C8/C6: certified planar near cluster coefficients alpha_1, alpha_2 = nu(2) and the elder-failure constant C_fail^{B,K} in [0.0027244, 0.0027312] (about 1.5e5 below the cap route), for the reference kernel and the torus model with every L >= 10 (v1.2); conditional d = 3 corollary; scientific effect NONE

.github/workflows/c8-elder-failure-coefficient.yml

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+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 $FLAGS certificate.py --check --procs "$(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)"

frontiers/c8_elder_failure_coefficient_20261001/README.md

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+# 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.