C8 Theorem L′: the radial ledger on the torus with a vanishing remainder — |A_r/A_0 − 1| ≤ a r² + β r against the same kernel's A_0 (β₂ ≤ 1.04e-9, β₃ ≤ 1.92e-7; β = 0 for the Gaussian kernel), removing #215's ε_d (scientific effect NONE)

.github/workflows/c8-torus-vanishing-remainder.yml

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+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

frontiers/c8_torus_vanishing_remainder_20261001/README.md

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+# 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.