From ee6127f99667a4fda35816eaf9dcb0fc65973ee2 Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 10:33:02 +0000 Subject: [PATCH 1/4] Hard-direction factorization: the Gaussian-kernel near cluster coefficients in every dimension are the planar ones times R_d(b) New packet frontiers/c6_hard_direction_factorization_20260930 (PROOF.md, factorization.py, RESULTS.json, README.md, SOURCE_MAP.json, SOURCE_FILES.json) and its replay workflow. Theorem 1: for the unit Gaussian kernel and every functional g of the soft cubic (s, a, beta, c), integral g dM^(d) = R_d(b) integral g dM^(2) with R_d(b) = N_d(b) m_(2,b) / m_(d,b), N_d a shifted-GOE hard-eigenvalue integral with weight prod h_j^3 and m_(d,b) = E[(det A)^2 1{A negative definite}] for A = -bI + GOE. Lemma 1 (conditional jet law given the contact frame: A = -bI + GOE(2,1), odd jets isotropic, independent, variances 6/2/1) checked exactly in rational arithmetic for d = 2..5; Lemma 2 (z0^(d) = 36 k^2 m_(d,b)) corroborated by finite-r Monte Carlo of Z_r / r^2 in d = 2, 3. Corollaries: a_j^(d) = R_d(b) a_j^(2), C_*^(d) = R_d(b) C_*^(2), a_2/a_1 and every normalized microscopic shape law dimension-independent; closed form R_3(0) = (32 + 28 sqrt2)/17. Values of R_d(b) for d = 2..5, b = 0, 1 (quadrature converged to 1e-14, Monte Carlo cross-checks) and the d = 3, 4, 5 near coefficients from Math-#168's planar values (floating, not certified). Continuum kernel; torus image corrections stated, not quantified. Scientific effect NONE; author-side candidate; nonauthor review requested. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c6-hard-direction-factorization.yml | 68 ++ .../PROOF.md | 276 +++++++++ .../README.md | 19 + .../RESULTS.json | 525 ++++++++++++++++ .../SOURCE_FILES.json | 39 ++ .../SOURCE_MAP.json | 27 + .../factorization.py | 579 ++++++++++++++++++ 7 files changed, 1533 insertions(+) create mode 100644 .github/workflows/c6-hard-direction-factorization.yml create mode 100644 frontiers/c6_hard_direction_factorization_20260930/PROOF.md create mode 100644 frontiers/c6_hard_direction_factorization_20260930/README.md create mode 100644 frontiers/c6_hard_direction_factorization_20260930/RESULTS.json create mode 100644 frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json create mode 100644 frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json create mode 100644 frontiers/c6_hard_direction_factorization_20260930/factorization.py diff --git a/.github/workflows/c6-hard-direction-factorization.yml b/.github/workflows/c6-hard-direction-factorization.yml new file mode 100644 index 000000000..ae14f3255 --- /dev/null +++ b/.github/workflows/c6-hard-direction-factorization.yml @@ -0,0 +1,68 @@ +name: C6 hard-direction factorization (replay) + +'on': + pull_request: + paths: + - 'frontiers/c6_hard_direction_factorization_20260930/**' + - '.github/workflows/c6-hard-direction-factorization.yml' + - 'frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md' + - 'frontiers/radial_rate_20260930/PROOF.md' + - 'frontiers/spectral_cluster_closure_20260929/PROOF.md' + - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' + - 'frontiers/planar_cubic_cluster_20260929/PROOF.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + workflow_dispatch: + +permissions: + contents: read + +jobs: + exact-controls-and-replay: + runs-on: ubuntu-latest + timeout-minutes: 20 + 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 the exact lemma, closed forms and tables, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c6_hard_direction_factorization_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 / 'factorization.py') + for flags in (['-B', '-S'], ['-B', '-O', '-S']): + out = subprocess.run([sys.executable, *flags, script, '--check'], cwd=root, capture_output=True, timeout=900) + if out.returncode != 0: + raise RuntimeError('check failed under ' + ' '.join(flags) + ': ' + out.stdout.decode()[-400:]) + for m in ['drop-vandermonde', 'hard-power-two', 'drop-spectral-constant']: + rc = subprocess.run([sys.executable, *flags, script, '--check', '--mutant', m], cwd=root, capture_output=True, timeout=900).returncode + if rc != 1: + raise RuntimeError('mutant ' + m + ' not rejected') + print(json.dumps({'scientific_effect': 'NONE', 'certified': False})) + PY + git diff --exit-code diff --git a/frontiers/c6_hard_direction_factorization_20260930/PROOF.md b/frontiers/c6_hard_direction_factorization_20260930/PROOF.md new file mode 100644 index 000000000..23cc65dd8 --- /dev/null +++ b/frontiers/c6_hard_direction_factorization_20260930/PROOF.md @@ -0,0 +1,276 @@ +# Hard-direction factorization: the near cluster coefficients of the Gaussian kernel in every dimension are the planar ones times an explicit factor R_d(b) + +**Object:** CL-C6-HARD-DIRECTION-FACTORIZATION-20260930-v1. +**Author:** Anthropic Claude (Claude Code `session_015wNj8LPTKXsaT68G3DgPPh`), foreground owner-authorized continuation. +**Disposition:** AUTHOR-SIDE CANDIDATE; NONAUTHOR REVIEW OPEN. Scientific effect NONE: no register, catalog, GRAPH, +STATUS or source change; nothing in any consumed source is revalidated or promoted. + +## 0. Statement + +Fix `d >= 2`, `m = d - 1`, `b` real, `k > 0`, and the unit Gaussian kernel `K(x) = exp(-|x|^2/2)` on `R^d` +(the continuum limit of the periodized parent kernel of [LP] section 2; the size of the torus corrections is +stated in section 6). Let `dM^(d)` be the limiting near cluster measure [SC] (17) on `(s, h, O, tau)`, with the +spectral constant `c_m` of [SC] (8), the full normalizer `z0^(d) = lim Z_r / r^2` of [LP], the raw conditional +jet density `h_0` of [SC] section 2, and the endpoint weight `w(s, a, beta) = 9k^2 (4s^2 - B^2)` on the typed +domain `s < -|B|/2`, `B = beta - a^2/(12k)`. For `d = 2` it is the planar measure `z0^(-1) w h_0(0, a, beta, c) +ds da dbeta dc` of [CUB] section 4 / [SC] (17) with empty hard products. + +**Theorem 1 (hard-direction factorization).** For every nonnegative measurable `g` of the soft cubic +coordinates `(s, a, beta, c)`, + + integral g(s, a, beta, c) dM^(d)(s, h, O, tau) = R_d(b) integral g(s, a, beta, c) dM^(2)(s, a, beta, c), (1) + +with + + R_d(b) = N_d(b) m_(2,b) / m_(d,b), (2) + N_d(b) = c_m (2 pi)^(-m(m-1)/4) integral_(0 < h_2 < ... < h_m) prod_j h_j^3 phi_2(h_j - b) prod_(i= 1`, the shape region measure `Q du dv` of [R] (R9)–(R10), the Pareto-11 radius law and the constants +`E eta = 5771/7062`, `I, J, D, E0` and all ratios of [R], [E], [Q], [ELD] — are the same in every dimension +`d >= 2`. The radial-tail constant of [R] (R13) satisfies `C_*^(d) = R_d(b) C_*^(2)`, since (R12)'s `J_cusp` +integrand has the same hard-direction factor (section 3). + +**Corollary 2 (conditional, elder statements).** Where a source identifies `lim r^(-3) (1 - p_r) = a_1 + a_2` +(#170 Theorem S in `d = 2`, #175 Theorem F in `d >= 3`, both unmerged drafts), the elder-failure coefficient in +dimension `d` is `R_d(b) (alpha_1 + alpha_2)^(2)`, and the conditional lifetime-fraction and shape laws given +failure (#170 section 8.8, #175 M1, #179/#181's `S \ O2` laws) are dimension-independent. This corollary inherits +those sources' conditional status and promotes nothing. + +**Closed form.** `m_(3,0) = (7 - 4 sqrt2)/2`, `N_3(0) = 2 sqrt2`, hence + + R_3(0) = 4 sqrt2 / (7 - 4 sqrt2) = (32 + 28 sqrt2)/17 = 4.21164586743804... (3) + +Numerical values of `R_d(b)` for `d = 3, 4, 5` and `b = 0, 1`, the resulting `a_1, a_2, z0` in `d = 3, 4, 5`, and +the Monte Carlo and finite-`r` controls are in section 5 and `RESULTS.json`. + +## 1. Sources and exposure + +- [SC] `frontiers/spectral_cluster_closure_20260929/PROOF.md` (main, blob `16c56821`): section 2 (4), (7), (8), + (9); section 5 (17); section 6 (19)–(20). Consumed as the definition of `dM^(d)`, `c_m`, `h_0`, `a_j`. +- [LP] `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` (blob `dfed3b8d`): the model, + `Q_r`, `W_r = F_d(H_M) F_(d-1)(H_S)`, `Z_r / r^2 -> z0 > 0`, the moment bounds used in Lemma 2. +- [CUB] `frontiers/planar_cubic_cluster_20260929/PROOF.md` (blob `bb446d08`): the classifier `n(s, a, beta, c)` + and the planar contact structure. +- [R] `frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md` (blob `0f14417e`): (R11)–(R13). +- [Q] `frontiers/radial_rate_20260930/PROOF.md` (blob `65f24b1f`, merged with #176): (21) the planar odd + variances `v_a = v_beta = 2`, `v_c = 6` in the continuum, matching Lemma 1. +- [NUM] `frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json` (blob `2648314f`, Math-#168): the + planar values `alpha_1, alpha_2` (GH60, floating, not certified) multiplied in section 5. +- Unmerged, quoted only for Corollary 2 and the list of dimension-independent laws: [E] Math-#169 `PROOF.md` + `19a6044f`, [ELD] Math-#181 `PROOF.md` `91f88e39` and Math-#179 `PROOF.md` `1310dc11`, #170 `ef2aa579`, + #175 `923d3236`. + +Exposure: I authored [NUM] (Math-#168) and the numerical packets Math-#174/#178, and [CL] (Math-#159); I authored +none of [SC], [LP], [CUB], [R], [Q], [E], [ELD]. Same GitHub account as every lane; zero organizational-independence +credit. Lemma 2 in `d = 2` is the normalizer identification of Math-#168 (accepted as an identity in the xAI read +5360559751); its `d >= 3` form is new here. + +## 2. Lemma 1: the conditional jet law at contact + +Let `U_0 = (f, grad f, f_xx, f_(x y_i) (i = 1..m), f_xxx)` at the midpoint, with values `(b, 0, 0, 0, 12k)`, and +let `A = (f_(y_i y_j))` be the transverse Hessian and `T` the list of third derivatives other than `f_xxx`, as in +[SC] section 2. For the Gaussian kernel, `Cov(d^alpha f(0), d^beta f(0)) = (-1)^|alpha| d^(alpha+beta) K(0)` and +`d^gamma K(0) = prod_i (-1)^(gamma_i/2) (gamma_i - 1)!!` when every `gamma_i` is even, `0` otherwise. + +**Lemma 1.** Conditionally on `U_0`: + +1. `A = -b I_m + G_m` with `G_m` as in section 0 (independent `N(0,2)` diagonal, `N(0,1)` off-diagonal entries); +2. `T` is centered Gaussian, independent of `A`, with independent components of variance `6` (`T_iii`), `2` + (`T_iij`, `i != j`, including `T_(xx y_i)` and `T_(x y_i y_i)`) and `1` (`T_ijk`, distinct indices); +3. the law of `(A, T)` is invariant under transverse rotations `y -> O y`, `O in O(m)`: `h_0(O A O^t, Rot_O T) = + h_0(A, T)`; and it does not depend on `k`. + +Proof. Parity: a covariance `(-1)^|alpha| d^(alpha+beta) K(0)` vanishes unless `alpha + beta` is even in every +coordinate, so the even jets `(f, f_xx, f_(x y_i), A)` and the odd jets `(grad f, f_xxx, T)` are independent, and +within each parity class the regression on `U_0` is the finite computation `Cov(target, U_0) Cov(U_0)^(-1)`. For +the even class the unconditional covariances are `Var f = 1`, `Var f_xx = Var A_ii = 3`, `Var f_(x y_i) = Var A_ij = +1` (`i != j`), `Cov(f, f_xx) = Cov(f, A_ii) = -1`, `Cov(f_xx, A_ii) = Cov(A_ii, A_jj) = 1`, all others zero; +conditioning on `(f, f_xx) = (b, 0)` (the `f_(x y_i)` are uncorrelated with `A`) gives `E[A_ii] = -b`, +`Var A_ii = 3 - 1 = 2`, `Cov(A_ii, A_jj) = 1 - 1 = 0`, and `A_ij` untouched: item 1. For the odd class, +`Cov(T_(x x y_i), f_(y_i)) = -1`, `Cov(T_(x y_i y_i), f_x) = -1`, `Cov(T_(y_i y_i y_i), f_(y_i)) = -3`, +`Cov(T_(y_i y_i y_j), f_(y_j)) = -1`, `Cov(T, f_xxx) = 0`, and the unconditional variances are `15`, `3`, `1` by +multiplicity; conditioning on `grad f = 0`, `f_xxx = 12k` leaves the means at `0` (the only nonzero observed values +`b`, `12k` have zero coefficients) and the variances `15 - 9 = 6`, `3 - 1 = 2`, `1`, with all cross terms +cancelling: item 2. Item 3: the three blocks of `T` (`T_(xx .)` an isotropic vector, `T_(x . .)` a GOE-type +matrix with the variances of item 2, `T_(. . .)` an isotropic symmetric 3-tensor) and `A` are each `O(m)`-invariant +and mutually independent, and no value of `U_0` other than `b` enters. The computation is carried out exactly in +rational arithmetic for `d = 2, 3, 4, 5` in `factorization.py` (`lemma_one`), including the vanishing of every +even/odd cross covariance and of every coefficient on `f` and `f_xxx` in the odd regression. In `d = 2` this is +Math-#168's exact contact structure (`A_0 ~ N(-b, 2)`, odd jets `N(0, diag(2, 2, 6))`), and [Q] (21) at `L = infinity`. QED + +## 3. Lemma 2: the full normalizer + +**Lemma 2.** `z0^(d) = lim_(r->0) Z_r / r^2 = 36 k^2 m_(d,b)`, with `m_(d,b) = E[(det A)^2 1{A negative +definite}]` for `A` as in Lemma 1. + +Proof. Under `Q_r` the fixed jets at the midpoint converge in law to their `U_0`-conditional law ([SC] section 2, +`h_r -> h_0` locally uniformly), and every fixed `C^q` moment is bounded uniformly in `r` ([LP], quoted as [SC] (6)). +With `K` the `C^4` bound, Taylor expansion from the midpoint gives, in the frame `(x, y)`, + + H_M = [[ -6kr + O(r^2 K), -(r/2) T_(xx .) + O(r^2 K) ], [ ., A + O(r K) ]], + H_S = [[ 6kr + O(r^2 K), (r/2) T_(xx .) + O(r^2 K) ], [ ., A + O(r K) ]], + +because `f_xx(0) = [f_x(S) - f_x(M)]/r + O(r^2 K) = O(r^2 K)` and `f_(x y_i)(0) = O(r^2 K)` are pinned by the +endpoint gradients while `f_xxx(0) = 12k + O(rK)`. Expanding the determinants along the first row, each +off-diagonal term carries two factors of order `r`, so `det H_M = -6kr det A + O(r^2 K^d)` and `det H_S = 6kr det A ++ O(r^2 K^d)`. On `{det A != 0}` (probability one) and for `r` small enough (depending on the realization), the +eigenvalue of `H_M` continuing the `x`-direction is `-6kr + O(r^2 K) < 0` and that of `H_S` is `6kr + O(r^2K) > 0`, +the remaining eigenvalues being those of `A` up to `O(rK)`; hence `1{index H_M = d} 1{index H_S = d - 1} -> +1{A negative definite}` and `W_r / r^2 -> 36 k^2 (det A)^2 1{A negative definite}` pointwise. Domination: +`W_r / r^2 <= C K^(2d)` ([LP], [SC] section 1) with all moments of `K` bounded uniformly in `r`, so dominated +convergence gives `Z_r / r^2 -> 36 k^2 E[(det A)^2 1{A negative definite}]` with `A` distributed as in Lemma 1. QED + +The `d = 2` case, `z0 = 36 k^2 m_(2,b)`, is Math-#168's identification. The Monte Carlo evaluation of `E_Q W_r / +r^2` under the exact finite-`r` endpoint regression (`f`, `grad f` at both pins; `r = 0.1, 0.05`; section 5) +agrees with `36 k^2 m_(d,b)` within statistical error in `d = 2` and `d = 3`. + +## 4. Proof of Theorem 1 + +In (17) write `tau = (a, beta, c, xi)` in the eigenframe `(x, e_soft, e_hard)`: `a = tau_(xxz)`, `beta = +tau_(xzz)`, `c = tau_(zzz)` and `xi` the remaining `(m+2)(m+1)m/6 + m(m+1)/2 + m - 3` tensor entries. By Lemma +1(3), `h_0(O diag(0, -h) O^t, Rot_O tau) = h_0(diag(0, -h), tau)`, and by Lemma 1(1)–(2) this equals + + p_A(diag(0, -h)) p_T(tau), p_A(diag(0, -h)) = phi_2(b) prod_(j=2..m) phi_2(h_j - b) (2 pi)^(-m(m-1)/4), + +the density of `-b I + G_m` in the upper-triangular coordinates at a matrix with diagonal `(0, -h_2, ..., -h_m)` +and zero off-diagonal entries (`phi_2(0 + b)` for the soft entry, `phi_2(-h_j + b)` for the hard ones, +`phi_1(0) = (2 pi)^(-1/2)` for each of the `m(m-1)/2` off-diagonal entries). The integrand of (17) depends on +`xi` only through `p_T`, and the `xi`-marginal of `p_T` is the planar odd density `p_odd(a, beta, c) = N(0, +diag(2, 2, 6))` (Lemma 1(2)). The normalized Haar integral is `1`. What remains is + + integral g dM^(d) = (c_m / z0^(d)) phi_2(b) (2 pi)^(-m(m-1)/4) + integral_(0=2) h_j^3 prod_(2<=i= 3` products inherit it. `R_d(b)` itself is a + quadrature value converged to `1e-14`, and exact for `(d, b) = (3, 0)`. +- The Monte Carlo columns are consistency checks with reported standard errors, not enclosures. + +## 7. Verification and review obligations + +`python -B -S factorization.py --check` and `-B -O -S` exit `0`: exact Lemma 1 in `d = 2..5`; `c_1 = 1`, `c_2 = +pi`; eigenvalue-density normalization; `m_(2,0) = 1`, `m_(3,0) = (7 - 4 sqrt2)/2`, `N_3(0) = 2 sqrt2`, `R_3(0) = +(32 + 28 sqrt2)/17`, `R_2 = 1`; replay of every `N_d, m_d, R_d` against `RESULTS.json` to `1e-9`; the Monte Carlo +and finite-`r` entries within `4` standard errors (plus the stated bin and `O(r)` allowances). Mutants +`drop-vandermonde`, `hard-power-two`, `drop-spectral-constant` exit `1`. The full run regenerates `RESULTS.json` +deterministically (seeded) in about ten minutes. + +Requested bounded nonauthor read (OpenAI or xAI): **Slice A** Lemma 1 and Lemma 2 (the conditional jet law and +the normalizer identification in `d >= 3`, including the index bookkeeping of `H_M`, `H_S`); **Slice B** the proof +of Theorem 1 (the `p_A` density at `diag(0, -h)` in the upper-triangular coordinates, the trivial Haar integral, the +`xi`-marginal, the `c_m` bookkeeping against [SC] (8)–(9) and (17)) and the closed form (3); **Slice C** whether +the corollaries and non-claims are stated at the right conditional scope. The author will not merge. diff --git a/frontiers/c6_hard_direction_factorization_20260930/README.md b/frontiers/c6_hard_direction_factorization_20260930/README.md new file mode 100644 index 000000000..31062e2cc --- /dev/null +++ b/frontiers/c6_hard_direction_factorization_20260930/README.md @@ -0,0 +1,19 @@ +# Hard-direction factorization of the near cluster coefficients (Gaussian kernel) + +Object `CL-C6-HARD-DIRECTION-FACTORIZATION-20260930-v1`. Scientific effect NONE; not certified; nonauthor review open. + +`PROOF.md` proves that for the unit Gaussian kernel the [SC] (17) near cluster measure in dimension `d` is the +planar measure times an explicit factor: for every functional `g` of the soft cubic `(s, a, beta, c)`, +`integral g dM^(d) = R_d(b) integral g dM^(2)` with `R_d(b) = N_d(b) m_(2,b) / m_(d,b)` (a shifted-GOE eigenvalue +integral over a conditioned determinant moment). Consequently `a_j^(d) = R_d(b) a_j^(2)`, `C_*^(d) = R_d(b) C_*^(2)`, +`a_2/a_1` and every normalized microscopic shape law are dimension-independent, and `R_3(0) = (32 + 28 sqrt2)/17`. + +`factorization.py` (standard library) checks Lemma 1 exactly in `d = 2..5`, evaluates `R_d(b)` for `d = 2..5`, +`b = 0, 1`, cross-checks by Monte Carlo and by a finite-`r` evaluation of `Z_r / r^2`, and assembles the `d = 3, 4, 5` +near coefficients from Math-#168's planar values. + + python -B -S factorization.py --check # rc 0 (also with -O); under ten seconds + python -B -S factorization.py --check --mutant drop-vandermonde # rc 1 (also hard-power-two, drop-spectral-constant) + python -B -S factorization.py # regenerates RESULTS.json (about ten minutes) + +`SOURCE_MAP.json` pins the consumed sources on `main`; `SOURCE_FILES.json` is the packet manifest. diff --git a/frontiers/c6_hard_direction_factorization_20260930/RESULTS.json b/frontiers/c6_hard_direction_factorization_20260930/RESULTS.json new file mode 100644 index 000000000..c24366a1b --- /dev/null +++ b/frontiers/c6_hard_direction_factorization_20260930/RESULTS.json @@ -0,0 +1,525 @@ +{ + "assembled_near_coefficients": { + "k=1,b=0,d=2": { + "R_d": 1.0, + "a1": 1.32906, + "a1_plus_a2": 1.3545691, + "a2": 0.0255091, + "a2_over_a1": 0.019193339653589755, + "planar_source": "Math-#168 RESULTS.json prefactor/k=1,b=0 (GH60, not certified)", + "z0_d": 36.00000000000001 + }, + "k=1,b=0,d=3": { + "R_d": 4.211645867438039, + "a1": 5.597530056577199, + "a1_plus_a2": 5.704965352174263, + "a2": 0.10743529559706368, + "a2_over_a1": 0.019193339653589755, + "planar_source": "Math-#168 RESULTS.json prefactor/k=1,b=0 (GH60, not certified)", + "z0_d": 24.176623509137166 + }, + "k=1,b=0,d=4": { + "R_d": 11.666954393023346, + "a1": 15.506082405591608, + "a1_plus_a2": 15.80369591189868, + "a2": 0.29761350630707184, + "a2_over_a1": 0.019193339653589755, + "planar_source": "Math-#168 RESULTS.json prefactor/k=1,b=0 (GH60, not certified)", + 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"main_at_creation": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", + "note": "Exact reduction of the [SC] (17) near cluster measure in dimension d to the planar one for the unit Gaussian kernel (continuum). Consumes [SC] (8), (17), (20), [LP]'s Q_r / W_r / z0 interface, [CUB]'s classifier and [R] (R11)-(R13) at their stated scopes; multiplies Math-#168's floating planar values. No source theorem is revalidated or promoted; elder corollaries are conditional on unmerged #170/#175/#179/#181 and are quoted only.", + "review_status": "AUTHOR_SIDE_CANDIDATE", + "nonauthor_acceptance": false, + "governing_status_changed": false, + "nonauthor_reads": [], + "sources": [ + {"id": "SC", "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", "bytes": 25006, "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", "commit": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", "role": "section 2 (4), (7)-(9): spectral coordinates, c_m, h_0; section 5 (17): the near measure dM; section 6 (19)-(20): the coefficients a_1, a_2", "current_required": true, "on_main": true}, + {"id": "LP", "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", "bytes": 40261, "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", "commit": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", "role": "the model and kernel (section 2), the endpoint regression Q_r, the weight W_r = F_d(H_M) F_(d-1)(H_S), the full normalizer Z_r / r^2 -> z0 and the uniform C^q moment bounds used in Lemma 2", "current_required": true, "on_main": true}, + {"id": "CUB", "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", "bytes": 19889, "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", "commit": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", "role": "the exact classifier n(s, a, beta, c) in {0, 1, 2} and the planar contact structure (G6)-(G11)", "current_required": true, "on_main": true}, + {"id": "R", "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", "bytes": 12533, "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", "commit": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", "role": "(R10)-(R13): the root-resolved representation and C_* = (216/11) k^7 (c_m/z0) I J_cusp, reduced by the same factor", "current_required": true, "on_main": true}, + {"id": "Q", "path": "frontiers/radial_rate_20260930/PROOF.md", "bytes": 15755, "sha256": "a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710", "git_blob": "65f24b1f9a7b80522a2ee059f5f6877c9f970f0d", "commit": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", "role": "(21): the aligned planar periodic odd variances, whose continuum limit (2, 2, 6) is Lemma 1(2) in d = 2", "current_required": true, "on_main": true}, + {"id": "NUM", "path": "frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json", "bytes": 4095, "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb", "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f", "commit": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", "role": "Math-#168: the planar values alpha_1, alpha_2 (GH60, floating, not certified) and m_(2,b), z0^(2), multiplied by R_d(b) in section 5; read at run time by factorization.py", "current_required": true, "on_main": true} + ], + "companions_unmerged": [ + {"id": "E", "pr": 169, "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", "use": "shape constants I, J, D and the whole-cluster law: dimension-independent by Corollary 1; quoted only"}, + {"id": "ELDER2", "pr": 170, "path": "frontiers/local_elder_geometry_20260930/PROOF.md", "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", "use": "Theorem S: r^-3 (1 - p_r) -> alpha_1 + alpha_2 in d = 2 (Corollary 2, conditional)"}, + {"id": "ELDERD", "pr": 175, "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", "use": "Theorem F / M1 in d >= 3 (Corollary 2, conditional)"}, + {"id": "ELD", "pr": 181, "path": "frontiers/elder_partner_extremes_20260930/PROOF.md", "git_blob": "91f88e392fb8aa947dac4fc81c2b45f783fa66cd", "use": "elder-selected laws (S \\ O2, E0, lifetime density): dimension-independent by Corollary 1, conditional; quoted only"}, + {"id": "ELD179", "pr": 179, "path": "frontiers/elder_selected_extremes_20260930/PROOF.md", "git_blob": "1310dc11bae684a9889548d1e764e83381b39d57", "use": "shared elder-selected base (same statement as ELD sections 1-3); quoted only"} + ] +} diff --git a/frontiers/c6_hard_direction_factorization_20260930/factorization.py b/frontiers/c6_hard_direction_factorization_20260930/factorization.py new file mode 100644 index 000000000..500a957cc --- /dev/null +++ b/frontiers/c6_hard_direction_factorization_20260930/factorization.py @@ -0,0 +1,579 @@ +#!/usr/bin/env python3 +"""Hard-direction factorization of the near cluster coefficients for the Gaussian kernel (Math-, object +CL-C6-HARD-DIRECTION-FACTORIZATION-20260930-v1). Standard library only. + + python -B -S factorization.py # full run: writes RESULTS.json (about two minutes) + python -B -S factorization.py --check # exact lemma, closed forms and replay against RESULTS.json (rc 0) + python -B -S factorization.py --check --mutant NAME # must exit 1 + +Objects (see PROOF.md). For the unit Gaussian kernel K(x) = exp(-|x|^2/2) in dimension d = m + 1, the [SC] (17) +measure factors through the hard directions: for every nonnegative measurable g of the soft cubic (s, a, beta, c), + + integral g dM^(d) = R_d(b) integral g dM^(2), R_d(b) = N_d(b) m_(2,b) / m_(d,b), + + N_d(b) = c_m (2 pi)^(-m(m-1)/4) integral_(0 < h_2 < ... < h_m) prod h_j^3 phi_2(h_j - b) prod_(i= 5 else 60 + + +def N_d(d, b, n=None, L=14.0): + """Hard-direction factor: the m - 1 hard eigenvalues h > 0 with weight h^3 phi_2(h - b) and their Vandermonde, + times c_m (2 pi)^(-m(m-1)/4); the soft eigenvalue sits at 0 and contributes phi_2(b) on both sides of the ratio.""" + m = d - 1 + n = n or default_order(d) + key = ('N', d, b, n, L, MUT) + if key in _CACHE: + return _CACHE[key] + pref = c_m(m) * (2 * pi) ** (-m * (m - 1) / 4) + if m == 1: + _CACHE[key] = pref + return pref + power = 2 if MUT == 'hard-power-two' else 3 + rule = gauss_legendre(n) + k = m - 1 + + def rec(prev, depth): + if depth == k: + return 1.0 + + def f(h): + v = h ** power * phi2(h - b) + if MUT != 'drop-vandermonde': + for p in prev: + v *= (h - p) + return v * rec(prev + [h], depth + 1) + return gl(f, prev[-1] if prev else 0.0, L, rule) + _CACHE[key] = pref * rec([], 0) + return _CACHE[key] + + +def m_d(d, b, n=None, L=14.0): + """E[(det A)^2 1{A negative definite}] for A = -b I_m + GOE_m, by the ordered eigenvalue density.""" + m = d - 1 + if m == 1: + return m2(b) + n = n or default_order(d) + key = ('m', d, b, n, L, MUT) + if key not in _CACHE: + _CACHE[key] = ordered_integral(m, lambda l: l * l * phi2(l + b), -L, 0.0, gauss_legendre(n)) + return _CACHE[key] + + +def normalization(d, b, n=None, L=14.0): + """Total mass of the eigenvalue density: must be 1 (tests c_m, the (2 pi) power and the quadrature).""" + m = d - 1 + n = n or default_order(d) + return ordered_integral(m, lambda l: phi2(l + b), -L - abs(b), L + abs(b), gauss_legendre(n)) + + +def R_d(d, b, n=None): + return N_d(d, b, n) * m2(b) / m_d(d, b, n) + + +def closed_forms(): + """b = 0, d = 3: m_(3,0) = (7 - 4 sqrt2)/2, N_3(0) = 2 sqrt2, R_3(0) = (32 + 28 sqrt2)/17; d = 2: R_2 = 1.""" + return {'m_3_0': (7 - 4 * sqrt(2)) / 2, 'N_3_0': 2 * sqrt(2), 'R_3_0': (32 + 28 * sqrt(2)) / 17, + 'E_absA3_negative_b0': 4 / sqrt(pi)} + + +# ----------------------------------------------------------------------------- Monte Carlo controls +def det(M): + n = len(M) + A = [r[:] for r in M] + dt = 1.0 + for c in range(n): + p = max(range(c, n), key=lambda r: abs(A[r][c])) + if A[p][c] == 0: + return 0.0 + if p != c: + A[c], A[p] = A[p], A[c] + dt = -dt + dt *= A[c][c] + for r in range(c + 1, n): + f = A[r][c] / A[c][c] + for j in range(c, n): + A[r][j] -= f * A[c][j] + return dt + + +def n_negative(M): + """Negative eigenvalues of a symmetric matrix from the signs of its leading principal minors (generic case).""" + n = len(M) + seq = [1.0] + [det([row[:k] for row in M[:k]]) for k in range(1, n + 1)] + return sum(1 for i in range(n) if seq[i] * seq[i + 1] < 0) + + +def mc_m_d(d, b, n, seed): + rng = random.Random(seed) + m = d - 1 + acc = acc2 = 0.0 + for _ in range(n): + A = [[0.0] * m for _ in range(m)] + for i in range(m): + A[i][i] = -b + sqrt(2) * rng.gauss(0, 1) + for j in range(i + 1, m): + A[i][j] = A[j][i] = rng.gauss(0, 1) + v = det(A) ** 2 if n_negative(A) == m else 0.0 + acc += v + acc2 += v * v + mu = acc / n + return {'value': mu, 'se': sqrt(max(acc2 / n - mu * mu, 0.0) / n), 'samples': n} + + +def mc_N_3(b, n, seed, eps=0.05): + """N_3(b) phi_2(b) = lim E[l_other^2 1{|l_max| < eps}]/(2 eps) for A = -b I_2 + GOE_2 (density of the largest + eigenvalue at 0 times the conditional second moment of the other one).""" + rng = random.Random(seed) + acc = acc2 = 0.0 + for _ in range(n): + p = -b + sqrt(2) * rng.gauss(0, 1) + q = -b + sqrt(2) * rng.gauss(0, 1) + g = rng.gauss(0, 1) + mid, rad = 0.5 * (p + q), sqrt(0.25 * (p - q) ** 2 + g * g) + lmax, lother = mid + rad, mid - rad + v = lother * lother / (2 * eps) if abs(lmax) < eps else 0.0 + acc += v + acc2 += v * v + mu = acc / n + return {'value': mu / phi2(b), 'se': sqrt(max(acc2 / n - mu * mu, 0.0) / n) / phi2(b), 'samples': n, 'eps': eps} + + +# ----------------------------------------------------------------------------- finite-r normalizer Z_r / r^2 (Monte Carlo) +def hermite(n, x): + if n == 0: + return 1.0 + h0, h1 = 1.0, x + for k in range(1, n): + h0, h1 = h1, x * h1 - k * h0 + return h1 + + +def kernel_derivative(gam, y): + """d^gam K(y) for K(y) = exp(-|y|^2/2): prod (-1)^gam_i He_gam_i(y_i) exp(-y_i^2/2).""" + v = 1.0 + for g, yi in zip(gam, y): + v *= (-1) ** g * hermite(g, yi) * exp(-yi * yi / 2) + return v + + +def cov_two_point(al, s, be, t): + """Cov(d^al f(s), d^be f(t)) = (-1)^|al| d^(al+be) K(t - s).""" + return (-1) ** sum(al) * kernel_derivative(add(al, be), [ti - si for si, ti in zip(s, t)]) + + +def cholesky(S): + n = len(S) + L = [[0.0] * n for _ in range(n)] + for i in range(n): + for j in range(i + 1): + s = S[i][j] - sum(L[i][k] * L[j][k] for k in range(j)) + if i == j: + if s <= 0: + raise ValueError('not positive definite') + L[i][i] = sqrt(s) + else: + L[i][j] = s / L[j][j] + return L + + +def jacobi_eigen(S, sweeps=100): + """Eigen decomposition of a small symmetric matrix by cyclic Jacobi rotations: returns (eigenvalues, V) with S = V diag V^T.""" + n = len(S) + A = [row[:] for row in S] + V = [[float(i == j) for j in range(n)] for i in range(n)] + for _ in range(sweeps): + off = sum(A[i][j] ** 2 for i in range(n) for j in range(n) if i != j) + if off < 1e-30: + break + for p in range(n - 1): + for q in range(p + 1, n): + if abs(A[p][q]) < 1e-300: + continue + theta = 0.5 * math.atan2(2 * A[p][q], A[q][q] - A[p][p]) + c, s_ = math.cos(theta), math.sin(theta) + for k in range(n): + akp, akq = A[k][p], A[k][q] + A[k][p], A[k][q] = c * akp - s_ * akq, s_ * akp + c * akq + for k in range(n): + apk, aqk = A[p][k], A[q][k] + A[p][k], A[q][k] = c * apk - s_ * aqk, s_ * apk + c * aqk + for k in range(n): + vkp, vkq = V[k][p], V[k][q] + V[k][p], V[k][q] = c * vkp - s_ * vkq, s_ * vkp + c * vkq + return [A[i][i] for i in range(n)], V + + +def sym_sqrt(S): + """Symmetric square root with negative rounding eigenvalues clipped to zero (S = L L^T).""" + lam, V = jacobi_eigen(S) + scale = max(abs(l) for l in lam) + for l in lam: + if l < -1e-9 * scale: + raise ValueError('conditional covariance has a genuinely negative eigenvalue: %r' % l) + root = [sqrt(max(l, 0.0)) for l in lam] + n = len(S) + return [[sum(V[i][k] * root[k] * V[j][k] for k in range(n)) for j in range(n)] for i in range(n)] + + +def solve_spd(S, B): + """Solve S X = B (S symmetric positive definite, B a list of columns) by Cholesky.""" + L = cholesky(S) + n = len(S) + out = [] + for col in B: + y = [0.0] * n + for i in range(n): + y[i] = (col[i] - sum(L[i][k] * y[k] for k in range(i))) / L[i][i] + x = [0.0] * n + for i in reversed(range(n)): + x[i] = (y[i] - sum(L[k][i] * x[k] for k in range(i + 1, n))) / L[i][i] + out.append(x) + return out + + +def finite_r_normalizer(d, r, b, k, n, seed): + """Monte Carlo E_Q[F_d(H_M) F_(d-1)(H_S)] / r^2 under the endpoint regression Q_r (pins f, grad f at M and S), + compared with 36 k^2 m_(d,b) = lim Z_r / r^2 (Lemma 2 of PROOF.md).""" + M = [-r / 2] + [0.0] * (d - 1) + S = [r / 2] + [0.0] * (d - 1) + zero = (0,) * d + obs = [(zero, M)] + [(unit(d, i), M) for i in range(d)] + [(zero, S)] + [(unit(d, i), S) for i in range(d)] + vals = [b] + [0.0] * d + [b - k * r ** 3] + [0.0] * d + pairs = [(i, j) for i in range(d) for j in range(i, d)] + targets = [(add(unit(d, i), unit(d, j)), M) for i, j in pairs] + [(add(unit(d, i), unit(d, j)), S) for i, j in pairs] + Soo = [[cov_two_point(a, s, c, t) for (c, t) in obs] for (a, s) in obs] + Sto = [[cov_two_point(a, s, c, t) for (c, t) in obs] for (a, s) in targets] + Stt = [[cov_two_point(a, s, c, t) for (c, t) in targets] for (a, s) in targets] + W = solve_spd(Soo, Sto) # W[i] = Soo^{-1} Sto[i] + mean = [sum(W[i][j] * vals[j] for j in range(len(obs))) for i in range(len(targets))] + cc = [[Stt[i][j] - sum(W[i][l] * Sto[j][l] for l in range(len(obs))) for j in range(len(targets))] for i in range(len(targets))] + # the conditional covariance is nearly singular at small r (f_xx at the two pins is pinned to O(r^2) precision), + # so sample through a symmetric square root with the tiny negative rounding eigenvalues clipped to zero + L = sym_sqrt(cc) + rng = random.Random(seed) + nt = len(targets) + acc = acc2 = 0.0 + for _ in range(n): + z = [rng.gauss(0, 1) for _ in range(nt)] + x = [mean[i] + sum(L[i][j] * z[j] for j in range(nt)) for i in range(nt)] + HM = [[0.0] * d for _ in range(d)] + HS = [[0.0] * d for _ in range(d)] + for idx, (i, j) in enumerate(pairs): + HM[i][j] = HM[j][i] = x[idx] + HS[i][j] = HS[j][i] = x[len(pairs) + idx] + v = abs(det(HM)) * abs(det(HS)) if (n_negative(HM) == d and n_negative(HS) == d - 1) else 0.0 + acc += v + acc2 += v * v + mu = acc / n + return {'d': d, 'r': r, 'b': b, 'k': k, 'Z_r_over_r2': mu / r ** 2, 'se': sqrt(max(acc2 / n - mu * mu, 0.0) / n) / r ** 2, + 'limit_36k2_m_d': 36 * k * k * m_d(d, b), 'samples': n} + + +# ----------------------------------------------------------------------------- assembly with Math-#168's planar values +def planar_values(): + path = os.path.join(HERE, '..', 'c6_cluster_coefficients_numerics_20260930', 'RESULTS.json') + with open(path) as fh: + ref = json.load(fh) + return ref['prefactor'] + + +def assemble(table): + pre = planar_values() + out = {} + for key, row in pre.items(): + kstr, bstr = key.split(',') + b = float(bstr.split('=')[1]) + for d in DIMS: + R = table['R_d']['d=%d' % d]['b=%s' % b] + out['%s,d=%d' % (key, d)] = {'a1': R * row['alpha1_gh60'], 'a2': R * row['alpha2_gh60'], + 'a1_plus_a2': R * (row['alpha1_gh60'] + row['alpha2_gh60']), + 'a2_over_a1': row['alpha2_gh60'] / row['alpha1_gh60'], 'R_d': R, + 'z0_d': 36 * float(Fr(kstr.split('=')[1])) ** 2 * m_d(d, b), + 'planar_source': 'Math-#168 RESULTS.json prefactor/%s (GH60, not certified)' % key} + return out + + +def full_run(fast=False): + res = {'schema': 1, 'object': 'CL-C6-HARD-DIRECTION-FACTORIZATION-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, + 'kernel': 'exp(-|x|^2/2), continuum (torus images neglected; see PROOF.md section 6)', 'mutant': MUT, + 'python_requirement': '>= 3.11'} + res['lemma_one'] = {'d=%d' % d: lemma_one(d) for d in (2, 3, 4, 5)} + res['c_m'] = {'m=%d' % m: c_m(m) for m in (1, 2, 3, 4)} + res['closed_forms'] = closed_forms() + tab = {'N_d': {}, 'm_d': {}, 'R_d': {}, 'normalization': {}, 'm_2': {'b=%s' % b: m2(b) for b in BS}} + for d in DIMS: + for key in ('N_d', 'm_d', 'R_d', 'normalization'): + tab[key].setdefault('d=%d' % d, {}) + for b in BS: + tab['N_d']['d=%d' % d]['b=%s' % b] = N_d(d, b) + tab['m_d']['d=%d' % d]['b=%s' % b] = m_d(d, b) + tab['R_d']['d=%d' % d]['b=%s' % b] = R_d(d, b) + tab['normalization']['d=%d' % d]['b=%s' % b] = normalization(d, b) + res['table'] = tab + res['quadrature_orders'] = {'d=%d,b=0.0' % d: {'n=%d' % n: {'N_d': N_d(d, 0.0, n), 'm_d': m_d(d, 0.0, n)} for n in ((30, 40, 50) if d == 5 else (40, 60, 80))} for d in (3, 4, 5)} + nmc = 20000 if fast else 200000 + res['monte_carlo'] = {'m_d': {'d=%d,b=%s' % (d, b): mc_m_d(d, b, nmc if d < 5 else nmc // 2, 100 + d) for d in (3, 4, 5) for b in BS}, + 'N_3': {'b=%s' % b: mc_N_3(b, 5 * nmc, 200) for b in BS}} + nz = 4000 if fast else 400000 + res['finite_r_normalizer'] = {'d=%d,r=%s,b=%s' % (d, r, b): finite_r_normalizer(d, r, b, 1.0, nz, 300 + int(100 * r)) + for d in (2, 3) for r in (0.1, 0.05) for b in BS} + res['assembled_near_coefficients'] = assemble(res['table']) + return res + + +def check_run(): + for d in (2, 3, 4, 5): + lemma_one(d) + cf = closed_forms() + require(abs(c_m(1) - 1) < 1e-14 and abs(c_m(2) - pi) < 1e-13, 'c_1 = 1, c_2 = pi') + require(abs(m2(0.0) - 1) < 1e-14 and abs(m2(1.0) - 2.7201411062) < 1e-9, 'm_(2,0) = 1, m_(2,1) = 2.72014') + require(abs(N_d(3, 0.0) - cf['N_3_0']) < 1e-10, 'N_3(0) = 2 sqrt2') + require(abs(m_d(3, 0.0) - cf['m_3_0']) < 1e-10, 'm_(3,0) = (7 - 4 sqrt2)/2') + require(abs(R_d(3, 0.0) - cf['R_3_0']) < 1e-9, 'R_3(0) = (32 + 28 sqrt2)/17') + require(abs(R_d(2, 0.0) - 1) < 1e-12 and abs(R_d(2, 1.0) - 1) < 1e-12, 'R_2 = 1 (planar case reproduces itself)') + for d in DIMS: + for b in BS: + # the full-line normalization integral is the least well resolved piece at order 40 in four dimensions + tol = 1e-5 if d >= 5 else 1e-9 + require(abs(normalization(d, b) - 1) < tol, 'eigenvalue density normalizes to 1 (d=%d, b=%s)' % (d, b)) + require(N_d(4, 0.0) > N_d(3, 0.0) and m_d(4, 0.0) < m_d(3, 0.0), 'monotonicity sanity d = 3 -> 4 at b = 0') + path = os.path.join(HERE, 'RESULTS.json') + require(os.path.exists(path), 'RESULTS.json present') + with open(path) as fh: + ref = json.load(fh) + for d in DIMS: + for b in BS: + for key, fn in (('N_d', N_d), ('m_d', m_d), ('R_d', R_d)): + want = ref['table'][key]['d=%d' % d]['b=%s' % b] + require(abs(fn(d, b) - want) <= REPLAY_TOL * max(abs(want), 1.0), 'replay %s d=%d b=%s' % (key, d, b)) + for key, row in ref['monte_carlo']['m_d'].items(): + d = int(key.split(',')[0][2:]) + b = float(key.split('=')[-1]) + require(abs(row['value'] - m_d(d, b)) <= 4 * row['se'] + 1e-12, 'Monte Carlo m_d within 4 s.e. (%s)' % key) + for key, row in ref['monte_carlo']['N_3'].items(): + b = float(key.split('=')[-1]) + require(abs(row['value'] - N_d(3, b)) <= 4 * row['se'] + 0.02 * N_d(3, b), 'Monte Carlo N_3 within 4 s.e. + 2%% bin bias (%s)' % key) + for key, row in ref['finite_r_normalizer'].items(): + require(abs(row['Z_r_over_r2'] - row['limit_36k2_m_d']) <= 4 * row['se'] + 0.15 * row['r'] * row['limit_36k2_m_d'] + 1e-9, + 'finite-r normalizer within 4 s.e. + O(r) of 36 k^2 m_d (%s)' % key) + for key, row in ref['assembled_near_coefficients'].items(): + d = int(key.split('d=')[1]) + b = float(key.split('b=')[1].split(',')[0]) + require(abs(row['R_d'] - R_d(d, b)) <= REPLAY_TOL * row['R_d'], 'assembled R_d replay (%s)' % key) + print(json.dumps({'check': 'ok', 'R_3_0': R_d(3, 0.0), 'closed_form': cf['R_3_0'], 'mutant': MUT})) + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--fast', action='store_true', help='fewer Monte Carlo samples (development only)') + ap.add_argument('--mutant', choices=MUTANTS) + args = ap.parse_args() + MUT = args.mutant + if args.check: + check_run() + return + res = full_run(fast=args.fast) + with open(os.path.join(HERE, 'RESULTS.json'), 'w') as fh: + json.dump(res, fh, indent=1, sort_keys=True) + fh.write('\n') + print('wrote RESULTS.json') + + +if __name__ == '__main__': + main() From 7dc9ee5107dc7ccf8a66527bb06eaea1af0c717d Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 10:50:44 +0000 Subject: [PATCH 2/4] Hard-direction factorization: four Codex repairs (joint-regression wording in Lemma 1, actual d = 5 normalization tolerance, no finite-torus error claim, stable tail evaluation of m_(2,b)); numbers unchanged Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../PROOF.md | 38 ++++++++++++------- .../SOURCE_FILES.json | 12 +++--- .../factorization.py | 15 +++++++- 3 files changed, 45 insertions(+), 20 deletions(-) diff --git a/frontiers/c6_hard_direction_factorization_20260930/PROOF.md b/frontiers/c6_hard_direction_factorization_20260930/PROOF.md index 23cc65dd8..823bd4438 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/PROOF.md +++ b/frontiers/c6_hard_direction_factorization_20260930/PROOF.md @@ -97,10 +97,16 @@ the even class the unconditional covariances are `Var f = 1`, `Var f_xx = Var A_ conditioning on `(f, f_xx) = (b, 0)` (the `f_(x y_i)` are uncorrelated with `A`) gives `E[A_ii] = -b`, `Var A_ii = 3 - 1 = 2`, `Cov(A_ii, A_jj) = 1 - 1 = 0`, and `A_ij` untouched: item 1. For the odd class, `Cov(T_(x x y_i), f_(y_i)) = -1`, `Cov(T_(x y_i y_i), f_x) = -1`, `Cov(T_(y_i y_i y_i), f_(y_i)) = -3`, -`Cov(T_(y_i y_i y_j), f_(y_j)) = -1`, `Cov(T, f_xxx) = 0`, and the unconditional variances are `15`, `3`, `1` by -multiplicity; conditioning on `grad f = 0`, `f_xxx = 12k` leaves the means at `0` (the only nonzero observed values -`b`, `12k` have zero coefficients) and the variances `15 - 9 = 6`, `3 - 1 = 2`, `1`, with all cross terms -cancelling: item 2. Item 3: the three blocks of `T` (`T_(xx .)` an isotropic vector, `T_(x . .)` a GOE-type +`Cov(T_(y_i y_i y_j), f_(y_j)) = -1`, `Cov(T_(x y_i y_i), f_xxx) = 3` (the only nonzero covariance of a component +of `T` with `f_xxx`), `Cov(f_x, f_xxx) = -3`, `Var f_xxx = 15`, and the unconditional variances of `T` are `15`, +`3`, `1` by multiplicity. Regress jointly on `(grad f, f_xxx)`: for `T_(x y_i y_i)` the coefficients on +`(f_x, f_xxx)` are `(-1, 3) [[1, -3], [-3, 15]]^(-1) = (-1, 0)`, so the coefficient on `f_xxx` vanishes although the +raw covariance does not, the observed value `12k` never enters, and the conditional variance is `3 - 1 = 2`; +for `T_(y_i y_i y_i)` the coefficient on `f_(y_i)` is `-3` and the variance `15 - 9 = 6`; for `T_(y_i y_i y_j)` (`i != j`; unconditional +variance `3`) the coefficient on `f_(y_j)` is `-1` and the variance `3 - 1 = 2`; `T_(xx y_i)` has `Cov(., f_(y_i)) = -1`, variance `3 - 1 = 2`; the +distinct-index components `T_(ijk)` are uncorrelated with `U_0` and keep variance `1`. Every conditional mean is +`0` because the coefficients on the two nonzero observations `f = b` and `f_xxx = 12k` are all zero, and every +conditional cross covariance vanishes: item 2. Item 3: the three blocks of `T` (`T_(xx .)` an isotropic vector, `T_(x . .)` a GOE-type matrix with the variances of item 2, `T_(. . .)` an isotropic symmetric 3-tensor) and `A` are each `O(m)`-invariant and mutually independent, and no value of `U_0` other than `b` enters. The computation is carried out exactly in rational arithmetic for `d = 2, 3, 4, 5` in `factorization.py` (`lemma_one`), including the vanishing of every @@ -183,7 +189,9 @@ reproduces both closed forms to `1e-10` by quadrature. `c_1 = 1`, `c_2 = pi`, `c_3 = 19.7392088`, `c_4 = 194.818182` ([SC] (8) through Mehta's integral, `H_m = (2 pi)^(m/2) prod_(j<=m) Gamma(1 + j/2) / (Gamma(3/2)^m m!)`); the eigenvalue densities normalize to `1` to `1e-9` -for every `(d, b)` used. +for `d <= 4` and to `1.9e-8` (`b = 0`) and `4.6e-6` (`b = 1`) for `d = 5`, where the full-line four-dimensional +integral is evaluated at order `40`; the one-sided integrals `N_5`, `m_5` themselves are converged to `1e-14` +across orders `30/40/50` (`quadrature_orders`). | `d` | `b` | `N_d(b)` | `m_(d,b)` | `m_(d,b)` (Monte Carlo) | `R_d(b)` | |---|---|---|---|---|---| @@ -246,11 +254,13 @@ times the conditional second moment of the other eigenvalue, bin half-width `0.0 ## 6. Non-claims and the torus -- The theorem is for the continuum kernel. The [SC] measure for the periodized field on the torus `T_L^d` uses - the one-site jet covariances of `K_L`, which differ from the continuum values by image terms; Math-#168 measured - the largest relative deviation of the planar one-site table at `8.8e-4` (`L = 6`), `2e-14` (`L = 12`), `1.4e-13` - (`L = 24`). At finite `L` the transverse rotation invariance of Lemma 1(3) is broken to the lattice symmetry and - the factorization holds only up to corrections of that size; they are not quantified here beyond that table. +- The theorem is for the continuum kernel only. The [SC] measure for the periodized field on the torus `T_L^d` + uses the one-site jet covariances of `K_L`, which differ from the continuum values by image terms, and at finite + `L` the transverse rotation invariance of Lemma 1(3) is broken to the lattice symmetry, so the exact step that + extracts a common hard-direction factor is not available. No finite-`L` statement is made and no perturbation + bound is proved. The only finite-`L` data quoted are Math-#168's measured deviations of the planar one-site + covariance table from the continuum (`8.8e-4` relative at `L = 6`, `2e-14` at `L = 12`, `1.4e-13` at `L = 24`); + they are not an error estimate for the factorization in any dimension. - Nothing is said about the remote singleton coefficient `beta_far` ([SC] (22), two-site) or the contact kernel `Lambda(x)`; the dimension dependence of `nu(1) = a_1 + beta_far` is not reduced by this note. - No rate, no finite-`r` statement, no uniformity in `d`, `k`, `b` or `L`. The sources' theorems are consumed at @@ -263,9 +273,11 @@ times the conditional second moment of the other eigenvalue, bin half-width `0.0 ## 7. Verification and review obligations `python -B -S factorization.py --check` and `-B -O -S` exit `0`: exact Lemma 1 in `d = 2..5`; `c_1 = 1`, `c_2 = -pi`; eigenvalue-density normalization; `m_(2,0) = 1`, `m_(3,0) = (7 - 4 sqrt2)/2`, `N_3(0) = 2 sqrt2`, `R_3(0) = -(32 + 28 sqrt2)/17`, `R_2 = 1`; replay of every `N_d, m_d, R_d` against `RESULTS.json` to `1e-9`; the Monte Carlo -and finite-`r` entries within `4` standard errors (plus the stated bin and `O(r)` allowances). Mutants +pi`; eigenvalue-density normalization to `1e-9` for `d <= 4` and to `1e-5` for `d = 5` (observed `4.6e-6` at +`b = 1`, section 5); `m_(2,0) = 1`, `m_(3,0) = (7 - 4 sqrt2)/2`, `N_3(0) = 2 sqrt2`, `R_3(0) = (32 + 28 sqrt2)/17`, +`R_2 = 1`; `m_(2,b) > 0` at `b = -12` (stable tail evaluation); replay of every `N_d, m_d, R_d` against +`RESULTS.json` to `1e-9` relative; the Monte Carlo and finite-`r` entries within `4` standard errors (plus the +stated bin and `O(r)` allowances). Mutants `drop-vandermonde`, `hard-power-two`, `drop-spectral-constant` exit `1`. The full run regenerates `RESULTS.json` deterministically (seeded) in about ten minutes. diff --git a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json index 7f5ce19fc..18dfaa89d 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json +++ b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json @@ -7,9 +7,9 @@ "files": [ { "path": "PROOF.md", - "bytes": 20322, - "sha256": "1155fc8c991d0052a68b2f5a2633778aef2d19b4c41d550caff18dab92e0a8c4", - "git_blob": "23cc65dd80a02f8b97e9dc18939a16eac0ac973f" + "bytes": 21659, + "sha256": "0d8eeeac272201bf2e355fba4c834305e05af89c0d252139d8c90799e7efb9d2", + "git_blob": "823bd4438749f245798029f57b378b0e0c91d760" }, { "path": "README.md", @@ -31,9 +31,9 @@ }, { "path": "factorization.py", - "bytes": 24056, - "sha256": "a585be22148dd41edde0598da52ea2168005fd1fca3f60cf5cfe5f4d6dd08a1a", - "git_blob": "500a957cc6e27243056a50475057327e76fd793d" + "bytes": 24761, + "sha256": "1fb7868b061b172b7f5349409d65cf03a601907f728cf2d6bad4270b41870db2", + "git_blob": "69a725580a30e7aa64e05d769c4ff987cb16a574" } ] } diff --git a/frontiers/c6_hard_direction_factorization_20260930/factorization.py b/frontiers/c6_hard_direction_factorization_20260930/factorization.py index 500a957cc..69a725580 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/factorization.py +++ b/frontiers/c6_hard_direction_factorization_20260930/factorization.py @@ -179,7 +179,18 @@ def c_m(m): def m2(b): - """E[A^2 1{A < 0}], A ~ N(-b, 2) (Math-#168's m_(2,b)).""" + """E[A^2 1{A < 0}], A ~ N(-b, 2) (Math-#168's m_(2,b)). The closed form (mu^2 + s^2) Phi(-mu/s) - mu s phi(mu/s) + cancels catastrophically when the mean -b is far above zero (b <= -4: the two terms agree to their leading + digits), so there the moment is integrated directly as a positive integrand.""" + mu, s = -b, sqrt(2) + t = mu / s + if t > 2.5: + rule = gauss_legendre(200) + return gl(lambda a: a * a * phi((a - mu) / s) / s, -(abs(mu) + 40.0), 0.0, rule) + return (mu * mu + s * s) * Phi(-t) - mu * s * phi(t) + + +def _m2_closed(b): mu, s = -b, sqrt(2) t = mu / s return (mu * mu + s * s) * Phi(-t) - mu * s * phi(t) @@ -521,6 +532,8 @@ def check_run(): cf = closed_forms() require(abs(c_m(1) - 1) < 1e-14 and abs(c_m(2) - pi) < 1e-13, 'c_1 = 1, c_2 = pi') require(abs(m2(0.0) - 1) < 1e-14 and abs(m2(1.0) - 2.7201411062) < 1e-9, 'm_(2,0) = 1, m_(2,1) = 2.72014') + require(m2(-12.0) > 0 and abs(m2(-3.6) - _m2_closed(-3.6)) < 1e-8 * _m2_closed(-3.6), + 'm_(2,b) positive at b = -12 and the direct tail integral matches the closed form at b = -3.6') require(abs(N_d(3, 0.0) - cf['N_3_0']) < 1e-10, 'N_3(0) = 2 sqrt2') require(abs(m_d(3, 0.0) - cf['m_3_0']) < 1e-10, 'm_(3,0) = (7 - 4 sqrt2)/2') require(abs(R_d(3, 0.0) - cf['R_3_0']) < 1e-9, 'R_3(0) = (32 + 28 sqrt2)/17') From d29e6741404f1e7bc06eb1ad91b547399715fb0a Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 10:53:19 +0000 Subject: [PATCH 3/4] Hard-direction factorization: finite-L scope sentence replaced per the Astra Slice C read (5365199521); positive finite normalizing mass and continuum-model caveat on the elder corollary; read recorded Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../PROOF.md | 25 ++-- .../SOURCE_FILES.json | 12 +- .../SOURCE_MAP.json | 123 ++++++++++++++++-- 3 files changed, 133 insertions(+), 27 deletions(-) diff --git a/frontiers/c6_hard_direction_factorization_20260930/PROOF.md b/frontiers/c6_hard_direction_factorization_20260930/PROOF.md index 823bd4438..91d22fce0 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/PROOF.md +++ b/frontiers/c6_hard_direction_factorization_20260930/PROOF.md @@ -33,17 +33,21 @@ not depend on `k`. **Corollary 1 (near coefficients).** The [SC] (20) coefficients satisfy `a_j^(d)(b, k) = R_d(b) a_j^(2)(b, k)`, `j = 1, 2`. Hence `a_2 / a_1` (the near doublet-to-singleton ratio, `nu_near(2)/nu_near(1)`), and every law -obtained by normalizing a function of `(s, a, beta, c)` under `dM` — the conditional law of the soft cubic given +obtained by normalizing a function of `(s, a, beta, c)` under `dM` by a positive finite mass (`0 < integral g dM +< infinity`; the factor `R_d(b)` is positive and finite and cancels) — the conditional law of the soft cubic given `n >= 1`, the shape region measure `Q du dv` of [R] (R9)–(R10), the Pareto-11 radius law and the constants `E eta = 5771/7062`, `I, J, D, E0` and all ratios of [R], [E], [Q], [ELD] — are the same in every dimension -`d >= 2`. The radial-tail constant of [R] (R13) satisfies `C_*^(d) = R_d(b) C_*^(2)`, since (R12)'s `J_cusp` +`d >= 2`, for the continuum-kernel measures `dM^(d)`. The radial-tail constant of [R] (R13) satisfies `C_*^(d) = R_d(b) C_*^(2)`, since (R12)'s `J_cusp` integrand has the same hard-direction factor (section 3). **Corollary 2 (conditional, elder statements).** Where a source identifies `lim r^(-3) (1 - p_r) = a_1 + a_2` (#170 Theorem S in `d = 2`, #175 Theorem F in `d >= 3`, both unmerged drafts), the elder-failure coefficient in dimension `d` is `R_d(b) (alpha_1 + alpha_2)^(2)`, and the conditional lifetime-fraction and shape laws given failure (#170 section 8.8, #175 M1, #179/#181's `S \ O2` laws) are dimension-independent. This corollary inherits -those sources' conditional status and promotes nothing. +those sources' conditional status and promotes nothing. It concerns the sources' statements read for the same +continuum-kernel model as Theorem 1; the sources are stated for the periodized field on `T_L^d`, and a finite-`L` +theorem does not transfer to the continuum measures by replacing its covariance in notation, so no new +actual-field theorem is claimed here. **Closed form.** `m_(3,0) = (7 - 4 sqrt2)/2`, `N_3(0) = 2 sqrt2`, hence @@ -255,12 +259,15 @@ times the conditional second moment of the other eigenvalue, bin half-width `0.0 ## 6. Non-claims and the torus - The theorem is for the continuum kernel only. The [SC] measure for the periodized field on the torus `T_L^d` - uses the one-site jet covariances of `K_L`, which differ from the continuum values by image terms, and at finite - `L` the transverse rotation invariance of Lemma 1(3) is broken to the lattice symmetry, so the exact step that - extracts a common hard-direction factor is not available. No finite-`L` statement is made and no perturbation - bound is proved. The only finite-`L` data quoted are Math-#168's measured deviations of the planar one-site - covariance table from the continuum (`8.8e-4` relative at `L = 6`, `2e-14` at `L = 12`, `1.4e-13` at `L = 24`); - they are not an error estimate for the factorization in any dimension. + uses the one-site jet covariances of `K_L`, which differ from the continuum values by image terms. At finite `L` + the transverse rotational invariance of Lemma 1(3) is broken, so the exact step that extracts a common + hard-direction factor is not available. The quoted planar one-site deviations of Math-#168 (`8.8e-4` relative at + `L = 6`, `2e-14` at `L = 12`, `1.4e-13` at `L = 24`) are diagnostics for those covariance entries only. No + finite-`L` factorization identity, coefficient error bound, relative accuracy or higher-dimensional correction + is established in this packet. Obtaining one requires a separate perturbation estimate for the relevant + conditional covariance, the numerator integrals and the full normalizer; a small covariance perturbation does + not give a uniform relative error over arbitrary nonnegative `g` (tail indicators `1{x >= a}` under `N(0, 1)` + against `N(0, 1 + delta)` have unbounded relative error as `a -> infinity`). - Nothing is said about the remote singleton coefficient `beta_far` ([SC] (22), two-site) or the contact kernel `Lambda(x)`; the dimension dependence of `nu(1) = a_1 + beta_far` is not reduced by this note. - No rate, no finite-`r` statement, no uniformity in `d`, `k`, `b` or `L`. The sources' theorems are consumed at diff --git a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json index 18dfaa89d..01e08dabf 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json +++ b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json @@ -7,9 +7,9 @@ "files": [ { "path": "PROOF.md", - "bytes": 21659, - "sha256": "0d8eeeac272201bf2e355fba4c834305e05af89c0d252139d8c90799e7efb9d2", - "git_blob": "823bd4438749f245798029f57b378b0e0c91d760" + "bytes": 22456, + "sha256": "26743f82bf4b86d4ae4c7dfb085d4d15bf36014aa627dee5edfc86a75ca64c05", + "git_blob": "91d22fce0fd7d4630bfe2e2c93eab289babc97f2" }, { "path": "README.md", @@ -25,9 +25,9 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 4959, - "sha256": "f05da68f0069a2d4171be6db51f86cacd1e990f03c6606dac7c992f2d3d516ea", - "git_blob": "05205c4cae03a707768070eb272cac5a85f1eb8c" + "bytes": 6212, + "sha256": "9524b2a9ac42d9544cce41f567fbd8f9b58c3fbf66158679c69c08be3b2d5bc3", + "git_blob": "51387c696c68b19ecec49cbe838ae2ab8d2e7f91" }, { "path": "factorization.py", diff --git a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json index 05205c4ca..51387c696 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json +++ b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json @@ -8,20 +8,119 @@ "review_status": "AUTHOR_SIDE_CANDIDATE", "nonauthor_acceptance": false, "governing_status_changed": false, - "nonauthor_reads": [], + "nonauthor_reads": [ + { + "lane": "OpenAI/GPT-6 Astra Pro", + "review_id": 5365199521, + "head": "ee6127f99667a4fda35816eaf9dcb0fc65973ee2", + "date": "2026-09-30", + "slice": "C (scope)", + "disposition": "AMEND the finite-L scope sentence (the planar one-site deviations are not an error estimate for the factorization; applied in the successor commit with the suggested replacement) and two non-blocking precisions (positive finite normalizing mass; elder corollaries refer to the continuum model), also applied; the continuum-only conditional corollaries have the correct structural form subject to Slices A/B; the Lemma 1 raw-covariance sentence corrected at 7dc9ee5. 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read at run time by factorization.py", + "current_required": true, + "on_main": true + } ], "companions_unmerged": [ - {"id": "E", "pr": 169, "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", "use": "shape constants I, J, D and the whole-cluster law: dimension-independent by Corollary 1; quoted only"}, - {"id": "ELDER2", "pr": 170, "path": "frontiers/local_elder_geometry_20260930/PROOF.md", "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", "use": "Theorem S: r^-3 (1 - p_r) -> alpha_1 + alpha_2 in d = 2 (Corollary 2, conditional)"}, - {"id": "ELDERD", "pr": 175, "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", "use": "Theorem F / M1 in d >= 3 (Corollary 2, conditional)"}, - {"id": "ELD", "pr": 181, "path": "frontiers/elder_partner_extremes_20260930/PROOF.md", "git_blob": "91f88e392fb8aa947dac4fc81c2b45f783fa66cd", "use": "elder-selected laws (S \\ O2, E0, lifetime density): dimension-independent by Corollary 1, conditional; quoted only"}, - {"id": "ELD179", "pr": 179, "path": "frontiers/elder_selected_extremes_20260930/PROOF.md", "git_blob": "1310dc11bae684a9889548d1e764e83381b39d57", "use": "shared elder-selected base (same statement as ELD sections 1-3); quoted only"} + { + "id": "E", + "pr": 169, + "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", + "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", + "use": "shape constants I, J, D and the whole-cluster law: dimension-independent by Corollary 1; quoted only" + }, + { + "id": "ELDER2", + "pr": 170, + "path": "frontiers/local_elder_geometry_20260930/PROOF.md", + "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113", + "use": "Theorem S: r^-3 (1 - p_r) -> alpha_1 + alpha_2 in d = 2 (Corollary 2, conditional)" + }, + { + "id": "ELDERD", + "pr": 175, + "path": "frontiers/concave_fibre_elder_20260930/PROOF.md", + "git_blob": "923d3236b5291a29bd162937f172190db84dcef5", + "use": "Theorem F / M1 in d >= 3 (Corollary 2, conditional)" + }, + { + "id": "ELD", + "pr": 181, + "path": "frontiers/elder_partner_extremes_20260930/PROOF.md", + "git_blob": "91f88e392fb8aa947dac4fc81c2b45f783fa66cd", + "use": "elder-selected laws (S \\ O2, E0, lifetime density): dimension-independent by Corollary 1, conditional; quoted only" + }, + { + "id": "ELD179", + "pr": 179, + "path": "frontiers/elder_selected_extremes_20260930/PROOF.md", + "git_blob": "1310dc11bae684a9889548d1e764e83381b39d57", + "use": "shared elder-selected base (same statement as ELD sections 1-3); quoted only" + } ] } From d37ff5d0315a4f71453d37f0cae2da1f40fdef3c Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 13:03:48 +0000 Subject: [PATCH 4/4] Hard-direction factorization: record the Sol Slice C reassessment (5366261768) and Slices A/B read (5366497205) in the source map; no proof or numerical change Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../SOURCE_FILES.json | 6 +++--- .../SOURCE_MAP.json | 18 +++++++++++++++++- 2 files changed, 20 insertions(+), 4 deletions(-) diff --git a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json index 01e08dabf..f56d11a86 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json +++ b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_FILES.json @@ -25,9 +25,9 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 6212, - "sha256": "9524b2a9ac42d9544cce41f567fbd8f9b58c3fbf66158679c69c08be3b2d5bc3", - "git_blob": "51387c696c68b19ecec49cbe838ae2ab8d2e7f91" + "bytes": 7776, + "sha256": "55045563420eafd7731c0012e0676ce9953eb569455b0aebca3c9934e27c76fd", + "git_blob": "33d77c707440999f13fb15dba21a703891b11148" }, { "path": "factorization.py", diff --git a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json index 51387c696..33d77c707 100644 --- a/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json +++ b/frontiers/c6_hard_direction_factorization_20260930/SOURCE_MAP.json @@ -4,7 +4,7 @@ "scientific_effect": "NONE", "source_repository": "d6g8k5htny-coder/Math-", "main_at_creation": "7ca3eb00534d0a0d6dad4895f019f86bc2b2bf70", - "note": "Exact reduction of the [SC] (17) near cluster measure in dimension d to the planar one for the unit Gaussian kernel (continuum). Consumes [SC] (8), (17), (20), [LP]'s Q_r / W_r / z0 interface, [CUB]'s classifier and [R] (R11)-(R13) at their stated scopes; multiplies Math-#168's floating planar values. No source theorem is revalidated or promoted; elder corollaries are conditional on unmerged #170/#175/#179/#181 and are quoted only.", + "note": "Exact reduction of the [SC] (17) near cluster measure in dimension d to the planar one for the unit Gaussian kernel (continuum). Consumes [SC] (8), (17), (20), [LP]'s Q_r / W_r / z0 interface, [CUB]'s classifier and [R] (R11)-(R13) at their stated scopes; multiplies Math-#168's floating planar values. No source theorem is revalidated or promoted; elder corollaries are conditional on unmerged #170/#175/#179/#181 and are quoted only. Nonauthor reads: Astra 5365199521 (Slice C AMEND at ee6127f, applied at d29e674), Sol 5366261768 (Slice C ACCEPT at d29e674), Sol 5366497205 (Slices A/B ACCEPT at d29e674); review_status unchanged (author-side candidate; no acceptance of numerical values; organizational independence zero).", "review_status": "AUTHOR_SIDE_CANDIDATE", "nonauthor_acceptance": false, "governing_status_changed": false, @@ -16,6 +16,22 @@ "date": "2026-09-30", "slice": "C (scope)", "disposition": "AMEND the finite-L scope sentence (the planar one-site deviations are not an error estimate for the factorization; applied in the successor commit with the suggested replacement) and two non-blocking precisions (positive finite normalizing mass; elder corollaries refer to the continuum model), also applied; the continuum-only conditional corollaries have the correct structural form subject to Slices A/B; the Lemma 1 raw-covariance sentence corrected at 7dc9ee5. Same account, credit 0." + }, + { + "lane": "OpenAI/GPT-5.6 Sol", + "review_id": 5366261768, + "head": "d29e6741404f1e7bc06eb1ad91b547399715fb0a", + "date": "2026-09-30", + "slice": "C (scope), successor reread", + "disposition": "ACCEPT: the two blocking defects of 5365199521 resolved (continuum-only finite-L scope, fail-closed; positive finite normalizing mass; continuum-model reading of the elder corollary; Lemma 1 raw covariance repaired). Same account, credit 0." + }, + { + "lane": "OpenAI/GPT-5.6 Sol", + "review_id": 5366497205, + "head": "d29e6741404f1e7bc06eb1ad91b547399715fb0a", + "date": "2026-09-30", + "slice": "A and B", + "disposition": "ACCEPT Lemma 1 (conditional jet law, joint gradient/f_xxx regression, Hermite residual covariances, O(m) invariance, k-independence), Lemma 2 (endpoint expansion, type limit, domination, z_0 = 36 k^2 m_(d,b)), Theorem 1 (hard/soft separation, c_m/Haar/Vandermonde ledger, measure identity for arbitrary nonnegative soft g, k cancels) and the closed form R_3(0) = (32 + 28 sqrt2)/17, as identities for the continuum Gaussian-kernel near measure; numerical tables corroborative only; no finite-L or actual torus theorem accepted. Source-exposed (SC/R author), same account, credit 0." } ], "sources": [