    U3_2_ = U3_2
+    termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * sq(EUt(8)))
+    EoUt3 = sum(binom(3, l) * et1 ** l * E1o[6 + l] for l in range(4)) + Fk ** 3 * To[6]
+    EvUt3 = sum(binom(3, l) * et1 ** l * E1v[6 + l] for l in range(4)) + Fk ** 3 * Tv[6]
+    termBp = aK2_hi ** 3 / 3 * (EoUt3 + (ndo(2) + epsc4 * no4) * U3_2_)
+    if MUT == 'no-om-term':
+        termBp = Fr(0)
+    termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt3
+                                         + ((1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nbar('nu', 4) ** 2) * U3_2_
+                                         + r1 * nT4 / six_klo * ebar4 * U3_2_)
+    # density of lambda = -w: pbar = its supremum on [0, inf); on the failure window [0, rU] also p(lambda) <= p0 + lambda Lp
+    # with p0 = sup over the box of the density at 0 and Lp = sup |p'| <= phi(1)/sigma^2, phi(1) < 0.24198
+    mlo = max(Fr(0), mu['w'].lo)
+    pbar = (phi_pt(mlo / sw_hi) / sw_lo).hi
+    m0 = Fr(0) if mu['w'].lo <= 0 <= mu['w'].hi else min(abs(mu['w'].lo), abs(mu['w'].hi))
+    p0 = (phi_pt(m0 / sw_hi) / sw_lo).hi
+    Lp = Fr(24198, 100000) / (sw_lo * sw_lo)
+    # ---- normalizer floor Z_r / (36 k^2 r^2) >= El2 - corr
+    m_lam = -mu['w']
+    El2 = elam2_pos(m_lam.lo, sw_lo).lo
+    Elp = elam_pos(m_lam.hi, sw_hi).hi
+    lam_n = lambda p: muw_abs + sw_hi * znorm(pow2(p))
+    sdT, sdtau = law.sdQ['T'].hi, law.sdQ['tau'].hi
+    q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1))
+    Eom_g = mabs['om'] + law.sdg['om'].hi * Fr(4, 5)
+    Evg2 = mabs['v'] ** 2 + law.sdg['v'].hi ** 2
+    E_om_lam = Eom_g * Elp + bw['om'] * (muw_abs * lam_n(1) + lam_n(2) ** 2)
+    E_v2_lam = 2 * Evg2 * Elp + 2 * bw['v'] ** 2 * (sw_hi * znorm(4)) ** 2 * lam_n(2)
+    corr = (lam_n(4) ** 2 * iroot(q_bad, 2) if q_bad > 0 else Fr(0)) \
+        + (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) * lam_n(4) ** 2 / six_klo \
+        + r1 * (E_v2_lam / (3 * klo) + E_om_lam)
+    if MUT == 'no-zfloor-correction':
+        corr = Fr(0)
+    zt = El2 - corr
+    require(zt > 0, 'normalizer floor not positive')
+    main = min(pbar * (termA + termB + termC), p0 * (termA + termB + termC) + r1 * Lp * (termAp + termBp + termCp)) / zt
+    # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q) (Hoelder), ||W 1_typed||_q / r^2 <= g1 g2,
+    # for each q in QS; the smallest resulting bound is used
+    six_khi = 6 * khi
+
+    def wnorm(q):
+        p4 = 4 * q
+        nTQ = gauss_norm(mabs['T'], sdT, p4)
+        ntauQ = gauss_norm(mabs['tau'], sdtau, p4)
+        g1 = (six_khi + r1 * nTQ) * lam_n(p4)
+        g2 = (six_khi + r1 * nTQ + r1 * r1 * ntauQ) * (lam_n(p4) + r1 * gauss_norm(mabs['om'], law.sdQ['om'].hi, p4)) \
+            + r1 * (r1 * gauss_norm(mabs['nu'], law.sdQ['nu'].hi, p4) + gauss_norm(mabs['v'], law.sdQ['v'].hi, p4)) ** 2
+        return g1 * g2
+    thr = 3 * klo / (10 * r1)
+    qA2 = Fr(0)
+    args = []
+    for n in JET4:
+        args.append((thr * Fr(7, 8) - mabs[n]) / law.sdQ[n].hi)
+        qA2 += gauss_tail2(mabs[n], law.sdQ[n].hi, thr * Fr(7, 8))
+        qA2 += (law.S[n] * (sqE + znorm(32)) / (thr / 8)) ** 32
+    a_hi = Fr(4, 3) / klo * a_fac
+    qfar = Fr(0)
+    qx = Fr(0)
+    if bbar > 0:
+        lam_far = 1 / (4 * a_hi * r1 * bbar * bbar)
+        args.append((lam_far - muw_abs) / sw_hi)
+        qfar = gauss_tail1(muw_abs, sw_hi, lam_far)
+    for i in (1, 2, 3, 4):                         # x_i > X0  <=>  y_i' > X0 / (a r1 beta_i)
+        if bbars[i] == 0:
+            continue
+        tx = x0 / (a_hi * r1 * bbars[i])
+        if i == 1:
+            cb = 12 * khi + bbars[1] * muw_abs
+            qx += Fr(1) if tx <= cb else (r1 * nDelta(1, 32) / (tx - cb)) ** 32
+        else:
+            s_i = {2: s2, 3: s3, 4: s4}[i]
+            args.append(tx / (2 * s_i))
+            qx += gauss_tail2(0, s_i, tx / 2) + (nd(i, 32) * Klo / (tx / 2)) ** 32
+    qs = [min(Fr(1), x) for x in (qA2, qfar, qx)]
+    tail_num = min(wnorm(q) * sum(pow_tail(x, q) for x in qs) for q in QS)
+    if last_band:
+        # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian tail argument scales at least like 1/r and is >= 3 at R,
+        # so each Gaussian piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32; after the
+        # power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of tail / r^3
+        # over (0, R] is attained on [R/2, R]
+        require(min(args) >= 3, 'last-band tail arguments')
+        # no piece is capped at 1, so the factor-64 decrease of every piece per halving carries over to each sum
+        require(max(qA2, qfar, qx) < 1, 'last-band tail pieces')
+        rdiv = r1 / 2
+    else:
+        rdiv = r0
+    tail = tail_num / (36 * klo * klo * zt * rdiv ** 3)
+    total = main + tail
+    return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'pbar': pbar, 'termA': termA, 'termB': termB,
+            'termC': termC, 'qA2': qA2, 'qfar': qfar, 'qx': qx, 'El2': El2, 'corr': corr, 'min_arg': min(args)}
+
+
+def pow_tail(x, q):
+    """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two)."""
+    if x <= 0:
+        return Fr(0)
+    return iroot(rup_rel(x, 64) ** (q - 1), q)
+
+
+def phibar_upper(x):
+    """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound
+    phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor."""
+    x = Fr(x)
+    if x <= 12:
+        return Phibar_pt(x).hi
+    y = x * x / 2
+    m = 40
+    return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2))      # sqrt(2 pi) > 5/2
+
+
+def gauss_tail2(mu_abs, sd, t):
+    """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd."""
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_tail1(mu_abs, sd, t):
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), phibar_upper((t - mu_abs) / sd))
+
+
+# ----------------------------------------------------------------------------- bands, boxes, runs
+B_BOXES = ((Fr(0), Fr(1, 64)), (Fr(1, 64), Fr(1, 16)), (Fr(1, 16), Fr(1, 4)), (Fr(1, 4), Fr(1)))
+K_BOXES = ((Fr(1, 2), Fr(1)), (Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(7, 4)), (Fr(7, 4), Fr(15, 8)), (Fr(15, 8), Fr(31, 16)),
+           (Fr(31, 16), Fr(63, 32)), (Fr(63, 32), Fr(2)))
+HIGH_OCTAVES = 3           # octaves [2^-(j+1), 2^-j], j = 6..8 (r up to 1/64), split into 8 sub-bands
+TOP_OCTAVES = 4            # octaves j = 9..12, split into 16 sub-bands
+LOW_OCTAVES = 16           # octaves j = 13..28, split into 4 sub-bands
+R_LAST = Fr(1, 2 ** 29)    # last band [0, 2^-29]
+
+
+def bands():
+    """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST]."""
+    out = []
+    for j in range(6, 9 + TOP_OCTAVES + LOW_OCTAVES):
+        hi = Fr(1, 2 ** j)
+        lo = hi / 2
+        m = 8 if j < 9 else (16 if j < 9 + TOP_OCTAVES else 4)
+        for i in range(m):
+            out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m))
+    out.append((Fr(0), R_LAST))
+    require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover')
+    return sorted(out)
+
+
+def up(x, digits=6):
+    """Decimal string of a rational rounded UP to `digits` significant digits (as a string float)."""
+    x = Fr(x)
+    if x <= 0:
+        return '0'
+    e = len(str(x.numerator // x.denominator)) if x >= 1 else -len(str(x.denominator // x.numerator))
+    scale = Fr(10) ** (digits - e)
+    n = -((-x.numerator * scale.numerator) // (x.denominator * scale.denominator))
+    return str(Fr(n) / scale) if scale.denominator == 1 else '%s' % (Fr(n) / scale)
+
+
+def dec_down(x, places=4):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_up(x, places=4):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+BOXES = tuple((bb, kb) for bb in B_BOXES for kb in K_BOXES)      # the order of the per-box lists in a band record
+# (x0, theta) choices; each gives a valid bound and the record keeps the smaller one
+COMBOS = ((Fr(1, 64), Fr(1, 32)), (Fr(1, 1024), Fr(1, 256)))
+# the fixed-axis SIDE24 point of [CAP] section 1 (b = 6/5, kappa = 1/6), a single (b, k) point outside the band
+SIDE24_B, SIDE24_K = (Fr(6, 5), Fr(6, 5)), (Fr(1, 6), Fr(1, 6))
+CG_02_LO, CG_SIDE24_LO = Fr('5324360.4426'), Fr('35736.7352')      # certified lower ends of c_G at (0, 2), (6/5, 1/6) ([G])
+
+
+def best_box(law, bb, kb, r0, r1, last):
+    best = None
+    for x0, theta in COMBOS:
+        res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=x0, theta=theta)
+        if best is None or res['total'] < best['total']:
+            best = res
+    return best
+
+
+def band_record(args):
+    """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down,
+    6 places) for each of the 28 boxes in BOXES order, the largest tail part (rounded up, 12 places), and the same two
+    numbers at the SIDE24 point."""
+    r0, r1 = args
+    law = band_law(r0, r1)
+    last = (r0 == 0)
+    bound, zfloor, tails = [], [], []
+    for bb, kb in BOXES:
+        res = best_box(law, bb, kb, r0, r1, last)
+        bound.append(dec_up(res['total']))
+        zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6))
+        tails.append(res['tail'])
+    s24 = best_box(law, SIDE24_B, SIDE24_K, r0, r1, last)
+    return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12),
+            'side24': [dec_up(s24['total']), dec_down(36 * SIDE24_K[0] ** 2 * s24['zt'], 6)]}
+
+
+def full_run(procs):
+    bl = bands()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, bl)
+    return assemble(recs)
+
+
+def assemble(recs):
+    table = {}
+    for rs in R_STARS:
+        mx, arg, zmin = Fr(0), None, None
+        for rec in recs:
+            if Fr(rec['r'][1]) <= rs:
+                for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']):
+                    v = Fr(bs)
+                    if v > mx:
+                        mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]}
+                    z = Fr(zs)
+                    zmin = z if zmin is None or z < zmin else zmin
+        table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG02': dec_up(mx / CG_02_LO, 6),
+                          'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    side = {}
+    for rs in R_STARS:
+        rows = [rec for rec in recs if Fr(rec['r'][1]) <= rs]
+        mx = max(Fr(rec['side24'][0]) for rec in rows)
+        arg = [rec['r'] for rec in rows if Fr(rec['side24'][0]) == mx][0]
+        zmin = min(Fr(rec['side24'][1]) for rec in rows)
+        side[str(rs)] = {'C': str(mx), 'argmax_r': arg, 'C_over_cG': dec_up(mx / CG_SIDE24_LO, 6),
+                         'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-PLANAR-20260930-v1', 'scientific_effect': 'NONE',
+            'certified': True, 'mutant': MUT, 'L_min': L_MIN,
+            'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]},
+            'side24_point': {'b': str(SIDE24_B[0]), 'k': str(SIDE24_K[0])},
+            'parameters': params(), 'C_table': table, 'C_table_side24': side, 'bands': recs}
+
+
+def params():
+    return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBOS_X0_THETA': [[str(a), str(b)] for a, b in COMBOS], 'THETA_E': str(THETA_E), 'QS': list(QS),
+            'R_STARS': [str(x) for x in R_STARS], 'HIGH_OCTAVES': HIGH_OCTAVES, 'TOP_OCTAVES': TOP_OCTAVES,
+            'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST),
+            'B_BOXES': [[str(a), str(b)] for a, b in B_BOXES], 'K_BOXES': [[str(a), str(b)] for a, b in K_BOXES]}
+
+
+def results_text(res):
+    """RESULTS.json layout: the summary indented, then one line per band record."""
+    head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True)
+    lines = ',\n'.join('  ' + json.dumps(rec, sort_keys=True) for rec in res['bands'])
+    return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n'
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def replay_sample():
+    """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band."""
+    bl = bands()
+    idx = []
+    for rs in R_STARS:
+        j = max(i for i, b in enumerate(bl) if b[1] == rs)
+        idx.append(j)
+    idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))]
+    idx.append(0)
+    return sorted(set(idx))
+
+
+def check_run(procs, full=False):
+    ref = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout')
+    require(ref['parameters'] == params(), 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json produced under a mutant')
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure')
+    bl = bands()
+    require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs')
+    idx = list(range(len(bl))) if full else replay_sample()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, [bl[i] for i in idx])
+    for i, rec in zip(idx, recs):
+        require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r']))
+    # the table is the maximum over the stored records
+    asm = assemble(ref['bands'])
+    require(asm['C_table'] == ref['C_table'] and asm['C_table_side24'] == ref['C_table_side24'],
+            'C tables are not the maxima of the stored bounds')
+    # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G(0, 2) of Math-#203 (a lower bound for every
+    # cap-route constant valid for all L >= 10, since c_G^{L,R} -> c_G^{ref} as L -> infinity)
+    for rs, row in ref['C_table'].items():
+        require(Fr(row['C']) > CG_02_LO, 'C below the sharp cap coefficient')
+    for rs, row in ref['C_table_side24'].items():
+        require(Fr(row['C']) > CG_SIDE24_LO, 'SIDE24 C below the sharp cap coefficient')
+    print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()},
+                      'C_table_side24': {rs: row['C'] for rs, row in ref['C_table_side24'].items()}, 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.mc:
+        rr, bb, kk, nn, sd = args.mc
+        print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd))))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({'band': {rs: row['C'] for rs, row in res['C_table'].items()},
+                      'side24': {rs: row['C'] for rs, row in res['C_table_side24'].items()}}))
+
+
+
+# ----------------------------------------------------------------------------- floating control (not part of the certificate)
+def mc_event(r, b, k, n, seed, deg=8, grid=9):
+    """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel (floating point): the Taylor jet of
+    order deg at the midpoint is sampled under Q, the field near D is rebuilt from it, W_r and G_r (M_3, M_4 on a grid of D)
+    are evaluated directly from the definitions."""
+    import random
+    pins, _ = functionals()
+    P = [pins[nm] for nm in PIN_NAMES]
+    jets = [(i, m - i) for m in range(deg + 1) for i in range(m, -1, -1)]
+    J = [[term(1, 0, a, bb, X0PT)] for (a, bb) in jets]
+
+    def ev(F, G, key):
+        ser, w, _ = pair_series(F, G, key)
+        return sum(float(c) * r ** p for p, c in ser.items())
+    Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(6)] for i in range(6)]
+    Sjp = [[ev(J[i], P[j], ('jp', deg, i, j)) for j in range(6)] for i in range(len(J))]
+    Sjj = [[ev(J[i], J[j], ('jj', deg, i, j)) for j in range(len(J))] for i in range(len(J))]
+
+    def solve(A, bv):
+        m = len(A)
+        M = [A[i][:] + [bv[i]] for i in range(m)]
+        for c in range(m):
+            p = max(range(c, m), key=lambda i: abs(M[i][c]))
+            M[c], M[p] = M[p], M[c]
+            for i in range(c + 1, m):
+                f = M[i][c] / M[c][c]
+                for j in range(c, m + 1):
+                    M[i][j] -= f * M[c][j]
+        x = [0.0] * m
+        for i in reversed(range(m)):
+            x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i]
+        return x
+    vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k, 0, 0, 0]
+    Wt = [solve(Spp, Sjp[i]) for i in range(len(J))]
+    mu = [sum(Wt[i][j] * vr[j] for j in range(6)) for i in range(len(J))]
+    C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(6)) for j in range(len(J))] for i in range(len(J))]
+    # symmetric square root (Jacobi)
+    m = len(J)
+    A = [row[:] for row in C]
+    V = [[float(i == j) for j in range(m)] for i in range(m)]
+    for _ in range(100):
+        off = sum(A[p][q] ** 2 for p in range(m) for q in range(p + 1, m))
+        if off < 1e-26:
+            break
+        for p in range(m - 1):
+            for q in range(p + 1, m):
+                if abs(A[p][q]) < 1e-300:
+                    continue
+                th = 0.5 * math.atan2(2 * A[p][q], A[q][q] - A[p][p])
+                c, s = math.cos(th), math.sin(th)
+                for kk in range(m):
+                    akp, akq = A[kk][p], A[kk][q]
+                    A[kk][p], A[kk][q] = c * akp - s * akq, s * akp + c * akq
+                for kk in range(m):
+                    apk, aqk = A[p][kk], A[q][kk]
+                    A[p][kk], A[q][kk] = c * apk - s * aqk, s * apk + c * aqk
+                for kk in range(m):
+                    vkp, vkq = V[kk][p], V[kk][q]
+                    V[kk][p], V[kk][q] = c * vkp - s * vkq, s * vkp + c * vkq
+    lam = [max(A[i][i], 0.0) for i in range(m)]
+    S = [[sum(V[i][kk] * math.sqrt(lam[kk]) * V[j][kk] for kk in range(m)) for j in range(m)] for i in range(m)]
+    fact = [math.factorial(i) for i in range(deg + 2)]
+    pts = [(-2 * r + 4 * r * i / (grid - 1), -2 * r + 4 * r * j / (grid - 1)) for i in range(grid) for j in range(grid)]
+
+    def deriv(cf, a0, b0, x, y):
+        tot = 0.0
+        for idx, (a, bb) in enumerate(jets):
+            if a >= a0 and bb >= b0:
+                tot += cf[idx] * x ** (a - a0) * y ** (bb - b0) / (fact[a - a0] * fact[bb - b0])
+        return tot
+    rng = random.Random(seed)
+    num = den = 0.0
+    acap = 4 / (3 * k)
+    for _ in range(n):
+        z = [rng.gauss(0, 1) for _ in jets]
+        cf = [mu[i] + sum(S[i][j] * z[j] for j in range(m)) for i in range(m)]
+        hM = [deriv(cf, 2, 0, -r / 2, 0.0), deriv(cf, 1, 1, -r / 2, 0.0), deriv(cf, 0, 2, -r / 2, 0.0)]
+        hS = [deriv(cf, 2, 0, r / 2, 0.0), deriv(cf, 1, 1, r / 2, 0.0), deriv(cf, 0, 2, r / 2, 0.0)]
+        dM = hM[0] * hM[2] - hM[1] ** 2
+        dS = hS[0] * hS[2] - hS[1] ** 2
+        if not (hM[0] < 0 and dM > 0 and dS < 0):
+            continue
+        Wv = abs(dM * dS)
+        den += Wv
+        lamv = -hM[2]
+        M3 = max(abs(deriv(cf, a, 3 - a, x, y)) for (x, y) in pts for a in range(4))
+        M4 = max(abs(deriv(cf, a, 4 - a, x, y)) for (x, y) in pts for a in range(5))
+        if not (lamv > acap * r * M3 * M3 and r * M4 <= 3 * k / 10):
+            num += Wv
+    return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3, 'Z_over_r2': den / n / r ** 2}
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py
new file mode 100644
index 000000000..267890232
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/engine_e3.py
@@ -0,0 +1,1714 @@
+#!/usr/bin/env python3
+"""Explicit cap-route constants for Theorem A of [LP] in dimension d = 3, for the torus kernel K_L with L >= 10 in every
+frame: for 0 < r <= r_*, b in [0, 1], k in [1/2, 2],
+
+    1 - p_r(b, k, R)  <=  Q^W(G_r^c)  <=  C3(r_*) r^3,        Z_r / r^2 >= z3(r_*),
+
+with G_r the good event of [LP] section 7 / [CAP] (1) (partial-block operator norms on the cylinder
+D = [-2r, 2r] x ball(0, 2r)) and C3(r_*), z3(r_*) certified here (NOTE.md Theorem E3).  The proof bounds
+Q^W(G_r^c) = E_Q[W_r 1{G_r^c}] / Z_r from above by an exact Gaussian regression near r = 0:
+
+  * covariances of the eight preconditioned pins and the 31 targets are exact power series in r, obtained from the Taylor
+    coefficients of the kernel at 0 (every negative power cancels identically, checked); the torus remainder is bounded
+    uniformly in r and in the frame by Cauchy estimates on a polydisc in C^3;
+  * the failure event is reduced to {lambda_1 <= (4/(3k)) r M_3^2} u {r M_4 > 3k/10}; M_3 is bounded blockwise through exact
+    averaging identities of the pinned jets and M_4 through the Taylor series at the midpoint (Frobenius norms);
+  * the main term is integrated over the transverse Hessian B = -D_y^2 f(M) in eigenvalue coordinates after an exact
+    regression on its three entries, against an angle-free Gaussian majorant of its density; the remaining expectations are
+    closed-form incomplete Gaussian and chi moments plus Hoelder-controlled perturbations;
+  * the normalizer Z_r is bounded below through a Loewner-order coupling with the contact law, whose moments
+    E[det(B)^a tr(B)^c 1{B > 0}] are in closed form; the rare branches are Gaussian and chi tails.
+All arithmetic is exact rational interval arithmetic (outward rounding to 2^-160); standard library only.
+"""
+import argparse
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-delta', 'cap-half', 'no-zfloor-correction', 'no-om-term', 'no-lambda2')
+MUT = None
+PBITS = 160
+L_MIN = 10
+NX = 60                    # Taylor order in x of the covariance series (exact); the rest is a bounded tail
+NB = 24                    # truncation order of the Taylor tails of the 4-jets at the midpoint (the rest is bounded)
+X0 = Fr(1, 64)             # default threshold for the quadratic near-branch analysis (records use COMBOS)
+THETA = Fr(1, 32)          # default splitting parameter in the max-moment lemma (records use COMBOS)
+THETA_E = Fr(1, 16)        # splitting parameter for the ||r nu - v||^2 factor
+QS = (2, 4, 8, 16)         # Hoelder exponents of the rare branches: E[W 1_E] <= ||W||_q P(E)^(1 - 1/q), smallest bound used
+R0_COUPLING = 12           # radius of the Loewner coupling of the transverse Hessian with the contact law (floor, section 6)
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+BAND_B = (Fr(0), Fr(1))
+BAND_K = (Fr(1, 2), Fr(2))
+
+
+def require(cond, msg):
+    if not cond:
+        print(json.dumps({'check': 'FAIL', 'reason': msg, 'mutant': MUT}))
+        sys.exit(1)
+
+
+# ----------------------------------------------------------------------------- rational intervals (from Math-#195 / #203 / #206)
+def rdown(x):
+    return Fr(x.numerator * (1 << PBITS) // x.denominator, 1 << PBITS)
+
+
+def rup(x):
+    return Fr(-((-x.numerator) * (1 << PBITS) // x.denominator), 1 << PBITS)
+
+
+class IV:
+    __slots__ = ('lo', 'hi')
+
+    def __init__(self, lo, hi=None):
+        lo = Fr(lo)
+        hi = lo if hi is None else Fr(hi)
+        if hi < lo:
+            raise ValueError('empty interval')
+        self.lo, self.hi = rdown(lo), rup(hi)
+
+    def __repr__(self):
+        return 'IV(%s, %s)' % (float(self.lo), float(self.hi))
+
+    def __add__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return IV(self.lo + o.lo, self.hi + o.hi)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return IV(-self.hi, -self.lo)
+
+    def __sub__(self, o):
+        return self + (-(o if isinstance(o, IV) else IV(o)))
+
+    def __rsub__(self, o):
+        return IV(o) - self
+
+    def __mul__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        c = (self.lo * o.lo, self.lo * o.hi, self.hi * o.lo, self.hi * o.hi)
+        return IV(min(c), max(c))
+    __rmul__ = __mul__
+
+    def inv(self):
+        if self.lo <= 0 <= self.hi:
+            raise ZeroDivisionError('interval contains zero')
+        return IV(1 / self.hi, 1 / self.lo)
+
+    def __truediv__(self, o):
+        o = o if isinstance(o, IV) else IV(o)
+        return self * o.inv()
+
+    def __rtruediv__(self, o):
+        return IV(o) * self.inv()
+
+    def sq(self):
+        if self.lo >= 0:
+            return IV(self.lo * self.lo, self.hi * self.hi)
+        if self.hi <= 0:
+            return IV(self.hi * self.hi, self.lo * self.lo)
+        return IV(0, max(self.lo * self.lo, self.hi * self.hi))
+
+    def meet(self, o):
+        lo, hi = max(self.lo, o.lo), min(self.hi, o.hi)
+        if hi < lo:
+            raise ValueError('inconsistent enclosures')
+        return IV(lo, hi)
+
+    def width(self):
+        return self.hi - self.lo
+
+    def mid(self):
+        return (self.lo + self.hi) / 2
+
+    def contains(self, x):
+        return self.lo <= x <= self.hi
+
+
+def isqrt_iv(x):
+    """Enclosure of sqrt(x) for an interval x with x.lo > 0."""
+    if x.lo <= 0:
+        raise ValueError('sqrt of interval reaching zero: %r' % x)
+
+    def lo_root(v):
+        n = v.numerator * (1 << (2 * PBITS)) // v.denominator
+        s = math.isqrt(n)
+        return Fr(s, 1 << PBITS)
+
+    def hi_root(v):
+        n = -((-v.numerator * (1 << (2 * PBITS))) // v.denominator)
+        s = math.isqrt(n) + 1
+        return Fr(s, 1 << PBITS)
+    return IV(lo_root(x.lo), hi_root(x.hi))
+
+
+def exp_neg_point(t):
+    """Enclosure of exp(-t) for a rational t >= 0: halve t to t <= 1/2, evaluate the alternating series in fixed point
+    (ulp = 2^-(PBITS+40)) with an integer bound on the accumulated rounding error, then square back."""
+    t = Fr(t)
+    m = 0
+    while t > Fr(1, 2):
+        t /= 2
+        m += 1
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = t.numerator * ONE // t.denominator            # X / ONE <= t < (X + 1) / ONE
+    # series at x_lo = X / ONE: exp(-x_lo) = sum (-1)^n x_lo^n / n!  (alternating, terms decreasing for x_lo <= 1/2)
+    S = ONE
+    T = ONE
+    E = 0                                             # error bound of T in ulps
+    Etot = 0
+    n = 0
+    while T > 0:
+        n += 1
+        T = (T * X) // (ONE * n)
+        E = (E * X) // (ONE * n) + 2
+        S += (-1) ** n * T
+        Etot += E
+    Etot += E + 1                                     # first omitted exact term is below (T_n + E) <= E + 1 ulps
+    # exp(-t) in [exp(-(X + 1)/ONE), exp(-X/ONE)] and exp(-(X + 1)/ONE) >= exp(-X/ONE) - 1/ONE (Lipschitz 1)
+    v = IV(Fr(S - Etot - 1, ONE), Fr(S + Etot, ONE))
+    for _ in range(m):
+        v = v.sq()
+    return v
+
+
+def exp_neg(t):
+    """exp(-t) on an interval t with t.lo >= 0 (monotone decreasing)."""
+    return IV(exp_neg_point(t.hi).lo, exp_neg_point(t.lo).hi)
+
+
+def _atan_inv(n):
+    """Enclosure of arctan(1/n), n >= 2, alternating series."""
+    x = Fr(1, n)
+    s = Fr(0)
+    term = x
+    k = 0
+    while True:
+        s += term / (2 * k + 1)
+        k += 1
+        term = term * (-x * x)
+        if abs(term) / (2 * k + 1) < Fr(1, 1 << (PBITS + 8)):
+            break
+    return IV(s - abs(term), s + abs(term))
+
+
+PI = 4 * (4 * _atan_inv(5) - _atan_inv(239))
+SQRT2 = isqrt_iv(IV(2))
+SQRT2PI = isqrt_iv(2 * PI)
+
+
+def phi_point(x):
+    """Enclosure of Phi(x) = P(N(0,1) <= x) for a rational x with |x| <= 12, from the series
+    Phi(x) = 1/2 + (1/sqrt(2 pi)) sum_n (-1)^n x^(2n+1) / (2^n n! (2n+1)), evaluated in fixed point (ulp = 2^-(PBITS+40))
+    with an explicit integer bound E on the accumulated rounding error (in ulps) and the alternating-tail bound."""
+    x = Fr(x)
+    if x < 0:
+        p = phi_point(-x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 12:
+        raise ValueError('phi_point range')
+    P2 = PBITS + 40
+    ONE = 1 << P2
+    X = x.numerator * ONE // x.denominator          # x_lo = X / ONE <= x < (X + 1) / ONE ; |Phi(x) - Phi(x_lo)| <= 1 ulp
+    if X == 0:                                       # 0 <= x < 1 ulp: Phi(x) in [1/2, 1/2 + ulp]
+        return IV(Fr(1, 2), Fr(1, 2) + Fr(1, ONE))
+    X2u = (X + 1) * (X + 1)                          # upper bound of x^2 ONE^2
+    S = 0
+    T = X                                            # T_n ~ x^(2n+1) / (2^n n!) in ulps
+    E = 1                                            # error bound of T_n in ulps
+    Etot = 1                                         # error of S in ulps (starts with the x rounding)
+    n = 0
+    while T > 0:
+        S += (-1) ** n * (T // (2 * n + 1))
+        Etot += E + 1
+        n += 1
+        T = (T * X * X) // (ONE * ONE * 2 * n)
+        E = (E * X2u) // (ONE * ONE * 2 * n) + 2
+    # once T == 0 in fixed point the remaining terms are each below 1 ulp; if x^2 >= 2 n they are still growing in
+    # exact arithmetic, so continue counting until they decrease: bound the tail by the first omitted exact term
+    # <= (T_n + E) ulps <= (E + 1) ulps when T_n == 0 ... but growth may resume; require the decreasing regime.
+    if x * x > 2 * n:
+        raise ValueError('series terminated before the decreasing regime')
+    Etot += E + 1
+    core = IV(Fr(S - Etot, ONE), Fr(S + Etot, ONE)) / SQRT2PI
+    return IV(Fr(1, 2) + core.lo, Fr(1, 2) + core.hi)
+
+
+def Phi(x):
+    """Phi on an interval (monotone); arguments clipped to [-12, 12] (clipping only widens)."""
+    lo = min(max(x.lo, Fr(-12)), Fr(12))
+    hi = max(min(x.hi, Fr(12)), Fr(-12))
+    if x.lo < -12:
+        plo = Fr(0)
+    else:
+        plo = phi_point(lo).lo
+    if x.hi > 12:
+        phi_ = Fr(1)
+    else:
+        phi_ = phi_point(hi).hi
+    return IV(plo, phi_)
+
+
+def phi_pt(x):
+    """phi(x) for rational x (interval)."""
+    x = Fr(x)
+    return exp_neg_point(x * x / 2) / SQRT2PI
+
+
+def Phibar_pt(x):
+    """1 - Phi(x) for rational x (interval)."""
+    x = Fr(x)
+    if abs(x) <= 12:
+        p = phi_point(x)
+        return IV(1 - p.hi, 1 - p.lo)
+    if x > 0:
+        return IV(0, (phi_pt(x) / x).hi)            # Mills: Phibar(x) <= phi(x)/x
+    return IV(1 - (phi_pt(-x) / (-x)).hi, 1)
+
+
+def dfact(n):
+    """(n)!! for n >= -1."""
+    return math.prod(range(n, 0, -2)) if n > 0 else 1
+
+
+def gauss_moment(j):
+    """E[Z^j] for Z ~ N(0, 1)."""
+    return 0 if j % 2 else dfact(j - 1)
+
+
+def upper_moments(x, J):
+    """[M_0(x), ..., M_J(x)], M_j(x) = integral_x^inf t^j phi(t) dt, for rational x (intervals).  For x < 0,
+    M_j(x) = E[Z^j] - (-1)^j M_j(-x)."""
+    x = Fr(x)
+    if x < 0:
+        M = upper_moments(-x, J)
+        return [IV(gauss_moment(j)) - M[j] * (-1) ** j for j in range(J + 1)]
+    ph = phi_pt(x)
+    M = [Phibar_pt(x), ph]
+    for j in range(2, J + 1):
+        M.append(ph * x ** (j - 1) + M[j - 2] * (j - 1))
+    return M[:J + 1]
+
+
+def rup_rel(x, bits=200):
+    """Round a rational x >= 0 up to a dyadic with `bits` significant bits (upper bounds stay upper bounds)."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr(-((-x.numerator << sh) // x.denominator), 1 << sh)
+    return Fr(-((-x.numerator) // (x.denominator << -sh)) << -sh)
+
+
+def rdown_rel(x, bits=200):
+    """Round a rational x >= 0 down to a dyadic with `bits` significant bits."""
+    x = Fr(x)
+    if x <= 0:
+        return x
+    e = x.numerator.bit_length() - x.denominator.bit_length()
+    sh = bits - e
+    if sh >= 0:
+        return Fr((x.numerator << sh) // x.denominator, 1 << sh)
+    return Fr((x.numerator // (x.denominator << -sh)) << -sh)
+
+
+def sqrt_up(x):
+    """Rational upper bound of sqrt(x), x >= 0 rational, relative excess below 2^-198 (no absolute rounding floor): the root
+    step adds less than 2^-209 and rup_rel(., 200) less than 2^-199.  (The earlier '2^-200' was too sharp: at x = 49/100 the
+    excess is (8/7) 2^-200, review C59-DOC-01.)"""
+    x = Fr(x)
+    if x == 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rup_rel(Fr(math.isqrt(nd << (2 * m)) + 1, d << m))
+
+
+def sqrt_down(x):
+    """Rational lower bound of sqrt(x), x >= 0 rational."""
+    x = Fr(x)
+    if x <= 0:
+        return Fr(0)
+    n, d = x.numerator, x.denominator
+    nd = n * d
+    m = max(0, (420 - nd.bit_length()) // 2 + 1)
+    return rdown_rel(Fr(math.isqrt(nd << (2 * m)), d << m))
+
+
+def iroot(x, p):
+    """Upper bound (rational) of x^(1/p) for rational x >= 0 and p a power of two."""
+    v = Fr(x)
+    while p > 1:
+        v = sqrt_up(v)
+        p //= 2
+    return v
+
+
+def pow2(p):
+    q = 1
+    while q < p:
+        q *= 2
+    return q
+
+
+ZN = {}
+
+
+def znorm(p):
+    """Upper bound of ||N(0,1)||_p for p a power of two (p = 1 uses sqrt(2/pi) < 4/5)."""
+    if p == 1:
+        return Fr(4, 5)
+    if p not in ZN:
+        ZN[p] = iroot(Fr(dfact(p - 1)), p)
+    return ZN[p]
+
+
+def znorm1(p):
+    """max(1, ||N(0,1)||_p): the factor in ||(|mu| + ||Sigma^(1/2) z||)||_p <= |mu| + sqrt(tr Sigma) znorm1(p)."""
+    return Fr(1) if p <= 2 else znorm(p)
+
+
+def frob_norm(mu_abs, tr_hi, p):
+    """Upper bound of || ||X|| ||_p for a Gaussian vector X with ||E X|| <= mu_abs and tr Cov X <= tr_hi (Minkowski in
+    L^(p/2) on the eigen-decomposition for p >= 2; Lyapunov for p <= 2)."""
+    return rup_rel(mu_abs + sqrt_up(tr_hi) * znorm1(pow2(p)))
+
+
+def gamma_half(n2):
+    """Gamma(n2 / 2) for an integer n2 >= 1, as an interval (sqrt(pi) for odd n2)."""
+    if n2 % 2 == 0:
+        return IV(math.factorial(n2 // 2 - 1))
+    n = (n2 - 1) // 2                                 # Gamma(n + 1/2) = (2n - 1)!! sqrt(pi) / 2^n
+    return isqrt_iv(PI) * Fr(dfact(2 * n - 1), 2 ** n)
+
+
+def chi_moment(nu, j):
+    """E[chi_nu^j] = 2^(j/2) Gamma((nu + j)/2) / Gamma(nu/2) (interval), integer j >= 0."""
+    if j % 2 == 0:
+        return IV(math.prod(nu + 2 * i for i in range(j // 2)))
+    out = gamma_half(nu + j) / gamma_half(nu)
+    for _ in range(j):
+        out = out * SQRT2
+    return out
+
+
+CN = {}
+
+
+def chi_norm(nu, p):
+    """Upper bound of ||chi_nu||_p for p a power of two."""
+    if p == 1:
+        return sqrt_up(Fr(nu))
+    key = (nu, p)
+    if key not in CN:
+        CN[key] = iroot(Fr(math.prod(nu + 2 * i for i in range(p // 2))), p)
+    return CN[key]
+
+
+CC = {}
+
+
+def chi_const(nu):
+    """C_nu sqrt(2 pi) with C_nu = 1 / (2^(nu/2 - 1) Gamma(nu/2)): E[chi_nu^j 1{chi > x}] = C_nu sqrt(2 pi) M_(j+nu-1)(x)."""
+    if nu not in CC:
+        p2 = IV(Fr(2) ** ((nu - 2) // 2)) if nu % 2 == 0 else IV(Fr(2) ** ((nu - 3) // 2)) * SQRT2
+        CC[nu] = SQRT2PI / (p2 * gamma_half(nu))
+    return CC[nu]
+
+
+def chi_upper_moments(nu, x, J):
+    """[E[chi_nu^j 1{chi_nu > x}] for j = 0..J] (intervals), rational x >= 0."""
+    M = upper_moments(x, J + nu - 1)
+    c = chi_const(nu)
+    return [M[j + nu - 1] * c for j in range(J + 1)]
+
+
+def chi_tail(nu, x):
+    """Rational upper bound of P(chi_nu > x) for rational x >= 0 (Mills-type bound beyond the series range)."""
+    x = Fr(x)
+    if x <= 0:
+        return Fr(1)
+    if x <= 12:
+        return min(Fr(1), chi_upper_moments(nu, x, 0)[0].hi)
+    # x > 12: P(chi_nu > x) = c_nu sqrt(2pi) M_(nu-1)(x), M_j(x) <= phi(x) x^(j-1) (1 + 2 (j-1)/x^2) for x^2 >= 2 j, and
+    # phi(x) <= m!/(x^2/2)^m / sqrt(2 pi) (m = 40)
+    y = x * x / 2
+    ph = Fr(math.factorial(40)) / y ** 40 * Fr(2, 5)          # 1/sqrt(2 pi) < 2/5
+    j = nu - 1
+    Mj = ph * (x ** (j - 1) if j >= 1 else 1 / x) * (1 + Fr(2 * max(j - 1, 1)) / (x * x))
+    return min(Fr(1), Mj * Fr(51, 20))                          # c_nu sqrt(2 pi) <= sqrt(2 pi) < 51/20
+
+
+def phibar_upper(x):
+    """Rational upper bound of 1 - Phi(x) for rational x >= 0: the series value for x <= 12, and for x > 12 the Mills bound
+    phi(x)/x with exp(-y) <= m!/y^m (m = 40), which stays exact far below the 2^-160 rounding floor."""
+    x = Fr(x)
+    if x <= 12:
+        return Phibar_pt(x).hi
+    y = x * x / 2
+    m = 40
+    return Fr(math.factorial(m)) / y ** m / (x * Fr(5, 2))      # sqrt(2 pi) > 5/2
+
+
+def gauss_tail2(mu_abs, sd, t):
+    """Upper bound of P(|X| > t) for X Gaussian with |mean| <= mu_abs, sd <= sd."""
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), 2 * phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_tail1(mu_abs, sd, t):
+    if t <= mu_abs:
+        return Fr(1)
+    return min(Fr(1), phibar_upper((t - mu_abs) / sd))
+
+
+def gauss_norm(mu_abs, sd, p):
+    """Upper bound of ||X||_p for X Gaussian with |mean| <= mu_abs and sd <= sd."""
+    return rup_rel(mu_abs + sd * znorm(pow2(p)))
+
+
+def absup(x):
+    return max(abs(x.lo), abs(x.hi))
+
+
+def pow_tail(x, q):
+    """Upper bound of x^(1 - 1/q) for 0 <= x <= 1 (q a power of two)."""
+    if x <= 0:
+        return Fr(0)
+    return iroot(rup_rel(x, 64) ** (q - 1), q)
+
+
+# ----------------------------------------------------------------------------- functionals and exact covariance series (d = 3)
+HALF = Fr(1, 2)
+XM, XS, X0PT = -HALF, HALF, Fr(0)
+JET4_IDX = {'X%d%d%d' % (a, b1, 4 - a - b1): (a, b1, 4 - a - b1) for a in range(5) for b1 in range(5 - a)}
+JET4 = tuple(sorted(JET4_IDX, key=lambda n: (-JET4_IDX[n][0], -JET4_IDX[n][1])))
+BLOCKS = ((4, 0), (3, 1), (2, 2), (1, 3), (0, 4))           # partial blocks (a, c) of the fourth derivatives, a + c = 4
+
+
+def block_members(a, c):
+    """[(name, Frobenius multiplicity c!/beta!)] of the 4-jet block d_x^a D_y^c f(0)."""
+    out = []
+    for b1 in range(c, -1, -1):
+        b2 = c - b1
+        out.append(('X%d%d%d' % (a, b1, b2), math.factorial(c) // (math.factorial(b1) * math.factorial(b2))))
+    return out
+
+
+def term(c, p, a, b1, b2, x0):
+    """c r^p d_x^a d_y1^b1 d_y2^b2 f(x0 r, 0, 0)."""
+    return (Fr(c), p, a, b1, b2, Fr(x0))
+
+
+def functionals():
+    pins = {
+        'p1': [term(HALF, 0, 0, 0, 0, XM), term(HALF, 0, 0, 0, 0, XS)],
+        'p2': [term(-1, -1, 0, 0, 0, XM), term(1, -1, 0, 0, 0, XS)],
+        'p3p': [term(6, -2, 1, 0, 0, XM), term(6, -2, 1, 0, 0, XS), term(12, -3, 0, 0, 0, XM), term(-12, -3, 0, 0, 0, XS)],
+        'p4': [term(-1, -1, 1, 0, 0, XM), term(1, -1, 1, 0, 0, XS)],
+    }
+    for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))):
+        pins['q1%d' % j] = [term(HALF, 0, 0, e1, e2, XM), term(HALF, 0, 0, e1, e2, XS)]
+        pins['q2%d' % j] = [term(-1, -1, 0, e1, e2, XM), term(1, -1, 0, e1, e2, XS)]
+    targets = {
+        'a11': [term(1, 0, 0, 2, 0, XM)], 'a22': [term(1, 0, 0, 0, 2, XM)], 'a12': [term(1, 0, 0, 1, 1, XM)],
+        'o11': [term(1, -1, 0, 2, 0, XS), term(-1, -1, 0, 2, 0, XM)],
+        'o22': [term(1, -1, 0, 0, 2, XS), term(-1, -1, 0, 0, 2, XM)],
+        'o12': [term(1, -1, 0, 1, 1, XS), term(-1, -1, 0, 1, 1, XM)],
+        'T': [term(1, -2, 2, 0, 0, XM), term(4, -3, 1, 0, 0, XM), term(2, -3, 1, 0, 0, XS), term(6, -4, 0, 0, 0, XM),
+              term(-6, -4, 0, 0, 0, XS)],
+        'tau': [term(1, -3, 2, 0, 0, XS), term(-1, -3, 2, 0, 0, XM), term(-6, -4, 1, 0, 0, XM), term(-6, -4, 1, 0, 0, XS),
+                term(-12, -5, 0, 0, 0, XM), term(12, -5, 0, 0, 0, XS)],
+    }
+    for j, (e1, e2) in ((1, (1, 0)), (2, (0, 1))):
+        targets['v%d' % j] = [term(1, -1, 1, e1, e2, XM), term(1, -2, 0, e1, e2, XM), term(-1, -2, 0, e1, e2, XS)]
+        targets['nu%d' % j] = [term(1, -2, 1, e1, e2, XM), term(1, -2, 1, e1, e2, XS), term(2, -3, 0, e1, e2, XM),
+                               term(-2, -3, 0, e1, e2, XS)]
+    for b1 in range(3, -1, -1):
+        targets['y%s' % ('1' * b1 + '2' * (3 - b1))] = [term(1, 0, 0, b1, 3 - b1, X0PT)]
+    for n, (a, b1, b2) in JET4_IDX.items():
+        targets[n] = [term(1, 0, a, b1, b2, X0PT)]
+    return pins, targets
+
+
+PIN_NAMES = ('p1', 'p2', 'p3p', 'p4', 'q11', 'q21', 'q12', 'q22')
+BNAMES = ('a11', 'a22', 'a12')                              # the transverse Hessian A = D_y^2 f(M) = -B
+VNAMES = ('v1', 'v2')
+ONAMES = ('o11', 'o22', 'o12')
+YNAMES = ('y111', 'y112', 'y122', 'y222')
+FREE = VNAMES + ONAMES + YNAMES                              # the free third-derivative blocks, in Cholesky order
+FREE_W = (1, 1, 1, 1, 2, 1, 3, 3, 1)                         # Frobenius multiplicities (Euclidean norms in these weights)
+TARGET_NAMES = BNAMES + FREE + ('T', 'tau', 'nu1', 'nu2') + JET4
+NP = len(PIN_NAMES)
+
+
+def weight(F):
+    """Scaling weight a - (power of r), common to all terms of a preconditioned functional."""
+    ws = {t[2] - t[1] for t in F}
+    require(len(ws) == 1, 'functional not homogeneous')
+    return ws.pop()
+
+
+def hcoef(n):
+    """d^n/dx^n exp(-x^2/2) at 0."""
+    if n % 2:
+        return 0
+    return (-1) ** (n // 2) * dfact(n - 1)
+
+
+PAIR_CACHE = {}
+
+
+def pair_series(F, G, key):
+    """Exact data of Cov_K(F, G) = sum_gamma kappa_gamma pi_gamma r^(gx - wF - wG), gamma = (gx, gy1, gy2):
+    returns (reference series {power: rational}, weight w, list of (|c c'|, a+a', gy1, gy2) for the remainder bounds)."""
+    if key in PAIR_CACHE:
+        return PAIR_CACHE[key]
+    wF, wG = weight(F), weight(G)
+    w = wF + wG
+    pi = {}
+    absterms = []
+    for (cf, pf, af, bf1, bf2, xf) in F:
+        for (cg, pg, ag, bg1, bg2, xg) in G:
+            g1, g2 = bf1 + bg1, bf2 + bg2
+            sgn = (-1) ** (ag + bg1 + bg2)
+            d = xf - xg
+            absterms.append((abs(cf * cg), af + ag, g1, g2))
+            for gx in range(af + ag, NX + 1):
+                m = gx - af - ag
+                val = cf * cg * sgn * d ** m / math.factorial(m)
+                if val != 0:
+                    pi[(gx, g1, g2)] = pi.get((gx, g1, g2), Fr(0)) + val
+    for (gx, g1, g2), v in pi.items():
+        require(v == 0 or gx - w >= 0, 'negative power of r in a covariance (regularity fails)')
+    ser = {}
+    for (gx, g1, g2), v in pi.items():
+        kap = hcoef(gx) * hcoef(g1) * hcoef(g2)
+        if kap and v:
+            ser[gx - w] = ser.get(gx - w, Fr(0)) + kap * v
+    out = ({p: c for p, c in ser.items() if c != 0}, w, absterms)
+    PAIR_CACHE[key] = out
+    return out
+
+
+E_UP = Fr(27183, 10000)                                      # > e
+SQRT3_UP = Fr(7, 4)                                          # > sqrt(3)
+
+
+def torus_M1(L):
+    """Bound of |sum_{n != 0} G(z + p_n)| on the polydisc |z_1|, |z_2|, |z_3| <= 1 (every frame, |p_n| = |n| L):
+    exp(3/2) sum_{j >= 1} (24 j^2 + 2) exp(-(j L - sqrt 3)^2 / 2); 24 j^2 + 2 lattice points have sup-norm j."""
+    L = Fr(L)
+    e32 = exp_neg_point(Fr(3, 2)).inv().hi
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += (24 * j * j + 2) * exp_neg_point((j * L - SQRT3_UP) ** 2 / 2).hi
+    # j >= 6: (24 j^2 + 2) exp(-(jL - 7/4)^2/2) decreases by more than 1/2 per step for L >= 10
+    tail = 2 * (24 * 36 + 2) * exp_neg_point((6 * L - SQRT3_UP) ** 2 / 2).hi
+    return e32 * (tot + tail)
+
+
+def theta_minus_one(L):
+    L = Fr(L)
+    tot = Fr(0)
+    for j in range(1, 6):
+        tot += (24 * j * j + 2) * exp_neg_point((j * L) ** 2 / 2).hi
+    return tot + 2 * (24 * 36 + 2) * exp_neg_point((6 * L) ** 2 / 2).hi
+
+
+def cov_band(F, G, key, r0, r1, M1):
+    """Enclosure of the covariance (unnormalized torus kernel sum_n G(z + p_n)) of F and G, for r in [r0, r1]."""
+    ser, w, absterms = pair_series(F, G, key)
+    val = IV(0)
+    for p, c in ser.items():
+        val = val + IV(r0 ** p, r1 ** p) * c
+    A = max((t[1] for t in absterms), default=0)
+    # Taylor truncation tail of the reference kernel (orders gx > NX)
+    tail = Fr(0)
+    for (acc, aa, g1, g2) in absterms:
+        tail += acc * abs(hcoef(g1) * hcoef(g2))
+    tail *= 2 * Fr(NX + 1) ** A * r1 ** (NX + 1 - w)
+    # torus remainder: Cauchy estimates |rho_gamma| <= gx! gy1! gy2! M1 on the unit polydisc
+    tor = Fr(0)
+    for (acc, aa, g1, g2) in absterms:
+        tor += acc * math.factorial(g1) * math.factorial(g2) * Fr(max(w, 1)) ** aa
+    tor = tor * M1 / (1 - E_UP * r1)
+    err = tail + tor
+    return val + IV(-err, err)
+
+
+# ----------------------------------------------------------------------------- interval linear algebra
+def solve_iv(A, B):
+    """Interval Gaussian elimination with partial pivoting on midpoints: X[c] solves A X[c] = B[c]."""
+    n = len(A)
+    M = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    for col in range(n):
+        p = max(range(col, n), key=lambda i: abs(M[i][col].mid()))
+        M[col], M[p] = M[p], M[col]
+        piv = M[col][col]
+        for i in range(col + 1, n):
+            f = M[i][col] / piv
+            for j in range(col, len(M[i])):
+                M[i][j] = M[i][j] - f * M[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            s = M[i][n + c]
+            for j in range(i + 1, n):
+                s = s - M[i][j] * x[j]
+            x[i] = s / M[i][i]
+        X.append(x)
+    return X
+
+
+def cholesky(C):
+    """Interval Cholesky of an n x n SPD interval matrix (pivots must be positive)."""
+    n = len(C)
+    L = [[IV(0)] * n for _ in range(n)]
+    for i in range(n):
+        for j in range(i + 1):
+            s = C[i][j] - sum((L[i][l] * L[j][l] for l in range(j)), IV(0))
+            if i == j:
+                require(s.lo > 0, 'Cholesky pivot not positive')
+                L[i][j] = isqrt_iv(s)
+            else:
+                L[i][j] = s / L[j][j]
+    return L
+
+
+def opnorm_lower_tri(L, rows, cols):
+    """Upper bound of the operator norm of the block L[rows][cols] of an interval lower-triangular matrix:
+    max |diagonal| + Frobenius norm of the rest (square diagonal blocks), Frobenius norm otherwise."""
+    if rows == cols:
+        dg = max(absup(L[i][i]) for i in rows)
+        off = sum(absup(L[i][j]) ** 2 for i in rows for j in cols if j < i)
+        return rup_rel(dg + sqrt_up(off))
+    return sqrt_up(sum(absup(L[i][j]) ** 2 for i in rows for j in cols))
+
+
+def sym3_eig_bounds(C):
+    """(lower bound of lambda_min, upper bound of lambda_max) of a symmetric 3 x 3 interval matrix (Gershgorin)."""
+    lo = min(C[i][i].lo - sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3))
+    hi = max(C[i][i].hi + sum(absup(C[i][j]) for j in range(3) if j != i) for i in range(3))
+    return lo, hi
+
+
+def det3(C):
+    a, b, c = C[0]
+    d, e, f = C[1]
+    g, h, i = C[2]
+    return a * (e * i - f * h) - b * (d * i - f * g) + c * (d * h - e * g)
+
+
+# ----------------------------------------------------------------------------- the conditional law on an r-band
+class Law:
+    pass
+
+
+def band_law(r0, r1, L=L_MIN):
+    """Interval enclosures, uniform over r in [r0, r1], every frame and every torus side >= L, of the Q-law of the
+    targets: mean = b alpha + k beta, covariance C (normalized kernel), and the pin energy coefficients; then the
+    regression on the transverse Hessian (three entries) and the g-law of the remaining targets."""
+    pins, targets = functionals()
+    P = [pins[n] for n in PIN_NAMES]
+    T = [targets[n] for n in TARGET_NAMES]
+    M1 = torus_M1(L)
+    th1 = theta_minus_one(L)
+    nt = len(T)
+    Spp = [[None] * NP for _ in range(NP)]
+    for i in range(NP):
+        for j in range(i, NP):
+            Spp[i][j] = Spp[j][i] = cov_band(P[i], P[j], ('p', i, j), r0, r1, M1)
+    Stp = [[cov_band(T[i], P[j], ('tp', i, j), r0, r1, M1) for j in range(NP)] for i in range(nt)]
+    Stt = [[None] * nt for _ in range(nt)]
+    for i in range(nt):
+        for j in range(i, nt):
+            Stt[i][j] = Stt[j][i] = cov_band(T[i], T[j], ('tt', i, j), r0, r1, M1)
+    r = IV(r0, r1)
+    kvec = [-(r * r * r) * HALF, -(r * r), IV(12)] + [IV(0)] * (NP - 3)
+    e1 = [IV(1)] + [IV(0)] * (NP - 1)
+    W = solve_iv(Spp, Stp)
+    xs = solve_iv(Spp, [e1, kvec])
+    law = Law()
+    law.r0, law.r1, law.L, law.M1, law.th1 = r0, r1, L, M1, th1
+    law.alpha = {n: W[i][0] for i, n in enumerate(TARGET_NAMES)}
+    law.beta = {n: sum((W[i][j] * kvec[j] for j in range(NP)), IV(0)) for i, n in enumerate(TARGET_NAMES)}
+    scale = IV(1 / (1 + th1), 1)          # K_L = (unnormalized) / Theta, Theta in [1, 1 + th1]
+    C = {}
+    for i, a in enumerate(TARGET_NAMES):
+        for j in range(i, nt):
+            c = TARGET_NAMES[j]
+            v = Stt[i][j] - sum((W[i][l] * Stp[j][l] for l in range(NP)), IV(0))
+            C[(a, c)] = C[(c, a)] = v * scale
+    law.C = C
+    escale = IV(1, 1 + th1)
+    law.e_bb = xs[0][0] * escale
+    law.e_bk = sum((kvec[j] * xs[0][j] for j in range(NP)), IV(0)) * escale
+    law.e_kk = sum((kvec[j] * xs[1][j] for j in range(NP)), IV(0)) * escale
+    regress_on_B(law)
+    law.S = {blk: taylor_tail_coefficient(blk, r1, M1) for blk in BLOCKS}
+    law.net_var = {blk: net_max_variance(law.C, blk) for blk in BLOCKS if blk[1] >= 2}
+    return law
+
+
+NET_M = 128                                # rational unit vectors ((1 - t^2), 2 t)/(1 + t^2), t = j/NET_M, and their rotations
+
+
+def circle_net():
+    """Rational unit vectors u_j covering angles [0, pi) with consecutive angular gaps <= 2/NET_M (theta = 2 arctan t has
+    d theta/dt <= 2), together with the rotated copies (-u2, u1)."""
+    pts = []
+    for j in range(NET_M + 1):
+        t = Fr(j, NET_M)
+        pts.append(((1 - t * t) / (1 + t * t), 2 * t / (1 + t * t)))
+    return pts + [(-u2, u1) for (u1, u2) in pts]
+
+
+NET = circle_net()
+
+
+def block_form_weights(blk, u):
+    """w_beta = (c!/beta!) u1^beta1 u2^beta2: d_x^a d_u^c f(0) = sum_beta w_beta X_(a,beta)."""
+    a, c = blk
+    return [(n, m * u[0] ** int(n[2]) * u[1] ** int(n[3])) for n, m in block_members(a, c)]
+
+
+def net_max_variance(C, blk):
+    """Upper bound of max_j Var_Q(d_x^a d_(u_j)^c f(0)) over the net."""
+    best = Fr(0)
+    for u in NET:
+        w = block_form_weights(blk, u)
+        v = sum(abs(wi) * abs(wj) * absup(C[(ni, nj)]) if ni != nj else wi * wi * C[(ni, ni)].hi
+                for ni, wi in w for nj, wj in w)
+        best = max(best, v)
+    return best
+
+
+FROB_B = (IV(1), IV(1), SQRT2)               # (B11, B22, B12) -> Frobenius-isometric coordinates (B11, B22, sqrt2 B12)
+
+
+def regress_on_B(law):
+    """Regression of every target on the three entries of the transverse Hessian, in the Frobenius-isometric coordinates
+    b~ = (B11, B22, sqrt2 B12) of B = -A: F = F_g + beta~_F . (b~ - mu~), with F_g independent of B."""
+    C = law.C
+    # covariance of b~ (B = -A: signs cancel in the covariance)
+    Cb = [[C[(BNAMES[i], BNAMES[j])] * FROB_B[i] * FROB_B[j] for j in range(3)] for i in range(3)]
+    law.Cb = Cb
+    law.detCb = det3(Cb)
+    require(law.detCb.lo > 0, 'transverse Hessian covariance not positive')
+    lo, hi = sym3_eig_bounds(Cb)
+    require(lo > 0, 'transverse Hessian covariance eigenvalue bound')
+    law.sB_lo2, law.sB_hi2 = lo, hi
+    law.sB_lo = sqrt_down(lo)
+    # regression coefficients of F on b~: beta~_F = Cov(F, b~) Cb^-1, Cov(F, b~_i) = -Cov(F, A_i) FROB_B[i]
+    others = [n for n in TARGET_NAMES if n not in BNAMES]
+    rhs = [[-C[(n, BNAMES[i])] * FROB_B[i] for i in range(3)] for n in others]
+    sol = solve_iv(Cb, rhs)
+    law.bt = {n: sol[i] for i, n in enumerate(others)}
+    law.bnorm = {n: sqrt_up(sum(absup(x) ** 2 for x in law.bt[n])) for n in others}
+    Cg = {}
+    for a in others:
+        for c in others:
+            Cg[(a, c)] = C[(a, c)] - sum((law.bt[a][i] * (-C[(c, BNAMES[i])] * FROB_B[i]) for i in range(3)), IV(0))
+    law.Cg = Cg
+    # the free blocks in Frobenius coordinates: v, omega = (o11, o22, sqrt2 o12), Y~ = (y111, sqrt3 y112, sqrt3 y122, y222)
+    fw = [isqrt_iv(IV(w)) if w != 1 else IV(1) for w in FREE_W]
+    law.Lfree = cholesky([[Cg[(FREE[i], FREE[j])] * fw[i] * fw[j] for j in range(9)] for i in range(9)])
+    law.fw = fw
+    law.sdg = {n: sqrt_up(max(Cg[(n, n)].hi, Fr(0))) for n in others}
+    law.sdQ = {n: sqrt_up(C[(n, n)].hi) for n in TARGET_NAMES}
+
+
+def sd0_sq(g, M1):
+    """Upper bound of Var d^g f(0) = |d^(2g) K_L(0)| (normalized kernel, every frame, L >= L_MIN), g = (gx, gy1, gy2):
+    the reference value prod (2 g_i - 1)!! plus the Cauchy bound (2g)! M1 of the torus remainder."""
+    return math.prod(dfact(2 * x - 1) for x in g) + math.prod(math.factorial(2 * x) for x in g) * M1
+
+
+def taylor_tail_coefficient(blk, r1, M1):
+    """S_(a,c)(r1) = sum_{gamma != 0} sigma_(a,c),gamma (2 r1)^|gamma| / gamma!, sigma^2 = sum_beta (c!/beta!) s^2_(a+gx, beta+gy):
+    the Frobenius norm of the Taylor tail of the block d_x^a D_y^c f at the midpoint over D."""
+    a, c = blk
+    tot = Fr(0)
+    for n in range(1, NB + 1):
+        for gx in range(n + 1):
+            for g1 in range(n - gx + 1):
+                g2 = n - gx - g1
+                s2 = Fr(0)
+                for b1 in range(c + 1):
+                    b2 = c - b1
+                    mult = math.factorial(c) // (math.factorial(b1) * math.factorial(b2))
+                    s2 += mult * sd0_sq((a + gx, b1 + g1, b2 + g2), M1)
+                tot += sqrt_up(s2) * (2 * r1) ** n / (math.factorial(gx) * math.factorial(g1) * math.factorial(g2))
+    # tail |gamma| = n > NB: s_delta <= 2^|delta| delta! sqrt(1 + M1) and (alpha + gamma)!/gamma! <= (n + 4)^4, so
+    # sigma (2 r1)^n / gamma! <= 2^(c/2) 2^(4 + n) (n + 4)^4 (2 r1)^n sqrt(1 + M1); (n+1)(n+2)/2 multi-indices of order n;
+    # the ratio of consecutive bounds is below 2 q for these n (q = 4 r1 <= 1/16)
+    q = 4 * r1
+    require(q <= Fr(1, 16), 'Taylor tail ratio too large')
+    n = NB + 1
+    base = 2 * 2 ** 4 * sqrt_up(Fr(2) ** c * (1 + M1))
+    tail = base * Fr((n + 1) * (n + 2), 2) * Fr(n + 4) ** 4 * q ** n / (1 - 2 * q)
+    return rup_rel(tot + tail)
+
+
+# ----------------------------------------------------------------------------- closed forms for the contact transverse Hessian
+# B = m I + GOE_2 (diagonal N(0, 2), off-diagonal N(0, 1)): its eigenvalues are s -+ rho with s = m + (G11 + G22)/2 ~ N(m, 1)
+# and rho = sqrt(((G11 - G22)/2)^2 + G12^2) Rayleigh, independent; det B = s^2 - rho^2, tr B = 2 s, B > 0 iff s > rho.
+def upper_moments_iv(x, J):
+    """[M_0(x), ..., M_J(x)] over an interval x whose interior does not contain 0 (each M_j is monotone there)."""
+    if x.lo == x.hi:
+        return upper_moments(x.lo, J)
+    require(not (x.lo < 0 < x.hi), 'incomplete moment over an interval containing 0')
+    A, B = upper_moments(x.lo, J), upper_moments(x.hi, J)
+    return [IV(min(a.lo, b.lo), max(a.hi, b.hi)) for a, b in zip(A, B)]
+
+
+def es_pos(m, n):
+    """[E[s^j 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m (intervals)."""
+    M = upper_moments(-Fr(m), n)
+    return [sum((M[i] * (math.comb(j, i) * Fr(m) ** (j - i)) for i in range(j + 1)), IV(0)) for j in range(n + 1)]
+
+
+def isqrt2_pow(i):
+    """2^(-i/2) (interval)."""
+    return IV(Fr(1, 2 ** (i // 2))) if i % 2 == 0 else IV(Fr(1, 2 ** (i // 2))) / SQRT2
+
+
+def es_pos_exp(m, n):
+    """[E[s^j exp(-s^2/2) 1{s > 0}] for j = 0..n], s ~ N(m, 1), rational m:
+    = exp(-m^2/4)/sqrt2 . E[(m/2 + U/sqrt2)^j 1{U > -m/sqrt2}], U ~ N(0, 1)."""
+    m = Fr(m)
+    M = upper_moments_iv(-(IV(m) / SQRT2), n)
+    pre = exp_neg_point(m * m / 4) / SQRT2
+    out = []
+    for j in range(n + 1):
+        tot = IV(0)
+        for i in range(j + 1):
+            tot = tot + M[i] * (math.comb(j, i) * (m / 2) ** (j - i)) * isqrt2_pow(i)
+        out.append(pre * tot)
+    return out
+
+
+def inner_rho_polys(a):
+    """integral_0^s (s^2 - rho^2)^a rho exp(-rho^2/2) d rho = P_a(s) - exp(-s^2/2) Q_a(s) (exact polynomial coefficient
+    dicts {power: Fraction}); with u = rho^2/2 it is integral_0^(s^2/2) (s^2 - 2u)^a e^(-u) du."""
+    P, Q = {}, {}
+    for i in range(a + 1):
+        c = Fr(math.comb(a, i) * (-2) ** i * math.factorial(i))
+        P[2 * (a - i)] = P.get(2 * (a - i), Fr(0)) + c
+        for l in range(i + 1):
+            pw = 2 * (a - i) + 2 * l
+            Q[pw] = Q.get(pw, Fr(0)) + c / (2 ** l * math.factorial(l))
+    return P, Q
+
+
+def contact_moment(a, c, m):
+    """E[det(B)^a tr(B)^c 1{B > 0}] for B = m I + GOE_2, rational m (interval)."""
+    P, Q = inner_rho_polys(a)
+    n = 2 * a + c
+    E1 = es_pos(m, n)
+    E2 = es_pos_exp(m, n)
+    tot = IV(0)
+    for pw, cf in P.items():
+        tot = tot + E1[pw + c] * (cf * 2 ** c)
+    for pw, cf in Q.items():
+        tot = tot - E2[pw + c] * (cf * 2 ** c)
+    return tot
+
+
+def m3(m):
+    """m_(3,m) = E[det(B)^2 1{B > 0}] = E[(s^4 - 4 s^2 + 8) 1{s > 0}] - 4 sqrt2 exp(-m^2/4) Phi(m/sqrt2)."""
+    return contact_moment(2, 0, m)
+
+
+def gauss_half_moment(j, m, sig):
+    """J_j(m, sig) = integral_0^inf lambda^j exp(-(lambda - m)^2/(2 sig^2)) d lambda for rational m, sig > 0 (interval):
+    sig sqrt(2 pi) E[(m + sig Z)^j 1{m + sig Z > 0}]."""
+    m, sig = Fr(m), Fr(sig)
+    M = upper_moments(-m / sig, j)
+    tot = sum((M[i] * (math.comb(j, i) * m ** (j - i) * sig ** i) for i in range(j + 1)), IV(0))
+    return tot * SQRT2PI * sig
+
+
+# ----------------------------------------------------------------------------- moment calculus (normalized by K = 12 k)
+def binom(n, k):
+    return math.comb(n, k)
+
+
+def pos_part_moments_chi(s, c, nu, nmax, extra_max):
+    """P[(n, e)] = upper bound of E[u^e (u^n - c^n)_+], u = s chi_nu, for 1 <= n <= nmax, 0 <= e <= extra_max."""
+    x = c / s
+    M = chi_upper_moments(nu, x, nmax + extra_max)
+    out = {}
+    for n in range(1, nmax + 1):
+        for e in range(extra_max + 1):
+            out[(n, e)] = rup_rel((M[n + e] * s ** (n + e) - M[e] * c ** n * s ** e).hi)
+    return out
+
+
+NU = {2: 2, 3: 3, 4: 4}                      # chi degrees of freedom of the free blocks v (2), omega (3), Y~ (4)
+
+
+def normalized_moments(q, nmax, theta):
+    """Component moments, normalized by K = 12 k (K >= Klo), for n <= nmax, under the product of the g-law and nu:
+      E1[n], E1o[n], E1v[n] : upper bounds of E[y1^n], E[o0 y1^n], E[v0^2 y1^n], y1 = 1 + cmt1 + (r1/Klo) D1;
+      Tn[n], To[n], Tv[n]   : upper bounds of sum_{i=2,3,4} E[(y_i^n - c^n)_+] (and with the factors o0, v0^2),
+                              y_i <= u_i + delta_i (u_i = s_i chi_nu_i independent), threshold c = q['c'].
+    o0 = Klo u_3 and v0^2 = Klo^2 u_2^2 / 4 are unnormalized."""
+    th = theta
+    cp = q['c'] * (1 - th)                  # <= c (1 + th)^{-(n-1)/n} for every n (Bernoulli)
+    PP = {i: pos_part_moments_chi(q['s'][i], cp, NU[i], nmax, 2) for i in (2, 3, 4)}
+    Eo0 = (chi_moment(3, 1) * q['s3']).hi
+    Ev0sq = (chi_moment(2, 2) * q['s2'] * q['s2']).hi / 4
+    n_o0_2 = q['s3'] * chi_norm(3, 2)
+    n_v0sq_2 = q['s2'] * q['s2'] * chi_norm(2, 4) ** 2 / 4
+    Klo = q['Klo']
+    E1, E1o, E1v, Tn, To, Tv = {}, {}, {}, {}, {}, {}
+    base = rup_rel(1 + q['cmt1'])
+    bp = [Fr(1)]
+    for _ in range(nmax):
+        bp.append(rup_rel(bp[-1] * base))
+    # (r/K) D1 terms: ||(rK D1)^j||_1 <= (rK ||D1||_pow2(j))^j and ||(rK D1)^j||_2 <= (rK ||D1||_pow2(2j))^j
+    d1 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(j))) ** j) for j in range(1, nmax + 1)]
+    d2 = [Fr(1)] + [rup_rel((q['rK'] * q['nD1'](pow2(2 * j))) ** j) for j in range(1, nmax + 1)]
+    for n in range(0, nmax + 1):
+        ey = bp[n]
+        eo = bp[n] * Eo0
+        ev = bp[n] * Ev0sq
+        for j in range(1, n + 1):
+            coef = binom(n, j) * bp[n - j]
+            ey += coef * d1[j]
+            eo += coef * n_o0_2 * d2[j]
+            ev += coef * n_v0sq_2 * d2[j]
+        E1[n], E1o[n], E1v[n] = rup_rel(ey), rup_rel(eo), rup_rel(ev)
+        if n == 0:
+            continue
+        f1 = (1 + th) ** (n - 1)
+        f2 = (1 + 1 / th) ** (n - 1)
+        tn = to = tv = Fr(0)
+        for i in (2, 3, 4):
+            P = PP[i]
+            dn = rup_rel(q['nd'](i, pow2(n)) ** n)
+            dn2 = rup_rel(q['nd'](i, pow2(2 * n)) ** n)
+            tn += f1 * P[(n, 0)] + f2 * dn
+            to += (f1 * Klo * P[(n, 1)] if i == 3 else f1 * Eo0 * P[(n, 0)]) + f2 * n_o0_2 * dn2
+            tv += (f1 * Klo * Klo / 4 * P[(n, 2)] if i == 2 else f1 * Ev0sq * P[(n, 0)]) + f2 * n_v0sq_2 * dn2
+        Tn[n], To[n], Tv[n] = rup_rel(tn), rup_rel(to), rup_rel(tv)
+    return E1, E1o, E1v, Tn, To, Tv
+
+
+def half_gauss_moments(m, sig, J):
+    """[J_j(m, sig) for j = 0..J], J_j = integral_0^inf l^j exp(-(l - m)^2/(2 sig^2)) dl, rational m >= 0, sig > 0 (intervals):
+    J_0 = sig sqrt(2 pi) Phi(m/sig), J_1 = m J_0 + sig^2 exp(-m^2/(2 sig^2)), J_(j+1) = m J_j + j sig^2 J_(j-1)
+    (integration by parts; every term is positive for m >= 0)."""
+    m, sig = Fr(m), Fr(sig)
+    require(m >= 0 and sig > 0, 'half-line Gaussian moments')
+    s2 = sig * sig
+    out = [SQRT2PI * sig * (1 - Phibar_pt(m / sig))]
+    out.append(out[0] * m + exp_neg_point(m * m / (2 * s2)) * s2)
+    for j in range(1, J):
+        nxt = out[j] * m + out[j - 1] * (j * s2)
+        out.append(IV(rdown_rel(nxt.lo), rup_rel(nxt.hi)))
+    return out
+
+
+def lam2_moments(b_lo, b_hi, eps, sig):
+    """Bounds on the lambda_2-integrals of the density majorant g(l) = exp(-(l - b)^2/(2 sig^2) + eps |l - b|/sig^2),
+    uniformly over b in [b_lo, b_hi] (b >= 0): I_hi[j] >= integral_0^inf l^j g(l) dl (j = 0..JM), I3_lo <= integral l^3 g.
+    For l >= 0, eps |l - b| <= eps (l + b), so g(l) <= exp((4 b eps + eps^2)/(2 sig^2)) exp(-(l - b - eps)^2/(2 sig^2));
+    each J_j(m, sig) increases with m; and g(l) >= exp(-(l - b)^2/(2 sig^2))."""
+    JM = 3 + 64
+    shift = (4 * b_hi * eps + eps * eps) / (2 * sig * sig)
+    fac = exp_neg_point(shift).inv().hi
+    Jh = half_gauss_moments(b_hi + eps, sig, JM)
+    I_hi = [rup_rel(x.hi * fac) for x in Jh]
+    I3_lo = rdown_rel(half_gauss_moments(b_lo, sig, 3)[3].lo)
+    return I_hi, I3_lo
+
+
+DETN = {}
+
+
+def det_norm(p, bpp, muB, sig):
+    """Upper bound of ||det(B) 1{B > 0}||_p (p a power of two) for B = mu_B + Gaussian with Loewner coupling
+    B <= (b'' + eta ||z||) I + GOE (b'' rounded up to a 1/1024 grid): the contact moment E[det^p 1{B > 0}] at b'' on
+    {||z|| <= R0}, plus E[(||B||_F^2/2)^p 1{||z|| > R0}] <= 2^-p E[(||mu_B|| + sig chi_3)^(2p) 1{chi_3 > R0}]."""
+    bg = Fr(math.ceil(bpp * 1024), 1024)
+    key = (p, bg)
+    if key not in DETN:
+        DETN[key] = contact_moment(p, 0, bg).hi
+    cm = DETN[key]
+    R0 = Fr(R0_COUPLING)
+    U = chi_upper_moments(3, R0, 2 * p)
+    out = sum((U[i] * (binom(2 * p, i) * muB ** (2 * p - i) * sig ** i) for i in range(2 * p + 1)), IV(0)).hi / 2 ** p
+    return iroot(rup_rel(cm + out), p)
+
+
+def box_bound(law, bb, kb, r0, r1, last_band=False, x0=None, theta=None):
+    """Certified upper bound of sup_{r in [r0, r1]} Q^W(G_r^c) / r^3 over b in bb, k in kb, with its pieces, for the near-branch
+    threshold x0 (X_0 of NOTE section 4) and the split parameter theta (section 5).  The bound is proved for 0 < x0 < 2/9 and
+    0 < theta < 1: then 3 x0 < 1, x0 <= 1/4 (root bounds), kappa = 3/(1 - 3 x0), c = 1 - kappa x0/2 > 0 and
+    (1 + kappa x0) c^2 <= 1.  No claim is made outside that domain.  COMBO = (1/64, 1/32), also used by Math-#215's caller,
+    lies inside it (review C61)."""
+    x0 = X0 if x0 is None else Fr(x0)
+    theta = THETA if theta is None else Fr(theta)
+    b = IV(bb[0], bb[1])
+    k = IV(kb[0], kb[1])
+    blo, bhi = bb
+    klo, khi = kb[0], kb[1]
+    require(blo >= 0, 'b >= 0 assumed by the density majorant')
+    Klo = 12 * klo
+    mu = {n: b * law.alpha[n] + k * law.beta[n] for n in TARGET_NAMES}
+    mabs = {n: absup(mu[n]) for n in TARGET_NAMES}
+    E = b * b * law.e_bb + b * k * law.e_bk * 2 + k * k * law.e_kk
+    require(E.hi > 0, 'energy')
+    sqE = sqrt_up(E.hi)
+    nl2 = MUT != 'no-lambda2'
+    # ---- the transverse Hessian B = -A: mean in Frobenius coordinates, its distance to b I, the density majorant
+    dmu = [b * (-law.alpha['a11'] - 1) + k * (-law.beta['a11']), b * (-law.alpha['a22'] - 1) + k * (-law.beta['a22']),
+           (b * law.alpha['a12'] + k * law.beta['a12']) * SQRT2]
+    eps = sqrt_up(sum(absup(x) ** 2 for x in dmu))                 # sup over the box of ||mu_B - b I||_F
+    muB = rup_rel(bhi * sqrt_up(Fr(2)) + eps)                       # sup ||mu_B||_F
+    sig = sqrt_up(law.sB_hi2)                                       # rational, >= sqrt(lambda_max(Cov b~))
+    sB_lo = law.sB_lo                                               # <= sqrt(lambda_min(Cov b~))
+    KB = rup_rel((PI * SQRT2 / (2 * PI * SQRT2PI) / isqrt_iv(law.detCb)).hi)   # pi sqrt2 (2 pi)^(-3/2) det^(-1/2)
+    s2g = sig * sig
+    ex0 = blo * blo / (2 * s2g) - eps * bhi / s2g          # g(0; b) <= exp(-ex0) over the box
+    g0 = exp_neg_point(ex0).hi if ex0 >= 0 else exp_neg_point(-ex0).inv().hi
+    gbar = exp_neg_point(eps * eps / (2 * s2g)).inv().hi
+    # slope of the linear density bound g(l) <= g0 + l Lg on l >= 0: the global |g'| <= exp(-1/2) gbar / sig, or the chord
+    # of min(gbar, g0 exp(c l)) with c = (b_hi + eps)/sig^2 (g(l; b) <= g(0; b) exp((b + eps) l / sig^2) for l >= 0):
+    # g0 c (x - 1)/log x with log x = log(gbar/g0) = y (rational), (x - 1)/log x increasing in x
+    Lg = rup_rel(exp_neg_point(Fr(1, 2)).hi * gbar / sig)
+    y = (eps * eps + blo * blo - 2 * eps * bhi) / (2 * s2g)
+    if y > 0:
+        cc = (bhi + eps) / s2g
+        Lg = min(Lg, rup_rel(g0 * cc * (exp_neg_point(y).inv().hi - 1) / y))
+    else:
+        Lg = Fr(0)                                    # gbar <= g0: the bound g <= gbar alone is the smaller one
+    I_hi, I3_lo = lam2_moments(blo, bhi, eps, sig)
+    require(I3_lo > 0, 'lambda_2 moment')
+
+    def l2norm(p):                                  # ||lambda_2||_p under nu (and under nu'), p a power of two
+        if not nl2:
+            return Fr(0)
+        return iroot(rup_rel(I_hi[3 + p] / I3_lo), p)
+    # ---- regression on B and the coefficients bar-beta_i (NOTE section 4)
+    bn = law.bnorm
+    bv = sqrt_up(bn['v1'] ** 2 + bn['v2'] ** 2)
+    bo = sqrt_up(bn['o11'] ** 2 + bn['o22'] ** 2 + 2 * bn['o12'] ** 2)
+    by = sqrt_up(bn['y111'] ** 2 + 3 * bn['y112'] ** 2 + 3 * bn['y122'] ** 2 + bn['y222'] ** 2)
+    bnu = sqrt_up(bn['nu1'] ** 2 + bn['nu2'] ** 2)
+    a_fac = Fr(1, 2) if MUT == 'cap-half' else Fr(1)      # the cap constant 4/(3k) (mutant: halved)
+
+    def Bblk(blk):
+        return rup_rel(sqrt_up(sum(m * bn[n] ** 2 for n, m in block_members(*blk))) + law.S[blk] / sB_lo)
+    Bb = {blk: Bblk(blk) for blk in BLOCKS}
+    bb1 = rup_rel(Fr(5, 2) * Bb[(4, 0)] + 2 * Bb[(3, 1)])           # bar-beta_1 / r
+    bbars = {1: rup_rel(r1 * bb1),
+             2: rup_rel(2 * bv + r1 * (Fr(5, 2) * Bb[(3, 1)] + 2 * Bb[(2, 2)])),
+             3: rup_rel(bo + r1 * (Fr(5, 2) * Bb[(2, 2)] + 2 * Bb[(1, 3)])),
+             4: rup_rel(by + r1 * (2 * Bb[(1, 3)] + 2 * Bb[(0, 4)]))}
+    bbar = max(bbars.values())
+    # ---- free blocks: Cholesky factors in Frobenius coordinates (v | omega | Y~)
+    Lf = law.Lfree
+    Iv, Io, Iy = (0, 1), (2, 3, 4), (5, 6, 7, 8)
+    Lvv, Low, Loo = opnorm_lower_tri(Lf, Iv, Iv), opnorm_lower_tri(Lf, Io, Iv), opnorm_lower_tri(Lf, Io, Io)
+    Lyv, Lyo, Lyy = opnorm_lower_tri(Lf, Iy, Iv), opnorm_lower_tri(Lf, Iy, Io), opnorm_lower_tri(Lf, Iy, Iy)
+    mu_v = sqrt_up(mabs['v1'] ** 2 + mabs['v2'] ** 2)
+    mu_o = sqrt_up(mabs['o11'] ** 2 + mabs['o22'] ** 2 + 2 * mabs['o12'] ** 2)
+    mu_y = sqrt_up(mabs['y111'] ** 2 + 3 * mabs['y112'] ** 2 + 3 * mabs['y122'] ** 2 + mabs['y222'] ** 2)
+    mu_nu = sqrt_up(mabs['nu1'] ** 2 + mabs['nu2'] ** 2)
+    s2, s3, s4 = 2 * Lvv, Loo, Lyy
+
+    memo = {}
+
+    def blk_g_norm(blk, p):                         # || ||X_blk,g||_F ||_p
+        mem = block_members(*blk)
+        m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem))
+        tr = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in mem)
+        return frob_norm(m_, tr, p)
+
+    def nA(blk, p):                                 # ||A^g_blk||_p, A^g = ||X_blk,g||_F + T^g_blk
+        key = ('A', blk, p)
+        if key not in memo:
+            memo[key] = rup_rel(blk_g_norm(blk, p) + law.S[blk] * (sqE + znorm1(pow2(p))))
+        return memo[key]
+
+    def nDelta(i, p):
+        key = ('D', i, p)
+        if key not in memo:
+            memo[key] = rup_rel(nDelta0(i, p))
+        return memo[key]
+
+    def nDelta0(i, p):
+        if MUT == 'no-delta':
+            return Fr(0)
+        if i == 1:
+            return Fr(5, 2) * nA((4, 0), p) + 2 * nA((3, 1), p)
+        if i == 2:
+            return Fr(5, 2) * nA((3, 1), p) + 2 * nA((2, 2), p)
+        if i == 3:
+            return Fr(5, 2) * nA((2, 2), p) + 2 * nA((1, 3), p)
+        return 2 * nA((1, 3), p) + 2 * nA((0, 4), p)
+
+    def nd(i, p, lam=True):
+        key = ('d', i, p, lam)
+        if key not in memo:
+            memo[key] = rup_rel(nd0(i, p, lam))
+        return memo[key]
+
+    def nd0(i, p, lam):     # ||delta_i||_p / Klo, delta_i including bar-beta_i (||mu_B|| + lambda_2)
+        cm = bbars[i] * (muB + (l2norm(p) if lam else 0))
+        if i == 2:
+            v = 2 * mu_v + cm + r1 * nDelta(2, p)
+        elif i == 3:
+            v = mu_o + Low * chi_norm(2, pow2(p)) + cm + r1 * nDelta(3, p)
+        else:
+            v = mu_y + Lyv * chi_norm(2, pow2(p)) + Lyo * chi_norm(3, pow2(p)) + cm + r1 * nDelta(4, p)
+        return v / Klo
+
+    kappa = 3 / (1 - 3 * x0)
+    Fk = 1 + kappa * x0                      # (1 + kappa x_i) <= Fk on the event {x_i <= x0}
+    cth = 1 - kappa * x0 / 2                 # <= Fk^{-1/2}
+    NMAX = 24
+    q = {'cmt1': bbars[1] * muB / Klo, 'rK': r1 / Klo,
+         'nD1': lambda p: rup_rel(nDelta(1, p) + bb1 * l2norm(pow2(p))),
+         's': {2: s2 / Klo, 3: s3 / Klo, 4: s4 / Klo}, 'nd': nd, 's3': s3, 's2': s2, 'c': cth, 'Klo': Klo}
+    E1, E1o, E1v, Tn, To, Tv = normalized_moments(q, NMAX, theta)
+    et1 = rup_rel(16 * kappa * r1 * bbars[1] * a_fac)     # eta_1 K, k-free
+    aK2_hi = 192 * khi * a_fac                    # a K^2 = 192 k
+
+    def EUt(p):                                   # upper bound of E[Ut^p], Ut = U/(a K^2) <= max(Z1, Fk y_i^2)
+        key = ('U', p)
+        if key not in memo:
+            memo[key] = rup_rel(sum(binom(p, l) * et1 ** l * E1[2 * p + l] for l in range(p + 1)) + Fk ** p * Tn[2 * p])
+        return memo[key]
+
+    def EoUt(p):
+        return sum(binom(p, l) * et1 ** l * E1o[2 * p + l] for l in range(p + 1)) + Fk ** p * To[2 * p]
+
+    def EvUt(p):
+        return sum(binom(p, l) * et1 ** l * E1v[2 * p + l] for l in range(p + 1)) + Fk ** p * Tv[2 * p]
+
+    def sq(x):
+        return sqrt_up(x)
+
+    def nUt(m, p):                                # ||Ut^m||_p <= Fk^m ||max_i y_i||_2mp^2m (on the event x_i <= X0)
+        require(2 * m * p <= NMAX and pow2(p) == p, 'norm order')
+        key = ('nUt', m, p)
+        if key not in memo:
+            memo[key] = rup_rel(Fk ** m * iroot(E1[2 * m * p] + Tn[2 * m * p], p))
+        return memo[key]
+
+    def nU(p):
+        return rup_rel(aK2_hi * nUt(1, p))
+
+    def ndel(p):                                   # || ||B - mu_B||_F ||_p <= ||mu_B|| + lambda_1 + lambda_2 <= ||mu_B|| + 2 lambda_2
+        return rup_rel(muB + 2 * l2norm(p))
+
+    def nbar(n, p):                                # ||F_bar||_p, F_bar = |F_g| + ||beta_F|| ||B - mu_B||_F
+        return rup_rel(gauss_norm(mabs[n], law.sdg[n], p) + bn[n] * ndel(p))
+
+    six_klo = 6 * klo
+    nT2, nT4, nt2, nt4 = nbar('T', 2), nbar('T', 4), nbar('tau', 2), nbar('tau', 4)
+    epsc2 = (2 * r1 * nT2 + r1 * r1 * nt2) / six_klo + (r1 * nT4) * (r1 * nT4 + r1 * r1 * nt4) / six_klo ** 2
+    nT8, nt8 = nbar('T', 8), nbar('tau', 8)
+    epsc4 = (2 * r1 * nT4 + r1 * r1 * nt4) / six_klo + (r1 * nT8) * (r1 * nT8 + r1 * r1 * nt8) / six_klo ** 2
+    U2_2, U3_2, U4_2 = sq(EUt(4)), sq(EUt(6)), sq(EUt(8))
+    U2_4, U3_4 = nUt(2, 4), nUt(3, 4)
+    # the Omega majorant: Omega_bar = o0 + d_o, o0 = s3 chi_3 (shares u_3), d_o = rest
+    ndo = lambda p: rup_rel(mu_o + Low * chi_norm(2, pow2(p)) + bo * ndel(p))
+    no4 = rup_rel(s3 * chi_norm(3, 4) + ndo(4))
+    # the ||r nu - v||^2 majorant (Term C)
+    th = THETA_E
+    dv = lambda p: rup_rel(mu_v + bv * ndel(p))
+    nnu = lambda p: rup_rel(frob_norm(mu_nu, law.Cg[('nu1', 'nu1')].hi + law.Cg[('nu2', 'nu2')].hi, p)
+                            + bnu * ndel(p))
+    ebar4 = rup_rel((r1 * nnu(8) + Lvv * chi_norm(2, 8) + dv(8)) ** 2)        # ||e_bar||_4
+    ebar8 = rup_rel((r1 * nnu(16) + Lvv * chi_norm(2, 16) + dv(16)) ** 2)     # ||e_bar||_8
+    cC = (1 + th) * (1 + 1 / th) * dv(4) ** 2 + (1 + 1 / th) * r1 * r1 * nnu(4) ** 2
+    # ---- Term A, B, C (one power of U per lambda_1 integration), normalized by 36 k^2 r^5 and by the lambda_2 weight
+    termA = aK2_hi ** 3 / 3 * (EUt(3) + epsc2 * U3_2)
+    termB = aK2_hi ** 2 / 2 * (EoUt(2) + (ndo(2) + epsc4 * no4) * U2_2)
+    termC = aK2_hi ** 2 / (12 * klo) * ((1 + th) ** 2 * EvUt(2) + cC * U2_2 + r1 * nT4 / six_klo * ebar4 * U2_2)
+    termAp = aK2_hi ** 4 / 4 * (EUt(4) + epsc2 * U4_2)
+    termBp = aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2)
+    termCp = aK2_hi ** 3 / (18 * klo) * ((1 + th) ** 2 * EvUt(3) + cC * U3_2 + r1 * nT4 / six_klo * ebar4 * U3_2)
+    if MUT == 'no-om-term':
+        termB = termBp = Fr(0)
+    # ---- the Jacobian-correction terms: the part r Omega_bar of (lambda_2 + r Omega_bar), integrated against nu'
+    # D  = E[Omega_bar (A + B + C integrands)],  D' = E[Omega_bar (A' + B' + C' integrands)]
+    nT16 = nbar('T', 16)
+    termD = (aK2_hi ** 3 / 3 * (EoUt(3) + (ndo(2) + epsc4 * no4) * U3_2)
+             + aK2_hi ** 2 / 2 * no4 * no4 * (U2_2 + epsc4 * U2_4)
+             + aK2_hi ** 2 / (12 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U2_4)
+    termDp = (aK2_hi ** 4 / 4 * (EoUt(4) + (ndo(2) + epsc4 * no4) * U4_2)
+              + aK2_hi ** 3 / 3 * no4 * no4 * (U3_2 + epsc4 * U3_4)
+              + aK2_hi ** 3 / (18 * klo) * no4 * (1 + r1 * nT16 / six_klo) * ebar8 * U3_4)
+    if MUT == 'no-om-term':
+        termD = termDp = Fr(0)
+    I2, I3 = I_hi[2], I_hi[3]
+    lead = I3 * (termA + termB + termC) + r1 * I2 * termD
+    leadp = I3 * (termAp + termBp + termCp) + r1 * I2 * termDp
+    # ---- normalizer floor Z_r / (36 k^2 r^2) >= m3(b'_lo) - corrections (NOTE section 6)
+    Lb = cholesky([[IV(x.lo, x.hi) for x in row] for row in law.Cb])
+    eta = sqrt_up(sum(absup(Lb[i][j] - (SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1)))
+    R0 = Fr(R0_COUPLING)
+    bp_lo = blo - eps - eta * R0
+    bp_hi = bhi + eps + eta * R0
+    bp_lo = Fr(math.floor(bp_lo * 2 ** 40), 2 ** 40)
+    bp_hi = Fr(math.ceil(bp_hi * 2 ** 40), 2 ** 40)
+    pR0 = chi_tail(3, R0)
+    m3lo = m3(bp_lo).lo
+    tail_c = sqrt_up(contact_moment(4, 0, bp_lo).hi) * sqrt_up(pR0)
+    # upper bounds of E[h(B)] for Loewner-monotone h through b'' = bp_hi, plus the outside-ball part
+    nBF16 = rup_rel(muB + sig * chi_norm(3, 16))                 # || ||B||_F ||_16
+    out_ball = sqrt_up(pR0)
+
+    def up_moment(a, c):                    # E[det^a tr^c 1{B > 0}] <= contact value at b'' + outside-ball part
+        crude = rup_rel(nBF16 ** (2 * a + c))      # det <= ||B||^2, tr <= sqrt2 ||B|| <= 2 ||B||: crude, times P^(1/2)
+        return rup_rel(contact_moment(a, c, bp_hi).hi + 2 ** c * crude * out_ball)
+    F2 = up_moment(4, 0)                    # E[(det B)^4 1{B > 0}]
+    DT, TDT, TTDT = up_moment(1, 1), up_moment(1, 2), up_moment(1, 3)
+    sdT, sdtau = law.sdQ['T'], law.sdQ['tau']
+    q_bad = gauss_tail2(mabs['T'], sdT, 3 * klo / (2 * r1)) + gauss_tail2(mabs['tau'], sdtau, 3 * klo / (2 * r1 * r1))
+    nF = sqrt_up(F2)
+    corr_G0 = nF * sqrt_up(q_bad) if q_bad > 0 else Fr(0)
+    corr_eps = (2 * r1 * gauss_norm(mabs['T'], sdT, 2) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, 2)) / six_klo * nF
+    trO = sum(m * max(law.Cg[(n, n)].hi, Fr(0)) for n, m in zip(ONAMES, (1, 1, 2)))
+    trV = law.Cg[('v1', 'v1')].hi + law.Cg[('v2', 'v2')].hi
+    Eog = rup_rel(mu_o + sqrt_up(trO))
+    corr_om = Eog * DT + bo * (TDT + muB * DT)
+    Evg2 = mu_v ** 2 + trV
+    corr_v = (Fr(17, 16) * Evg2 * DT + 17 * bv ** 2 * (TTDT + 2 * muB * TDT + muB * muB * DT)) / (4 * klo)
+    corr = tail_c + corr_G0 + corr_eps + r1 * (corr_om + corr_v)
+    if MUT == 'no-zfloor-correction':
+        corr = Fr(0)
+    zt = m3lo - corr
+    require(zt > 0, 'normalizer floor not positive')
+    main = KB * min(gbar * lead, g0 * lead + r1 * Lg * leadp) / zt
+    # ---- rare branches at the band top: E[W 1_E] <= ||W 1_typed||_q P(E)^(1 - 1/q), ||W 1_typed||_q / r^2 <= g1 g2
+    six_khi = 6 * khi
+    trB = sum(x.hi for x in (law.Cb[0][0], law.Cb[1][1], law.Cb[2][2]))
+    trOQ = sum(m * law.C[(n, n)].hi for n, m in zip(ONAMES, (1, 1, 2)))
+    trVQ = law.C[('v1', 'v1')].hi + law.C[('v2', 'v2')].hi
+    trNQ = law.C[('nu1', 'nu1')].hi + law.C[('nu2', 'nu2')].hi
+
+    # W_r / r^2 <= |c1| D [|c2| (D + r Om tr + r^2 Om^2) + r ||e||^2 (tr + r Om)] on the typed support, D = det B 1{B > 0},
+    # tr = tr B, Om = ||Omega'||_F (NOTE section 7)
+    sd_tr = sqrt_up(law.C[('a11', 'a11')].hi + law.C[('a22', 'a22')].hi + 2 * absup(law.C[('a11', 'a22')]))
+    m_tr = absup(mu['a11'] + mu['a22'])
+
+    def wnorm(qq):
+        p4, p8 = 4 * qq, 8 * qq
+        nD = lambda p: det_norm(p, bp_hi, muB, sig)
+        ntr = lambda p: gauss_norm(m_tr, sd_tr, p)
+        no = lambda p: frob_norm(mu_o, trOQ, p)
+        nT = lambda p: gauss_norm(mabs['T'], sdT, p)
+        nc2 = lambda p: six_khi + r1 * nT(p) + r1 * r1 * gauss_norm(mabs['tau'], sdtau, p)
+        ne = lambda p: r1 * frob_norm(mu_nu, trNQ, p) + frob_norm(mu_v, trVQ, p)
+        g1 = (six_khi + r1 * nT(p4)) * nD(p4)
+        g2 = (nc2(p4) * (nD(p4) + r1 * no(p8) * ntr(p8) + r1 * r1 * no(p8) ** 2)
+              + r1 * ne(p8) ** 2 * (ntr(p4) + r1 * no(p4)))
+        return g1 * g2
+    thr = 3 * klo / (10 * r1)
+    qA2 = Fr(0)
+    args = []
+    for blk in BLOCKS:
+        a_, c_ = blk
+        mem = block_members(*blk)
+        m_ = sqrt_up(sum(m * mabs[n] ** 2 for n, m in mem))       # |mean of d_x^a d_u^c f(0)| <= ||mean||_F for unit u
+        best = None
+        for frac in (Fr(7, 8), Fr(15, 16)):                        # split t = frac t + (1 - frac) t between jet and Taylor tail
+            if c_ == 0:                                            # scalar block
+                x_ = (thr * frac - m_) / law.sdQ[mem[0][0]]
+                pj = gauss_tail2(m_, law.sdQ[mem[0][0]], thr * frac)
+            elif c_ == 1:                                          # vector block: ||X - mean|| <= sqrt(lambda_max) chi_2
+                lmax = max(sum(absup(law.C[(ni, nj)]) for nj, mj in mem) for ni, mi in mem)
+                x_ = (thr * frac - m_) / sqrt_up(lmax)
+                pj = chi_tail(2, x_) if x_ > 0 else Fr(1)
+            else:                                                  # ||X||_op = max_theta |Q(theta)|, Q a trigonometric
+                rho = 1 - Fr(c_, NET_M)                            # polynomial of degree c: Bernstein |Q'| <= c max|Q| and
+                sdn = sqrt_up(law.net_var[blk])                    # net gaps <= 2/NET_M give max|Q| <= max_j |Q_j| / rho
+                x_ = (thr * frac * rho - m_) / sdn
+                pj = min(Fr(1), len(NET) * gauss_tail2(m_, sdn, thr * frac * rho))
+            pj += (law.S[blk] * (sqE + znorm(32)) / (thr * (1 - frac))) ** 32
+            if best is None or pj < best[0]:
+                best = (pj, x_)
+        args.append(best[1])
+        qA2 += best[0]
+    a_hi = Fr(4, 3) / klo * a_fac
+    qfar = Fr(0)
+    qx = Fr(0)
+    sdB11 = sqrt_up(law.C[('a11', 'a11')].hi)
+    if bbar > 0:
+        lam_far = 1 / (4 * a_hi * r1 * bbar * bbar)
+        args.append((lam_far - absup(mu['a11'])) / sdB11)
+        qfar = gauss_tail1(absup(mu['a11']), sdB11, lam_far)
+
+    def lam2_tail(s):                       # P(lambda_2 > s) <= P(sig chi_3 > s - ||mu_B||)
+        x_ = (s - muB) / sig
+        args.append(x_)
+        return chi_tail(3, x_) if x_ > 0 else Fr(1)
+    for i in (1, 2, 3, 4):                         # x_i > X0  <=>  y_i' + bar-beta_i lambda_2 > X0 / (a r1 bar-beta_i)
+        if bbars[i] == 0:
+            continue
+        tx = x0 / (a_hi * r1 * bbars[i])
+        if i == 1:
+            cb = 12 * khi + bbars[1] * muB
+            if tx <= cb:
+                qx += Fr(1)
+            else:
+                qx += (r1 * nDelta(1, 32) / ((tx - cb) / 2)) ** 32
+                qx += lam2_tail((tx - cb) / (2 * bbars[1]))
+        else:
+            s_i = {2: s2, 3: s3, 4: s4}[i]
+            x_ = tx / (3 * s_i)
+            args.append(x_)
+            qx += chi_tail(NU[i], x_) + (nd(i, 32, lam=False) * Klo / (tx / 3)) ** 32 + lam2_tail(tx / (3 * bbars[i]))
+    qs = [min(Fr(1), x) for x in (qA2, qfar, qx)]
+    tail_num = min(wnorm(qq) * sum(pow_tail(x, qq) for x in qs) for qq in QS)
+    if last_band:
+        # (0, R] is the union of [R 2^-(j+1), R 2^-j]; every Gaussian or chi tail argument scales at least like 1/r and is
+        # >= 3 at R, so each such piece drops by more than 64 when r halves; every Markov piece is a power r^m with m >= 32;
+        # after the power 1 - 1/q (q >= 2) each piece still drops by more than 8, so for the q chosen at R the supremum of
+        # tail / r^3 over (0, R] is attained on [R/2, R]
+        require(min(args) >= 3, 'last-band tail arguments')
+        require(max(qA2, qfar, qx) < 1, 'last-band tail pieces')
+        rdiv = r1 / 2
+    else:
+        rdiv = r0
+    tail = tail_num / (36 * klo * klo * zt * rdiv ** 3)
+    total = main + tail
+    return {'total': total, 'main': main, 'tail': tail, 'zt': zt, 'tail_num': tail_num, 'KB': KB, 'g0': g0, 'gbar': gbar, 'I3': I3,
+            'termA': termA, 'termB': termB, 'termC': termC, 'termD': termD, 'qA2': qA2, 'qfar': qfar, 'qx': qx,
+            'm3lo': m3lo, 'corr': corr, 'eps': eps, 'eta': eta, 'min_arg': min(args)}
+# ----------------------------------------------------------------------------- the (k, b) boxes of the sweep
+F_ = Fr
+GRID = (
+    ((F_(1, 2), F_(3, 4)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(3, 4), F_(1)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(1), F_(5, 4)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(5, 4), F_(3, 2)), (F_(0), F_(1, 2), F_(1))),
+    ((F_(3, 2), F_(13, 8)), (F_(0), F_(243, 512), F_(1))),
+    ((F_(13, 8), F_(7, 4)), (F_(0), F_(309, 1024), F_(899, 1024), F_(1))),
+    ((F_(7, 4), F_(15, 8)), (F_(0), F_(37, 256), F_(105, 256), F_(933, 1024), F_(1))),
+    ((F_(15, 8), F_(31, 16)), (F_(0), F_(37, 512), F_(207, 1024), F_(225, 512), F_(113, 128), F_(1))),
+    ((F_(31, 16), F_(63, 32)), (F_(0), F_(37, 1024), F_(103, 1024), F_(111, 512), F_(439, 1024), F_(421, 512), F_(1))),
+    ((F_(63, 32), F_(2)), (F_(0), F_(1, 512), F_(5, 1024), F_(5, 512), F_(19, 1024), F_(35, 1024), F_(1, 16), F_(29, 256),
+                           F_(209, 1024), F_(377, 1024), F_(685, 1024), F_(1))),
+)
+
+
+# ----------------------------------------------------------------------------- bands, boxes, runs
+TOP_OCTAVES = 7            # octaves [2^-(j+1), 2^-j], j = 6..12 (r up to 1/64), split into 8 sub-bands
+LOW_OCTAVES = 16           # octaves j = 13..28, split into 4 sub-bands
+R_LAST = Fr(1, 2 ** 29)    # last band [0, 2^-29]
+COMBO = (Fr(1, 64), Fr(1, 32))     # (X_0, theta) of NOTE sections 4-5
+
+
+def bands():
+    """Sub-bands [r0, r1] covering (0, 1/64]; each octave split into equal sub-bands; the last band is [0, R_LAST]."""
+    out = []
+    for j in range(6, 6 + TOP_OCTAVES + LOW_OCTAVES):
+        hi = Fr(1, 2 ** j)
+        lo = hi / 2
+        m = 8 if j < 6 + TOP_OCTAVES else 4
+        for i in range(m):
+            out.append((lo + (hi - lo) * i / m, lo + (hi - lo) * (i + 1) / m))
+    out.append((Fr(0), R_LAST))
+    require(min(b[0] for b in out) == 0 and max(b[1] for b in out) == Fr(1, 64), 'band cover')
+    out = sorted(out)
+    require(all(out[i][1] == out[i + 1][0] for i in range(len(out) - 1)), 'bands contiguous')
+    return out
+
+
+def dec_down(x, places=4):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_up(x, places=4):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def grid_boxes():
+    out = []
+    for kb, bg in GRID:
+        require(bg[0] == 0 and bg[-1] == 1 and all(bg[i] < bg[i + 1] for i in range(len(bg) - 1)), 'b grid')
+        for i in range(len(bg) - 1):
+            out.append(((Fr(bg[i]), Fr(bg[i + 1])), (Fr(kb[0]), Fr(kb[1]))))
+    ks = [kb for kb, _ in GRID]
+    require(ks[0][0] == BAND_K[0] and ks[-1][1] == BAND_K[1] and all(ks[i][1] == ks[i + 1][0] for i in range(len(ks) - 1)),
+            'k boxes cover K')
+    return tuple(out)
+
+
+CG3_02_LO = Fr('22424320.655069')     # certified lower end of c_G^(3)(0, 2) (Math-#208)
+
+
+def band_record(args):
+    """Per band: the certified bound of sup Q^W(G_r^c)/r^3 (rounded up, 4 places) and the floor of Z_r/r^2 (rounded down,
+    6 places) for each box in BOXES order, and the largest tail part (rounded up, 12 places)."""
+    r0, r1 = args
+    law = band_law(r0, r1)
+    last = (r0 == 0)
+    bound, zfloor, tails = [], [], []
+    for bb, kb in BOXES:
+        res = box_bound(law, bb, kb, r0, r1, last_band=last, x0=COMBO[0], theta=COMBO[1])
+        bound.append(dec_up(res['total']))
+        zfloor.append(dec_down(36 * kb[0] * kb[0] * res['zt'], 6))
+        tails.append(res['tail'])
+    return {'r': [str(r0), str(r1)], 'bound': bound, 'zfloor': zfloor, 'tail_max': dec_up(max(tails), 12)}
+
+
+def full_run(procs):
+    bl = bands()
+    with multiprocessing.Pool(procs) as pool:
+        recs = pool.map(band_record, bl)
+    return assemble(recs)
+
+
+def assemble(recs):
+    table = {}
+    for rs in R_STARS:
+        mx, arg, zmin = Fr(0), None, None
+        for rec in recs:
+            if Fr(rec['r'][1]) <= rs:
+                for (bb, kb), bs, zs in zip(BOXES, rec['bound'], rec['zfloor']):
+                    v = Fr(bs)
+                    if v > mx:
+                        mx, arg = v, {'r': rec['r'], 'b': [str(bb[0]), str(bb[1])], 'k': [str(kb[0]), str(kb[1])]}
+                    z = Fr(zs)
+                    zmin = z if zmin is None or z < zmin else zmin
+        table[str(rs)] = {'C': str(mx), 'argmax': arg, 'C_over_cG3_02': dec_up(mx / CG3_02_LO, 6),
+                          'C_rstar3': dec_up(mx * rs ** 3, 10), 'z_star': str(zmin)}
+    return {'schema': 1, 'object': 'CL-C8-THEOREM-A-CONSTANTS-D3-20261001-v1', 'scientific_effect': 'NONE',
+            'certified': True, 'mutant': MUT, 'L_min': L_MIN, 'dimension': 3,
+            'band': {'b': [str(BAND_B[0]), str(BAND_B[1])], 'k': [str(BAND_K[0]), str(BAND_K[1])]},
+            'parameters': params(), 'C_table': table, 'bands': recs}
+
+
+def params():
+    return {'PBITS': PBITS, 'NX': NX, 'NB': NB, 'COMBO_X0_THETA': [str(COMBO[0]), str(COMBO[1])], 'THETA_E': str(THETA_E),
+            'QS': list(QS), 'R0_COUPLING': R0_COUPLING, 'NET_M': NET_M, 'R_STARS': [str(x) for x in R_STARS],
+            'TOP_OCTAVES': TOP_OCTAVES, 'LOW_OCTAVES': LOW_OCTAVES, 'R_LAST': str(R_LAST),
+            'GRID': [[[str(kb[0]), str(kb[1])], [str(x) for x in bg]] for kb, bg in GRID]}
+
+
+def results_text(res):
+    """RESULTS.json layout: the summary indented, then one line per band record."""
+    head = json.dumps({key: val for key, val in res.items() if key != 'bands'}, indent=1, sort_keys=True)
+    lines = ',\n'.join('  ' + json.dumps(rec, sort_keys=True) for rec in res['bands'])
+    return head[:-2] + ',\n "bands": [\n' + lines + '\n ]\n}\n'
+
+
+def write_results(res):
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        return json.load(fh)
+
+
+def replay_sample():
+    """Band indices replayed by --check: the band ending at each r_*, the bands ending at 2^-14, 2^-20, 2^-26, the last band."""
+    bl = bands()
+    idx = []
+    for rs in R_STARS:
+        j = max(i for i, b in enumerate(bl) if b[1] == rs)
+        idx.append(j)
+    idx += [i for i, b in enumerate(bl) if b[1] in (Fr(1, 2 ** 14), Fr(1, 2 ** 20), Fr(1, 2 ** 26))]
+    idx.append(0)
+    return sorted(set(idx))
+
+
+def check_run(procs, full=False):
+    ref = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(ref), 'RESULTS.json is not in the canonical layout')
+    require(ref['parameters'] == params(), 'parameters differ from RESULTS.json')
+    require(ref['mutant'] is None, 'RESULTS.json produced under a mutant')
+    require(abs(PI.mid() - Fr('3.14159265358979323846')) < Fr(1, 10 ** 19) and PI.width() < Fr(1, 10 ** 30), 'pi enclosure')
+    require(abs(Phi(IV(1)).mid() - Fr('0.841344746068542948585')) < Fr(1, 10 ** 19), 'Phi(1) enclosure')
+    require(abs(m3(0).mid() - (Fr(7, 2) - 2 * SQRT2.mid())) < Fr(1, 10 ** 40), 'm_(3,0) closed form')
+    bl = bands()
+    require([rec['r'] for rec in ref['bands']] == [[str(a), str(b)] for a, b in bl], 'band list differs')
+    idx = list(range(len(bl))) if full else replay_sample()
+    with multiprocessing.Pool(procs) as pool:          # in order; the first differing band exits (and stops the pool)
+        for i, rec in zip(idx, pool.imap(band_record, [bl[i] for i in idx])):
+            require(rec == ref['bands'][i], 'band %d (%s) differs from RESULTS.json' % (i, rec['r']))
+    # the table is the maximum over the stored records
+    asm = assemble(ref['bands'])
+    require(asm['C_table'] == ref['C_table'], 'C table is not the maximum of the stored bounds')
+    # sanity: every C exceeds the sharp reference-kernel cap coefficient c_G^(3)(0, 2) of Math-#208 (a lower bound for every
+    # d = 3 cap-route constant valid at (0, 2) for the reference kernel, which this certificate covers)
+    for rs, row in ref['C_table'].items():
+        require(Fr(row['C']) > CG3_02_LO, 'C below the sharp cap coefficient')
+    print(json.dumps({'check': 'ok', 'bands_replayed': len(idx), 'C_table': {rs: row['C'] for rs, row in ref['C_table'].items()},
+                      'z_star': {rs: row['z_star'] for rs, row in ref['C_table'].items()}, 'mutant': MUT}))
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--mc', nargs=5, metavar=('R', 'B', 'K', 'N', 'SEED'), help='floating end-to-end control')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.mc:
+        rr, bb, kk, nn, sd = args.mc
+        print(json.dumps(mc_event(float(Fr(rr)), float(Fr(bb)), float(Fr(kk)), int(nn), int(sd))))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    res = full_run(args.procs)
+    write_results(res)
+    print(json.dumps({'C': {rs: row['C'] for rs, row in res['C_table'].items()},
+                      'z': {rs: row['z_star'] for rs, row in res['C_table'].items()}}))
+BOXES = grid_boxes()
+
+
+# ----------------------------------------------------------------------------- floating control (not part of the certificate)
+def mc_event(r, b, k, n, seed, deg=7, nang=24):
+    """End-to-end Monte Carlo of Q^W(G_r^c) / r^3 and Z_r / r^2 for the reference kernel in d = 3 (floating point): the
+    Taylor jet of order deg at the midpoint is sampled under Q (exact pin series evaluated in floating point), W_r is
+    evaluated from the Hessians at M and S, and M_3, M_4 are the partial-block operator norms ([CAP] (1)) maximized over a
+    grid of the cylinder D (binary forms maximized over nang angles; floating, so slightly below the true suprema)."""
+    import random
+    pins, _ = functionals()
+    P = [pins[nm] for nm in PIN_NAMES]
+    jets = [(a, b1, m - a - b1) for m in range(deg + 1) for a in range(m, -1, -1) for b1 in range(m - a, -1, -1)]
+    J = [[term(1, 0, a, b1, b2, X0PT)] for (a, b1, b2) in jets]
+    nj = len(J)
+
+    def ev(F, G, key):
+        ser, w, _ = pair_series(F, G, key)
+        return sum(float(c) * r ** p for p, c in ser.items())
+    Spp = [[ev(P[i], P[j], ('p', i, j)) for j in range(NP)] for i in range(NP)]
+    Sjp = [[ev(J[i], P[j], ('jp3', deg, i, j)) for j in range(NP)] for i in range(nj)]
+    Sjj = [[ev(J[i], J[j], ('jj3', deg, i, j)) for j in range(nj)] for i in range(nj)]
+
+    def solve(A, bv):
+        m = len(A)
+        M = [A[i][:] + [bv[i]] for i in range(m)]
+        for c in range(m):
+            p = max(range(c, m), key=lambda i: abs(M[i][c]))
+            M[c], M[p] = M[p], M[c]
+            for i in range(c + 1, m):
+                f = M[i][c] / M[c][c]
+                for j in range(c, m + 1):
+                    M[i][j] -= f * M[c][j]
+        x = [0.0] * m
+        for i in reversed(range(m)):
+            x[i] = (M[i][m] - sum(M[i][j] * x[j] for j in range(i + 1, m))) / M[i][i]
+        return x
+    vr = [b - k * r ** 3 / 2, -k * r * r, 12 * k] + [0.0] * (NP - 3)
+    Wt = [solve(Spp, Sjp[i]) for i in range(nj)]
+    mu = [sum(Wt[i][j] * vr[j] for j in range(NP)) for i in range(nj)]
+    C = [[Sjj[i][j] - sum(Wt[i][l] * Sjp[j][l] for l in range(NP)) for j in range(nj)] for i in range(nj)]
+    # Cholesky with clamped pivots (the conditional covariance is positive semidefinite up to rounding)
+    L = [[0.0] * nj for _ in range(nj)]
+    for i in range(nj):
+        for j in range(i + 1):
+            s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j))
+            if i == j:
+                L[i][i] = math.sqrt(s) if s > 1e-300 else 0.0
+            else:
+                L[i][j] = s / L[j][j] if L[j][j] > 0 else 0.0
+    index = {al: i for i, al in enumerate(jets)}
+    fact = [math.factorial(i) for i in range(deg + 2)]
+
+    def deriv(cf, be, p):
+        """d^be f(p) from the jet (Taylor at 0)."""
+        tot = 0.0
+        for al, i in index.items():
+            if al[0] >= be[0] and al[1] >= be[1] and al[2] >= be[2]:
+                e = (al[0] - be[0], al[1] - be[1], al[2] - be[2])
+                tot += cf[i] * (p[0] ** e[0] * p[1] ** e[1] * p[2] ** e[2]) / (fact[e[0]] * fact[e[1]] * fact[e[2]])
+        return tot
+    angs = [(math.cos(math.pi * t / nang), math.sin(math.pi * t / nang)) for t in range(nang)]
+    pts = [(x, 0.0, 0.0) for x in (-2 * r, -r, 0.0, r, 2 * r)]
+    pts += [(x, 2 * r * math.cos(2 * math.pi * t / 6), 2 * r * math.sin(2 * math.pi * t / 6)) for x in (-2 * r, 0.0, 2 * r)
+            for t in range(6)]
+
+    def block_norm(vals, c):
+        """Operator norm of the symmetric c-form with components vals[(b1, b2)] (b1 + b2 = c) over unit vectors."""
+        if c == 0:
+            return abs(vals[(0, 0)])
+        best = 0.0
+        for (u1, u2) in angs:
+            q = sum(math.comb(c, b1) * vals[(b1, c - b1)] * u1 ** b1 * u2 ** (c - b1) for b1 in range(c + 1))
+            best = max(best, abs(q))
+        return best
+
+    def Mj(cf, j):
+        best = 0.0
+        for p in pts:
+            for a in range(j + 1):
+                c = j - a
+                vals = {(b1, c - b1): deriv(cf, (a, b1, c - b1), p) for b1 in range(c + 1)}
+                best = max(best, block_norm(vals, c))
+        return best
+
+    def hess(cf, p):
+        return [[deriv(cf, tuple(int(x == i) + int(x == j) for x in range(3)), p) for j in range(3)] for i in range(3)]
+
+    def det3f(H):
+        return (H[0][0] * (H[1][1] * H[2][2] - H[1][2] * H[2][1]) - H[0][1] * (H[1][0] * H[2][2] - H[1][2] * H[2][0])
+                + H[0][2] * (H[1][0] * H[2][1] - H[1][1] * H[2][0]))
+
+    def neg_def(H):
+        return H[0][0] < 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and det3f(H) < 0
+
+    def index2(H):
+        """Number of negative eigenvalues of a symmetric 3 x 3 matrix is 2: det > 0 and not positive definite."""
+        d = det3f(H)
+        pd = H[0][0] > 0 and H[0][0] * H[1][1] - H[0][1] ** 2 > 0 and d > 0
+        return d > 0 and not pd
+    rng = random.Random(seed)
+    num = den = 0.0
+    acap = 4 / (3 * k)
+    for _ in range(n):
+        z = [rng.gauss(0, 1) for _ in range(nj)]
+        cf = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(nj)]
+        HM, HS = hess(cf, (-r / 2, 0.0, 0.0)), hess(cf, (r / 2, 0.0, 0.0))
+        if not (neg_def(HM) and index2(HS)):
+            continue
+        Wv = abs(det3f(HM) * det3f(HS))
+        den += Wv
+        a11, a22, a12 = -HM[1][1], -HM[2][2], -HM[1][2]
+        lam1 = (a11 + a22) / 2 - math.sqrt(((a11 - a22) / 2) ** 2 + a12 * a12)
+        M3 = Mj(cf, 3)
+        if lam1 <= acap * r * M3 * M3:
+            num += Wv
+            continue
+        if r * Mj(cf, 4) > 3 * k / 10:
+            num += Wv
+    return {'r': r, 'b': b, 'k': k, 'n': n, 'seed': seed, 'QW_Gc_over_r3': num / den / r ** 3 if den else None,
+            'Z_over_r2': den / n / r ** 2}
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py
new file mode 100644
index 000000000..36b39dbd3
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_b.py
@@ -0,0 +1,1166 @@
+#!/usr/bin/env python3
+"""Explicit constants for [LP] Theorem B on the C8 band, d = 2 and d = 3 (NOTE.md Theorem L and its corollaries).
+
+For the radial ledger A_r = 12 pi_r(R; v_r) Z_r / r^2 of [LP] (10.2) the certificate proves, for 0 < r <= R, every frame,
+every torus side L >= 10 and the reference kernel,
+
+    -(a_dn r^2 + eps) <= A_r(b, k, u) / A_0^ref(b, k) - 1 <= a_up r^2 + eps         (b in [0, 1], k in [1/2, 2]),
+
+with A_0^ref the closed-form contact integrand of the Gaussian kernel, and assembles from it two-sided second-order bounds for
+nu_cand and nu_eld, the difference constant C_{B,K} of [LP] (1.2), and the constants of (12.1) and (12.3).
+
+Standard library only; exact rational interval arithmetic throughout.
+- Taylor models in r (fixed-point integer coefficients, every rounding carried in the remainder) for the pin covariance, its
+  determinant, the pin energies and the law of the transverse Hessian and of Omega' (section 2 of NOTE.md).
+- The interval band laws and the box-bound engines of Math-#206 (d = 2) and Math-#212 (d = 3), as byte-identical pinned
+  copies engine_e2.py and engine_e3.py (section 2; SOURCE_MAP.json).
+
+  python3 -B -S theorem_b.py --procs N              regenerate RESULTS.json
+  python3 -B -S theorem_b.py --check --procs N      replay: Taylor models, rate sweep, reference integrals, a sample of
+                                                    cap bands, and the assembled tables (exact comparison)
+  python3 -B -S theorem_b.py --check-full --procs N replay every cap band as well
+  python3 -B -S theorem_b.py --mutant NAME --check  a semantic mutation; the replay must then fail (exit 1)
+  python3 -B -S theorem_b.py --controls             floating controls (not part of the certificate)
+"""
+import argparse
+import importlib.util
+import json
+import math
+import multiprocessing
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+MUTANTS = ('no-om-term', 'no-bad-set', 'no-det', 'no-torus', 'cap-half')
+MUT = None
+
+R_STARS = (Fr(1, 4096), Fr(1, 2048), Fr(1, 1024), Fr(1, 512), Fr(1, 256), Fr(1, 128), Fr(1, 64))
+TM_S = 6                       # Taylor models on [0, 2^-TM_S] = [0, 1/64]
+TM_N = 8                       # Taylor-model degree
+TM_P = 288                     # fixed-point bits of the Taylor-model coefficients
+R0C = Fr(12)                   # Loewner coupling radius (d = 3)
+TINY = Fr(1, 10 ** 60)         # polynomial coefficients below this are rounding residue of exact zeros
+SUB_B = tuple((Fr(i, 8), Fr(i + 1, 8)) for i in range(8))
+SUB_K = tuple((Fr(1, 2) + Fr(j, 8), Fr(1, 2) + Fr(j + 1, 8)) for j in range(12))
+SUBBOXES = tuple((bb, kb) for kb in SUB_K for bb in SUB_B)
+NH = 4096                      # b-cells of the reference integrals
+BLOCK = 16                     # cells per second-derivative block
+CAP_B = tuple((Fr(i, 4), Fr(i + 1, 4)) for i in range(4))
+CAP_K = tuple((Fr(1, 2) + Fr(i, 32), Fr(1, 2) + Fr(i + 1, 32)) for i in range(8)) \
+    + tuple((Fr(3, 4) + Fr(i, 16), Fr(3, 4) + Fr(i + 1, 16)) for i in range(4)) + ((Fr(1), Fr(3, 2)), (Fr(3, 2), Fr(2)))
+CAP_BOXES = tuple((bb, kb) for kb in CAP_K for bb in CAP_B)
+C_REF_197 = Fr('0.0023869380211529483091019117')    # Math-#197, reference c_{B,K} in d = 2 (cross-check only)
+C_REF_205 = Fr('0.0009743823660242504624710565')    # Math-#205, reference c_{B,K} in d = 3 (cross-check only)
+
+
+def require(cond, msg):
+    if not cond:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def load_engine(name, fn):
+    spec = importlib.util.spec_from_file_location(name, os.path.join(HERE, fn))
+    mod = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    return mod
+
+
+E2 = load_engine('engine_e2', 'engine_e2.py')
+E3 = load_engine('engine_e3', 'engine_e3.py')
+ENG = {2: E2, 3: E3}
+
+
+def exp_minus(M, t):
+    """Enclosure of exp(-t) for rational t of either sign."""
+    t = Fr(t)
+    return M.exp_neg_point(t) if t >= 0 else M.exp_neg_point(-t).inv()
+
+
+def grid_up(x, g=2 ** 40):
+    return Fr(math.ceil(Fr(x) * g), g)
+
+
+def grid_dn(x, g=2 ** 40):
+    return Fr(math.floor(Fr(x) * g), g)
+
+
+def dec_up(x, places):
+    x = Fr(x)
+    n = -((-x.numerator * 10 ** places) // x.denominator)
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+def dec_down(x, places):
+    x = Fr(x)
+    n = (x.numerator * 10 ** places) // x.denominator
+    s = str(abs(n)).rjust(places + 1, '0')
+    return ('-' if n < 0 else '') + s[:-places] + '.' + s[-places:]
+
+
+# ============================================================================= 1. Taylor models in r
+def cdiv(a, b):
+    return -((-a) // b)
+
+
+class TM:
+    """f(r) = sum_{j <= N} (C_j / 2^P) r^j + theta (E / 2^P), |theta| <= 1, for every r in [0, 2^-s]."""
+    __slots__ = ('C', 'E', 's', 'N')
+
+    def __init__(self, C, E, s, N):
+        self.C, self.E, self.s, self.N = C, E, s, N
+
+    @classmethod
+    def from_fracs(cls, coeffs, e, s, N):
+        items = coeffs.items() if isinstance(coeffs, dict) else enumerate(coeffs)
+        C = [0] * (N + 1)
+        E = Fr(e)
+        for p, x in items:
+            x = Fr(x)
+            if x == 0:
+                continue
+            if p <= N:
+                v = round(x * (1 << TM_P))
+                C[p] += v
+                E += abs(x - Fr(v, 1 << TM_P)) / Fr(2) ** (s * p)
+            else:
+                E += abs(x) / Fr(2) ** (s * p)
+        return cls(C, cdiv(E.numerator << TM_P, E.denominator), s, N)
+
+    @classmethod
+    def const(cls, x, s, N):
+        return cls.from_fracs({0: Fr(x)}, 0, s, N)
+
+    def polybound(self):
+        return sum(cdiv(abs(c), 1 << (self.s * j)) for j, c in enumerate(self.C) if c)
+
+    def bound(self):
+        return self.polybound() + self.E
+
+    def frac_coeffs(self):
+        return [Fr(c, 1 << TM_P) for c in self.C]
+
+    def frac_e(self):
+        return Fr(self.E, 1 << TM_P)
+
+    def __add__(self, o):
+        if not isinstance(o, TM):
+            o = TM.const(o, self.s, self.N)
+        return TM([a + b for a, b in zip(self.C, o.C)], self.E + o.E, self.s, self.N)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return TM([-a for a in self.C], self.E, self.s, self.N)
+
+    def __sub__(self, o):
+        if not isinstance(o, TM):
+            o = TM.const(o, self.s, self.N)
+        return self + (-o)
+
+    def scale(self, x):
+        x = Fr(x)
+        C = []
+        E = Fr(self.E, 1 << TM_P) * abs(x)
+        for j, c in enumerate(self.C):
+            v = Fr(c, 1 << TM_P) * x
+            w = round(v * (1 << TM_P))
+            C.append(w)
+            E += abs(v - Fr(w, 1 << TM_P)) / Fr(2) ** (self.s * j)
+        return TM(C, cdiv(E.numerator << TM_P, E.denominator), self.s, self.N)
+
+    def __mul__(self, o):
+        if not isinstance(o, TM):
+            return self.scale(o)
+        N, s = self.N, self.s
+        S = [0] * (2 * N + 1)
+        for i, a in enumerate(self.C):
+            if a:
+                for j, b in enumerate(o.C):
+                    if b:
+                        S[i + j] += a * b
+        C, E = [], 0
+        for k in range(N + 1):
+            v = S[k] >> TM_P                                  # floor: error below one unit, times r^k <= 1
+            C.append(v)
+            if S[k] - (v << TM_P):
+                E += 1
+        for k in range(N + 1, 2 * N + 1):
+            if S[k]:
+                E += cdiv(abs(S[k]), 1 << (TM_P + s * k))
+        E += cdiv(self.E * o.polybound() + o.E * self.polybound() + self.E * o.E, 1 << TM_P)
+        return TM(C, E, s, N)
+    __rmul__ = __mul__
+
+    def inv(self):
+        c0 = Fr(self.C[0], 1 << TM_P)
+        require(c0 != 0, 'Taylor-model inverse at a zero constant term')
+        h = (self - TM([self.C[0]] + [0] * self.N, 0, self.s, self.N)).scale(1 / c0)
+        H = Fr(h.bound(), 1 << TM_P)
+        require(H < Fr(1, 2), 'Taylor-model inverse: |h| too large')
+        acc = TM.const(1, self.s, self.N)
+        p = TM.const(1, self.s, self.N)
+        mh = -h
+        for _ in range(self.N):
+            p = p * mh
+            acc = acc + p
+        tail = H ** (self.N + 1) / (1 - H)
+        acc = TM(acc.C, acc.E + cdiv(tail.numerator << TM_P, tail.denominator), self.s, self.N)
+        return acc.scale(1 / c0)
+
+    def enclose(self, r0, r1):
+        lo = hi = Fr(0)
+        for j, c in enumerate(self.C):
+            if c:
+                x = Fr(c, 1 << TM_P)
+                a, b = Fr(r0) ** j, Fr(r1) ** j
+                lo += x * (a if x >= 0 else b)
+                hi += x * (b if x >= 0 else a)
+        e = self.frac_e()
+        return lo - e, hi + e
+
+
+def cov_tm(M, F, G, key, s, N, M1):
+    """Taylor model on [0, 2^-s] of the covariance of F, G under the unnormalized torus kernel: the engine's exact series,
+    its truncation tail and its torus remainder (as in the engines' cov_band, at r1 = 2^-s)."""
+    R = Fr(1, 2 ** s)
+    ser, w, absterms = M.pair_series(F, G, key)
+    A = max((t[1] for t in absterms), default=0)
+    tail = tor = Fr(0)
+    for t in absterms:
+        acc, aa, gys = t[0], t[1], t[2:]
+        hprod = fprod = 1
+        for g in gys:
+            hprod *= M.hcoef(g)
+            fprod *= math.factorial(g)
+        tail += acc * abs(hprod)
+        tor += acc * fprod * Fr(max(w, 1)) ** aa
+    tail *= 2 * Fr(M.NX + 1) ** A * R ** (M.NX + 1 - w)
+    tor = tor * M1 / (1 - Fr(27183, 10000) * R)
+    return TM.from_fracs(ser, tail + tor, s, N)
+
+
+def solve_tm(A, B):
+    """Gaussian elimination without pivoting (A(0) is positive definite): X[c] solves A X[c] = B[c]; also det A."""
+    n = len(A)
+    Mx = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    det, invs = None, []
+    for col in range(n):
+        piv = Mx[col][col]
+        det = piv if det is None else det * piv
+        ip = piv.inv()
+        invs.append(ip)
+        for i in range(col + 1, n):
+            f = Mx[i][col] * ip
+            for j in range(col, len(Mx[i])):
+                Mx[i][j] = Mx[i][j] - f * Mx[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            acc = Mx[i][n + c]
+            for j in range(i + 1, n):
+                acc = acc - Mx[i][j] * x[j]
+            x[i] = acc * invs[i]
+        X.append(x)
+    return X, det
+
+
+class TMLaw:
+    pass
+
+
+TM_NAMES = {2: ('w', 'om'), 3: ('a11', 'a22', 'a12', 'o11', 'o22', 'o12')}
+
+
+def tm_law(d, s=TM_S, N=TM_N, L=10):
+    """Taylor-model law over r in [0, 2^-s] of the targets TM_NAMES[d] (every frame, every torus side >= L, the reference
+    kernel): alpha, beta (mean = b alpha + k beta), the unnormalized covariance block, the unnormalized pin energies, det."""
+    M = ENG[d]
+    names = TM_NAMES[d]
+    pins, targets = M.functionals()
+    P = [pins[nm] for nm in M.PIN_NAMES]
+    idx = [M.TARGET_NAMES.index(nm) for nm in names]
+    T = [targets[nm] for nm in names]
+    M1 = Fr(0) if MUT == 'no-torus' else M.torus_M1(L)
+    n, nt = len(P), len(T)
+    Spp = [[None] * n for _ in range(n)]
+    for i in range(n):
+        for j in range(i, n):
+            Spp[i][j] = Spp[j][i] = cov_tm(M, P[i], P[j], ('p', i, j), s, N, M1)
+    Stp = [[cov_tm(M, T[a], P[j], ('tp', idx[a], j), s, N, M1) for j in range(n)] for a in range(nt)]
+    Stt = [[None] * nt for _ in range(nt)]
+    for a in range(nt):
+        for b in range(a, nt):
+            i, j = idx[a], idx[b]
+            if i <= j:
+                Stt[a][b] = Stt[b][a] = cov_tm(M, T[a], T[b], ('tt', i, j), s, N, M1)
+            else:
+                Stt[a][b] = Stt[b][a] = cov_tm(M, T[b], T[a], ('tt', j, i), s, N, M1)
+    zero = TM.const(0, s, N)
+    kvec = [TM.from_fracs({3: Fr(-1, 2)}, 0, s, N), TM.from_fracs({2: Fr(-1)}, 0, s, N), TM.const(12, s, N)] + [zero] * (n - 3)
+    e1 = [TM.const(1, s, N)] + [zero] * (n - 1)
+    X, det = solve_tm(Spp, Stp + [e1, kvec])
+    W, xs0, xs1 = X[:nt], X[nt], X[nt + 1]
+    law = TMLaw()
+    law.d, law.M1, law.th1, law.s, law.N = d, M1, (Fr(0) if MUT == 'no-torus' else M.theta_minus_one(L)), s, N
+    law.alpha = {nm: W[a][0] for a, nm in enumerate(names)}
+    law.beta = {nm: sum((W[a][j] * kvec[j] for j in range(n)), zero) for a, nm in enumerate(names)}
+    law.Cu = {}
+    for a, na in enumerate(names):
+        for b, nb in enumerate(names):
+            law.Cu[(na, nb)] = Stt[a][b] - sum((W[a][l] * Stp[b][l] for l in range(n)), zero)
+    law.e_bb = xs0[0]
+    law.e_bk = sum((kvec[j] * xs0[j] for j in range(n)), zero)
+    law.e_kk = sum((kvec[j] * xs1[j] for j in range(n)), zero)
+    law.det = det
+    if d == 3:
+        # E[O - mu_O | B] = sum_l G[O][l] (B_l - mu_B_l) in natural coordinates (B11, B22, B12), B = -A:
+        # G = -Cu(O, A) Cu(A, A)^-1
+        an = ('a11', 'a22', 'a12')
+        Xg, _ = solve_tm([[law.Cu[(x, y)] for y in an] for x in an], [[law.Cu[(o, y)] for y in an] for o in ('o11', 'o22', 'o12')])
+        law.G = {o: [-Xg[c][l] for l in range(3)] for c, o in enumerate(('o11', 'o22', 'o12'))}
+    return law
+
+
+def tm_summary(law):
+    """Exact low-order coefficients (decimal, 30 places) and remainders of the Taylor models, for the record."""
+    def row(t):
+        c = t.frac_coeffs()
+        return {'c': [dec_down(x, 30) if x >= 0 else dec_up(x, 30) for x in c[:5]], 'rem': dec_up(t.frac_e(), 30)}
+    out = {'det': row(law.det), 'e_bb': row(law.e_bb), 'e_bk': row(law.e_bk), 'e_kk': row(law.e_kk)}
+    for nm in law.alpha:
+        out['alpha_' + nm] = row(law.alpha[nm])
+        out['beta_' + nm] = row(law.beta[nm])
+    names = TM_NAMES[law.d]
+    for i, a in enumerate(names):
+        for b in names[i:]:
+            out['C_%s_%s' % (a, b)] = row(law.Cu[(a, b)])
+    return out
+
+
+# ============================================================================= 2. (a, eps) forms and Taylor-model readers
+def ae_add(*xs):
+    return (sum(x[0] for x in xs), sum(x[1] for x in xs))
+
+
+def ae_mul(x, c):
+    return (x[0] * c, x[1] * c)
+
+
+def ae_at(x, R):
+    return x[0] * R * R + x[1]
+
+
+def exp_up(M, x, R):
+    """exp(a0 r^2 + e0) - 1 <= (a0 r^2 + e0) exp(a0 R^2 + e0) on (0, R]."""
+    f = exp_minus(M, -ae_at(x, R)).hi
+    return (x[0] * f, x[1] * f)
+
+
+def tm_dev(t, ref, R):
+    """(a, eps): |t(r) - ref| <= a r^2 + eps for 0 <= r <= R."""
+    c = t.frac_coeffs()
+    eps = abs(c[0] - ref) + abs(c[1]) * R + t.frac_e()
+    a, Rp = Fr(0), Fr(1)
+    for x in c[2:]:
+        a += abs(x) * Rp
+        Rp *= R
+    return (a, eps)
+
+
+def tm_lin(t, R):
+    """(s, eps): |t(r)| <= s r + eps for 0 <= r <= R."""
+    c = t.frac_coeffs()
+    s, Rp = Fr(0), Fr(1)
+    for x in c[1:]:
+        s += abs(x) * Rp
+        Rp *= R
+    return (s, abs(c[0]) + t.frac_e())
+
+
+def tm_shift(t, ref, R):
+    """(lo, hi, eps): t(r) - ref in [lo r^2 - eps, hi r^2 + eps] for 0 <= r <= R."""
+    c = t.frac_coeffs()
+    eps = abs(c[0] - ref) + abs(c[1]) * R + t.frac_e()
+    lo = hi = c[2]
+    Rp = R
+    for x in c[3:]:
+        v = x * Rp
+        lo += min(v, 0)
+        hi += max(v, 0)
+        Rp *= R
+    return (lo, hi, eps)
+
+
+def assert_flat(t, ref, what):
+    c = t.frac_coeffs()
+    require(abs(c[0] - ref) <= TINY and all(abs(x) <= TINY for x in c[1:]),
+            'the reference law of %s is not the contact law for every r' % what)
+
+
+def gersh_min(C):
+    n = len(C)
+    return min(C[i][i].lo - sum(max(abs(C[i][j].lo), abs(C[i][j].hi)) for j in range(n) if j != i) for i in range(n))
+
+
+def wq_bound(M, Cg, rows, vecs):
+    """Upper bound of ||V' Cg^-1 V|| (rows of V given as magnitude lists).  With D the lower endpoints of the diagonal,
+    Cg = D^(1/2)(I + H + E)D^(1/2) exactly, where H >= 0 is diagonal (the actual diagonal's surplus over D) and E has zero
+    diagonal with ||E||_F <= Ef; so Cg >= (1 - Ef) D and Cg^-1 <= D^-1/(1 - Ef) (review C58: the surplus H only helps)."""
+    n = len(rows)
+    D = [Cg[(rows[i], rows[i])].lo for i in range(n)]
+    Ef2 = Fr(0)
+    for i in range(n):
+        for j in range(n):
+            if i != j:
+                c = max(abs(Cg[(rows[i], rows[j])].lo), abs(Cg[(rows[i], rows[j])].hi))
+                Ef2 += c * c / (D[i] * D[j])
+    Ef = M.sqrt_up(Ef2)
+    require(Ef < Fr(1, 2), 'diagonal scaling of the conditional covariance')
+    return sum(sum(x * x for x in vecs[i]) / D[i] for i in range(n)) / (1 - Ef)
+
+
+def half_moments(M, c):
+    """H_j = int_0^inf u^j exp(-u^2/(2c)) du, j = 0..3: sqrt(pi c/2), c, c sqrt(pi c/2), 2 c^2 (upper bounds)."""
+    s = M.isqrt_iv(M.PI * c / 2).hi
+    return [s, Fr(c), c * s, 2 * Fr(c) ** 2]
+
+
+# ============================================================================= 3. the ratio A_r / A_0^ref on a (b, k) box
+def pi_ratio(M, tml, n, Q, R):
+    """log(pi_r / pi_0^ref) in [-dn, up], with dn, up in (a, eps) form; pi_0^ref has det 12 and energy 3b^2/2 + 24k^2."""
+    b0, b1, k0, k1 = Q
+    th1 = tml.th1
+    dlo, dhi, de = tm_shift(tml.det, 12, R)
+    if MUT == 'no-det':
+        dlo = dhi = de = Fr(0)
+    xm = (max(abs(dlo), abs(dhi)) * R * R + de) / 12
+    require(xm < Fr(1, 2), 'pin determinant deviation')
+    half_log_up = (max(Fr(0), -dlo) / (24 * (1 - xm)), de / (24 * (1 - xm)))
+    half_log_dn = (max(Fr(0), dhi) / 24, de / 24)
+    parts = ((tml.e_bb, Fr(3, 2), (b0 * b0, b1 * b1)), (tml.e_bk, Fr(0), (2 * b0 * k0, 2 * b1 * k1)),
+             (tml.e_kk, Fr(24), (k0 * k0, k1 * k1)))
+    Elo = Ehi = Ee = Eu = Fr(0)
+    for t, ref, (wlo, whi) in parts:
+        lo, hi, e = tm_shift(t, ref, R)
+        Elo += min(wlo * lo, whi * lo)
+        Ehi += max(wlo * hi, whi * hi)
+        Ee += whi * e
+        Eu += whi * (abs(ref) + max(abs(lo), abs(hi)) * R * R + e)
+    up = ae_add(half_log_up, (max(Fr(0), -Elo / 2), Ee / 2 + n * th1 / 2))
+    dn = ae_add(half_log_dn, (max(Fr(0), Ehi / 2), (Ee + th1 * Eu) / 2))
+    return up, dn
+
+
+def combine(M, p_up, p_dn, z_up, z_dn, R):
+    pu = exp_up(M, p_up, R)
+    zR = ae_at(z_up, R)
+    up = (pu[0] + z_up[0] + pu[0] * zR, pu[1] + z_up[1] + pu[1] * zR)
+    return up, ae_add(p_dn, z_dn)
+
+
+def bounds_d2(Q, R, tml, il):
+    """(up, dn): A_r/A_0^ref - 1 <= up(r), 1 - A_r/A_0^ref <= dn(r) on the box, for 0 < r <= R (NOTE sections 3-5)."""
+    M = E2
+    b0, b1, k0, k1 = Q
+    th1 = tml.th1
+    assert_flat(tml.alpha['w'], -1, 'alpha_w')
+    assert_flat(tml.beta['w'], 0, 'beta_w')
+    assert_flat(tml.Cu[('w', 'w')], 2, 'Var w')
+    # the law of lambda = -w: mean b + D, variance 2 + S (remainder only)
+    Dae = ae_add(ae_mul(tm_dev(tml.alpha['w'], -1, R), b1), ae_mul(tm_dev(tml.beta['w'], 0, R), k1))
+    aC = tm_dev(tml.Cu[('w', 'w')], 2, R)
+    Sae = (aC[0], aC[1] + (2 + ae_at(aC, R)) * th1)
+    Dm, Sm = ae_at(Dae, R), ae_at(Sae, R)
+    r2 = M.isqrt_iv(M.IV(2))
+    s_lo, s_hi = M.isqrt_iv(M.IV(2 - Sm)).lo, M.isqrt_iv(M.IV(2 + Sm)).hi
+    sa, sb = min(s_lo, r2.lo), max(s_hi, r2.hi)
+    m_lo, m_hi = b0 - Dm, b1 + Dm
+    mmax = max(abs(m_lo), abs(m_hi))
+    Fmin = M.elam2_pos(m_lo, sa).lo
+    Fmax = M.elam2_pos(m_hi, sb).hi
+    F1max = M.elam_pos(m_hi, sb).hi
+    # the variance multiplier (3/4) 2 sb = (3/2) sb only needs to be >= 1 (dF/d(sigma^2) = Phi(mu/sigma) <= 1); it is not
+    # a claim that Phi <= 3/4, which fails at mu = 1, sigma^2 = 2 (review C60)
+    Fae = ae_add(ae_mul(Dae, 2 * F1max / Fmin), ae_mul(Sae, 2 * sb / Fmin * Fr(3, 4)))
+    Fref = M.elam2_pos(b0, r2.lo).lo
+    # secondary law: the engine's interval law over [0, R]
+    b, k = M.IV(b0, b1), M.IV(k0, k1)
+    ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES}
+    gn = lambda nm, p: M.gauss_norm(ma[nm], il.sdQ[nm].hi, p)
+    lam = lambda p: M.gauss_norm(mmax, s_hi, p)
+    T, tau, om, v, nu = (lambda p, x=x: gn(x, p) for x in ('T', 'tau', 'om', 'v', 'nu'))
+    kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p)
+    norm36 = 36 * k0 * k0 * Fref
+    # first order: r E[lambda_+ om] = r [mu_om F_1 - bw_om (F - m F_1)], with mu_om, bw_om = O(r) (Taylor models)
+    sa_, ea_ = tm_lin(tml.alpha['om'], R)
+    sb_, eb_ = tm_lin(tml.beta['om'], R)
+    sc_, ec_ = tm_lin(tml.Cu[('om', 'w')], R)
+    cwlo = 2 - ae_at(aC, R)
+    T2 = (((b1 * sa_ + k1 * sb_) * F1max + sc_ / cwlo * (Fmax + mmax * F1max)) / Fref,
+          R * ((b1 * ea_ + k1 * eb_) * F1max + ec_ / cwlo * (Fmax + mmax * F1max)) / Fref)
+    if MUT == 'no-om-term':
+        T2 = (Fr(0), Fr(0))
+    # second order (explicit r^2). kappa = 6k tau - T^2 - r T tau with E[tau | w] = mu_tau + bw_tau (w - mu_w):
+    # 6k mu_tau F + 6k bw_tau E[(w - mu_w) lambda_+^2] - E[T^2 lambda_+^2] - r E[T tau lambda_+^2] (signed in the leading part)
+    mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(Fmin, Fmax)
+    side = 6 * k1 * M.absup(il.bw['tau']) * s_hi * lam(4) ** 2 + R * T(4) * tau(4) * lam(4) ** 2
+    T3_up = max(Fr(0), mt.hi + side) / norm36
+    T3_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * lam(4) ** 2) / norm36
+    if MUT == 'no-om-term':
+        T3_up = T3_dn = Fr(0)
+    T4 = R * kap(4) * lam(4) * om(2)
+    xi2 = 6 * k1 * (R * nu(4) ** 2 + 2 * nu(4) * v(4)) + T(4) * (R * R * nu(8) ** 2 + 2 * R * nu(8) * v(8) + 2 * v(8) ** 2) \
+        + R * tau(4) * v(8) ** 2
+    T5 = lam(2) * xi2
+    T6 = v(8) ** 2 * (6 * k1 + R * T(4) + R * R * tau(4)) * om(4) + v(4) ** 2 * (R * nu(4) + v(4)) ** 2
+    # the thin set {0 < lambda <= r h} on G_0 (conditional density of lambda given (v, om, nu))
+    Yn = ('v', 'om', 'nu')
+    prec = 1 / (2 - Sm) + wq_bound(M, il.Cg, Yn, [[M.absup(il.bw[nm])] for nm in Yn])
+    pbar = Fr(2, 5) * M.sqrt_up(prec)
+    h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + om(p)
+    bad = R * pbar * h(2) * (Fr(15, 2) * k1 * h(4) + v(8) ** 2) * (9 * k1 * (h(4) + om(4)) + (R * nu(8) + v(8)) ** 2)
+    if MUT == 'no-bad-set':
+        bad = Fr(0)
+    # G_0^c (not r^2-scaled; astronomically small)
+    q = M.gauss_tail2(ma['T'], il.sdQ['T'].hi, 3 * k0 / (2 * R)) + M.gauss_tail2(ma['tau'], il.sdQ['tau'].hi, 3 * k0 / (2 * R * R))
+    P1n = (6 * k1 + R * T(8)) * lam(8) + R * v(8) ** 2
+    P2n = (6 * k1 + R * T(8) + R * R * tau(8)) * (lam(8) + R * om(8)) + R * (R * nu(8) + v(8)) ** 2
+    g0c = P1n * P2n * M.sqrt_up(q) if q > 0 else Fr(0)
+    X = ae_add(T2, ((T4 + T5 + T6 + bad) / norm36, g0c / norm36))
+    p_up, p_dn = pi_ratio(M, tml, 6, Q, R)
+    return combine(M, p_up, p_dn, ae_add(exp_up(M, Fae, R), X, (T3_up, Fr(0))), ae_add(Fae, X, (T3_dn, Fr(0))), R)
+
+
+def bounds_d3(Q, R, tml, il):
+    """(up, dn) in d = 3 (NOTE sections 3-5)."""
+    M = E3
+    b0, b1, k0, k1 = Q
+    th1 = tml.th1
+    names = ('a11', 'a22', 'a12')
+    for nm, ref in (('a11', -1), ('a22', -1), ('a12', 0)):
+        assert_flat(tml.alpha[nm], ref, 'alpha_' + nm)
+        assert_flat(tml.beta[nm], 0, 'beta_' + nm)
+    for (x, y), ref in ((('a11', 'a11'), 2), (('a22', 'a22'), 2), (('a12', 'a12'), 1), (('a11', 'a22'), 0),
+                        (('a11', 'a12'), 0), (('a22', 'a12'), 0)):
+        assert_flat(tml.Cu[(x, y)], ref, 'Cov(%s, %s)' % (x, y))
+    # Loewner coupling with the contact law at the same b: eps = sup ||mu_B - b I||_F, eta = ||chol(Cb) - sqrt2 I||
+    comps = []
+    for nm, ref, f in (('a11', -1, 1), ('a22', -1, 1), ('a12', 0, 2)):
+        comps.append(ae_at(ae_add(ae_mul(tm_dev(tml.alpha[nm], ref, R), b1), ae_mul(tm_dev(tml.beta[nm], 0, R), k1)), R) * f)
+    eps = M.sqrt_up(sum(x * x for x in comps))
+    Fw = (Fr(1), Fr(1), M.SQRT2)
+    Cb = []
+    for i in range(3):
+        row = []
+        for j in range(3):
+            lo, hi = tml.Cu[(names[i], names[j])].enclose(0, R)
+            row.append(M.IV(lo, hi) * Fw[i] * Fw[j] * M.IV(1 / (1 + th1), 1))
+        Cb.append(row)
+    Lb = M.cholesky(Cb)
+    eta = M.sqrt_up(sum(M.absup(Lb[i][j] - (M.SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1)))
+    Dl = M.rup_rel(eps + eta * R0C)
+    cm = M.contact_moment
+    bA, bB = grid_dn(b0 - Dl), grid_up(b1 + Dl)
+    m3b0 = cm(2, 0, b0).lo
+    L3up = 2 * cm(1, 1, bB).hi / m3b0
+    L3dn = 2 * cm(1, 1, b1).hi / cm(2, 0, bA).lo
+    pR0 = M.chi_tail(3, R0C)
+    muB = M.rup_rel(b1 * M.sqrt_up(Fr(2)) + eps)
+    sig = M.sqrt_up(max(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2]))
+                    + 2 * max(M.absup(Cb[i][j]) for i in range(3) for j in range(3) if i != j))
+    U = M.chi_upper_moments(3, R0C, 4)
+    out_D2 = sum((U[i] * (M.binom(4, i) * muB ** (4 - i) * sig ** i) for i in range(5)), M.IV(0)).hi / 4
+    tail_c = M.sqrt_up(cm(4, 0, b1).hi) * M.sqrt_up(pR0)
+    nBF16 = M.rup_rel(muB + sig * M.chi_norm(3, 16))
+    out_ball = M.sqrt_up(pR0)
+
+    def up_moment(a, c):
+        return M.rup_rel(cm(a, c, bB).hi + 2 ** c * M.rup_rel(nBF16 ** (2 * a + c)) * out_ball)
+    ED2, ED4, ED2T2, ET4, DT = up_moment(2, 0), up_moment(4, 0), up_moment(2, 2), up_moment(0, 4), up_moment(1, 1)
+    trCb = sum(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2]))
+    BF = lambda p: M.frob_norm(muB, trCb, p)
+    # secondary law (engine interval law over [0, R])
+    b, k = M.IV(b0, b1), M.IV(k0, k1)
+    ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES}
+    T = lambda p: M.gauss_norm(ma['T'], il.sdQ['T'], p)
+    tau = lambda p: M.gauss_norm(ma['tau'], il.sdQ['tau'], p)
+    mu_o = M.sqrt_up(ma['o11'] ** 2 + ma['o22'] ** 2 + 2 * ma['o12'] ** 2)
+    trO = sum(m * il.C[(nm, nm)].hi for nm, m in zip(M.ONAMES, (1, 1, 2)))
+    Om = lambda p: M.frob_norm(mu_o, trO, p)
+    mu_v = M.sqrt_up(ma['v1'] ** 2 + ma['v2'] ** 2)
+    v = lambda p: M.frob_norm(mu_v, il.C[('v1', 'v1')].hi + il.C[('v2', 'v2')].hi, p)
+    mu_nu = M.sqrt_up(ma['nu1'] ** 2 + ma['nu2'] ** 2)
+    nu = lambda p: M.frob_norm(mu_nu, il.C[('nu1', 'nu1')].hi + il.C[('nu2', 'nu2')].hi, p)
+    e = lambda p: R * nu(p) + v(p)
+    kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p)
+    c1 = lambda p: 6 * k1 + R * T(p)
+    c2 = lambda p: 6 * k1 + R * T(p) + R * R * tau(p)
+    norm36 = 36 * k0 * k0 * m3b0
+    # the mean of Omega': ||mu_Omega'||_F <= s_mu r + e_mu (Taylor models; beta_O = r^2 on the diagonal)
+    s_mu2 = e_mu2 = Fr(0)
+    for nm, mult in zip(M.ONAMES, (1, 1, 2)):
+        sa_, ea_ = tm_lin(tml.alpha[nm], R)
+        sb_, eb_ = tm_lin(tml.beta[nm], R)
+        s_mu2 += mult * (b1 * sa_ + k1 * sb_) ** 2
+        e_mu2 += mult * (b1 * ea_ + k1 * eb_) ** 2
+    muO = M.sqrt_up(s_mu2) * R + M.sqrt_up(e_mu2)                      # sup over (0, R] (it multiplies r^2 or more)
+    # the regression of Omega' on B: E[Omega' - mu | B] = beta (B - mu_B) + DY, beta = (r - r^3/4)/2 + ..., DY remainder-only
+    G = tml.G
+    beta_t = G['o11'][0]
+    bc = beta_t.frac_coeffs()
+    require(abs(bc[0]) <= TINY, 'regression of Omega on B at r = 0')
+    for (o, l) in (('o22', 1), ('o12', 2)):
+        assert_flat(G[o][l] - beta_t, 0, 'regression of %s on B' % o)
+    for o, ls in (('o11', (1, 2)), ('o22', (0, 2)), ('o12', (0, 1))):
+        for l in ls:
+            assert_flat(G[o][l], 0, 'off-diagonal regression of %s on B' % o)
+    dG2 = Fr(0)
+    for o, mult in zip(('o11', 'o22', 'o12'), (1, 1, 2)):
+        for l in range(3):
+            t = G[o][l] - beta_t if (o, l) in (('o11', 0), ('o22', 1), ('o12', 2)) else G[o][l]
+            dG2 += mult * Fr(t.bound(), 1 << TM_P) ** 2
+    dGn = M.sqrt_up(dG2)
+    b1c = bc[1]                                                         # = 1/2
+    require(b1c > 0, 'sign of the first-order regression coefficient of Omega on B')
+    sb_beta = sum(abs(x) * R ** (j - 2) for j, x in enumerate(bc) if j >= 2)
+    e_beta = beta_t.frac_e()
+    # E[D tr(A* Omega') 1] = beta (2 E[D^2 1] - E[D tr(A* mu_B) 1]) + R1, tr(A* mu_B) = b tr B + tr(A* dmu), ||dmu||_F <= eps
+    DTlo = cm(1, 1, bA).lo - M.sqrt_up(cm(2, 2, b1).hi) * M.sqrt_up(pR0)
+    m3up = cm(2, 0, b1).hi
+    rho_lo = exp_minus(M, Dl * L3dn).lo - tail_c / m3b0                 # m3_r / m3(b)
+    rho_hi = exp_minus(M, -(Dl * L3up)).hi + out_D2 / m3b0
+    tau_lo = b0 * max(DTlo, Fr(0)) / m3up                               # b E[D tr B 1] / m3(b)
+    tau_hi = b1 * DT / m3b0
+    X3lo = 2 * rho_lo - tau_hi - eps * DT / m3b0
+    X3hi = 2 * rho_hi - tau_lo + eps * DT / m3b0
+    X3m = max(abs(X3lo), abs(X3hi))
+    # |R1|/m3(b) <= (s_mu r + e_mu) DT/m3 + ||DG|| sqrt(E[D^2 tr^2 1]) sqrt(tr Cb)/m3, so r |R1|/m3 <= r^2 s_mu DT/m3 + R (...)
+    R1a = M.sqrt_up(s_mu2) * DT / m3b0
+    R1e = (M.sqrt_up(e_mu2) * DT + dGn * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0
+    # -r E[D tr(A* Omega') 1]/m3(b) = -(b1c r^2 + (r beta - b1c r^2)) X3 - r R1/m3(b), |r beta - b1c r^2| <= sb_beta R r^2 + e_beta r
+    U3_up = (max(Fr(0), -b1c * X3lo) + sb_beta * R * X3m + R1a, R * (e_beta * X3m + R1e))
+    U3_dn = (max(Fr(0), b1c * X3hi) + sb_beta * R * X3m + R1a, R * (e_beta * X3m + R1e))
+    if MUT == 'no-om-term':
+        U3_up = U3_dn = (Fr(0), Fr(0))
+    # explained parts of Omega' by B (for U4) and by zeta = (B, v, nu) (for U6, U7): traces of C_Oz C_zz^-1 C_zO
+    zB = ('a11', 'a22', 'a12')
+    zA = zB + tuple(M.VNAMES) + ('nu1', 'nu2')
+    def cov_Oz(o, z):
+        if z in zB:
+            s_, e_ = tm_lin(tml.Cu[(o, z)], R)
+            return s_ * R + e_
+        return M.absup(il.C[(o, z)])
+    def explained(zs):
+        rows = []
+        for o, mult in zip(M.ONAMES, (1, 1, 2)):
+            rows.append([cov_Oz(o, z) for z in zs])
+        D = [il.C[(z, z)].lo for z in zs]
+        Ef2 = Fr(0)
+        for i in range(len(zs)):
+            for j in range(len(zs)):
+                if i != j:
+                    c = M.absup(il.C[(zs[i], zs[j])])
+                    Ef2 += c * c / (D[i] * D[j])
+        Ef = M.sqrt_up(Ef2)
+        require(Ef < Fr(1, 2), 'diagonal scaling (explained part of Omega)')
+        return sum(mult * sum(x * x / Dz for x, Dz in zip(row, D)) for row, mult in zip(rows, (1, 1, 2))) / (1 - Ef)
+    tB, tK = explained(zB), explained(zA)
+    YB4, YK4 = M.frob_norm(0, tB, 4), M.frob_norm(0, tK, 4)
+    detX = M.absup(il.Cg[('o11', 'o22')] - il.Cg[('o12', 'o12')])     # |E[det(Omega' - E[Omega' | B])]|
+    E1 = up_moment(1, 0)
+    # second order (explicit r^2)
+    # signed kappa term: 6k mu_tau E[D^2 1] + 6k E[(tau - mu_tau) D^2 1] - E[T^2 D^2 1] - r E[T tau D^2 1]
+    ED2lo = max(Fr(0), cm(2, 0, bA).lo - tail_c)
+    mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(ED2lo, ED2)
+    side = 6 * k1 * il.bnorm['tau'] * M.sqrt_up(trCb) * M.sqrt_up(ED4) + R * T(4) * tau(4) * M.sqrt_up(ED4)
+    U2_up = max(Fr(0), mt.hi + side) / norm36
+    U2_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * M.sqrt_up(ED4)) / norm36
+    if MUT == 'no-om-term':
+        U2_up = U2_dn = Fr(0)
+    U2 = Fr(0)
+    U3r = R * kap(4) * M.sqrt_up(ED2T2) * Om(4)
+    # U4 = r^2 E[c1 c2 D det Omega' 1]: E[det Omega' | B] = det(mu + Y_B) + E[det X'] (36 k^2 cancels; relative)
+    U4rel = (detX * E1 + M.sqrt_up(ED2) * (muO + YB4) ** 2 / 2) / m3b0
+    U4 = R * R * kap(4) * M.sqrt_up(ED2) * Om(8) ** 2 / 2
+    U5 = M.sqrt_up(ED2T2) * ((2 * T(4) + R * tau(4)) * v(8) ** 2 + 2 * c1(4) * nu(8) * v(8) + R * c1(4) * nu(8) ** 2)
+    # U6, U7: the 6k parts through E[Omega' | B, v, nu] = mu + Y_K (Omega' - E[Omega' | zeta] has conditional mean 0)
+    U67rel = (M.sqrt_up(ED2) * (muO * e(4) ** 2 + e(8) ** 2 * YK4) + M.sqrt_up(ET4) * (muO * v(4) ** 2 + v(8) ** 2 * YK4)) / (6 * k0 * m3b0)
+    U6 = R * T(8) * M.sqrt_up(ED2) * Om(8) * e(8) ** 2
+    U7 = R * (T(8) + R * tau(8)) * M.sqrt_up(ET4) * Om(8) * v(8) ** 2
+    U8 = R * c2(8) * Om(16) ** 2 / 2 * M.iroot(ET4, 4) * v(16) ** 2
+    U9 = M.sqrt_up(ET4) * v(8) ** 2 * e(8) ** 2
+    U10 = R * M.iroot(ET4, 4) * v(8) ** 2 * Om(4) * e(8) ** 2
+    # the thin set {0 < lambda_1 <= r h} on G_0: conditional law of B given Y = (v, Omega', nu), eigenvalue coordinates
+    Yn = tuple(M.VNAMES) + tuple(M.ONAMES) + ('nu1', 'nu2')
+    lmB = gersh_min(Cb)
+    rows = []
+    for nm in Yn:
+        if nm in M.ONAMES:
+            cov = [(tm_lin(tml.Cu[(nm, an)], R)[0] * R + tm_lin(tml.Cu[(nm, an)], R)[1]) * f for an, f in zip(names, (1, 1, 2))]
+            rows.append([M.sqrt_up(sum(x * x for x in cov)) / lmB])
+        else:
+            rows.append([M.absup(x) for x in il.bt[nm]])
+    Wt = wq_bound(M, il.Cg, Yn, rows)
+    c_hi = sig * sig
+    cl = 1 / (1 / lmB + Wt)
+    KBp = M.rup_rel((M.PI * M.SQRT2 / (2 * M.PI * M.SQRT2PI)).hi / (cl * M.sqrt_down(cl)))
+    m1 = muB + c_hi * M.sqrt_up(Wt) * M.chi_norm(7, 8)
+    Hs = half_moments(M, c_hi)
+    Ja = m1 ** 4 / 4 + sum(M.binom(3, i) * m1 ** (3 - i) * Hs[i] for i in range(4))
+    Jb = m1 ** 3 / 3 + sum(M.binom(2, i) * m1 ** (2 - i) * Hs[i] for i in range(3))
+    h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + Om(p)
+    nhg = h(4) * (Fr(15, 2) * k1 * h(8) + v(16) ** 2) * (9 * k1 * (h(8) + Om(8)) + e(16) ** 2)
+    nhgO = h(8) * (Fr(15, 2) * k1 * h(16) + v(32) ** 2) * (9 * k1 * (h(16) + Om(16)) + e(32) ** 2) * Om(8)
+    bad = R * KBp * (Ja * nhg + R * Jb * nhgO)
+    if MUT == 'no-bad-set':
+        bad = Fr(0)
+    q = M.gauss_tail2(ma['T'], il.sdQ['T'], 3 * k0 / (2 * R)) + M.gauss_tail2(ma['tau'], il.sdQ['tau'], 3 * k0 / (2 * R * R))
+    P1n = c1(8) * BF(16) ** 2 + R * BF(8) * v(16) ** 2
+    P2n = c2(8) * (BF(16) + R * Om(16)) ** 2 + R * (BF(8) + R * Om(8)) * e(16) ** 2
+    g0c = P1n * P2n * M.sqrt_up(q) if q > 0 else Fr(0)
+    X = ((U2 + U3r + U4 + U5 + U6 + U7 + U8 + U9 + U10 + bad) / norm36 + U4rel + U67rel, g0c / norm36)
+    z_up = ae_add(exp_up(M, (Fr(0), Dl * L3up), R), (Fr(0), out_D2 / m3b0), U3_up, X, (U2_up, Fr(0)))
+    z_dn = ae_add((Fr(0), Dl * L3dn), (Fr(0), tail_c / m3b0), U3_dn, X, (U2_dn, Fr(0)))
+    p_up, p_dn = pi_ratio(M, tml, 8, Q, R)
+    return combine(M, p_up, p_dn, z_up, z_dn, R)
+
+
+def rate_sweep(d, tml):
+    """For every R in R_STARS and every sub-box: (a_up, eps_up, a_dn, eps_dn) as decimal strings (rounded up)."""
+    M = ENG[d]
+    fn = bounds_d2 if d == 2 else bounds_d3
+    out = {}
+    for R in R_STARS:
+        il = M.band_law(Fr(0), R)
+        rows = []
+        for bb, kb in SUBBOXES:
+            up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, tml, il)
+            rows.append([dec_up(up[0], 6), dec_up(up[1], 20), dec_up(dn[0], 6), dec_up(dn[1], 20)])
+        out[str(R)] = {'boxes': rows,
+                       'a_up': max((r[0] for r in rows), key=Fr), 'eps_up': max((r[1] for r in rows), key=Fr),
+                       'a_dn': max((r[2] for r in rows), key=Fr), 'eps_dn': max((r[3] for r in rows), key=Fr)}
+    return out
+
+
+# ============================================================================= 4. certified reference integrals
+PREC = 240
+
+
+def iroot_int(N, p):
+    """Largest y >= 0 with y^p <= N."""
+    if N < 2:
+        return N
+    y = 1 << ((N.bit_length() + p - 1) // p)
+    while True:
+        z = ((p - 1) * y + N // y ** (p - 1)) // p
+        if z >= y:
+            break
+        y = z
+    while y ** p > N:
+        y -= 1
+    while (y + 1) ** p <= N:
+        y += 1
+    return y
+
+
+def pow_frac(M, x, num, den):
+    """Enclosure of x^(num/den), rational x > 0."""
+    x = Fr(x) ** num
+    S = 1 << (den * PREC)
+    lo_n = (x.numerator * S) // x.denominator
+    hi_n = -((-x.numerator * S) // x.denominator)
+    ylo, yhi = iroot_int(lo_n, den), iroot_int(hi_n, den)
+    if yhi ** den < hi_n:
+        yhi += 1
+    return M.IV(Fr(ylo, 1 << PREC), Fr(yhi, 1 << PREC))
+
+
+def lower_gamma(M, an, ad, x):
+    """gamma(a, x) = x^a e^-x sum_n x^n / (a (a+1) ... (a+n)), a = an/ad > 0, rational x > 0 (series with a geometric tail)."""
+    a, x = Fr(an, ad), Fr(x)
+    term = s = 1 / a
+    n = 0
+    while True:
+        n += 1
+        term = term * x / (a + n)
+        s += term
+        if a + n + 1 > 2 * x and term < Fr(1, 10 ** 70) * s:
+            break
+    rho = x / (a + n + 1)
+    return pow_frac(M, x, an, ad) * M.exp_neg_point(x) * M.IV(s, s + term * rho / (1 - rho))
+
+
+def k_integral(M, qn, qd, k0, k1):
+    """int_{k0}^{k1} k^q exp(-12 k^2) dk = (1/2) 12^(-a) [gamma(a, 12 k1^2) - gamma(a, 12 k0^2)], a = (q + 1)/2."""
+    an, ad = qn + qd, 2 * qd
+    g = math.gcd(an, ad)
+    an, ad = an // g, ad // g
+    pre = pow_frac(M, Fr(1, 12), an, ad) * Fr(1, 2)
+    return pre * (lower_gamma(M, an, ad, 12 * Fr(k1) ** 2) - lower_gamma(M, an, ad, 12 * Fr(k0) ** 2))
+
+
+def m_ref(d, b):
+    """The reference contact moment: F(b, sqrt2) = E[(b + sqrt2 z)_+^2] (d = 2), m_3(b) = E[det(B)^2 1{B > 0}] (d = 3)."""
+    if d == 2:
+        s = E2.isqrt_iv(E2.IV(2))
+        return E2.IV(E2.elam2_pos(Fr(b), s.lo).lo, E2.elam2_pos(Fr(b), s.hi).hi)
+    return E3.contact_moment(2, 0, Fr(b))
+
+
+def m_ref_derivs(d, b):
+    """Upper bounds of m, m', m'' at b (each increasing in b)."""
+    if d == 2:
+        s = E2.isqrt_iv(E2.IV(2)).hi
+        return E2.elam2_pos(Fr(b), s).hi, 2 * E2.elam_pos(Fr(b), s).hi, Fr(2)
+    cm = E3.contact_moment
+    return cm(2, 0, Fr(b)).hi, 2 * cm(1, 1, Fr(b)).hi, 2 * cm(0, 2, Fr(b)).hi + 4 * cm(1, 0, Fr(b)).hi
+
+
+def b_cells(d):
+    """Enclosures of int over [j h, (j+1) h] of exp(-3b^2/4) m(b) db, h = 1/NH (midpoint rule, |g''| bounded per block)."""
+    M = ENG[d]
+    h = Fr(1, NH)
+    out = []
+    for blk in range(NH // BLOCK):
+        x0, x1 = blk * BLOCK * h, (blk + 1) * BLOCK * h
+        m, m1, m2 = m_ref_derivs(d, x1)
+        g2 = M.exp_neg_point(3 * x0 * x0 / 4).hi * ((Fr(9, 4) * x1 * x1 + Fr(3, 2)) * m + 3 * x1 * m1 + m2)
+        err = h ** 3 / 24 * g2
+        for j in range(BLOCK):
+            c = x0 + (j + Fr(1, 2)) * h
+            out.append(M.exp_neg_point(3 * c * c / 4) * m_ref(d, c) * h + M.IV(-err, err))
+    return out
+
+
+def b_integral(M, cells, b0, b1):
+    i0, i1 = Fr(b0) * NH, Fr(b1) * NH
+    require(i0.denominator == 1 and i1.denominator == 1, 'b-box off the cell grid')
+    return sum((cells[j] for j in range(int(i0), int(i1))), M.IV(0))
+
+
+def K_const(d):
+    """|S^(d-1)| 144 (2 pi)^(-(d+1)) / sqrt12: A_0^ref/(3 k^p) = K k^(2-p) e^(-12 k^2) e^(-3 b^2/4) m(b)."""
+    M = ENG[d]
+    tp = 2 * M.PI
+    pw = M.IV(1)
+    for _ in range(d + 1):
+        pw = pw * tp
+    return (tp if d == 2 else 2 * tp) * 144 / pw / M.isqrt_iv(M.IV(12))
+
+
+def reference(d):
+    M = ENG[d]
+    cells = b_cells(d)
+    K = K_const(d)
+    Ib = b_integral(M, cells, 0, 1)
+    c_ref = K * Ib * k_integral(M, 4, 3, Fr(1, 2), Fr(2))
+    J23, J43, J13 = {}, {}, {}
+    for kb in SUB_K:
+        J23[kb] = k_integral(M, 2, 3, kb[0], kb[1])
+        J43[kb] = k_integral(M, 4, 3, kb[0], kb[1])
+    for kb in CAP_K:
+        J13[kb] = k_integral(M, 1, 3, kb[0], kb[1])
+    Ibs = {bb: b_integral(M, cells, bb[0], bb[1]) for bb in set(SUB_B) | set(CAP_B)}
+    return {'c_ref': [dec_down(c_ref.lo, 24), dec_up(c_ref.hi, 24)],
+            'w_c': [dec_up((K * Ibs[bb] * J43[kb]).hi, 24) for bb, kb in SUBBOXES],          # int A_0/(3 k^(2/3))
+            'w_2': [dec_up((K * Ibs[bb] * J23[kb]).hi, 24) for bb, kb in SUBBOXES],          # int A_0/(3 k^(4/3))
+            'w_cap': [dec_up((K * Ibs[bb] * J13[kb]).hi, 24) for bb, kb in CAP_BOXES]}       # int A_0/(3 k^(5/3))
+
+
+# ============================================================================= 5. the refined cap sweep (engines of #206, #212)
+CAPM = {}
+
+
+def cap_init(d):
+    CAPM['M'] = ENG[d]
+    CAPM['d'] = d
+    if MUT == 'cap-half':
+        CAPM['M'].MUT = 'cap-half'
+
+
+def cap_band(args):
+    """Certified bound of sup over the band of Q^W(G_r^c)/r^3 on every refined box (rounded up, 4 places)."""
+    d, r0, r1 = args
+    M = ENG[d]
+    if MUT == 'cap-half':
+        M.MUT = 'cap-half'
+    law = M.band_law(r0, r1)
+    x0, th = (M.COMBOS[0] if d == 2 else M.COMBO)
+    return {'r': [str(r0), str(r1)],
+            'bound': [M.dec_up(M.box_bound(law, bb, kb, r0, r1, last_band=(r0 == 0), x0=x0, theta=th)['total'])
+                      for bb, kb in CAP_BOXES]}
+
+
+def cap_sweep(d, procs, which=None):
+    bl = ENG[d].bands()
+    idx = range(len(bl)) if which is None else which
+    jobs = [(d, bl[i][0], bl[i][1]) for i in idx]
+    if procs <= 1:
+        return [cap_band(j) for j in jobs]
+    with multiprocessing.Pool(procs, initializer=cap_init, initargs=(d,)) as pool:
+        return pool.map(cap_band, jobs, chunksize=1)
+
+
+def cap_sample(d):
+    """Band indices replayed by --check: the band ending at each r_*, the band ending at 2^-20 and the last band."""
+    bl = ENG[d].bands()
+    ends = set(R_STARS) | {Fr(1, 2 ** 20)}
+    return sorted({i for i, (r0, r1) in enumerate(bl) if r1 in ends} | {0})
+
+
+# ============================================================================= 6. assembly
+def assemble(d, rate, ref, cap):
+    """The C_{B,K}(r_*) table, the second-order density constants, and the (12.1), (12.3) constants."""
+    c_lo, c_hi = Fr(ref['c_ref'][0]), Fr(ref['c_ref'][1])
+    wc = [Fr(x) for x in ref['w_c']]
+    w2 = [Fr(x) for x in ref['w_2']]
+    wcap = [Fr(x) for x in ref['w_cap']]
+    out = {}
+    for R in R_STARS:
+        rr = rate[str(R)]
+        a_up, e_up, a_dn, e_dn = Fr(rr['a_up']), Fr(rr['eps_up']), Fr(rr['a_dn']), Fr(rr['eps_dn'])
+        c2_up = sum(Fr(row[0]) * w for row, w in zip(rr['boxes'], w2))
+        c2_dn = sum(Fr(row[2]) * w for row, w in zip(rr['boxes'], w2))
+        e_c_up = sum(Fr(row[1]) * w for row, w in zip(rr['boxes'], wc))
+        e_c_dn = sum(Fr(row[3]) * w for row, w in zip(rr['boxes'], wc))
+        Cbox = [Fr(0)] * len(CAP_BOXES)
+        for rec in cap:
+            if Fr(rec['r'][1]) <= R:
+                for i, x in enumerate(rec['bound']):
+                    Cbox[i] = max(Cbox[i], Fr(x))
+        S = sum(c * w for c, w in zip(Cbox, wcap))
+        CBK = (1 + a_up * R * R + e_up) * S
+        out[str(R)] = {'C_BK': dec_up(CBK, 4), 'C_121': dec_up(Fr(3, 5) * CBK, 4),
+                       'c2_up': dec_up(c2_up, 12), 'c2_dn': dec_up(c2_dn, 12),
+                       'epsc_up': dec_up(e_c_up, 30), 'epsc_dn': dec_up(e_c_dn, 30),
+                       'rel2_up': dec_up(c2_up / c_lo, 6), 'rel2_dn': dec_up(c2_dn / c_lo, 6),
+                       'ell_max': '%s' % (R ** 3 / 2)}
+    return out
+
+
+def params():
+    return {'R_STARS': [str(x) for x in R_STARS], 'TM': {'s': TM_S, 'N': TM_N, 'P': TM_P}, 'R0_COUPLING': str(R0C),
+            'SUB_B': [[str(a), str(b)] for a, b in SUB_B], 'SUB_K': [[str(a), str(b)] for a, b in SUB_K],
+            'NH': NH, 'BLOCK': BLOCK, 'CAP_B': [[str(a), str(b)] for a, b in CAP_B], 'CAP_K': [[str(a), str(b)] for a, b in CAP_K],
+            'band': {'b': ['0', '1'], 'k': ['1/2', '2']}, 'L_min': 10}
+
+
+def compute_core(d):
+    tml = tm_law(d)
+    return tm_summary(tml), rate_sweep(d, tml), reference(d)
+
+
+def full_run(procs):
+    res = {'schema': 1, 'object': 'CL-C8-LIFETIME-DIFFERENCE-CONSTANT-20261001-v1', 'scientific_effect': 'NONE',
+           'certified': True, 'mutant': None, 'parameters': params(), 'dims': {}}
+    for d in (2, 3):
+        summ, rate, ref = compute_core(d)
+        cap = cap_sweep(d, procs)
+        res['dims'][str(d)] = {'tm_law': summ, 'rate': rate, 'reference': ref, 'table': assemble(d, rate, ref, cap),
+                               'cap_boxes': [[str(bb[0]), str(bb[1]), str(kb[0]), str(kb[1])] for bb, kb in CAP_BOXES],
+                               'cap_bands': cap}
+    return res
+
+
+def results_text(res):
+    """Canonical layout: everything indented except the cap-band records, one line each."""
+    caps = {d: res['dims'][d].pop('cap_bands') for d in res['dims']}
+    head = json.dumps(res, indent=1, sort_keys=True)
+    for d in res['dims']:
+        res['dims'][d]['cap_bands'] = caps[d]
+    tail = ',\n "cap_bands": {\n' + ',\n'.join(
+        '  "%s": [\n' % d + ',\n'.join('   ' + json.dumps(rec, sort_keys=True) for rec in caps[d]) + '\n  ]'
+        for d in sorted(caps)) + '\n }\n}\n'
+    return head[:-2] + tail
+
+
+def load_results():
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        res = json.load(fh)
+    for d, recs in res.pop('cap_bands').items():
+        res['dims'][d]['cap_bands'] = recs
+    return res
+
+
+def check_run(procs, full=False):
+    res = load_results()
+    with open(os.path.join(HERE, 'RESULTS.json')) as fh:
+        require(fh.read() == results_text(res), 'RESULTS.json is not in canonical layout')
+    require(res['parameters'] == params(), 'parameters differ')
+    require(res['mutant'] is None and res['certified'] is True, 'header')
+    for d in (2, 3):
+        rd = res['dims'][str(d)]
+        tml = tm_law(d)
+        require(tm_summary(tml) == rd['tm_law'], 'Taylor-model law differs (d = %d)' % d)
+        rate = rate_sweep(d, tml)
+        require(rate == rd['rate'], 'rate sweep differs (d = %d)' % d)
+        ref = reference(d)
+        require(ref == rd['reference'], 'reference integrals differ (d = %d)' % d)
+        c_lo, c_hi = Fr(ref['c_ref'][0]), Fr(ref['c_ref'][1])
+        cx = C_REF_197 if d == 2 else C_REF_205
+        require(c_lo <= cx <= c_hi, 'reference c_{B,K} does not contain the value of Math-#197/#205 (d = %d)' % d)
+        require(rd['cap_boxes'] == [[str(bb[0]), str(bb[1]), str(kb[0]), str(kb[1])] for bb, kb in CAP_BOXES], 'cap boxes')
+        bl = ENG[d].bands()
+        require([rec['r'] for rec in rd['cap_bands']] == [[str(a), str(b)] for a, b in bl], 'cap band layout (d = %d)' % d)
+        which = None if full else cap_sample(d)
+        recs = cap_sweep(d, procs, which)
+        for i, rec in zip(range(len(bl)) if which is None else which, recs):
+            require(rec == rd['cap_bands'][i], 'cap band %d differs (d = %d)' % (i, d))
+        require(assemble(d, rate, ref, rd['cap_bands']) == rd['table'], 'assembled table differs (d = %d)' % d)
+    print('CHECK PASSED (%s)' % ('full' if full else 'sample'))
+
+
+# ============================================================================= 7. floating controls (not part of the certificate)
+def controls():
+    """Floating estimates of the reference-kernel cap-route benchmarks Gamma_{B,K} = |S| int A_0^ref c_G / (3 k^(5/3)) for comparison with C_{B,K}:
+    planar c_G = R(b) J(k) of Math-#203 with J(k) = 2359296 k^3 (1 + delta(k)), delta(1/2) = 0.0232, delta(3/4) = 1.3e-4;
+    d = 3 from the floating R_3(b) R(b) and J^(3)/J ratios of Math-#208."""
+    from math import exp, pi, sqrt, erf
+    out = {}
+    K2 = 2 * pi * 144 / (2 * pi) ** 3 / sqrt(12)
+    Ib2 = sqrt(pi) / 2 * erf(1) / (2 * sqrt(pi))                 # int_0^1 e^{-b^2} db / (2 sqrt(pi)) (m R = p_w(0))
+    def J2(k):
+        dl = 0.0232 + (k - 0.5) / 0.25 * (1.3e-4 - 0.0232) if k < 0.75 else 1.3e-4 * max(0.0, (1 - k) / 0.25)
+        return 2359296 * k ** 3 * (1 + dl)
+    n = 200000
+    Ik2 = sum(((0.5 + (i + 0.5) * 1.5 / n) ** (1 / 3)) * exp(-12 * (0.5 + (i + 0.5) * 1.5 / n) ** 2) * J2(0.5 + (i + 0.5) * 1.5 / n)
+              * 1.5 / n for i in range(n))
+    out['Gamma_2'] = K2 * Ib2 * Ik2
+    bs = [0, 0.25, 0.5, 0.75, 1.0]
+    m3 = [0.6715728752625627, 1.195532362807149, 2.05794812257308, 3.43079921823, 5.54806027367761]
+    r3r = [1.1880833639847637, 0.9085531055205767, 0.6853903187865128, 0.5096151204445851, 0.37314989157732215]
+    f = [exp(-0.75 * b * b) * m * r for b, m, r in zip(bs, m3, r3r)]
+    Ib3 = 0.25 / 3 * (f[0] + 4 * f[1] + 2 * f[2] + 4 * f[3] + f[4])
+    K3 = 4 * pi * 144 / (2 * pi) ** 4 / sqrt(12)
+    def J3(k):
+        dl = 0.1617 + (k - 0.5) / 0.25 * (0.0012 - 0.1617) if k < 0.75 else 0.0012 * max(0.0, (1 - k) / 0.25)
+        return 2359296 * k ** 3 * (1 + dl)
+    Ik3 = sum(((0.5 + (i + 0.5) * 1.5 / n) ** (1 / 3)) * exp(-12 * (0.5 + (i + 0.5) * 1.5 / n) ** 2) * J3(0.5 + (i + 0.5) * 1.5 / n)
+              * 1.5 / n for i in range(n))
+    out['Gamma_3'] = K3 * Ib3 * Ik3
+    return out
+
+
+def mc_ratio(d, n, seed=20261001):
+    """Floating Monte Carlo of (A_r/A_0^ref - 1)/r^2 for the reference kernel at r = 1/8, 1/16: the exact law at r from the
+    series (all r, not only r <= 1/64), Z_r/r^2 by sampling W_r/r^2 = P1 P2 1{typed}, and the exact pin ratio."""
+    import random
+    M = ENG[d]
+    names = ('w', 'v', 'om', 'T', 'tau', 'nu') if d == 2 else \
+        ('a11', 'a22', 'a12', 'v1', 'v2', 'o11', 'o22', 'o12', 'T', 'tau', 'nu1', 'nu2')
+    pins, targets = M.functionals()
+    P = [pins[x] for x in M.PIN_NAMES]
+    idx = [M.TARGET_NAMES.index(x) for x in names]
+    Tf = [targets[x] for x in names]
+    tml = tm_law(d)
+    def cov(F, G, key, r):
+        ser, w, _ = M.pair_series(F, G, key)
+        return sum(c * r ** p for p, c in ser.items())
+    def law(r, b, k):
+        n_ = len(P)
+        S = [[None] * n_ for _ in range(n_)]
+        for i in range(n_):
+            for j in range(i, n_):
+                S[i][j] = S[j][i] = cov(P[i], P[j], ('p', i, j), r)
+        Stp = [[cov(Tf[a], P[j], ('tp', idx[a], j), r) for j in range(n_)] for a in range(len(Tf))]
+        Mx = [list(S[i]) + [Stp[a][i] for a in range(len(Tf))] for i in range(n_)]
+        for col in range(n_):
+            for i in range(col + 1, n_):
+                f = Mx[i][col] / Mx[col][col]
+                for j in range(col, len(Mx[i])):
+                    Mx[i][j] -= f * Mx[col][j]
+        W = []
+        for a in range(len(Tf)):
+            x = [None] * n_
+            for i in reversed(range(n_)):
+                x[i] = (Mx[i][n_ + a] - sum(Mx[i][j] * x[j] for j in range(i + 1, n_))) / Mx[i][i]
+            W.append(x)
+        kv = [-r ** 3 / 2, -r ** 2, Fr(12)] + [Fr(0)] * (n_ - 3)
+        mu = [float(b * W[a][0] + k * sum(W[a][j] * kv[j] for j in range(n_))) for a in range(len(Tf))]
+        def stt(a, c):
+            i, j = idx[a], idx[c]
+            return cov(Tf[a], Tf[c], ('tt', i, j), r) if i <= j else cov(Tf[c], Tf[a], ('tt', j, i), r)
+        C = [[float(stt(a, c) - sum(W[a][l] * Stp[c][l] for l in range(n_))) for c in range(len(Tf))] for a in range(len(Tf))]
+        L = [[0.0] * len(C) for _ in C]
+        for i in range(len(C)):
+            for j in range(i + 1):
+                s = C[i][j] - sum(L[i][l] * L[j][l] for l in range(j))
+                L[i][j] = math.sqrt(max(s, 0.0)) if i == j else (s / L[j][j] if L[j][j] > 0 else 0.0)
+        return mu, L
+    rng = random.Random(seed)
+    out = []
+    for r in (Fr(1, 8), Fr(1, 16)):
+        rf = float(r)
+        for b, k in ((Fr(0), Fr(1, 2)), (Fr(0), Fr(2)), (Fr(1), Fr(1, 2)), (Fr(1), Fr(2))):
+            mu, L = law(r, b, k)
+            kf = float(k)
+            tot = tot2 = 0.0
+            for _ in range(n):
+                z = [rng.gauss(0.0, 1.0) for _ in mu]
+                x = [mu[i] + sum(L[i][j] * z[j] for j in range(i + 1)) for i in range(len(mu))]
+                if d == 2:
+                    w, v, om, T, tau, nu = x
+                    lam = -w
+                    c1, c2 = 6 * kf - rf * T, 6 * kf + rf * T + rf * rf * tau
+                    P1 = c1 * lam - rf * v * v
+                    P2 = c2 * (lam - rf * om) + rf * (rf * nu - v) ** 2
+                    Wv = P1 * P2 if (c1 > 0 and P1 > 0 and P2 > 0) else 0.0
+                else:
+                    a11, a22, a12, v1, v2, o11, o22, o12, T, tau, nu1, nu2 = x
+                    B = (-a11, -a22, -a12)
+                    S_ = (B[0] - rf * o11, B[1] - rf * o22, B[2] - rf * o12)
+                    detB, detS = B[0] * B[1] - B[2] ** 2, S_[0] * S_[1] - S_[2] ** 2
+                    c1, c2 = 6 * kf - rf * T, 6 * kf + rf * T + rf * rf * tau
+                    e1, e2 = rf * nu1 - v1, rf * nu2 - v2
+                    P1 = c1 * detB - rf * (B[1] * v1 * v1 - 2 * B[2] * v1 * v2 + B[0] * v2 * v2)
+                    P2 = c2 * detS + rf * (S_[1] * e1 * e1 - 2 * S_[2] * e1 * e2 + S_[0] * e2 * e2)
+                    Wv = 0.0
+                    if B[0] > 0 and detB > 0 and P1 > 0 and P2 > 0:
+                        # index(H_S) = 2: sign changes of the leading minors (-S_11, det S, r P2) of H_S reordered (y1, y2, x)
+                        m1, m2, m3 = -S_[0], detS, rf * P2
+                        if m1 != 0 and m2 != 0:
+                            neg = (m1 < 0) + (m2 / m1 < 0) + (m3 / m2 < 0)
+                            Wv = P1 * P2 if neg == 2 else 0.0
+                tot += Wv
+                tot2 += Wv * Wv
+            mean = tot / n
+            se = math.sqrt(max(tot2 / n - mean * mean, 0.0) / n)
+            mref = float(m_ref(d, b).mid())
+            zr, zse = mean / (36 * kf * kf * mref), se / (36 * kf * kf * mref)
+            ev = lambda t: sum(float(c) * rf ** j for j, c in enumerate(t.frac_coeffs()))
+            E = ev(tml.e_bb) * float(b) ** 2 + 2 * ev(tml.e_bk) * float(b) * kf + ev(tml.e_kk) * kf * kf
+            pr = math.sqrt(12 / ev(tml.det)) * math.exp(-(E - (1.5 * float(b) ** 2 + 24 * kf * kf)) / 2)
+            out.append({'d': d, 'r': str(r), 'b': str(b), 'k': str(k), 'n': n, 'Z_ratio': round(zr, 5), 'Z_se': round(zse, 5),
+                        'pin_ratio': round(pr, 6), 'coef': round((pr * zr - 1) / rf ** 2, 3), 'coef_se': round(pr * zse / rf ** 2, 3)})
+    return out
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser()
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--check-full', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    ap.add_argument('--procs', type=int, default=4)
+    ap.add_argument('--controls', action='store_true')
+    args = ap.parse_args()
+    MUT = args.mutant
+    if args.controls:
+        out = controls()
+        out['mc'] = mc_ratio(2, 200000) + mc_ratio(3, 100000)
+        print(json.dumps(out, indent=1))
+        return
+    if args.check or args.check_full:
+        check_run(args.procs, full=args.check_full)
+        return
+    require(MUT is None, 'a mutant cannot regenerate RESULTS.json')
+    res = full_run(args.procs)
+    with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh:
+        fh.write(results_text(res))
+    print(json.dumps({d: {R: row['C_BK'] for R, row in res['dims'][d]['table'].items()} for d in res['dims']}))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py
new file mode 100644
index 000000000..5fd6b75df
--- /dev/null
+++ b/frontiers/c8_torus_vanishing_remainder_20261001/theorem_v.py
@@ -0,0 +1,863 @@
+#!/usr/bin/env python3
+"""Theorem L' (NOTE.md): the radial ledger of [LP] Theorem B on the C8 band with a VANISHING remainder, d = 2 and d = 3.
+
+For the torus kernel (every side L >= 10, every frame) and for the reference kernel, b in [0, 1], k in [1/2, 2], 0 < r <= R:
+
+    -(a_dn r^2 + beta_dn r)  <=  A_r(b, k, u) / A_0(b, k, u) - 1  <=  a_up r^2 + beta_up r,
+
+where A_0 = lim_{r -> 0} A_r is the contact integrand of THE SAME kernel (for the torus: its own A_0^(L), not the Gaussian
+A_0^ref).  Math-#215 (Theorem L) proved the same shape against A_0^ref with an additive, r-independent eps_d; here every
+remainder carries a positive power of r.  The first-order coefficient beta collects the torus corrections only (about
+1e-12); for the reference kernel it is exactly 0 (torus=False gives M1 = 0, and the computed beta is the rational 0: its
+upward-rounded stored values are all 0).
+
+Method (NOTE.md sections 2-5):
+- Taylor models with INTERVAL coefficients and a remainder proportional to r^(N+1) (class TV): every rounding goes into a
+  coefficient radius, every truncation into the r^(N+1) remainder.  The value at r = 0 is the constant coefficient, so
+  t(r) - t(0) is controlled by the coefficients of r^1, r^2, ... only.
+- The torus part of each covariance has the coefficient majorant |tau_p| <= T0 e^p (Cauchy estimates and
+  g_x!/(g_x - A)! <= w^A e^(g_x - w), as in the engines of Math-#206/#212); it enters as radii T0 e^p on the coefficients
+  p <= N and as T0 e^(N+1)/(1 - e R) r^(N+1).
+- Increments against r = 0: the pin-density ratio pi_r/pi_0 (the kernel normalization Theta multiplies increments only),
+  the main normalizer term by the derivative bound dF = 2 E[lambda_+] dmu + Phi(mu/sigma) d(sigma^2) (d = 2) and by a Gaussian
+  gradient coupling with |grad det(B)^2 1{B > 0}| <= ||B||_F^3 (d = 3); first-order terms carry an explicit factor r; the
+  thin set and the far set G_0^c by Markov at a high moment (r^2).
+- The interval band laws, the second-order term bounds and the reference integrals are those of Math-#215's theorem_b.py and
+  the engines of Math-#206/#212, as byte-identical pinned copies (SOURCE_MAP.json).
+
+  python3 -B -S theorem_v.py --write        regenerate RESULTS.json
+  python3 -B -S theorem_v.py --check        replay and compare with RESULTS.json (exact)
+  python3 -B -S theorem_v.py --mutant NAME  a semantic mutation; the replay must then fail (exit 1)
+"""
+import argparse
+import importlib.util
+import json
+import math
+import os
+import sys
+from fractions import Fraction as Fr
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+OBJECT = 'CL-C8-TORUS-VANISHING-REMAINDER-20261001-v1'
+MUTANTS = ('torus-const-only', 'exp-radius', 'no-lin', 'no-om-term', 'coupling-half')
+MUT = None
+
+
+def require(cond, msg):
+    if not cond:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def _load(name, fn):
+    spec = importlib.util.spec_from_file_location(name, os.path.join(HERE, fn))
+    mod = importlib.util.module_from_spec(spec)
+    spec.loader.exec_module(mod)
+    return mod
+
+
+TB = _load('theorem_b', 'theorem_b.py')         # Math-#215, pinned byte-identical copy (loads engine_e2.py, engine_e3.py)
+E2, E3, ENG = TB.E2, TB.E3, TB.ENG
+R_STARS = TB.R_STARS
+SUBBOXES, SUB_K = TB.SUBBOXES, TB.SUB_K
+TM_S, TM_N, TM_P = TB.TM_S, TB.TM_N, TB.TM_P
+TINY = TB.TINY
+E_UP = Fr(27183, 10000)                          # > e
+SQRT2_UP = Fr(14143, 10000)                      # > sqrt(2)
+MARKOV_P = 64                                    # moment order of the far-set bound
+dec_up, dec_down, cdiv = TB.dec_up, TB.dec_down, TB.cdiv
+
+
+# ============================================================================= 1. interval-coefficient Taylor models
+class TV:
+    """f(r) = sum_{j <= N} c_j r^j + rho(r) for 0 <= r <= 2^-s, with FIXED (r-independent) coefficients
+    c_j in [(C_j - D_j)/2^P, (C_j + D_j)/2^P] and |rho(r)| <= (E/2^P) r^(N+1).  In particular f(0) = c_0.
+    Preconditions (the reuse boundary; reviews C56, C63): C and D are integer lists of length N + 1 with every D_j >= 0;
+    E is an integer >= 0; s >= 0 and N >= 0 are integers, with N >= 2 for the second-order readers (inc, absinc and
+    their callers read coeff(2)); the operands of +, -, * share (s, N); powers are nonnegative integers; reader radii
+    and scalars are exact rationals, with 0 < R <= 2^-s; the lists are not mutated after construction.  The certificate
+    meets all of them (s = TM_S, N = TM_N = 8); inputs outside them are not covered."""
+    __slots__ = ('C', 'D', 'E', 's', 'N')
+
+    def __init__(self, C, D, E, s, N):
+        self.C, self.D, self.E, self.s, self.N = C, D, E, s, N
+
+    @classmethod
+    def build(cls, centers, radii, e, s, N):
+        """centers, radii: {power: rational}; e: rational coefficient of r^(N+1) in the remainder bound."""
+        C, D = [0] * (N + 1), [0] * (N + 1)
+        R = Fr(1, 2 ** s)
+        E = Fr(e)
+        for p, x in centers.items():
+            x = Fr(x)
+            if x == 0:
+                continue
+            if p <= N:
+                v = round(x * (1 << TM_P))
+                C[p] += v
+                if x * (1 << TM_P) != v:
+                    D[p] += 1
+            else:
+                E += abs(x) * R ** (p - N - 1)
+        for p, x in radii.items():
+            x = Fr(x)
+            if p <= N:
+                D[p] += cdiv((x * (1 << TM_P)).numerator, (x * (1 << TM_P)).denominator)
+            else:
+                E += x * R ** (p - N - 1)
+        return cls(C, D, cdiv(E.numerator << TM_P, E.denominator), s, N)
+
+    @classmethod
+    def const(cls, x, s, N, rad=0):
+        return cls.build({0: Fr(x)}, {0: Fr(rad)} if rad else {}, 0, s, N)
+
+    def zero(self):
+        return TV([0] * (self.N + 1), [0] * (self.N + 1), 0, self.s, self.N)
+
+    def polybound(self):
+        """Upper bound (units 2^-P) of |sum c_j r^j| on [0, 2^-s]."""
+        return sum(cdiv(abs(c) + d, 1 << (self.s * j)) for j, (c, d) in enumerate(zip(self.C, self.D)) if c or d)
+
+    def __add__(self, o):
+        if not isinstance(o, TV):
+            o = TV.const(o, self.s, self.N)
+        return TV([a + b for a, b in zip(self.C, o.C)], [a + b for a, b in zip(self.D, o.D)], self.E + o.E, self.s, self.N)
+    __radd__ = __add__
+
+    def __neg__(self):
+        return TV([-a for a in self.C], list(self.D), self.E, self.s, self.N)
+
+    def __sub__(self, o):
+        if not isinstance(o, TV):
+            o = TV.const(o, self.s, self.N)
+        return self + (-o)
+
+    def scale(self, x):
+        x = Fr(x)
+        C, D = [], []
+        for c, d in zip(self.C, self.D):
+            v = c * x
+            w = round(v)
+            C.append(w)
+            D.append(cdiv((d * abs(x)).numerator, (d * abs(x)).denominator) + (1 if v != w else 0))
+        e = self.E * abs(x)
+        return TV(C, D, cdiv(e.numerator, e.denominator), self.s, self.N)
+
+    def __mul__(self, o):
+        if not isinstance(o, TV):
+            return self.scale(o)
+        N, s = self.N, self.s
+        S = [0] * (2 * N + 1)
+        Rd = [0] * (2 * N + 1)
+        for i, (a, da) in enumerate(zip(self.C, self.D)):
+            if a or da:
+                for j, (b, db) in enumerate(zip(o.C, o.D)):
+                    if b or db:
+                        S[i + j] += a * b
+                        Rd[i + j] += abs(a) * db + da * abs(b) + da * db
+        C, D, E = [], [], 0
+        for k in range(N + 1):
+            v = S[k] >> TM_P
+            C.append(v)
+            D.append(cdiv(Rd[k], 1 << TM_P) + (1 if S[k] - (v << TM_P) else 0))
+        for k in range(N + 1, 2 * N + 1):
+            if S[k] or Rd[k]:
+                E += cdiv(abs(S[k]) + Rd[k], 1 << (TM_P + s * (k - N - 1)))
+        E += cdiv(self.E * o.polybound() + o.E * self.polybound(), 1 << TM_P)
+        E += cdiv(self.E * o.E, 1 << (TM_P + s * (N + 1)))
+        return TV(C, D, E, s, N)
+    __rmul__ = __mul__
+
+    def c0(self):
+        return Fr(self.C[0] - self.D[0], 1 << TM_P), Fr(self.C[0] + self.D[0], 1 << TM_P)
+
+    def lin_bound(self):
+        """G with |f(r) - f(0)| <= G r on [0, 2^-s] (exact rational)."""
+        R = Fr(1, 2 ** self.s)
+        g = sum(Fr(abs(c) + d, 1 << TM_P) * R ** (j - 1) for j, (c, d) in enumerate(zip(self.C, self.D)) if j >= 1)
+        return g + Fr(self.E, 1 << TM_P) * R ** self.N
+
+    def inv(self):
+        lo, hi = self.c0()
+        require(lo > 0 or hi < 0, 'TV inverse at a constant term containing 0')
+        # centered at 1/C_0 (so that centers stay the reference values), radius covering [1/hi, 1/lo]
+        uc = Fr(1 << TM_P, self.C[0])
+        u = TV.build({0: uc}, {0: max(abs(1 / hi - uc), abs(1 / lo - uc))}, 0, self.s, self.N)
+        h = TV([0] + self.C[1:], [0] + self.D[1:], self.E, self.s, self.N)
+        g = h * u
+        require(g.C[0] == 0 and g.D[0] == 0, 'TV inverse: g(0) = 0')
+        G1 = g.lin_bound()
+        R = Fr(1, 2 ** self.s)
+        require(G1 * R < Fr(1, 2), 'TV inverse: |g| too large')
+        acc = TV.const(1, self.s, self.N)
+        p = TV.const(1, self.s, self.N)
+        mg = -g
+        for _ in range(self.N):
+            p = p * mg
+            acc = acc + p
+        tail = G1 ** (self.N + 1) / (1 - G1 * R)
+        acc = TV(acc.C, acc.D, acc.E + cdiv(tail.numerator << TM_P, tail.denominator), self.s, self.N)
+        return acc * u
+
+    # ----------------------------------------------------------------------------- readers
+    def coeff(self, j):
+        return Fr(self.C[j] - self.D[j], 1 << TM_P), Fr(self.C[j] + self.D[j], 1 << TM_P)
+
+    def high(self, R):
+        """H with |sum_{j >= 3} c_j r^j + rho(r)| <= H r^2 on (0, R]."""
+        h = sum(Fr(abs(c) + d, 1 << TM_P) * R ** (j - 2) for j, (c, d) in enumerate(zip(self.C, self.D)) if j >= 3)
+        return h + Fr(self.E, 1 << TM_P) * R ** (self.N - 1)
+
+    def inc(self, R):
+        """(l_lo, l_hi, q_lo, q_hi): f(r) - f(0) in [l_lo r + q_lo r^2, l_hi r + q_hi r^2] for 0 < r <= R."""
+        l_lo, l_hi = self.coeff(1)
+        q_lo, q_hi = self.coeff(2)
+        H = self.high(R)
+        if MUT == 'no-lin':
+            l_lo = l_hi = Fr(0)
+        return l_lo, l_hi, q_lo - H, q_hi + H
+
+    def absinc(self, R):
+        """(a, b): |f(r) - f(0)| <= a r^2 + b r."""
+        l_lo, l_hi, q_lo, q_hi = self.inc(R)
+        return max(abs(q_lo), abs(q_hi)), max(abs(l_lo), abs(l_hi))
+
+    def absval(self, R):
+        """(s, c): |f(r)| <= c + s r on [0, R], c = sup |c_0|."""
+        lo, hi = self.c0()
+        s = sum(Fr(abs(c) + d, 1 << TM_P) * R ** (j - 1) for j, (c, d) in enumerate(zip(self.C, self.D)) if j >= 1)
+        return s + Fr(self.E, 1 << TM_P) * R ** self.N, max(abs(lo), abs(hi))
+
+    def enclose(self, R):
+        """Interval of f(r) over r in [0, R]."""
+        lo, hi = self.c0()
+        for j in range(1, self.N + 1):
+            a, b = self.coeff(j)
+            Rj = R ** j
+            lo += min(a, 0) * Rj
+            hi += max(b, 0) * Rj
+        e = Fr(self.E, 1 << TM_P) * R ** (self.N + 1)
+        return lo - e, hi + e
+
+    def center(self, j):
+        return Fr(self.C[j], 1 << TM_P)
+
+
+def cov_tv(M, F, G, key, s, N, M1):
+    """TV on [0, 2^-s] of the covariance of F, G under the unnormalized torus kernel: the exact reference series (centers),
+    the Taylor truncation tail (gx > NX, proportional to r^(NX+1-w)) and the torus part sum_p tau_p r^p with
+    |tau_p| <= T0 e^p (radii T0 e^p for p <= N, remainder T0 e^(N+1)/(1 - e R) r^(N+1))."""
+    R = Fr(1, 2 ** s)
+    ser, w, absterms = M.pair_series(F, G, key)
+    A = max((t[1] for t in absterms), default=0)
+    tail = T0 = Fr(0)
+    for t in absterms:
+        acc, aa, gys = t[0], t[1], t[2:]
+        hprod = fprod = 1
+        for g in gys:
+            hprod *= M.hcoef(g)
+            fprod *= math.factorial(g)
+        tail += acc * abs(hprod)
+        T0 += acc * fprod * Fr(max(w, 1)) ** aa
+    require(M.NX + 1 - w >= N + 1, 'truncation tail below the Taylor-model order')
+    tail_e = 2 * Fr(M.NX + 1) ** A * tail * R ** (M.NX + 1 - w - (N + 1))
+    T0 = T0 * M1
+    eu = Fr(1) if MUT == 'exp-radius' else E_UP
+    if MUT == 'torus-const-only':
+        radii = {0: T0}
+        tor_e = Fr(0)
+    else:
+        radii = {p: T0 * eu ** p for p in range(N + 1)}
+        tor_e = T0 * eu ** (N + 1) / (1 - eu * R)
+    return TV.build(ser, radii, tail_e + tor_e, s, N)
+
+
+def solve_tv(A, B):
+    n = len(A)
+    Mx = [[A[i][j] for j in range(n)] + [B[c][i] for c in range(len(B))] for i in range(n)]
+    det, invs = None, []
+    for col in range(n):
+        piv = Mx[col][col]
+        det = piv if det is None else det * piv
+        ip = piv.inv()
+        invs.append(ip)
+        for i in range(col + 1, n):
+            f = Mx[i][col] * ip
+            for j in range(col, len(Mx[i])):
+                Mx[i][j] = Mx[i][j] - f * Mx[col][j]
+    X = []
+    for c in range(len(B)):
+        x = [None] * n
+        for i in reversed(range(n)):
+            acc = Mx[i][n + c]
+            for j in range(i + 1, n):
+                acc = acc - Mx[i][j] * x[j]
+            x[i] = acc * invs[i]
+        X.append(x)
+    return X, det
+
+
+class Law:
+    pass
+
+
+def tv_law(d, torus=True, s=TM_S, N=TM_N, L=10):
+    """The TV law over [0, 2^-s] of the targets TB.TM_NAMES[d]: as TB.tm_law, with interval-coefficient models."""
+    M = ENG[d]
+    names = TB.TM_NAMES[d]
+    pins, targets = M.functionals()
+    P = [pins[nm] for nm in M.PIN_NAMES]
+    idx = [M.TARGET_NAMES.index(nm) for nm in names]
+    T = [targets[nm] for nm in names]
+    M1 = M.torus_M1(L) if torus else Fr(0)
+    n, nt = len(P), len(T)
+    Spp = [[None] * n for _ in range(n)]
+    for i in range(n):
+        for j in range(i, n):
+            Spp[i][j] = Spp[j][i] = cov_tv(M, P[i], P[j], ('p', i, j), s, N, M1)
+    Stp = [[cov_tv(M, T[a], P[j], ('tp', idx[a], j), s, N, M1) for j in range(n)] for a in range(nt)]
+    Stt = [[None] * nt for _ in range(nt)]
+    for a in range(nt):
+        for b in range(a, nt):
+            i, j = idx[a], idx[b]
+            if i <= j:
+                Stt[a][b] = Stt[b][a] = cov_tv(M, T[a], T[b], ('tt', i, j), s, N, M1)
+            else:
+                Stt[a][b] = Stt[b][a] = cov_tv(M, T[b], T[a], ('tt', j, i), s, N, M1)
+    zero = TV.const(0, s, N)
+    kvec = [TV.build({3: Fr(-1, 2)}, {}, 0, s, N), TV.build({2: Fr(-1)}, {}, 0, s, N), TV.const(12, s, N)] + [zero] * (n - 3)
+    e1 = [TV.const(1, s, N)] + [zero] * (n - 1)
+    X, det = solve_tv(Spp, Stp + [e1, kvec])
+    W, xs0, xs1 = X[:nt], X[nt], X[nt + 1]
+    law = Law()
+    law.d, law.M1, law.s, law.N = d, M1, s, N
+    law.th1 = M.theta_minus_one(L) if torus else Fr(0)
+    law.alpha = {nm: W[a][0] for a, nm in enumerate(names)}
+    law.beta = {nm: sum((W[a][j] * kvec[j] for j in range(n)), zero) for a, nm in enumerate(names)}
+    law.Cu = {}
+    for a, na in enumerate(names):
+        for b, nb in enumerate(names):
+            law.Cu[(na, nb)] = Stt[a][b] - sum((W[a][l] * Stp[b][l] for l in range(n)), zero)
+    law.e_bb = xs0[0]
+    law.e_bk = sum((kvec[j] * xs0[j] for j in range(n)), zero)
+    law.e_kk = sum((kvec[j] * xs1[j] for j in range(n)), zero)
+    law.det = det
+    if d == 3:
+        an = ('a11', 'a22', 'a12')
+        Xg, _ = solve_tv([[law.Cu[(x, y)] for y in an] for x in an], [[law.Cu[(o, y)] for y in an] for o in ('o11', 'o22', 'o12')])
+        law.G = {o: [-Xg[c][l] for l in range(3)] for c, o in enumerate(('o11', 'o22', 'o12'))}
+    return law
+
+
+def tv_summary(law):
+    """Low-order centers and radii (decimal, 30 places) and remainder coefficients of the TV law, for the record."""
+    def row(t):
+        out = {'c': [], 'rad': []}
+        for j in range(4):
+            x = t.center(j)
+            out['c'].append(dec_down(x, 30) if x >= 0 else dec_up(x, 30))
+            out['rad'].append(dec_up(Fr(t.D[j], 1 << TM_P), 30))
+        out['rem_r9'] = dec_up(Fr(t.E, 1 << TM_P), 30)
+        return out
+    out = {'det': row(law.det), 'e_bb': row(law.e_bb), 'e_bk': row(law.e_bk), 'e_kk': row(law.e_kk)}
+    for nm in law.alpha:
+        out['alpha_' + nm] = row(law.alpha[nm])
+        out['beta_' + nm] = row(law.beta[nm])
+    names = TB.TM_NAMES[law.d]
+    for i, a in enumerate(names):
+        for b in names[i:]:
+            out['C_%s_%s' % (a, b)] = row(law.Cu[(a, b)])
+    return out
+
+
+# ============================================================================= 2. (a, b) forms: a r^2 + b r
+def ab_add(*xs):
+    return (sum(x[0] for x in xs), sum(x[1] for x in xs))
+
+
+def ab_mul(x, c):
+    return (x[0] * c, x[1] * c)
+
+
+def ab_at(x, R):
+    return x[0] * R * R + x[1] * R
+
+
+def ab_norm(xs):
+    """Minkowski: || (a_i r^2 + b_i r)_i ||_2 <= sqrt(sum a_i^2) r^2 + sqrt(sum b_i^2) r."""
+    return (E3.sqrt_up(sum(x[0] * x[0] for x in xs)), E3.sqrt_up(sum(x[1] * x[1] for x in xs)))
+
+
+def exp_up_ab(M, x, R):
+    """exp(a r^2 + b r) - 1 <= (a r^2 + b r) exp(a R^2 + b R) on (0, R]."""
+    f = TB.exp_minus(M, -ab_at(x, R)).hi
+    return (x[0] * f, x[1] * f)
+
+
+def flat(t, ref, what):
+    """The reference part (centers) of t is the constant ref for every r (Lemma 1 of Math-#215)."""
+    require(abs(t.center(0) - ref) <= TINY and all(abs(t.center(j)) <= TINY for j in range(1, t.N + 1)),
+            'the reference law of %s is not flat in r' % what)
+
+
+def mean_inc(law, nm, b1, k1, R):
+    """|mu(r) - mu(0)| for mu = b alpha + k beta (b <= b1, k <= k1), in (a, b) form."""
+    return ab_add(ab_mul(law.alpha[nm].absinc(R), b1), ab_mul(law.beta[nm].absinc(R), k1))
+
+
+def pi_ratio_v(law, Q, R):
+    """log(pi_r / pi_0) in [-dn, up] for the SAME kernel: pi_r ~ det(S_r/Theta)^(-1/2) exp(-Theta E_r/2), so
+    log(pi_r/pi_0) = -(1/2) log(det S_r/det S_0) - Theta (E_r - E_0)/2; Theta in [1, 1 + th1] multiplies increments only."""
+    b0, b1, k0, k1 = Q
+    th1 = law.th1
+    l_lo, l_hi, q_lo, q_hi = law.det.inc(R)
+    d0lo = law.det.c0()[0]
+    require(d0lo > 0, 'pin determinant at r = 0')
+    xm = (max(abs(q_lo), abs(q_hi)) * R * R + max(abs(l_lo), abs(l_hi)) * R) / d0lo
+    require(xm < Fr(1, 2), 'pin determinant increment')
+    up = (max(Fr(0), -q_lo) / (2 * d0lo * (1 - xm)), max(Fr(0), -l_lo) / (2 * d0lo * (1 - xm)))
+    dn = (max(Fr(0), q_hi) / (2 * d0lo), max(Fr(0), l_hi) / (2 * d0lo))
+    parts = ((law.e_bb, (b0 * b0, b1 * b1)), (law.e_bk, (2 * b0 * k0, 2 * b1 * k1)), (law.e_kk, (k0 * k0, k1 * k1)))
+    EL = EH = EQ = EP = Fr(0)
+    for t, (wlo, whi) in parts:
+        a_lo, a_hi, c_lo, c_hi = t.inc(R)
+        EL += min(wlo * a_lo, whi * a_lo)
+        EH += max(wlo * a_hi, whi * a_hi)
+        EQ += min(wlo * c_lo, whi * c_lo)
+        EP += max(wlo * c_hi, whi * c_hi)
+    f = (1 + th1) / 2
+    up = ab_add(up, (f * max(Fr(0), -EQ), f * max(Fr(0), -EL)))
+    dn = ab_add(dn, (f * max(Fr(0), EP), f * max(Fr(0), EH)))
+    return up, dn
+
+
+def combine_v(M, p_up, p_dn, z_up, z_dn, R):
+    """A_r/A_0 = (pi_r/pi_0) (z_r/z_0): (1 + pu)(1 + zu) - 1 <= pu + zu + pu zu(R); 1 - (1 - pd)(1 - zd) <= pd + zd.
+    The lower sum is valid even when one lower estimate exceeds 1 (review C58): if pd + zd >= 1 it holds because both ratios
+    are positive; otherwise 0 <= 1 - pd, 1 - zd and the two lower factors may be multiplied."""
+    pu = exp_up_ab(M, p_up, R)
+    zR = ab_at(z_up, R)
+    up = (pu[0] + z_up[0] + pu[0] * zR, pu[1] + z_up[1] + pu[1] * zR)
+    return up, ab_add(p_dn, z_dn)
+
+
+def far_set(M, ma_T, sd_T, ma_tau, sd_tau, k0, R):
+    """q4 with P(G_0^c) <= q4 r^4 on (0, R]: G_0^c = {|T| > 3k/(2r)} u {|tau| > 3k/(2r^2)}, Markov at order p = MARKOV_P:
+    P(|X| > t) <= (||X||_p / t)^p, and r^p <= R^(p-4) r^4, r^(2p) <= R^(2p-4) r^4."""
+    p = MARKOV_P
+    nT = M.gauss_norm(ma_T, sd_T, p)
+    nt = M.gauss_norm(ma_tau, sd_tau, p)
+    c = Fr(2) / (3 * k0)
+    q4 = (nT * c) ** p * R ** (p - 4) + (nt * c) ** p * R ** (2 * p - 4)
+    return q4
+
+
+# ============================================================================= 3. the ratio A_r / A_0 on a (b, k) box
+def bounds_d2_v(Q, R, law, il):
+    """(up, dn) in (a, b) form: A_r/A_0 - 1 <= up(r), 1 - A_r/A_0 <= dn(r) on the box, 0 < r <= R (NOTE sections 3-4)."""
+    M = E2
+    b0, b1, k0, k1 = Q
+    th1 = law.th1
+    flat(law.alpha['w'], -1, 'alpha_w')
+    flat(law.beta['w'], 0, 'beta_w')
+    flat(law.Cu[('w', 'w')], 2, 'Var w')
+    # law of lambda = -w at r and at 0: mean mu = -(b alpha_w + k beta_w), variance Cu_ww / Theta
+    sA, cA = (law.alpha['w'] + 1).absval(R)
+    sB, cB = law.beta['w'].absval(R)
+    Dm = b1 * (cA + sA * R) + k1 * (cB + sB * R)                      # sup |mu - b| over [0, R]
+    sC, cC = (law.Cu[('w', 'w')] - 2).absval(R)
+    Sm = cC + sC * R
+    s_lo = M.isqrt_iv(M.IV((2 - Sm) / (1 + th1))).lo
+    s_hi = M.isqrt_iv(M.IV(2 + Sm)).hi
+    m_lo, m_hi = b0 - Dm, b1 + Dm
+    mmax = max(abs(m_lo), abs(m_hi))
+    Fmin = M.elam2_pos(m_lo, s_lo).lo                               # <= F_r and <= F_0
+    Fmax = M.elam2_pos(m_hi, s_hi).hi
+    F1max = M.elam_pos(m_hi, s_hi).hi
+    # main term: |F(mu_r, s_r) - F(mu_0, s_0)| <= 2 E[lambda_+] |dmu| + Phi(mu/s) |d(s^2)| along the segment (both <= sup)
+    dmu = mean_inc(law, 'w', b1, k1, R)
+    dS = law.Cu[('w', 'w')].absinc(R)                               # |d(s^2)| <= |dCu| / Theta, Theta >= 1
+    Fae = ab_mul(ab_add(ab_mul(dmu, 2 * F1max), dS), 1 / Fmin)
+    norm = 36 * k0 * k0 * Fmin                                      # 36 k^2 F_0 >= norm
+    b, k = M.IV(b0, b1), M.IV(k0, k1)
+    ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES}
+    gn = lambda nm, p: M.gauss_norm(ma[nm], il.sdQ[nm].hi, p)
+    lam = lambda p: M.gauss_norm(mmax, s_hi, p)
+    T, tau, om, v, nu = (lambda p, x=x: gn(x, p) for x in ('T', 'tau', 'om', 'v', 'nu'))
+    kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p)
+    # first order: r E[lambda_+ om] = r [mu_om F_1 - bw_om (F - m F_1)], |mu_om| <= c + s r, |bw_om| <= |Cu_om,w| / Cu_ww
+    sa_, ca_ = law.alpha['om'].absval(R)
+    sb_, cb_ = law.beta['om'].absval(R)
+    sc_, cc_ = law.Cu[('om', 'w')].absval(R)
+    cwlo = (2 - Sm) / (1 + th1)                                    # Var w = Cu_ww / Theta >= cwlo
+    # the regression coefficient Cu_om,w / Cu_ww is free of Theta (it cancels exactly); dividing |Cu_om,w| by the
+    # normalized floor cwlo is deliberately conservative (review C58)
+    # relative to 36 k^2 r^2 F_0: the 36 k^2 cancels, so the normalizer is F_0 >= Fmin
+    T2 = (((b1 * sa_ + k1 * sb_) * F1max + sc_ / cwlo * (Fmax + mmax * F1max)) / Fmin,
+          ((b1 * ca_ + k1 * cb_) * F1max + cc_ / cwlo * (Fmax + mmax * F1max)) / Fmin)
+    if MUT == 'no-om-term':
+        T2 = (Fr(0), Fr(0))
+    # second order (explicit r^2), as in Math-#215
+    mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(Fmin, Fmax)
+    side = 6 * k1 * M.absup(il.bw['tau']) * s_hi * lam(4) ** 2 + R * T(4) * tau(4) * lam(4) ** 2
+    T3_up = max(Fr(0), mt.hi + side) / norm
+    T3_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * lam(4) ** 2) / norm
+    T4 = R * kap(4) * lam(4) * om(2)
+    xi2 = 6 * k1 * (R * nu(4) ** 2 + 2 * nu(4) * v(4)) + T(4) * (R * R * nu(8) ** 2 + 2 * R * nu(8) * v(8) + 2 * v(8) ** 2) \
+        + R * tau(4) * v(8) ** 2
+    T5 = lam(2) * xi2
+    T6 = v(8) ** 2 * (6 * k1 + R * T(4) + R * R * tau(4)) * om(4) + v(4) ** 2 * (R * nu(4) + v(4)) ** 2
+    Yn = ('v', 'om', 'nu')
+    prec = 1 / cwlo + TB.wq_bound(M, il.Cg, Yn, [[M.absup(il.bw[nm])] for nm in Yn])
+    pbar = Fr(2, 5) * M.sqrt_up(prec)
+    h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + om(p)
+    bad = R * pbar * h(2) * (Fr(15, 2) * k1 * h(4) + v(8) ** 2) * (9 * k1 * (h(4) + om(4)) + (R * nu(8) + v(8)) ** 2)
+    # G_0^c: E[W 1{G_0^c}]/r^2 <= P1n P2n P(G_0^c)^(1/2) <= P1n P2n sqrt(q4) r^2
+    q4 = far_set(M, ma['T'], il.sdQ['T'].hi, ma['tau'], il.sdQ['tau'].hi, k0, R)
+    P1n = (6 * k1 + R * T(8)) * lam(8) + R * v(8) ** 2
+    P2n = (6 * k1 + R * T(8) + R * R * tau(8)) * (lam(8) + R * om(8)) + R * (R * nu(8) + v(8)) ** 2
+    g0c = P1n * P2n * M.sqrt_up(q4)
+    X = ab_add(T2, ((T4 + T5 + T6 + bad + g0c) / norm, Fr(0)))
+    p_up, p_dn = pi_ratio_v(law, Q, R)
+    z_up = ab_add(Fae, X, (T3_up, Fr(0)))
+    z_dn = ab_add(Fae, X, (T3_dn, Fr(0)))
+    return combine_v(M, p_up, p_dn, z_up, z_dn, R)
+
+
+def bounds_d3_v(Q, R, law, il):
+    """(up, dn) in (a, b) form in d = 3 (NOTE sections 3-4)."""
+    M = E3
+    b0, b1, k0, k1 = Q
+    th1 = law.th1
+    names = ('a11', 'a22', 'a12')
+    for nm, ref in (('a11', -1), ('a22', -1), ('a12', 0)):
+        flat(law.alpha[nm], ref, 'alpha_' + nm)
+        flat(law.beta[nm], 0, 'beta_' + nm)
+    for (x, y), ref in ((('a11', 'a11'), 2), (('a22', 'a22'), 2), (('a12', 'a12'), 1), (('a11', 'a22'), 0),
+                        (('a11', 'a12'), 0), (('a22', 'a12'), 0)):
+        flat(law.Cu[(x, y)], ref, 'Cov(%s, %s)' % (x, y))
+    Fw = (Fr(1), Fr(1), M.SQRT2)
+    # Loewner coupling with the contact law over the whole band [0, R] (as in Math-#215): moment bounds and m_0 >= m0lo
+    comps = []
+    for nm, ref, f in (('a11', -1, 1), ('a22', -1, 1), ('a12', 0, 2)):
+        sA, cA = (law.alpha[nm] - ref).absval(R)
+        sB, cB = law.beta[nm].absval(R)
+        comps.append((b1 * (cA + sA * R) + k1 * (cB + sB * R)) * f)
+    eps = M.sqrt_up(sum(x * x for x in comps))
+    Cb = []
+    for i in range(3):
+        row = []
+        for j in range(3):
+            lo, hi = law.Cu[(names[i], names[j])].enclose(R)
+            row.append(M.IV(lo, hi) * Fw[i] * Fw[j] * M.IV(1 / (1 + th1), 1))
+        Cb.append(row)
+    Lb = M.cholesky(Cb)
+    eta = M.sqrt_up(sum(M.absup(Lb[i][j] - (M.SQRT2 if i == j else 0)) ** 2 for i in range(3) for j in range(i + 1)))
+    Dl = M.rup_rel(eps + eta * TB.R0C)
+    cm = M.contact_moment
+    bA, bB = TB.grid_dn(b0 - Dl), TB.grid_up(b1 + Dl)
+    m3b0 = cm(2, 0, b0).lo
+    L3up = 2 * cm(1, 1, bB).hi / m3b0
+    L3dn = 2 * cm(1, 1, b1).hi / cm(2, 0, bA).lo
+    pR0 = M.chi_tail(3, TB.R0C)
+    muB = M.rup_rel(b1 * M.sqrt_up(Fr(2)) + eps)
+    sig = M.sqrt_up(max(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2]))
+                    + 2 * max(M.absup(Cb[i][j]) for i in range(3) for j in range(3) if i != j))
+    U = M.chi_upper_moments(3, TB.R0C, 4)
+    out_D2 = sum((U[i] * (M.binom(4, i) * muB ** (4 - i) * sig ** i) for i in range(5)), M.IV(0)).hi / 4
+    tail_c = M.sqrt_up(cm(4, 0, b1).hi) * M.sqrt_up(pR0)
+    nBF16 = M.rup_rel(muB + sig * M.chi_norm(3, 16))
+    out_ball = M.sqrt_up(pR0)
+
+    def up_moment(a, c):
+        return M.rup_rel(cm(a, c, bB).hi + 2 ** c * M.rup_rel(nBF16 ** (2 * a + c)) * out_ball)
+    ED2, ED4, ED2T2, ET4, DT = up_moment(2, 0), up_moment(4, 0), up_moment(2, 2), up_moment(0, 4), up_moment(1, 1)
+    trCb = sum(x.hi for x in (Cb[0][0], Cb[1][1], Cb[2][2]))
+    BF = lambda p: M.frob_norm(muB, trCb, p)
+    rho_lo = TB.exp_minus(M, Dl * L3dn).lo - tail_c / m3b0               # m3_r / m3(b) over [0, R], r = 0 included
+    rho_hi = TB.exp_minus(M, -(Dl * L3up)).hi + out_D2 / m3b0
+    require(rho_lo > 0, 'contact moment floor')
+    m0lo = m3b0 * rho_lo                                                 # m_0 = E_0[D^2 1{B > 0}] >= m0lo
+    rel = 1 / rho_lo                                                     # bounds relative to m3(b) -> relative to m_0
+    # main term: gradient coupling B_r = mu_r + C_r^(1/2) Z, B_0 = mu_0 + C_0^(1/2) Z (natural coordinates),
+    # |g(B_r) - g(B_0)| <= max(||B_r||, ||B_0||)^3 ||B_r - B_0||, g = det(B)^2 1{B > 0};
+    # ||C_r^(1/2) - C_0^(1/2)||_F <= ||C_r - C_0||_F / (2 sqrt(lmB)) (Sylvester equation in the eigenbases)
+    Fu = (Fr(1), Fr(1), SQRT2_UP)                                       # rational upper bounds of the weights Fw
+    dmuB = ab_norm([ab_mul(mean_inc(law, nm, b1, k1, R), f) for nm, f in zip(names, Fu)])
+    dC = ab_norm([ab_mul(law.Cu[(names[i], names[j])].absinc(R), Fu[i] * Fu[j]) for i in range(3) for j in range(3)])
+    lmB = TB.gersh_min(Cb)
+    require(lmB > 0, 'Gershgorin floor of the transverse covariance')
+    dS = ab_mul(dC, 1 / (2 * M.sqrt_down(lmB)))
+    K3 = M.sqrt_up(Fr(2)) * BF(8) ** 3 / m0lo
+    if MUT == 'coupling-half':
+        K3 = K3 / 2
+    main = ab_mul(ab_add(dmuB, dS), K3)
+    # secondary law (engine interval law over [0, R])
+    b, k = M.IV(b0, b1), M.IV(k0, k1)
+    ma = {nm: M.absup(b * il.alpha[nm] + k * il.beta[nm]) for nm in M.TARGET_NAMES}
+    T = lambda p: M.gauss_norm(ma['T'], il.sdQ['T'], p)
+    tau = lambda p: M.gauss_norm(ma['tau'], il.sdQ['tau'], p)
+    mu_o = M.sqrt_up(ma['o11'] ** 2 + ma['o22'] ** 2 + 2 * ma['o12'] ** 2)
+    trO = sum(m * il.C[(nm, nm)].hi for nm, m in zip(M.ONAMES, (1, 1, 2)))
+    Om = lambda p: M.frob_norm(mu_o, trO, p)
+    mu_v = M.sqrt_up(ma['v1'] ** 2 + ma['v2'] ** 2)
+    v = lambda p: M.frob_norm(mu_v, il.C[('v1', 'v1')].hi + il.C[('v2', 'v2')].hi, p)
+    mu_nu = M.sqrt_up(ma['nu1'] ** 2 + ma['nu2'] ** 2)
+    nu = lambda p: M.frob_norm(mu_nu, il.C[('nu1', 'nu1')].hi + il.C[('nu2', 'nu2')].hi, p)
+    e = lambda p: R * nu(p) + v(p)
+    kap = lambda p: 6 * k1 * tau(p) + T(2 * p) ** 2 + R * T(2 * p) * tau(2 * p)
+    c1 = lambda p: 6 * k1 + R * T(p)
+    c2 = lambda p: 6 * k1 + R * T(p) + R * R * tau(p)
+    norm36 = 36 * k0 * k0 * m3b0
+    # mean of Omega': ||mu_Omega'(r)||_F <= c_mu + s_mu r
+    s_mu2 = c_mu2 = Fr(0)
+    for nm, mult in zip(M.ONAMES, (1, 1, 2)):
+        sa_, ca_ = law.alpha[nm].absval(R)
+        sb_, cb_ = law.beta[nm].absval(R)
+        s_mu2 += mult * (b1 * sa_ + k1 * sb_) ** 2
+        c_mu2 += mult * (b1 * ca_ + k1 * cb_) ** 2
+    s_mu, c_mu = M.sqrt_up(s_mu2), M.sqrt_up(c_mu2)
+    muO = s_mu * R + c_mu
+    # regression of Omega' on B: E[Omega' - mu | B] = beta (B - mu_B) + DY
+    G = law.G
+    beta_t = G['o11'][0]
+    for (o, l) in (('o22', 1), ('o12', 2)):
+        flat(G[o][l] - beta_t, 0, 'regression of %s on B' % o)
+    for o, ls in (('o11', (1, 2)), ('o22', (0, 2)), ('o12', (0, 1))):
+        for l in ls:
+            flat(G[o][l], 0, 'off-diagonal regression of %s on B' % o)
+    require(abs(beta_t.center(0)) <= TINY, 'regression of Omega on B at r = 0 (reference part)')
+    dGs2 = dGc2 = Fr(0)
+    for o, mult in zip(('o11', 'o22', 'o12'), (1, 1, 2)):
+        for l in range(3):
+            t = G[o][l] - beta_t if (o, l) in (('o11', 0), ('o22', 1), ('o12', 2)) else G[o][l]
+            s_, c_ = t.absval(R)
+            dGs2 += mult * s_ ** 2
+            dGc2 += mult * c_ ** 2
+    dGs, dGc = M.sqrt_up(dGs2), M.sqrt_up(dGc2)
+    b1c = beta_t.center(1)                                               # = 1/2
+    require(b1c > 0, 'sign of the first-order regression coefficient of Omega on B')
+    # r beta(r) - b1c r^2 = c0 r + (c1 - b1c) r^2 + ...: |.| <= e_beta r + sb_beta r^2
+    b0lo, b0hi = beta_t.c0()
+    e_beta = max(abs(b0lo), abs(b0hi))
+    c1lo, c1hi = beta_t.coeff(1)
+    sb_beta = max(abs(c1lo - b1c), abs(c1hi - b1c)) + (max(abs(x) for x in beta_t.coeff(2)) + beta_t.high(R)) * R
+    DTlo = cm(1, 1, bA).lo - M.sqrt_up(cm(2, 2, b1).hi) * M.sqrt_up(pR0)
+    m3up = cm(2, 0, b1).hi
+    tau_lo = b0 * max(DTlo, Fr(0)) / m3up
+    tau_hi = b1 * DT / m3b0
+    X3lo = 2 * rho_lo - tau_hi - eps * DT / m3b0
+    X3hi = 2 * rho_hi - tau_lo + eps * DT / m3b0
+    X3m = max(abs(X3lo), abs(X3hi))
+    # r |R1|/m3(b) <= r (c_mu + s_mu r) DT/m3 + r (dGc + dGs r) sqrt(E[D^2 tr^2 1]) sqrt(tr Cb)/m3
+    R1a = (s_mu * DT + dGs * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0
+    R1b = (c_mu * DT + dGc * M.sqrt_up(ED2T2) * M.sqrt_up(trCb)) / m3b0
+    U3_up = (max(Fr(0), -b1c * X3lo) + sb_beta * X3m + R1a, e_beta * X3m + R1b)
+    U3_dn = (max(Fr(0), b1c * X3hi) + sb_beta * X3m + R1a, e_beta * X3m + R1b)
+    if MUT == 'no-om-term':
+        U3_up = U3_dn = (Fr(0), Fr(0))
+
+    def cov_Oz(o, z):
+        if z in names:
+            s_, c_ = law.Cu[(o, z)].absval(R)
+            return s_ * R + c_
+        return M.absup(il.C[(o, z)])
+
+    def explained(zs):
+        # with D the lower diagonal endpoints, C_zz = D^(1/2)(I + H + E)D^(1/2): H >= 0 diagonal (surplus), E zero-diagonal
+        # with ||E||_F <= Ef, so C_zz >= (1 - Ef) D; the surplus H only helps (review C58)
+        rows = [[cov_Oz(o, z) for z in zs] for o in M.ONAMES]
+        D = [il.C[(z, z)].lo for z in zs]
+        Ef2 = Fr(0)
+        for i in range(len(zs)):
+            for j in range(len(zs)):
+                if i != j:
+                    c = M.absup(il.C[(zs[i], zs[j])])
+                    Ef2 += c * c / (D[i] * D[j])
+        Ef = M.sqrt_up(Ef2)
+        require(Ef < Fr(1, 2), 'diagonal scaling (explained part of Omega)')
+        return sum(mult * sum(x * x / Dz for x, Dz in zip(row, D)) for row, mult in zip(rows, (1, 1, 2))) / (1 - Ef)
+    zA = names + tuple(M.VNAMES) + ('nu1', 'nu2')
+    tB, tK = explained(names), explained(zA)
+    YB4, YK4 = M.frob_norm(0, tB, 4), M.frob_norm(0, tK, 4)
+    detX = M.absup(il.Cg[('o11', 'o22')] - il.Cg[('o12', 'o12')])
+    E1 = up_moment(1, 0)
+    ED2lo = max(Fr(0), cm(2, 0, bA).lo - tail_c)
+    mt = M.IV(6 * k0, 6 * k1) * (b * il.alpha['tau'] + k * il.beta['tau']) * M.IV(ED2lo, ED2)
+    side = 6 * k1 * il.bnorm['tau'] * M.sqrt_up(trCb) * M.sqrt_up(ED4) + R * T(4) * tau(4) * M.sqrt_up(ED4)
+    U2_up = max(Fr(0), mt.hi + side) / norm36
+    U2_dn = max(Fr(0), -mt.lo + side + T(4) ** 2 * M.sqrt_up(ED4)) / norm36
+    U3r = R * kap(4) * M.sqrt_up(ED2T2) * Om(4)
+    U4rel = (detX * E1 + M.sqrt_up(ED2) * (muO + YB4) ** 2 / 2) / m3b0
+    U4 = R * R * kap(4) * M.sqrt_up(ED2) * Om(8) ** 2 / 2
+    U5 = M.sqrt_up(ED2T2) * ((2 * T(4) + R * tau(4)) * v(8) ** 2 + 2 * c1(4) * nu(8) * v(8) + R * c1(4) * nu(8) ** 2)
+    U67rel = (M.sqrt_up(ED2) * (muO * e(4) ** 2 + e(8) ** 2 * YK4) + M.sqrt_up(ET4) * (muO * v(4) ** 2 + v(8) ** 2 * YK4)) / (6 * k0 * m3b0)
+    U6 = R * T(8) * M.sqrt_up(ED2) * Om(8) * e(8) ** 2
+    U7 = R * (T(8) + R * tau(8)) * M.sqrt_up(ET4) * Om(8) * v(8) ** 2
+    U8 = R * c2(8) * Om(16) ** 2 / 2 * M.iroot(ET4, 4) * v(16) ** 2
+    U9 = M.sqrt_up(ET4) * v(8) ** 2 * e(8) ** 2
+    U10 = R * M.iroot(ET4, 4) * v(8) ** 2 * Om(4) * e(8) ** 2
+    Yn = tuple(M.VNAMES) + tuple(M.ONAMES) + ('nu1', 'nu2')
+    rows = []
+    for nm in Yn:
+        if nm in M.ONAMES:
+            cov = []
+            for an, f in zip(names, (1, 1, 2)):
+                s_, c_ = law.Cu[(nm, an)].absval(R)
+                cov.append((s_ * R + c_) * f)
+            rows.append([M.sqrt_up(sum(x * x for x in cov)) / lmB])
+        else:
+            rows.append([M.absup(x) for x in il.bt[nm]])
+    Wt = TB.wq_bound(M, il.Cg, Yn, rows)
+    c_hi = sig * sig
+    cl = 1 / (1 / lmB + Wt)
+    KBp = M.rup_rel((M.PI * M.SQRT2 / (2 * M.PI * M.SQRT2PI)).hi / (cl * M.sqrt_down(cl)))
+    m1 = muB + c_hi * M.sqrt_up(Wt) * M.chi_norm(7, 8)
+    Hs = TB.half_moments(M, c_hi)
+    Ja = m1 ** 4 / 4 + sum(M.binom(3, i) * m1 ** (3 - i) * Hs[i] for i in range(4))
+    Jb = m1 ** 3 / 3 + sum(M.binom(2, i) * m1 ** (2 - i) * Hs[i] for i in range(3))
+    h = lambda p: Fr(2) / (9 * k0) * v(2 * p) ** 2 + Om(p)
+    nhg = h(4) * (Fr(15, 2) * k1 * h(8) + v(16) ** 2) * (9 * k1 * (h(8) + Om(8)) + e(16) ** 2)
+    nhgO = h(8) * (Fr(15, 2) * k1 * h(16) + v(32) ** 2) * (9 * k1 * (h(16) + Om(16)) + e(32) ** 2) * Om(8)
+    bad = R * KBp * (Ja * nhg + R * Jb * nhgO)
+    q4 = far_set(M, ma['T'], il.sdQ['T'], ma['tau'], il.sdQ['tau'], k0, R)
+    P1n = c1(8) * BF(16) ** 2 + R * BF(8) * v(16) ** 2
+    P2n = c2(8) * (BF(16) + R * Om(16)) ** 2 + R * (BF(8) + R * Om(8)) * e(16) ** 2
+    g0c = P1n * P2n * M.sqrt_up(q4)
+    Xa = (U3r + U4 + U5 + U6 + U7 + U8 + U9 + U10 + bad + g0c) / norm36 + U4rel + U67rel
+    # everything above except `main` is relative to m3(b); m3(b)/m_0 <= 1/rho_lo
+    z_up = ab_add(main, ab_mul(ab_add(U3_up, (Xa, Fr(0)), (U2_up, Fr(0))), rel))
+    z_dn = ab_add(main, ab_mul(ab_add(U3_dn, (Xa, Fr(0)), (U2_dn, Fr(0))), rel))
+    p_up, p_dn = pi_ratio_v(law, Q, R)
+    return combine_v(M, p_up, p_dn, z_up, z_dn, R)
+
+
+def rate_sweep_v(d, law):
+    """For every R and every sub-box: (a_up, beta_up, a_dn, beta_dn), decimal strings rounded up."""
+    M = ENG[d]
+    fn = bounds_d2_v if d == 2 else bounds_d3_v
+    out = {}
+    for R in R_STARS:
+        il = M.band_law(Fr(0), R)
+        rows = []
+        for bb, kb in SUBBOXES:
+            up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, law, il)
+            rows.append([dec_up(up[0], 6), dec_up(up[1], 24), dec_up(dn[0], 6), dec_up(dn[1], 24)])
+        out[str(R)] = {'boxes': rows,
+                       'a_up': max((r[0] for r in rows), key=Fr), 'beta_up': max((r[1] for r in rows), key=Fr),
+                       'a_dn': max((r[2] for r in rows), key=Fr), 'beta_dn': max((r[3] for r in rows), key=Fr)}
+    return out
+
+
+# ============================================================================= 4. reference integrals and the corollary
+def k_weights(d):
+    """int_box A_0^ref/(3k) on every sub-box (the r^1 weight), as decimal upper bounds."""
+    M = ENG[d]
+    cells = TB.b_cells(d)
+    K = TB.K_const(d)
+    J11 = {kb: TB.k_integral(M, 1, 1, kb[0], kb[1]) for kb in SUB_K}
+    Ibs = {bb: TB.b_integral(M, cells, bb[0], bb[1]) for bb in set(bb for bb, _ in SUBBOXES)}
+    return [dec_up((K * Ibs[bb] * J11[kb]).hi, 24) for bb, kb in SUBBOXES]
+
+
+def assemble_v(d, rate, ref, w1, eps0):
+    """Corollary L'1: for 0 < ell < R^3/2,
+        ell^(1/3) nu_cand(ell) - c_{B,K}  in  [-(c2_dn ell^(2/3) + c1_dn ell^(1/3)), c2_up ell^(2/3) + c1_up ell^(1/3)],
+    c_{B,K} = int_{B x K x S^(d-1)} A_0(b, k, u)/(3 k^(2/3)) db dk dsigma(u) of the same kernel; the weights are
+    int_{box x S^(d-1)} A_0/(3k^(4/3)) and int_{box x S^(d-1)} A_0/(3k), bounded by (1 + eps0) times the reference
+    weights (eps0 = Math-#215's |A_0/A_0^ref - 1| bound).  The reference weights w_2, w_1 are full angular integrals
+    (theorem_b's K_const includes |S^(d-1)|); Theorem L' is uniform in u, and no isotropy is assumed."""
+    c_lo = Fr(ref['c_ref'][0])
+    w2 = [Fr(x) for x in ref['w_2']]
+    w1 = [Fr(x) for x in w1]
+    f = 1 + eps0
+    out = {}
+    for R in R_STARS:
+        rr = rate[str(R)]
+        c2_up = f * sum(Fr(row[0]) * w for row, w in zip(rr['boxes'], w2))
+        c2_dn = f * sum(Fr(row[2]) * w for row, w in zip(rr['boxes'], w2))
+        c1_up = f * sum(Fr(row[1]) * w for row, w in zip(rr['boxes'], w1))
+        c1_dn = f * sum(Fr(row[3]) * w for row, w in zip(rr['boxes'], w1))
+        cl = c_lo * (1 - eps0)
+        out[str(R)] = {'c2_up': dec_up(c2_up, 12), 'c2_dn': dec_up(c2_dn, 12),
+                       'c1_up': dec_up(c1_up, 30), 'c1_dn': dec_up(c1_dn, 30),
+                       'rel2_up': dec_up(c2_up / cl, 6), 'rel2_dn': dec_up(c2_dn / cl, 6),
+                       'rel1_up': dec_up(c1_up / cl, 30), 'rel1_dn': dec_up(c1_dn / cl, 30),
+                       'ell_max': '%s' % (R ** 3 / 2)}
+    return out
+
+
+def params():
+    return {'R_STARS': [str(x) for x in R_STARS], 'TV': {'s': TM_S, 'N': TM_N, 'P': TM_P}, 'E_UP': str(E_UP),
+            'MARKOV_P': MARKOV_P, 'R0_COUPLING': str(TB.R0C), 'L_min': 10,
+            'SUB_B': [[str(a), str(b)] for a, b in TB.SUB_B], 'SUB_K': [[str(a), str(b)] for a, b in SUB_K],
+            'band': {'b': ['0', '1'], 'k': ['1/2', '2']}}
+
+
+def eps0_215(d):
+    """Math-#215 Theorem L at r -> 0: |A_0/A_0^ref - 1| <= max(eps_up, eps_dn) over R (pinned RESULTS.json of #215,
+    recomputed here by TB.rate_sweep for R = 1/4096 and compared)."""
+    tml = TB.tm_law(d)
+    M = ENG[d]
+    R = R_STARS[0]
+    il = M.band_law(Fr(0), R)
+    fn = TB.bounds_d2 if d == 2 else TB.bounds_d3
+    e = Fr(0)
+    for bb, kb in SUBBOXES:
+        up, dn = fn((bb[0], bb[1], kb[0], kb[1]), R, tml, il)
+        e = max(e, up[1], dn[1])
+    return e
+
+
+def compute(d):
+    law_t = tv_law(d, torus=True)
+    law_r = tv_law(d, torus=False)
+    rate_t = rate_sweep_v(d, law_t)
+    rate_r = rate_sweep_v(d, law_r)
+    ref = TB.reference(d)
+    w1 = k_weights(d)
+    e0 = eps0_215(d)
+    return {'tv_law_torus': tv_summary(law_t), 'rate_torus': rate_t, 'rate_reference_kernel': rate_r,
+            'reference': {'c_ref': ref['c_ref'], 'w_c': ref['w_c'], 'w_2': ref['w_2'], 'w_1': w1},
+            'eps0_215': dec_up(e0, 24),
+            'corollary_torus': assemble_v(d, rate_t, ref, w1, Fr(dec_up(e0, 24))),
+            'corollary_reference_kernel': assemble_v(d, rate_r, ref, w1, Fr(0))}
+
+
+def full_run():
+    res = {'schema': 1, 'object': OBJECT, 'scientific_effect': 'NONE', 'certified': True, 'mutant': MUT,
+           'parameters': params(), 'dims': {}}
+    for d in (2, 3):
+        res['dims'][str(d)] = compute(d)
+    return res
+
+
+def text(res):
+    return json.dumps(res, indent=1, sort_keys=True) + '\n'
+
+
+def main():
+    global MUT
+    ap = argparse.ArgumentParser(description=__doc__, formatter_class=argparse.RawDescriptionHelpFormatter)
+    ap.add_argument('--write', action='store_true')
+    ap.add_argument('--check', action='store_true')
+    ap.add_argument('--mutant', choices=MUTANTS)
+    a = ap.parse_args()
+    MUT = a.mutant
+    res = full_run()
+    path = os.path.join(HERE, 'RESULTS.json')
+    if a.write:
+        require(MUT is None, 'no --write under a mutant')
+        with open(path, 'w') as fh:
+            fh.write(text(res))
+        print('RESULTS.json written')
+        return
+    with open(path) as fh:
+        stored = fh.read()
+    if MUT is not None:
+        res['mutant'] = None
+    require(text(res) == stored, 'replay differs from RESULTS.json')
+    print(json.dumps({'object': OBJECT, 'check': 'passed', 'scientific_effect': 'NONE'}))
+
+
+if __name__ == '__main__':
+    main()