Certified cusp coefficient c₁ of Theorem CU (Math-#207): Gaussian kernel d = 1, 2, 3 and, by Lemma S (v1.1), the [P] torus field for every L ≥ 24, so c₁/c_(3,24) ∈ [−5.07106215, −5.07106213] for SIDE24; every floating digit of #207 confirmed (scientific effect NONE)

.github/workflows/cusp-coefficient-certified.yml

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+name: Cusp coefficient c1 (certified replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/cusp_coefficient_certified_20261001/**'
+      - '.github/workflows/cusp-coefficient-certified.yml'
+      - 'coefficients/side24_v1/PROOF.md'
+      - 'coefficients/side24_v1/ENCLOSURE.json'
+      - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
+      - 'frontiers/cusp_second_order_20261001/PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certified-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 90
+    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/cusp_coefficient_certified_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/cusp_coefficient_certified_20261001
+          python $FLAGS certificate.py --check --procs "$(nproc)"
+          for m in kernel-A gl-weight bound-scale phase-sign tail-drop image-order6; do
+            set +e
+            python $FLAGS certificate.py --mutant "$m" > /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/cusp_coefficient_certified_20261001/README.md

@@ -0,0 +1,15 @@
+# Certified cusp coefficient `c1` (Math-#207, (CU.2)), Gaussian kernel
+
+`CL-CU-CUSP-COEFFICIENT-20261001-v1` (v1.1). Author-side certified numerics; scientific effect NONE. See `NOTE.md`.
+
+| `d` | `c1` | `c1 / c_(d,ref)` |
+|---|---|---|
+| `1` | `[-0.22760635877558, -0.22760635877504]` | |
+| `2` | `[-0.269398825674, -0.269398825672]` | `[-3.66993776909, -3.66993776907]` |
+| `3` | `[-0.211848347, -0.211848346]` | `[-5.07106215, -5.07106213]` |
+
+For the [P] torus field of every side `L >= 24` (Lemma S), including SIDE24: `c1/c_(3,24)` is in
+`[-5.07106215, -5.07106213]` and `c1/c_(2,24)` in `[-3.66993776909, -3.66993776907]`.
+
+Replay: `python3 -B -S certificate.py --check --procs N` (self-test, then a full recomputation against `RESULTS.json`).
+Standard library only.

frontiers/cusp_coefficient_certified_20261001/RESULTS.json

@@ -0,0 +1,285 @@
+{
+ "object": "CL-CU-CUSP-COEFFICIENT-20261001-v1",
+ "scientific_effect": "NONE",
+ "statement": "interval enclosures (outward decimals) of the cusp coefficient c1 of Math-#207 (CU.2) for the Gaussian kernel, d = 1, 2, 3; see NOTE.md",
+ "values": {
+  "d=1": {
+   "c1": [
+    "-0.2276063587755705",
+    "-0.2276063587750459"
+   ],
+   "I_cand = (3^(1/4)/2) c1": [
+    "-0.1497734069836378",
+    "-0.1497734069832868"
+   ],
+   "I_cand - c1": [
+    "0.07783295179170776",
+    "0.07783295179198402"
+   ]
+  },
+  "d=2": {
+   "c1": [
+    "-0.2693988256731888",
+    "-0.2693988256725268"
+   ],
+   "I_cand = (3^(1/4)/2) c1": [
+    "-0.1772743967942046",
+    "-0.1772743967937623"
+   ],
+   "I_cand - c1": [
+    "0.09212442887867683",
+    "0.09212442887907186"
+   ],
+   "c_(d,ref)": [
+    "0.07340691930595442",
+    "0.07340691930611396"
+   ],
+   "c1 / c_(d,ref)": [
+    "-3.669937769082580",
+    "-3.669937769072928"
+   ]
+  },
+  "d=3": {
+   "c1": [
+    "-0.2118483463910089",
+    "-0.2118483460317414"
+   ],
+   "I_cand = (3^(1/4)/2) c1": [
+    "-0.1394040516860850",
+    "-0.1394040514496684"
+   ],
+   "I_cand - c1": [
+    "0.07244429434593535",
+    "0.07244429494106150"
+   ],
+   "c_(d,ref)": [
+    "0.04177593184055288",
+    "0.04177593184064373"
+   ],
+   "c1 / c_(d,ref)": [
+    "-5.071062141698316",
+    "-5.071062133097566"
+   ]
+  },
+  "[P] field, every L >= 24 (Lemma S)": {
+   "d=2: c1": [
+    "-0.2693988256731890",
+    "-0.2693988256725267"
+   ],
+   "d=2: I_cand": [
+    "-0.1772743967942047",
+    "-0.1772743967937622"
+   ],
+   "d=2: I_cand - c1": [
+    "0.09212442887867663",
+    "0.09212442887907205"
+   ],
+   "d=2: c_(d,24) (side24_v1, consumed)": [
+    "0.07340691930596084",
+    "0.07340691930610771"
+   ],
+   "d=2: c1 / c_(d,24), L = 24": [
+    "-3.669937769082261",
+    "-3.669937769073238"
+   ],
+   "d=3: c1": [
+    "-0.2118483463910090",
+    "-0.2118483460317414"
+   ],
+   "d=3: I_cand": [
+    "-0.1394040516860851",
+    "-0.1394040514496684"
+   ],
+   "d=3: I_cand - c1": [
+    "0.07244429434593524",
+    "0.07244429494106161"
+   ],
+   "d=3: c_(d,24) (side24_v1, consumed)": [
+    "0.04177593184055655",
+    "0.04177593184064014"
+   ],
+   "d=3: c1 / c_(d,24), L = 24": [
+    "-5.071062141697873",
+    "-5.071062133098000"
+   ]
+  },
+  "parts": {
+   "d=2: int_0^oo g2(u) du": [
+    "0.7058330915814585",
+    "0.7058330915829000"
+   ],
+   "d=2: tail u > 15": [
+    "8.361094744137853E-10",
+    "8.361094817557882E-10"
+   ],
+   "d=2: P2": [
+    "-0.3816749722931025",
+    "-0.3816749722921806"
+   ],
+   "d=3: T3": [
+    "45.03930984738922",
+    "45.03930992375617"
+   ],
+   "d=3: boxes": [
+    "45.03930978790707",
+    "45.03930986381351"
+   ],
+   "d=3: tail t > 15": [
+    "5.948215822041717E-8",
+    "5.994264845299655E-8"
+   ],
+   "d=3: tail s > 3": [
+    "-4.940656458412466E-324",
+    "6.400000000006401E-279"
+   ],
+   "d=3: P3": [
+    "-0.004703632154885225",
+    "-0.004703632154874365"
+   ],
+   "K = pi/(2 Gamma(11/4) sin(7 pi/8))": [
+    "2.552096599674470",
+    "2.552096599679964"
+   ],
+   "Gamma(5/8)": [
+    "1.434518848089029",
+    "1.434518848092093"
+   ],
+   "Gamma(11/4)": [
+    "1.608359421983816",
+    "1.608359421987275"
+   ],
+   "Gamma(11/8)": [
+    "0.8889135691552759",
+    "0.8889135691571741"
+   ],
+   "Gamma(7/6)": [
+    "0.9277193336290466",
+    "0.9277193336310296"
+   ]
+  }
+ },
+ "rules": {
+  "d=2 panels [a, b, n, rho]": [
+   [
+    0.0,
+    0.4,
+    13,
+    4.0
+   ],
+   [
+    0.4,
+    0.7,
+    13,
+    4.0
+   ],
+   [
+    0.7,
+    0.9,
+    14,
+    4.0
+   ],
+   [
+    0.9,
+    1.0,
+    13,
+    4.0
+   ],
+   [
+    1.0,
+    1.1,
+    13,
+    4.0
+   ],
+   [
+    1.1,
+    1.3,
+    13,
+    4.0
+   ],
+   [
+    1.3,
+    1.6,
+    16,
+    3.0
+   ],
+   [
+    1.6,
+    2.0,
+    15,
+    3.0
+   ],
+   [
+    2.0,
+    2.6,
+    15,
+    3.0
+   ],
+   [
+    2.6,
+    3.4,
+    14,
+    3.0
+   ],
+   [
+    3.4,
+    4.5,
+    13,
+    3.0
+   ],
+   [
+    4.5,
+    6.0,
+    15,
+    2.4
+   ],
+   [
+    6.0,
+    8.0,
+    14,
+    2.4
+   ],
+   [
+    8.0,
+    11.0,
+    13,
+    2.4
+   ],
+   [
+    11.0,
+    15.0,
+    11,
+    2.4
+   ]
+  ],
+  "d=3": {
+   "boxes": 247,
+   "crude": 76,
+   "gauss_legendre": 171,
+   "nodes": 1406081,
+   "T_MAX": 15.0,
+   "S_MAX": 3.0
+  },
+  "Lemma S (exact rationals)": {
+   "d=2": {
+    "N": "9",
+    "E": "183921320046627603/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "mu": "1/4",
+    "eps": "1655291880419648427/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "M": "11/2",
+    "eta": "18208210684616132697/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+   },
+   "d=3": {
+    "N": "14",
+    "E": "183921320046627603/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "mu": "1/4",
+    "eps": "1287449240326393221/6250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "M": "8",
+    "eta": "1287449240326393221/97656250000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
+   }
+  }
+ },
+ "publication": {
+  "margin_relative": 1e-12,
+  "digits": 16
+ }
+}