d6g8k5htny-coder · Oct 1, 2026 · Oct 1, 2026 · Oct 1, 2026 · Oct 1, 2026 · Oct 1, 2026
@@ -0,0 +1,79 @@
+name: c2 exact closed forms (replay)
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/c2_exact_20261001/**'
+      - '.github/workflows/c2-exact.yml'
+      - 'frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md'
+      - 'coefficients/side24_v1/PROOF.md'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 30
+    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/c2_exact_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: Exact derivation, checks and replay against RESULTS.json, mutants, controls
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+          FLAGS: ${{ matrix.flags }}
+        run: |
+          set -euo pipefail
+          cd frontiers/c2_exact_20261001
+          python $FLAGS certificate.py --check
+          for m in target finite-part cone-sign truncation sphere; 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
+          python $FLAGS controls.py
+          cd "$GITHUB_WORKSPACE"
+          git diff --exit-code
+          test -z "$(git status --porcelain)"
@@ -0,0 +1,228 @@
+# The third-order coefficient `c2` in closed form: Gaussian kernel, `d = 1, 2, 3`
+
+**Object:** CL-C2-EXACT-20261001-v1.
+**Author:** Anthropic Claude (Claude Code session `session_015wNj8LPTKXsaT68G3DgPPh`), 1 October 2026.
+**Disposition:** exact closed forms, by an exact rational derivation replayed with the standard library, and certified
+interval enclosures of the coefficient `c2` that Math-#216 defines and Math-#218 (0.1) and Math-#220 use, together with
+the leading coefficient `c` from the same pipeline. Author-side. Nonauthor reads at `b43d36a`: OpenAI Slice B ACCEPT
+(review 5379878251) and Slice A PASS (review 5379979090) with amendment OA-223-A-01, applied in v1.1 (§§0–2: the analytic
+fixed-cone surrogate is written `A~_r`); xAI Slice C arithmetic check (5932294410). Scope claim: Math-#216 comment
+5931466669.
+**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-#214, #216, #218 or #220 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
+
+Math-#216 ([D], head `4f0927f`) defines, from the two-point kernel `A_r(b, k, u)` of [R] and its fold expansion
+`A_r = A_0 + r^2 A_2 + O(r^3)`, the coefficients below. Here `A_0`, `A_2` are the Taylor coefficients of the fixed-cone
+surrogate `A~_r` of §1, which is real-analytic in `r` (§2.3); with the consumed comparison `A_r = A~_r + O(r^3)` (§1) they
+are the coefficients of the fold expansion of `A_r` itself. No analyticity of the typed `A_r` is claimed.
+
+    c  = (1/3) |S^(d-1)| integral_R db integral_0^oo A_0(b, k) k^(-2/3) dk,
+    c2 = (1/3) |S^(d-1)| integral_R db  f.p. integral_0^oo A_2(b, k) k^(-4/3) dk
+       = (1/3) |S^(d-1)| integral_R db integral_0^oo [A_2(b, k) - A_2(b, 0)] k^(-4/3) dk.
+
+`c` is the leading coefficient of the short-lifetime law. `c2` is the coefficient of `ell^(1/3)`:
+- Math-#218 (Theorem T, candidate status) proves it for the candidate density;
+- Math-#220 (Theorem E3) proves it for the elder density, conditional on Math-#187;
+- [D] derives it formally and by Monte Carlo, and lists a certified value as not claimed.
+
+**Theorem (closed forms).** For the Gaussian kernel `K(z) = exp(-|z|^2/2)` on `R^d`:
+
+| `d` | `c` | `c2` |
+|---|---|---|
+| `1` | `Gamma(1/6) 12^(-1/6) / (6 pi^(3/2))` | `(3/4) 12^(1/6) Gamma(5/6) / pi^(3/2)` |
+| `2` | `Gamma(1/6) 12^(-1/6) / (9 pi^(3/2))` | `(13/18) 12^(1/6) Gamma(5/6) / pi^(3/2)` |
+| `3` | `(29/72 - sqrt6/12) Gamma(1/6) 12^(-1/6) / pi^(5/2)` | `(5/48)(33 - 7 sqrt6) 12^(1/6) Gamma(5/6) / pi^(5/2)` |
+
+In every dimension the ratio is an algebraic multiple of one transcendental factor:
+
+    c2/c = kappa_d Gamma(5/6) / (12^(2/3) Gamma(1/6)),     kappa_1 = 54,   kappa_2 = 78,   kappa_3 = 18 (141 - sqrt6)/25.
+
+Here `12^(1/6) = 2^(1/3) 3^(1/6)`. The certified values (interval arithmetic, outward to 16 digits, `RESULTS.json`) are:
+
+| `d` | `c` | `c2` | `c2 / c` |
+|---|---|---|---|
+| `1` | `[0.1101103789590410, 0.1101103789590625]` | `[0.2300445802661503, 0.2300445802661998]` | `[2.089217950577392, 2.089217950578207]` |
+| `2` | `[0.07340691930602738, 0.07340691930604167]` | `[0.2215244106266632, 0.2215244106267110]` | `[3.017759261945122, 3.017759261946299]` |
+| `3` | `[0.04177593184059426, 0.04177593184060272]` | `[0.1612340491269447, 0.1612340491269810]` | `[3.859496174547095, 3.859496174548665]` |
+
+At 30 digits (mpmath, a control only): `c2 = 0.230044580266174952088473542454`, `0.221524410626686990900011559`,
+`0.161234049126962737547589515632`.
+
+**Consequences.**
+- **[D]'s eight digits are confirmed and made exact.** [D] gives `c2 = 0.23004458`, `0.22152441`, `0.16123405` and
+  `c2/c = 2.089218`, `3.017759`, `3.859496` from an `r`-fit. Each is the correct rounding of the closed form (§7).
+- **The leading coefficient comes out exactly.** In `d = 2, 3` it equals the closed form `c_(d,ref)` of
+  `coefficients/side24_v1` (1). In `d = 1` it equals Math-#214's `C_0`, and `c2` equals #214's `2 B_2`. These are
+  identities of exact numbers, checked by the certificate (§4).
+- **The three-term law in closed form.** With [CU]'s `c1`, the relative correction is
+  `nu/(c ell^(-1/3)) - 1 = (c1/c) ell^(7/12) + (c2/c) ell^(2/3) + ...`. In `d = 3` it is
+  `-5.071062... ell^(7/12) + 3.859496174548... ell^(2/3)`, where `c1/c` is certified in Math-#219 and `c2/c` here.
+
+## 1. What is evaluated
+
+The kernel is that of [R] §§2–4 and [D] §0. Pins sit at `M = -r e1/2` and `S = r e1/2` with zero gradients and heights
+`b`, `b - k r^3`. The symmetric pin vector `U_r` has target `v_r = (b - k r^3/2, -k r^2, 0, 12k, 0, ..., 0)`, with rows
+as in [D]'s `c2_check.py` (`pseries.pin_rows`), and
+
+    A_r(b, k) = 12 pi_r(v_r) E[ |det H_M| |det H_S| 1{typed} | U_r = v_r ] / r^2,     u = e1 (isotropy).
+
+As in [D] §2.2, the typed indicator is:
+- nothing in `d = 1`, where the typed event has probability `1 - O(e^(-c k^2/r^2))`;
+- `1{f_yy(0) < 0}` in `d = 2`;
+- `1{D_y^2 f(0) < 0}` in `d = 3`.
+
+Write `A~_r` for the kernel with the typed indicator replaced in this way: the **fixed-cone surrogate**. Each
+replacement changes the kernel by `O(r^3)` at fixed admissible `(b, k)`, so
+
+    A_r(b, k) = A~_r(b, k) + O(r^3),        A~_r = A_0 + r^2 A_2 + O(r^3)  (§2.3).
+
+The first relation is consumed from [D] §2.2. Everything computed below is `A~_r`; analyticity is proved for `A~_r` only,
+and no claim is made that the typed `A_r` is analytic (OA-223-A-01). [T] Lemma F proves the fold expansion and identifies
+`A_2(b, 0) = -12 pi_0 E_0[Y^2 1{A < 0} | b]`, which is (0.1). Both are consumed, not reviewed. On the typed set,
+`|det H_M| |det H_S| = -det H_M det H_S`.
+
+## 2. Exact derivation (`pseries.py`, `exact.py`)
+
+**2.1 Series.**
+- Every quantity is a linear form in the jets at 0 with coefficients in `Q[[r]]`:
+  `d^beta f(x e1) = sum_j x^j/j! d^(beta + j e1) f(0)`, with `x = -+r/2`.
+- Jet covariances are the integers `(-1)^|b| d^(a+b) K(0)`.
+- The pin rows divide by `r`, `r` and `r^3`. Each division is exact, because the low coefficients vanish identically;
+  the code asserts this.
+- Keeping `r^0 .. r^10`, every derived series is exact through `r^7`. The integrand needs `r^6`.
+
+**2.2 Conditioning.**
+- The `r^0` pin covariance is invertible (`det = 12` in `d = 1, 2`), so the conditional means `G v_r` and covariances
+  are computed exactly in `Q[[r]]`.
+- In `d >= 2` the Hessian entries are conditioned further on the transverse block `T = D_y^2 f(0)`: on `a = x1` in
+  `d = 2`, and on `T = diag(x1, x2)` in `d = 3`.
+- `d = 3` uses transverse rotation invariance. The pins lie on `e1`, and transverse rotations preserve the law and
+  `det H_M det H_S`. With the Jacobian of the symmetric `2 x 2` eigenvalue map,
+  `integral_(T<0) g(T) p(T) dT = pi integral_(x2<x1<0) g p (x1 - x2) dx`.
+
+**2.3 The integrand.** `F = E[det H_M det H_S | T, U_r = v_r]` comes from Isserlis' theorem as an exact polynomial in
+`(r, b, k, x1, x2)`. Then
+
+    A~_r(b, k) = const . integral_cone exp(-(q + Q)/2) F w / r^2,    q = v_r^T S_r^-1 v_r,   Q = z^T C_TT^-1 z,
+
+where:
+- `const = -12 (2 pi)^(-(p + n_T)/2) (det S_0 det T_0)^(-1/2)`, times `pi` in `d = 3`;
+- `z = T - E[T | U_r]`;
+- `w = 1`, `da`, or `(x1 - x2) dx`.
+
+Writing `q = q_0 + dq` and `Q = Q_0 + dQ`, the factor `(det ratio)^(-1/2) exp(-(dq + dQ)/2) F` is expanded exactly in `r`.
+`A~_r` is real-analytic in `r` near 0, with Gaussian domination uniform in small `r` because the `r = 0` covariances are
+nondegenerate. Its Taylor coefficients are therefore the cone integrals of those of the integrand. This is a statement
+about the fixed cone; `A_r = A~_r + O(r^3)` (§1) then carries `A_0` and `A_2` over to the typed kernel.
+
+**2.4 Exact structural facts.** The certificate verifies each of these.
+- The integrand vanishes to order `r^2`.
+- The `r^3` coefficient vanishes, so `A~_r` has no `r^1` term ([D] T5).
+- `q_0 = 3b^2/2 + 24 k^2`, with no `b k` term.
+- `Q_0` has no `k`: the transverse block does not see `k`, by parity.
+- `A_0` and `A_2` contain only even powers of `k`.
+
+## 3. Integration
+
+**3.1 `k`, by Gamma finite parts.** `A_j(b, k) = exp(-12 k^2) sum_(even l) a_l(b, x) k^l`. For `-2 < s < 0`,
+`integral_0^oo (k^l e^(-12k^2) - [l = 0]) k^(s-1) dk = Gamma((l + s)/2) 12^(-(l+s)/2)/2`. With `s = 1/3` for `c` and
+`s = -1/3` for `c2`, every term is a rational multiple of `Gamma(1/6) 12^(-1/6)` or of `Gamma(5/6) 12^(-5/6)`. The
+`l = 0` finite part uses `Gamma(-1/6) = -6 Gamma(5/6)`.
+
+**3.2 `b`, by completing the square.** The `k`-free exponent is `a_b b^2 + b L(x) + R(x)`. The substitution
+`b -> b - L/(2 a_b)` leaves Gaussian moments `integral b^n e^(-a_b b^2) db` and the cone exponent `R - L^2/(4 a_b)`.
+
+**3.3 The cone.**
+- **`d = 2`:** half-line moments `integral_(-oo)^0 a^n e^(-g a^2) da`, with `g = 3/16`.
+- **`d = 3`:** with `x1 = -s` and `x2 = -s - t`, the exponent is `(3/10) s^2 + (3/10) s t + (1/5) t^2` and
+  `D = 4 al ga - be^2 = 3/20`. The quadrant moments `M(i, j)` follow from two integration-by-parts identities, in `s`
+  and in `t`. These form a `2 x 2` system with determinant `D`, starting from
+  `M(0, 0) = theta = (pi/2 - arctan(be/sqrt D))/sqrt D`.
+- Every `M(i+1, j)` with `j >= 1` is reached twice, and the two values agree exactly (asserted).
+- **The angle `theta` cancels exactly in both `c` and `c2`.** The results lie in `Q(sqrt6)` times powers of `pi`.
+
+Numbers are carried as `sum q pi^(h/2) sqrt(m) theta^e`, with `q` rational and `m` squarefree.
+
+## 4. Exact checks (`certificate.py`)
+
+1. **`theta` does not occur** in any result.
+2. **`d = 2, 3`, `c` equals `coefficients/side24_v1` (1).** Since `Gamma(7/6) = Gamma(1/6)/6` and
+   `(3/2)^(1/3) = sqrt3 12^(-1/6)`, the closed form `Gamma(7/6)(3/2)^(1/3) D_(d-1)/(2 sqrt3 pi^(d-1) sqrt pi)` equals
+   `Gamma(1/6) 12^(-1/6) D_(d-1)/(12 pi^(d-1/2))`, with `D_1 = 4/3` and `D_2 = 29/6 - sqrt6`. Equality is exact.
+3. **`d = 1`, `c = C_0` and `c2 = 2 B_2` of Math-#214 (D1.2).** For the Gaussian spectral moments `1, 3, 15, 105`:
+   - `D = 6`, `sigma_3 = sqrt6`, `p_12 = 1/(2 pi sqrt3)` and `Q/(120 lambda_2 lambda_4 D) = 3/4`;
+   - `72^(-1/6) 6^(2/3) = sqrt6 12^(-1/6)` and `3^(1/3) 6^(1/3) = 12 sqrt3 12^(-5/6)` give `C_0 = atom/(6 pi^(3/2))` and
+     `2 B_2 = 9 atom'/pi^(3/2)`;
+   - both are exact equalities, and the two reductions are also checked against #214's formulas in floating point.
+4. **Truncation stability.** The whole derivation is repeated with series order 12 and polynomial order 8, and every
+   result is identical.
+5. **The ratios.** `kappa_d Gamma(5/6)/(12^(2/3) Gamma(1/6))` must meet the interval quotient of the enclosures.
+
+## 5. Evaluation
+
+- `Gamma(1/6)` and `Gamma(5/6)` come from Stirling's series at `q + 10`, with the remainder bounded by the first
+  omitted term, and the exact product (`gam.py`: the code of Math-#219's `consts.py`).
+- `12^(rational)` is `exp(log)`; `pi` and `sqrt` come from `ia.py` and `elem.py` (Math-#219, byte-identical; `ia.py` is
+  also Math-#217's).
+- Outward one-ulp rounding is used throughout. Publication widens by `1e-14` relative and rounds outward to 16 digits.
+- `--check` re-derives everything and requires the regenerated document to equal `RESULTS.json`.
+
+## 6. Verification
+
+- `python3 -B -S certificate.py --check` takes about one second. It runs the exact derivation, the checks of §4 and the
+  evaluation, then compares with `RESULTS.json`.
+- Five seeded defects must each exit 1:
+  - `target`: the pin target `11 k` instead of `12 k`;
+  - `finite-part`: the `l = 0` term without the subtraction of `A_2(b, 0)`;
+  - `cone-sign`: the sign of the cone cross term;
+  - `truncation`: polynomial order 3;
+  - `sphere`: `|S^1| = pi`.
+- The workflow `.github/workflows/c2-exact.yml` runs, for `-B -S` and `-B -O -S`: the manifest and pin check,
+  `--check`, the mutants, the controls, and a clean-tree check.
+
+## 7. Floating controls (`controls.py`; not part of the certificate)
+
+1. **A finite-`r` kernel.**
+   - The fixed-cone surrogate `A~_r(b, k)` is computed at finite `r` directly from the closed-form kernel derivatives at
+     the displaced points,
+     `D^g K(z) = prod (-1)^(g_i) He_(g_i)(z_i) e^(-z_i^2/2)`, with 60-digit conditioning and no Taylor jets.
+   - The Richardson estimate `2 D(r) - D(2r)`, with `D(r) = (A~_r - A_0)/r^2`, converges to the exact `A_2(b, k)` at the
+     rate `O(r^2)`, at five points in `d = 1, 2, 3`.
+   - The relative errors at `r = 0.01, 0.005, 0.0025` are `1.3e-4`, `3.2e-5`, `8.0e-6` (`d = 1`), with ratios `3.75` to
+     `4.17` per halving everywhere.
+2. **[D]'s floating values.** All nine (`c`, `c2`, `c2/c` for `d = 1, 2, 3`) are correct roundings of the certified
+   midpoints. The largest difference, `4.0e-13`, is for `c(d = 3)`, against a half-unit of `5e-13`.
+
+## 8. What this does not do
+
+- **Gaussian kernel only.** The periodized covariance changes the jets by `1 + O(e^(-L^2/8))` ([D] §0). No torus
+  transfer of `c2` is made. The finite part and the typed indicator do not fit the homogeneity argument of Math-#219's
+  Lemma S.
+- **No `d >= 4`.** The method extends; the cone then needs the 3D eigenvalue reduction.
+- **The definition is consumed.**
+  - The identification of `c2` as the `ell^(1/3)` coefficient is [D]'s (formal), [T]'s (candidate density, candidate
+    status) and Math-#220's (elder density, conditional on #187), and is not reviewed here.
+  - So is the `O(r^3)` replacement of the typed indicator, `A_r = A~_r + O(r^3)` (§1). Analyticity is proved for the
+    surrogate `A~_r` only.
+- Not a review of any source. Same GitHub account as every lane; zero organizational-independence credit.
+  **I will not merge.**
+
+## 9. Provenance
+
+- **Sources** (`SOURCE_MAP.json`):
+  - [R] `frontiers/three_fronts_20260924/LIFETIME_REMAINDER.md` (the two-point kernel);
+  - [SIDE24] `coefficients/side24_v1/PROOF.md` (1).
+
+  Both are on `main` and verified by the workflow.
+- **Unmerged, recorded and not checked on this tree:**
+  - [D] Math-#216 `NOTE.md` and `c2_check.py` at head `4f0927f`;
+  - [T] Math-#218 `PROOF.md` at head `7f09d34`;
+  - [D1] Math-#214 `PROOF.md` at head `f4a58df`.
+- **Files:**
+  - the certificate: `certificate.py`, `exact.py`, `pseries.py`, `gam.py`, `ia.py`, `elem.py`;
+  - `controls.py`;
+  - `RESULTS.json`, `SOURCE_MAP.json`, `SOURCE_FILES.json`.
@@ -0,0 +1,14 @@
+# The third-order coefficient `c2` in closed form (Math-#216 / #218 / #220), Gaussian kernel
+
+`CL-C2-EXACT-20261001-v1`. Author-side exact closed forms with certified enclosures; scientific effect NONE. See `NOTE.md`.
+
+| `d` | `c2` | value |
+|---|---|---|
+| `1` | `(3/4) 12^(1/6) Gamma(5/6) / pi^(3/2)` (= Math-#214's `2 B_2`) | `0.2300445802661...` |
+| `2` | `(13/18) 12^(1/6) Gamma(5/6) / pi^(3/2)` | `0.2215244106266...` |
+| `3` | `(5/48)(33 - 7 sqrt6) 12^(1/6) Gamma(5/6) / pi^(5/2)` | `0.1612340491269...` |
+
+`c2/c = kappa_d Gamma(5/6)/(12^(2/3) Gamma(1/6))`, `kappa = 54, 78, 18(141 - sqrt6)/25`.
+
+Replay: `python3 -B -S certificate.py --check` (exact derivation, exact checks, interval evaluation; about one second).
+Controls: `python3 -B -S controls.py`. Standard library only.