From dc5ac51e236b264ba5f00f55d9ea0cff599d3690 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Wed, 30 Sep 2026 14:07:16 +0000 Subject: [PATCH 1/5] C6 remote coefficient numerics in d = 3: cone quadrature, holes, controls Add frontiers/c6_remote_coefficient_d3_numerics_20260930: an author-side numerical note evaluating the [RM] (13) contact kernel Lambda(x) in d = 3 for the Gaussian kernel with exact nested Gauss-Legendre quadrature over the transverse Hessian cone, weight-proportional sampling of the conditioned far-point Hessian under common random numbers, cylinder hole integrals (k = 1/2, 1, 3/2 at b = 0, 1), ray profiles, and exact far-field controls (ordered-eigenvalue GOE quadrature, BBKS level-integrated index densities reproduced to 1e-12). The assembled nu_1, nu_2 use the hard- direction factor R_3(b) from the companion factorization packet. Checker: stdlib-only remote3.py with --check (exact controls, sample-kernel replays at transverse, diagonal and axis points to 1e-9) and three mutants (hermite-sign, det-abs, weight-indicator), all rejected; RESULTS.json is byte-identical under -B -S and -B -O -S. Workflow replays the manifest, six main-resident pins, both check modes and the mutants. Scientific effect: NONE. Certified: false. No STATUS, PROOF_INDEX, GRAPH or catalog edits. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c6-remote-coefficient-d3-numerics.yml | 68 + .../NOTE.md | 132 ++ .../README.md | 15 + .../RESULTS.json | 1248 +++++++++++++++++ .../SOURCE_FILES.json | 40 + .../SOURCE_MAP.json | 23 + .../remote3.py | 588 ++++++++ 7 files changed, 2114 insertions(+) create mode 100644 .github/workflows/c6-remote-coefficient-d3-numerics.yml create mode 100644 frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md create mode 100644 frontiers/c6_remote_coefficient_d3_numerics_20260930/README.md create mode 100644 frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json create mode 100644 frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json create mode 100644 frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json create mode 100644 frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py diff --git a/.github/workflows/c6-remote-coefficient-d3-numerics.yml b/.github/workflows/c6-remote-coefficient-d3-numerics.yml new file mode 100644 index 000000000..657b344aa --- /dev/null +++ b/.github/workflows/c6-remote-coefficient-d3-numerics.yml @@ -0,0 +1,68 @@ +name: C6 remote coefficient numerics, d = 3 (replay) + +'on': + pull_request: + paths: + - 'frontiers/c6_remote_coefficient_d3_numerics_20260930/**' + - '.github/workflows/c6-remote-coefficient-d3-numerics.yml' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'frontiers/remote_window_20260924/PROOF.md' + - 'frontiers/c6_cluster_law_20260929/PROOF.md' + - 'frontiers/remote_collision_20260928/PROOF.md' + - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' + - 'frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json' + 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 exact controls and sample kernels, reject mutants + env: + PYTHONDONTWRITEBYTECODE: '1' + run: | + set -euo pipefail + python -B -S - <<'PY' + import hashlib, json, pathlib, subprocess, sys + root = pathlib.Path('frontiers/c6_remote_coefficient_d3_numerics_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 / 'remote3.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 ['hermite-sign', 'det-abs', 'weight-indicator']: + 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_remote_coefficient_d3_numerics_20260930/NOTE.md b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md new file mode 100644 index 000000000..ec2fb9ba3 --- /dev/null +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md @@ -0,0 +1,132 @@ +# Numerical values of the remote singleton coefficient k ∫_X Λ of ν(1) in dimension d = 3 (parent kernel; not certified) + +Object `CL-C6-REMOTE-COEFF-D3-NUMERICS-20260930-v1`. Author: Anthropic Claude (Claude Code +`session_015wNj8LPTKXsaT68G3DgPPh`). **Numerical evaluation only** (exact quadrature for the far field and the +pin weight, Monte Carlo with reported standard errors for the correlation hole). Not a proof, not an enclosure. +Scientific effect NONE: no register, catalog, GRAPH, STATUS or source change; nothing consumed is revalidated. + +## 1. What is evaluated + +[RM] (13), in every fixed `d >= 2`, defines the contact kernel + + Lambda_j(x; b, k, u) = p_(Y_x | U_0 = v_0)(0, b) / z_0 . E[w_0 F_j(H_x) | U_0 = v_0, Y_x = (0, b)], + +with `U_0 = (f, f_x, f_xx, f_xxx, f_y, f_z, f_xy, f_xz)(0)`, `v_0 = (b, 0, 0, 12k, 0, 0, 0, 0)`, `A_0 = D_y^2 f(0)` the +`2 x 2` transverse Hessian at the pin, `w_0 = (6k)^2 (det A_0)^2 1{A_0 < 0}` ([RM] (10)), `z_0 = E[w_0 | U_0 = v_0]` +([RM] (11)), `Y_x = (grad f(x), f(x))`, `H_x = D^2 f(x)` and `F_j(H) = |det H| 1{index H = j}`, `j = 0, ..., 3`. +`Lambda = sum_j Lambda_j` is the density in [SC] (22), `beta_far = k integral_(X \ {0}) Lambda dx`, so that the +cluster-law coefficients are `nu_1 = a_1 + beta_far`, `nu_2 = a_2` ([SC] (20)–(22); [CL] (1.4)). Math-#174 evaluated +this in `d = 2`; this note does `d = 3` for the continuum kernel `K(x) = exp(-|x|^2/2)`. + +Exact structure used (all checked in `remote3.py --check`): + +- **Contact law at the pin.** Given `U_0 = v_0`, `A_0 = -b I_2 + G_2` with `G_2` the GOE matrix of diagonal variance + `2` and off-diagonal variance `1` (exact rational regression; Math-#184 Lemma 1), so that + `z_0 = 36 k^2 m_(3,b)`, `m_(3,b) = E[(det A_0)^2 1{A_0 < 0}]`, `m_(3,0) = (7 - 4 sqrt2)/2` (Math-#184 Lemma 2, + companion, conditional; the finite-`r` normalizer `Z_r / r^2` was corroborated there by Monte Carlo). +- **Far field.** At separation `|x| -> infinity` the two sites decouple: `p_(Y_x | U_0)(0, b) -> phi(b) (2 pi)^(-3/2)`, + `E[w_0 F_j(H_x)] -> z_0 E[F_j(H_b)]` with `H_b = -b I_3 + G_3` (Hessian at a level-`b` critical point of the + unconditioned field), hence + + Lambda_inf_j(b) = phi(b) (2 pi)^(-3/2) E[|det(-b I_3 + G_3)| 1{index = j}], + + evaluated by the ordered-eigenvalue quadrature (Vandermonde density with `c_3` of [SC] (8)); at `b = 0` the split + is symmetric under `j <-> 3 - j`. Integrated over the level, `integral Lambda_inf_j(b) db` is the density of + index-`j` critical points of the unconditioned field, whose classical closed forms in this normalization (unit + kernel: `sigma_0 = 1`, per-component `sigma_1 = 1`, `sigma_2^2 = 15`) are `(29 sqrt3 - 18 sqrt2)/(72 pi^2)` for + maxima and for minima and `(29 sqrt3 + 18 sqrt2)/(72 pi^2)` for each saddle index (Bardeen–Bond–Kaiser–Szalay + 1986); the quadrature reproduces all four to `1e-12`, an exact control of the far-field machinery (the `d = 2` + analogue in Math-#174 was the Longuet-Higgins total `2/(pi sqrt3)`). +- **Symmetry.** `Lambda` depends on `(x_1, r)`, `r = |x_perp|`, only (rotations about the pin axis leave `U_0`, `v_0` + and the kernel invariant); the spatial integrals are cylindrical midpoint rules with weight `2 pi r`. + +## 2. Method + +At each `x = (x_1, r, 0)` the joint Gaussian law of the 21 jets (8 in `U_0`, 3 in `A_0`, 4 in `Y_x`, 6 in `H_x`) is +formed from `Cov(d^alpha f(s), d^beta f(t)) = (-1)^|alpha| d^(alpha+beta) K(t - s)` (products of probabilists' +Hermite polynomials), and `(A_0, H_x)` is conditioned on `(U_0, Y_x) = (v_0, 0, b)`; `p_(Y_x | U_0)(0, b)` is the +conditional Gaussian density. Then + + E[w_0 F_j(H_x) | .] = 36 k^2 integral_(A_0 < 0) (det A_0)^2 p(A_0) E[F_j(H_x) | A_0] dA_0 : + +the `A_0` integral is a nested Gauss–Legendre cone quadrature in `(a, c, e) = ((a_11 + a_22)/2, (a_11 - a_22)/2, +a_12)` (cone `a < 0`, `c^2 + e^2 < a^2`; `20 x 16 x 16` nodes; the integrand vanishes to second order on the cone +boundary, and against the exact `m_(3,b)` the rule is accurate to `1e-8`), and `E[F_j(H_x) | A_0]`, a six-dimensional +Gaussian expectation of `|det H| 1{index}`, is Monte Carlo: `H_x | A_0` is Gaussian with mean affine in `A_0`, the +budget of `4 x 40000` standard normals (four independent seeds) is allocated to the cone nodes in proportion to their +weights, and the same normals are used at every `x` (common random numbers). The correlation hole + + c_hole_j = integral (Lambda_j(x) - Lambda_j(x_far)) d^3x, x_far = (10, 0, 0), + +is estimated by the difference of the two common-random-number estimates, cell by cell, on the cylinder `|x_1| <= 6`, +`r <= 6` (`h = 0.4`); the standard error is the spread of the four batch means. `Lambda(x_far) = Lambda_inf` to +`1e-16`. The torus integral is then `integral_X Lambda = L^3 Lambda_inf + c_hole` for `L >= 12` (the depletion radius +is about `3`, section 3); no direct periodized integral is computed (at `L = 6` the additive form is not valid, as in +`d = 2`, and no value is given). + +## 3. Values (`RESULTS.json`; `k in {1/2, 1, 2}`, `b in {0, 1}`) + +Far field: `Lambda_inf(0) = 0.08574653`, `Lambda_inf(1) = 0.06501408` (index split `j = 0..3`: `0.001226, 0.041647, 0.041647, 0.001226` at `b = 0`, `0.000021, 0.006298, 0.047850, 0.010845` at `b = 1`); +`E|det G_3| = 3.3851375013`; level-integrated densities by index `0.0348624089, 0.1065073050, 0.1065073050, 0.0348624089` (closed forms `0.0348624089`, `0.1065073050`). +`m_(3,0) = 0.6715728753`, `m_(3,1) = 5.5480602736`; `z_0 = 36 k^2 m_(3,b)`. + +| `k` | `b` | `c_hole = integral (Lambda - Lambda_inf)` (± s.e.) | `integral_X Lambda`, `L = 12` | `L = 24` | `a_1^(3)` | `nu(1)` at `L = 24` | `nu(2)/nu(1)` at `L = 24` | +|---|---|---|---|---|---|---|---| +| `1/2` | `0` | -1.883 ± 0.025 | 146.29 | 1183.48 | 4.71 | 596.4 | 2.45e-04 | +| `1/2` | `1` | -1.231 ± 0.005 | 111.11 | 897.52 | 1.479 | 450.2 | 1.02e-04 | +| `1` | `0` | -3.133 ± 0.029 | 145.04 | 1182.23 | 5.598 | 1187.8 | 9.04e-05 | +| `1` | `1` | -2.309 ± 0.006 | 110.04 | 896.45 | 1.758 | 898.2 | 3.76e-05 | +| `2` | `0` | -4.575 ± 0.035 | 143.59 | 1180.78 | 9.106 | 2370.7 | 3.01e-05 | +| `2` | `1` | -3.473 ± 0.009 | 108.87 | 895.28 | 2.86 | 1793.4 | 1.25e-05 | + +Index decomposition of the hole at `k = 1`: `b = 0`: `-0.132 ± 0.006, -2.936 ± 0.017, -0.263 ± 0.023, 0.198 ± 0.006`; `b = 1`: `-0.001 ± 0.000, -0.282 ± 0.006, -1.823 ± 0.008, -0.202 ± 0.001`. + +The pin depletes a region of radius about `3` (as in `d = 2`): along the axis `Lambda/Lambda_inf` is below `1e-3` +at `|x_1| <= 2` and `0.0329` at `2.5`, `0.62` at `3`, `1.01` at `4`, `1` at `5`; transversally `0.182` at `0.5`, `0.237` at `1`, `0.644` at `2`, `0.998` at `3`, `1` at `4`; along the `45` degree ray `0.00917` at `1`, `0.777` at `2`, `1.07` at `2.5`, `1.03` at `3`, `1` at `4` (values at +`k = 1, b = 0`, `160000` samples per point, Monte Carlo error about `0.7 %` per point). The three profiles are within +Monte Carlo noise of `1` from `|x| = 4` on. + +**Reading.** `a_j^(3) = R_3(b) alpha_j^(2)` (Math-#184; `R_3(0) = (32 + 28 sqrt2)/17 = 4.2116458674`, `R_3(1) = 4.6201205312`) with +Math-#168's planar `alpha_j`. On the parent torus sizes the remote singletons dominate `nu(1)` by a factor of order +`10^2`–`10^3` (`L^3 Lambda_inf = 13824 Lambda_inf` at `L = 24`), so the two-point share of a nonempty window is +`nu(2)/(nu(1) + nu(2))` of order `1e-4` at `L = 12` and `1e-5` at `L = 24`, smaller than in `d = 2` (Math-#174) by +the factor `L Lambda_inf^(3) / Lambda_inf^(2)` up to the near terms. + +## 4. Precision (empirical, not bounds) + +- Far field: quadrature orders `30`, `40` agree to `1e-12`; the Monte Carlo at the decoupled point `x_far` reproduces + `z_0 E[|det| 1{j}]` within its statistical error (`1.052, 0.985, 1.001, 1.137` at `b = 0`, indices `j = 0..3`; the maxima/minima indices + carry `1.4 %` of the mass and have correspondingly larger errors). +- Cone quadrature for the pin weight: `1e-8` relative against the exact `m_(3,b)` at the far point; at the near point + `(0, 1, 0)` the `16 x 12 x 12` and `20 x 16 x 16` rules differ by `7e-5` relative. +- Monte Carlo: per-point relative error about `1.28 / sqrt(160000) = 0.3 %` on `Lambda`; the hole standard errors are + in the table (batch means of four independent seeds; common random numbers make the far cells contribute almost no + noise). +- Grid (`k = 1, b = 0`, two batches): hole `-3.155 ± 0.055` (`h = 0.6`), `-3.133 ± 0.029` (`h = 0.4`, tabulated), `-3.138 ± 0.057` (`h = 0.3`); + cutoff `Rc = 5` instead of `6`: `-3.164 ± 0.056`. These differences are measured, not bounds; the grid error of the + tabulated values is of the order of the difference between `h = 0.4` and `h = 0.3`. +- Near-pin points: the conditioning never lost positivity on the grid (`r >= 0.2`); `Lambda` is below `1e-20 + Lambda_inf` on the axis for `|x_1| <= 1.5`. + +## 5. What this does not do + +Not certified. Continuum kernel only: no periodized (finite-`L`) jets, so the values at `L = 12, 24` are the additive +approximation `L^3 Lambda_inf + c_hole` and no `L = 6` value is given. `d = 3` only; the pin direction is immaterial by +isotropy. The assembled `nu(1)`, `nu(2)` inherit Math-#168's empirical precision on `alpha_j` and are conditional on +Math-#184's factor `R_3(b)` (unmerged). No rate, no finite-`r` statement. Values of the formulas [RM] (13) and [SC] +(22); not a re-review of either. + +## 6. Verification + +`python -B -S remote3.py --check` and `-B -O -S`: exact rational contact law (`A_0 | U_0 = -b I + GOE(2, 1)`), +`c_2 = pi`, `m_(3,0) = (7 - 4 sqrt2)/2`, eigenvalue-density normalization, far-field index symmetry at `b = 0`, the +decoupled density `phi(b) (2 pi)^(-3/2)` at `x_far`, the far-point Monte Carlo against `z_0 E|det| 1{j}` within +`10 %` for the three largest indices, replay of the far field and of `R_3(b)` against `RESULTS.json`, and replay of +one kernel point per `b` with the stored seeds to `1e-9`. Mutants `hermite-sign`, `det-abs`, `weight-indicator` exit +`1`. The full run regenerates `RESULTS.json` deterministically (seeded) in about an hour. + +## 7. Provenance + +Pins (`SOURCE_MAP.json`, all on `main`): [RM] `b383bfcc`, [SC] `16c56821`, [CL] `ba492c8e`, [LP] `dfed3b8d`, +[NUM] `2648314f` (Math-#168), [NUM2] `3e4a7ddb` (Math-#174, merged). Unmerged companion: Math-#184 `PROOF.md` +`91d22fce` (Lemma 2 and `R_3(b)`, recomputed here from the displayed formulas). diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/README.md b/frontiers/c6_remote_coefficient_d3_numerics_20260930/README.md new file mode 100644 index 000000000..e58bba39e --- /dev/null +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/README.md @@ -0,0 +1,15 @@ +# Remote singleton coefficient in d = 3 (numerical note) + +Object `CL-C6-REMOTE-COEFF-D3-NUMERICS-20260930-v1`. Scientific effect NONE; not certified; nonauthor read open. + +`NOTE.md` evaluates the contact kernel `Lambda_j(x)` of [RM] (13) in dimension `d = 3` for the continuum parent kernel: +its far field (exact shifted-GOE quadrature), the correlation-hole constant `c_hole = integral (Lambda - Lambda_inf) d^3x` +(Monte Carlo with common random numbers, standard errors reported) and the torus integral +`k integral_X Lambda = k (L^3 Lambda_inf + c_hole)` for `L = 12, 24`, assembled with the `d = 3` near coefficients +`a_j^(3) = R_3(b) alpha_j^(2)` (Math-#184, conditional; Math-#168 planar values) into `nu(1)`, `nu(2)`. + + python -B -S remote3.py --check # rc 0 (also with -O); about half a minute + python -B -S remote3.py --check --mutant hermite-sign # rc 1 (also det-abs, weight-indicator) + python -B -S remote3.py # regenerates RESULTS.json (about an hour) + +`SOURCE_MAP.json` pins the consumed sources on `main`; `SOURCE_FILES.json` is the packet manifest. diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json new file mode 100644 index 000000000..02cc2bf1d --- /dev/null +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json @@ -0,0 +1,1248 @@ +{ + "R_3": { + "0.0": 4.211645867438047, + "1.0": 4.620120531230534 + }, + "assembled_nu": { + "k=0.5,b=0.0": { + "a1_d3": 4.7095045254278975, + "a2_d3": 0.14633532032248192, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", + "nu1_L": { + "L=12": 77.85325873143533, + "L=24": 596.4483000261176 + }, + "nu2_over_nu1_L": { + "L=12": 0.0018796299950305758, + "L=24": 0.0002453445173975248 + } + }, + "k=0.5,b=1.0": { + "a1_d3": 1.4791546886761118, + "a2_d3": 0.04596054323383355, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", + "nu1_L": { + "L=12": 57.03583271929747, + "L=24": 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b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json @@ -0,0 +1,23 @@ +{ + "schema": 1, + "object": "CL-C6-REMOTE-COEFF-D3-NUMERICS-20260930-v1", + "scientific_effect": "NONE", + "source_repository": "d6g8k5htny-coder/Math-", + "main_at_creation": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", + "note": "Numerical evaluation only: the remote contact kernel Lambda_j of [RM] (13) in dimension d = 3 for the continuum kernel exp(-|x|^2/2), its far field (exact shifted-GOE quadrature), the correlation-hole constant (Monte Carlo with common random numbers and reported standard errors) and the torus integral k (L^3 Lambda_inf + c_hole) entering nu(1) = a_1 + k integral Lambda ([SC] (22), [CL] (1.4)). The near coefficients a_j^(3) used in the assembly are the planar Math-#168 values times R_3(b) of Math-#184 (unmerged, conditional). Floating-point values with standard errors, not enclosures. No source theorem is consumed as accepted.", + "review_status": "AUTHOR_SIDE_NUMERICAL_NOTE", + "nonauthor_acceptance": false, + "governing_status_changed": false, + "nonauthor_reads": [], + "sources": [ + {"id": "RM", "path": "frontiers/remote_window_20260924/PROOF.md", "bytes": 18355, "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", "commit": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "sections 2, 4, 5: the contact frame U_0 (4), the weight w_0 = (6k)^2 (det A_0)^2 1{A_0 < 0} (10), the full normalizer (11) and the contact kernel Lambda_j (13) in every fixed d >= 2; the object evaluated", "current_required": true, "on_main": true}, + {"id": "SC", "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", "bytes": 25006, "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", "commit": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "(8) c_m; (20) the near coefficients a_1, a_2; (22) beta_far = k integral_X Lambda in every fixed d; nu_1 = a_1 + beta_far, nu_2 = a_2", "current_required": true, "on_main": true}, + {"id": "CL", "path": "frontiers/c6_cluster_law_20260929/PROOF.md", "bytes": 70131, "sha256": "96b8e2d59defee6059bf151a896b7fc7c91fc529b385a3db8a87574901197b38", "git_blob": "ba492c8e62e58bfc055fc8254c790e893d346ba5", "commit": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "(1.4) nu(1) = nu_near(1) + k integral_X Lambda; section 5.1 integrability at the pin", "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": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "section 2: the kernel exp(-|x|^2/2) and its periodization; the pins, W_r and z_0", "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": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "Math-#168: planar alpha_1, alpha_2 (GH60, floating, not certified), read at run time and multiplied by R_3(b) in the assembly", "current_required": true, "on_main": true}, + {"id": "NUM2", "path": "frontiers/c6_remote_coefficient_numerics_20260930/RESULTS.json", "bytes": 103899, "sha256": "e9af3b52761946b57d6f5fa50fdf6ccd8d26610461e471a2e97618da1611e48a", "git_blob": "3e4a7ddb850818c4477b99f678847e68ab2b92e4", "commit": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "Math-#174 (merged): the planar remote coefficient values quoted for comparison (far field, hole constants, depletion radius); not read at run time", "current_required": true, "on_main": true} + ], + "companions_unmerged": [ + {"id": "HDF", "pr": 184, "path": "frontiers/c6_hard_direction_factorization_20260930/PROOF.md", "git_blob": "91d22fce0fd7d4630bfe2e2c93eab289babc97f2", "use": "Lemma 2: z_0^(3) = 36 k^2 m_(3,b) with m_(3,b) = E[(det A)^2 1{A neg. def.}], A = -b I + GOE_2(2, 1); Theorem 1 / Corollary 1: a_j^(3) = R_3(b) alpha_j^(2); recomputed here from the same displayed formulas and marked conditional"} + ] +} diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py new file mode 100644 index 000000000..834b732f8 --- /dev/null +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py @@ -0,0 +1,588 @@ +#!/usr/bin/env python3 +"""Numerical evaluation of the remote contact kernel Lambda_j(x; b, k, u) of [RM] (13) in dimension d = 3 for the +continuum kernel K(x) = exp(-|x|^2/2): its far-field value (exact shifted-GOE quadrature), the correlation-hole +constant c_hole = integral (Lambda - Lambda_inf) d^3x (Monte Carlo with common random numbers), and the resulting +torus integral k integral_X Lambda dx = k (L^3 Lambda_inf + c_hole) entering nu(1) = a_1 + k integral Lambda +([SC] (22), [CL] (1.4)). Standard library only. Not a proof, not an enclosure. + + python -B -S remote3.py # full run, writes RESULTS.json (about 40 minutes) + python -B -S remote3.py --check # exact controls + replay of a small subset against RESULTS.json + python -B -S remote3.py --check --mutant NAME # must exit 1 + +Conventions ([RM] section 2, (4), (10), (13)). Coordinates x = (x1, y, z), x1 along the pin axis u. +U_0 = (f, f_x, f_xx, f_xxx, f_y, f_z, f_xy, f_xz)(0) = (b, 0, 0, 12k, 0, 0, 0, 0); A_0 = (f_yy, f_yz, f_zz)(0); +Y_x = (f_x, f_y, f_z, f)(x) = (0, 0, 0, b); H_x = the six Hessian entries at x. +w_0 = 36 k^2 (det A_0)^2 1{A_0 negative definite}; z_0 = E[w_0 | U_0] = 36 k^2 m_(3,b) (Math-#184 Lemma 2). +Lambda_j(x) = p_(Y_x | U_0)(0, b) E[w_0 F_j(H_x) | U_0, Y_x] / z_0, F_j(H) = |det H| 1{index H = j}, j = 0..3. +Cov(d^alpha f(s), d^beta f(t)) = (-1)^|alpha| d^(alpha+beta) K(t - s); d^n exp(-t^2/2) = (-1)^n He_n(t) exp(-t^2/2). +Lambda depends on (x1, r), r = sqrt(y^2 + z^2), by rotation invariance about the axis; the spatial integrals use +cylindrical midpoint rules with weight 2 pi r. +""" +import argparse +import json +import math +import os +import random +import sys +from fractions import Fraction as Fr +from math import erf, exp, factorial, gamma, pi, sqrt + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('hermite-sign', 'det-abs', 'weight-indicator') +RULES = (20, 16, 16) # nested Gauss-Legendre nodes for the A_0 cone quadrature (5120 nodes) +MUT = None +REPLAY_TOL = 1e-9 +D = 3 +Z3 = (0, 0, 0) + + +def require(cond, msg): + if not cond: + print(json.dumps({'check': 'FAIL', 'what': msg, 'mutant': MUT})) + sys.exit(1) + + +# ----------------------------------------------------------------------------- kernel and covariances +def hermite(n, t): + """Probabilists' Hermite polynomial He_n(t).""" + if n == 0: + return 1.0 + h0, h1 = 1.0, t + for m in range(1, n): + h0, h1 = h1, t * h1 - m * h0 + return h1 + + +def kernel_derivative(gam, y): + """d^gam K(y), K(y) = exp(-|y|^2/2): prod_i (-1)^gam_i He_gam_i(y_i) exp(-y_i^2/2).""" + v = 1.0 + for g, yi in zip(gam, y): + sign = (-1) ** g + if MUT == 'hermite-sign': + sign = 1.0 + v *= sign * hermite(g, yi) * exp(-yi * yi / 2) + return v + + +def cov(al, s, be, t): + return (-1) ** sum(al) * kernel_derivative(tuple(a + b for a, b in zip(al, be)), [ti - si for si, ti in zip(s, t)]) + + +def unit(i, n=1): + v = [0] * D + v[i] = n + return tuple(v) + + +def add(*vs): + return tuple(sum(v[i] for v in vs) for i in range(D)) + + +OBS0 = [Z3, unit(0), unit(0, 2), unit(0, 3), unit(1), unit(2), add(unit(0), unit(1)), add(unit(0), unit(2))] +A0J = [add(unit(1), unit(1)), add(unit(1), unit(2)), add(unit(2), unit(2))] +OBSX = [unit(0), unit(1), unit(2), Z3] +HXJ = [add(unit(i), unit(j)) for i in range(3) for j in range(i, 3)] # xx xy xz yy yz zz + + +def v0(b, k): + return [b, 0.0, 0.0, 12 * k, 0.0, 0.0, 0.0, 0.0] + + +# ----------------------------------------------------------------------------- exact one-site table (rational) +def c_even(n): + if n % 2: + return Fr(0) + v = Fr(1) + for j in range(1, n, 2): + v *= j + return v * (-1) ** (n // 2) + + +def cov_point_exact(al, be): + v = Fr((-1) ** sum(al)) + for a, b in zip(al, be): + v *= c_even(a + b) + return v + + +def mat_inv(M): + n = len(M) + A = [row[:] + [Fr(int(i == j)) for j in range(n)] for i, row in enumerate(M)] + for c in range(n): + p = next(r for r in range(c, n) if A[r][c] != 0) + A[c], A[p] = A[p], A[c] + pv = A[c][c] + A[c] = [x / pv for x in A[c]] + for r in range(n): + if r != c and A[r][c] != 0: + f = A[r][c] + A[r] = [x - f * y for x, y in zip(A[r], A[c])] + return [row[n:] for row in A] + + +def exact_contact_law(): + """A_0 | U_0 exactly: mean coefficients and conditional covariance (must be -b I + GOE(2, 1)).""" + Soo = [[cov_point_exact(a, b) for b in OBS0] for a in OBS0] + Sto = [[cov_point_exact(a, b) for b in OBS0] for a in A0J] + Stt = [[cov_point_exact(a, b) for b in A0J] for a in A0J] + inv = mat_inv(Soo) + reg = [[sum(Sto[i][k] * inv[k][j] for k in range(8)) for j in range(8)] for i in range(3)] + cc = [[Stt[i][j] - sum(reg[i][k] * Sto[j][k] for k in range(8)) for j in range(3)] for i in range(3)] + return reg, cc + + +# ----------------------------------------------------------------------------- linear algebra (floats) +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 solve(L, b): + n = len(L) + y = [0.0] * n + for i in range(n): + y[i] = (b[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] + return x + + +def det3(m): + return (m[0][0] * (m[1][1] * m[2][2] - m[1][2] * m[2][1]) - m[0][1] * (m[1][0] * m[2][2] - m[1][2] * m[2][0]) + + m[0][2] * (m[1][0] * m[2][1] - m[1][1] * m[2][0])) + + +def n_negative3(m): + d1 = m[0][0] + d2 = m[0][0] * m[1][1] - m[0][1] * m[1][0] + d3 = det3(m) + seq = [1.0, d1, d2, d3] + return sum(1 for i in range(3) if seq[i] * seq[i + 1] < 0) + + +# ----------------------------------------------------------------------------- quadrature and special functions +def phi(t): + return exp(-t * t / 2) / sqrt(2 * pi) + + +def Phi(t): + return 0.5 * (1 + erf(t / sqrt(2))) + + +def phi2(t): + return phi(t / sqrt(2)) / sqrt(2) + + +def gauss_legendre(n): + xs, ws = [], [] + for i in range(1, n + 1): + x = math.cos(math.pi * (i - 0.25) / (n + 0.5)) + for _ in range(100): + p0, p1 = 1.0, x + for m in range(2, n + 1): + p0, p1 = p1, ((2 * m - 1) * x * p1 - (m - 1) * p0) / m + dp = n * (x * p1 - p0) / (x * x - 1.0) + dx = p1 / dp + x -= dx + if abs(dx) < 1e-15: + break + xs.append(x) + ws.append(2 / ((1 - x * x) * dp * dp)) + return xs, ws + + +def gl(f, lo, hi, rule): + mid, half = 0.5 * (lo + hi), 0.5 * (hi - lo) + return half * sum(w * f(mid + half * x) for x, w in zip(*rule)) + + +def c_m(m): + """[SC] (8) spectral constant through Mehta's integral.""" + return pi ** (m * (m - 1) / 4) * factorial(m) * gamma(1.5) ** m / math.prod(gamma(1 + j / 2) for j in range(1, m + 1)) + + +def goe_functional(m, b, weight_fn, n=40, L=14.0): + """sum over sign patterns: E[weight_fn(eigs) 1{index j}] for A = -b I + GOE_m(2, 1), j = number of negative + eigenvalues, by the ordered eigenvalue density c_m (2 pi)^(-m(m-1)/4) prod phi2(l_i + b) Vandermonde.""" + rule = gauss_legendre(n) + pref = c_m(m) * (2 * pi) ** (-m * (m - 1) / 4) + out = [0.0] * (m + 1) + xs, ws = rule + + def rec(prev, depth, acc, nneg): + if depth == m: + out[nneg] += acc * weight_fn(prev) + return + last = prev[-1] if prev else None + for (lo, hi, neg) in (((last if last is not None else -L - abs(b)), 0.0, 1), (max(0.0, last if last is not None else 0.0), L + abs(b), 0)): + if hi <= lo: + continue + mid, half = 0.5 * (lo + hi), 0.5 * (hi - lo) + for xg, wg in zip(xs, ws): + l = mid + half * xg + v = phi2(l + b) * half * wg + for pp in prev: + v *= (l - pp) + rec(prev + [l], depth + 1, acc * v, nneg + neg) + rec([], 0, pref, 0) + return out + + +def m3(b, n=40): + """m_(3,b) = E[(det A)^2 1{A negative definite}], A = -b I_2 + GOE_2(2, 1) (Math-#184 Lemma 2 / closed form at b = 0).""" + return goe_functional(2, b, lambda l: (l[0] * l[1]) ** 2, n)[2] + + +def far_field(b, n=40): + """Lambda_inf_j(b) = phi(b) (2 pi)^(-3/2) E[|det(-b I + G_3)| 1{index j}]: the decoupled far-field kernel.""" + E = goe_functional(3, b, lambda l: abs(l[0] * l[1] * l[2]), n) + pref = phi(b) * (2 * pi) ** (-1.5) + return {'E_absdet_index': E, 'lambda_j': [pref * e for e in E], 'lambda': pref * sum(E)} + + +def R3(b): + """Math-#184's dimension factor R_3(b) = N_3(b) m_(2,b) / m_(3,b), quoted for the assembly (conditional on #184).""" + mu, s = -b, sqrt(2) + t = mu / s + m2 = (mu * mu + s * s) * Phi(-t) - mu * s * phi(t) + rule = gauss_legendre(200) + N3 = sqrt(pi / 2) * gl(lambda a: (-a) ** 3 * phi2(a + b), -40.0, 0.0, rule) + return N3 * m2 / m3(b) + + +# ----------------------------------------------------------------------------- conditioning at a point +class Conditioning: + """Conditional law of T = (A_0, H_x) given U_0 = v_0, Y_x = (0, b), and the density p_(Y_x | U_0)(0, b).""" + + def __init__(self, x, b, k): + obs = [(a, Z3) for a in OBS0] + [(a, x) for a in OBSX] + vals = v0(b, k) + [0.0, 0.0, 0.0, b] + tg = [(a, Z3) for a in A0J] + [(a, x) for a in HXJ] + Soo = [[cov(a, s, c, t) for (c, t) in obs] for (a, s) in obs] + Sto = [[cov(a, s, c, t) for (c, t) in obs] for (a, s) in tg] + Stt = [[cov(a, s, c, t) for (c, t) in tg] for (a, s) in tg] + L = cholesky(Soo) + W = [solve(L, row) for row in Sto] + self.mu = [sum(W[i][j] * vals[j] for j in range(len(vals))) for i in range(len(tg))] + cc = [[Stt[i][j] - sum(W[i][l] * Sto[j][l] for l in range(len(vals))) for j in range(len(tg))] for i in range(len(tg))] + self.cov = cc + SAA = [r[:3] for r in cc[:3]] + SHA = [r[:3] for r in cc[3:]] + SHH = [r[3:] for r in cc[3:]] + self.covA = SAA + self.LA = cholesky(SAA) + Bt = [solve(self.LA, [SHA[i][j] for j in range(3)]) for i in range(6)] # rows: Sigma_HA Sigma_AA^-1 + self.B = Bt + SHgA = [[SHH[i][j] - sum(Bt[i][l] * SHA[j][l] for l in range(3)) for j in range(6)] for i in range(6)] + self.LH = cholesky(SHgA) + self.rules = RULES + n0 = len(OBS0) + S00 = [r[:n0] for r in Soo[:n0]] + Sy0 = [r[:n0] for r in Soo[n0:]] + Syy = [r[n0:] for r in Soo[n0:]] + L0 = cholesky(S00) + Wy = [solve(L0, row) for row in Sy0] + muy = [sum(Wy[i][j] * vals[j] for j in range(n0)) for i in range(4)] + cy = [[Syy[i][j] - sum(Wy[i][l] * Sy0[j][l] for l in range(n0)) for j in range(4)] for i in range(4)] + Ly = cholesky(cy) + dy = [vals[n0 + i] - muy[i] for i in range(4)] + q = solve(Ly, dy) + q2 = sum(dy[i] * q[i] for i in range(4)) + logdet = 2 * sum(math.log(Ly[i][i]) for i in range(4)) + self.density = exp(-0.5 * q2 - 0.5 * logdet - 2 * math.log(2 * pi)) + self.k = k + + def cone_nodes(self, rules=None): + """Nodes of the nested Gauss-Legendre cone quadrature for the A_0 integral: (A_0 entries, weight, H-mean).""" + rules = rules or self.rules + na, nc, ne = rules + mu, k = self.mu, self.k + muA, muH = mu[:3], mu[3:] + LA, B, SA = self.LA, self.B, self.covA + logdetA = 2 * sum(math.log(LA[i][i]) for i in range(3)) + m_a = 0.5 * (muA[0] + muA[2]) + s_a = sqrt(max(0.25 * (SA[0][0] + SA[2][2] + 2 * SA[0][2]), 1e-300)) + a_lo = min(m_a - 7 * s_a, -1e-9) + a_hi = min(0.0, m_a + 7 * s_a) + nodes = [] + if a_hi <= a_lo: + return nodes + ra, rc, re = gauss_legendre(na), gauss_legendre(nc), gauss_legendre(ne) + mid_a, half_a = 0.5 * (a_lo + a_hi), 0.5 * (a_hi - a_lo) + for xa, wa in zip(*ra): + a = mid_a + half_a * xa + half_c = -a + for xc, wc in zip(*rc): + c = half_c * xc + half_e = sqrt(max(a * a - c * c, 0.0)) + for xe, we in zip(*re): + e = half_e * xe + A = [a + c, e, a - c] + d = [A[i] - muA[i] for i in range(3)] + q = solve(LA, d) + q2 = sum(d[i] * q[i] for i in range(3)) + dens = exp(-0.5 * q2 - 0.5 * logdetA - 1.5 * math.log(2 * pi)) + detA = a * a - c * c - e * e + if MUT == 'weight-indicator': + detA = detA + 0.5 * (c * c + e * e) + w = 2.0 * (half_a * wa) * (half_c * wc) * (half_e * we) * detA * detA * dens + if w <= 0.0: + continue + mH = [muH[i] + sum(B[i][j] * d[j] for j in range(3)) for i in range(6)] + nodes.append((w, mH)) + return nodes + + def weighted(self, zs, rules=None): + """E[w_0 F_j(H_x) | U_0, Y_x] = 36 k^2 integral_(A_0 neg. def.) (det A_0)^2 p(A_0) E[F_j(H_x) | A_0] dA_0. + + The A_0 integral is the nested Gauss-Legendre cone quadrature of cone_nodes (Jacobian 2; the cone is a < 0, + c^2 + e^2 < a^2 in (a, c, e) = ((a11 + a22)/2, (a11 - a22)/2, a12); the integrand (a^2 - c^2 - e^2)^2 vanishes to + second order on the cone boundary). E[F_j(H_x) | A_0] is Monte Carlo on the supplied 6-dim standard normals zs + (common random numbers across x), H_x | A_0 being Gaussian with mean affine in A_0 and fixed covariance; the + sample budget len(zs) is allocated to the nodes in proportion to their weights (at least one sample per node), + so the effective sample size is the whole budget.""" + if rules is None: + if getattr(self, '_nodes', None) is None: + self._nodes = self.cone_nodes() + nodes = self._nodes + else: + nodes = self.cone_nodes(rules) + if not nodes: + return [0.0] * 4 + total_w = sum(w for w, _ in nodes) + N = len(zs) + LH, k = self.LH, self.k + acc = [0.0] * 4 + pos = 0 + for w, mH in nodes: + n = max(1, int(round(N * w / total_w))) + block = zs[pos:pos + n] + pos += n + if not block: + # budget exhausted by rounding: reuse the first samples (deterministic, tiny weights only) + block = zs[:n] + part = [0.0] * 4 + for z in block: + t = [mH[i] + sum(LH[i][j] * z[j] for j in range(i + 1)) for i in range(6)] + H = [[t[0], t[1], t[2]], [t[1], t[3], t[4]], [t[2], t[4], t[5]]] + dH = det3(H) + part[n_negative3(H)] += (dH if MUT == 'det-abs' else abs(dH)) + for j in range(4): + acc[j] += w * part[j] / len(block) + return [36 * k * k * v for v in acc] + + +def standard_normals(n, seed, dim=6): + rng = random.Random(seed) + return [[rng.gauss(0.0, 1.0) for _ in range(dim)] for _ in range(n)] + + +def lambda_at(x, b, k, zs, z0): + c = Conditioning(x, b, k) + E = c.weighted(zs) + lam_j = [c.density * e / z0 for e in E] + return {'x': list(x), 'density': c.density, 'lambda_j': lam_j, 'lambda': sum(lam_j)} + + +# ----------------------------------------------------------------------------- spatial integrals +def hole_integral(b, k, zbatches, z0, Rc=6.0, h=0.4, x_far=(10.0, 0.0, 0.0)): + """c_hole_j = integral (Lambda_j - Lambda_j(x_far)) d^3x over the cylinder |x1| <= Rc, r <= Rc, by the midpoint rule in + (x1, r) with weight 2 pi r, estimated with common random numbers per batch; the far point is decoupled to 1e-16, so + Lambda(x_far) = Lambda_inf. zbatches is a list of independent sample arrays; batch means give the standard error.""" + n1 = int(round(2 * Rc / h)) + nr = int(round(Rc / h)) + h1, hr = 2 * Rc / n1, Rc / nr + far = Conditioning(x_far, b, k) + batches = len(zbatches) + far_b = [far.weighted(zs) for zs in zbatches] + tot_b = [[0.0] * 4 for _ in range(batches)] + skipped = 0 + for i in range(n1): + x1 = -Rc + (i + 0.5) * h1 + for j in range(nr): + r = (j + 0.5) * hr + vol = 2 * pi * r * h1 * hr + try: + c = Conditioning((x1, r, 0.0), b, k) + except ValueError: + skipped += 1 + continue + for bi, zs in enumerate(zbatches): + E = c.weighted(zs) + for q in range(4): + tot_b[bi][q] += vol * (c.density * E[q] - far.density * far_b[bi][q]) / z0 + means = [sum(tot_b[bi][q] for bi in range(batches)) / batches for q in range(4)] + ses = [sqrt(sum((tot_b[bi][q] - means[q]) ** 2 for bi in range(batches)) / (batches - 1) / batches) for q in range(4)] + tot = [sum(tot_b[bi]) for bi in range(batches)] + mean_t = sum(tot) / batches + se_t = sqrt(sum((t - mean_t) ** 2 for t in tot) / (batches - 1) / batches) + return {'hole_j': means, 'hole_j_se': ses, 'hole': mean_t, 'hole_se': se_t, + 'grid': {'Rc': Rc, 'h1': h1, 'hr': hr, 'points': n1 * nr, 'skipped_near_pin': skipped}, + 'far_point': list(x_far), 'far_weighted_mc': [sum(far_b[bi][q] for bi in range(batches)) / batches for q in range(4)], + 'samples_per_batch': len(zbatches[0]), 'batches': batches} + + +def ray_profile(b, k, zs, z0, lam_inf): + rays = {'axis': (1.0, 0.0), 'transverse': (0.0, 1.0), 'diag45': (1 / sqrt(2), 1 / sqrt(2))} + out = {} + for name, (c1, cr) in rays.items(): + rows = [] + for rho in (0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 8.0): + x = (c1 * rho, cr * rho, 0.0) + try: + res = lambda_at(x, b, k, zs, z0) + rows.append({'rho': rho, 'lambda': res['lambda'], 'ratio_to_far': res['lambda'] / lam_inf, 'lambda_j': res['lambda_j']}) + except ValueError: + rows.append({'rho': rho, 'lambda': None}) + out[name] = rows + return out + + +# ----------------------------------------------------------------------------- planar inputs (Math-#168) and assembly +def planar_alphas(): + path = os.path.join(HERE, '..', 'c6_cluster_coefficients_numerics_20260930', 'RESULTS.json') + with open(path) as fh: + return json.load(fh)['prefactor'] + + +def controls(zs): + reg, cc = exact_contact_law() + require(all(reg[i][0] == (Fr(-1) if i in (0, 2) else Fr(0)) for i in range(3)) and all(reg[i][j] == 0 for i in range(3) for j in range(1, 8)), + 'A_0 | U_0 has mean -b I (exact)') + require(cc == [[Fr(2), Fr(0), Fr(0)], [Fr(0), Fr(1), Fr(0)], [Fr(0), Fr(0), Fr(2)]], 'A_0 | U_0 is GOE(2, 1) (exact)') + require(abs(c_m(2) - pi) < 1e-13, 'c_2 = pi') + m30 = m3(0.0) + require(abs(m30 - (7 - 4 * sqrt(2)) / 2) < 1e-10, 'm_(3,0) = (7 - 4 sqrt2)/2') + ff0 = far_field(0.0) + E = ff0['E_absdet_index'] + require(abs(E[0] - E[3]) < 1e-10 and abs(E[1] - E[2]) < 1e-10, 'index symmetry j <-> 3 - j at b = 0') + tot = goe_functional(3, 0.0, lambda l: 1.0) + require(abs(sum(tot) - 1) < 1e-9, 'eigenvalue density normalizes to 1') + # decoupled far point: density equals phi(b) (2 pi)^(-3/2) and the weighted expectation equals z_0 E[|det| 1{j}] + out = {'m_3_0': m30, 'E_absdet_index_b0': E, 'far_field_b0': ff0['lambda'], 'far_field_b1': far_field(1.0)['lambda']} + for b in (0.0, 1.0): + c = Conditioning((10.0, 0.0, 0.0), b, 1.0) + require(abs(c.density - phi(b) * (2 * pi) ** (-1.5)) < 1e-12, 'far density decouples b = %r' % b) + z0 = 36 * m3(b) + Emc = c.weighted(zs) + Eq = far_field(b)['E_absdet_index'] + out['far_mc_over_quadrature_b%s' % b] = [Emc[j] / (z0 * Eq[j]) for j in range(4)] + # E|det G_3| at b = 0 (the level-0 factor) and the level-integrated far field: integral_b Lambda_inf_j(b) db is the + # density of index-j critical points of the unconditioned field, whose classical closed forms in this normalization + # (unit Gaussian kernel: sigma_0 = 1, per-component sigma_1 = 1, sigma_2^2 = 15) are (29 sqrt3 - 18 sqrt2)/(72 pi^2) + # for maxima and minima and (29 sqrt3 + 18 sqrt2)/(72 pi^2) for each saddle index (Bardeen-Bond-Kaiser-Szalay 1986) + out['E_absdet_G3_quadrature'] = sum(goe_functional(3, 0.0, lambda l: abs(l[0] * l[1] * l[2]))) + rule = gauss_legendre(60) + dens = [gl(lambda bb: far_field(bb, n=24)['lambda_j'][j], -8.0, 8.0, rule) for j in range(4)] + out['critical_point_density_by_index'] = dens + out['closed_form_extrema'] = (29 * sqrt(3) - 18 * sqrt(2)) / (72 * pi ** 2) + out['closed_form_saddles'] = (29 * sqrt(3) + 18 * sqrt(2)) / (72 * pi ** 2) + require(abs(dens[0] - out['closed_form_extrema']) < 1e-9 and abs(dens[3] - out['closed_form_extrema']) < 1e-9, + 'level-integrated far field: extrema density (29 sqrt3 - 18 sqrt2)/(72 pi^2)') + require(abs(dens[1] - out['closed_form_saddles']) < 1e-9 and abs(dens[2] - out['closed_form_saddles']) < 1e-9, + 'level-integrated far field: saddle density (29 sqrt3 + 18 sqrt2)/(72 pi^2)') + return out + + +def full_run(fast=False): + nb = 4 + per_batch = 3000 if fast else 40000 + zbatches = [standard_normals(per_batch, 2026 + i) for i in range(nb)] + zs = [z for zb in zbatches for z in zb] + res = {'schema': 1, 'object': 'CL-C6-REMOTE-COEFF-D3-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, + 'kernel': 'exp(-|x|^2/2), continuum, d = 3', 'mutant': MUT, 'python_requirement': '>= 3.11', + 'samples_per_batch': per_batch, 'batches': nb, 'rules': list(RULES), 'seeds': [2026 + i for i in range(nb)]} + res['controls'] = controls(zs) + res['far_field'] = {str(b): far_field(b) for b in (0.0, 1.0)} + res['z0'] = {} + res['hole'] = {} + res['ray_profiles'] = {} + res['remote_total'] = {} + res['R_3'] = {str(b): R3(b) for b in (0.0, 1.0)} + alphas = planar_alphas() + res['assembled_nu'] = {} + # grid and cutoff study at k = 1, b = 0 with two batches (same noise level as the difference estimator) + z00 = 36 * m3(0.0) + res['grid_convergence'] = {'k=1.0,b=0.0': { + 'hole_h=0.6,Rc=6': hole_integral(0.0, 1.0, zbatches[:2], z00, h=0.6 if not fast else 0.9), + 'hole_h=0.3,Rc=6': hole_integral(0.0, 1.0, zbatches[:2], z00, h=0.3 if not fast else 0.9), + 'hole_h=0.4,Rc=5': hole_integral(0.0, 1.0, zbatches[:2], z00, Rc=5.0, h=0.4 if not fast else 0.9)}} + for k in (0.5, 1.0, 2.0): + for b in (0.0, 1.0): + key = 'k=%s,b=%s' % (k, b) + z0 = 36 * k * k * m3(b) + res['z0'][key] = z0 + lam_inf = res['far_field'][str(b)]['lambda'] + H = hole_integral(b, k, zbatches, z0, h=0.6 if fast else 0.4) + res['hole'][key] = H + if k == 1.0: + res['ray_profiles'][key] = ray_profile(b, k, zs, z0, lam_inf) + res['remote_total'][key] = {'L=%d' % L: {'approx_L3_lambda_inf_plus_hole': L ** 3 * lam_inf + H['hole'], 'se': H['hole_se']} for L in (12, 24)} + kstr = {0.5: 'k=1/2', 1.0: 'k=1', 2.0: 'k=2'}[k] + a1 = alphas['%s,b=%d' % (kstr, int(b))]['alpha1_gh60'] * res['R_3'][str(b)] + a2 = alphas['%s,b=%d' % (kstr, int(b))]['alpha2_gh60'] * res['R_3'][str(b)] + res['assembled_nu'][key] = {'a1_d3': a1, 'a2_d3': a2, + 'nu1_L': {'L=%d' % L: a1 + k * (L ** 3 * lam_inf + H['hole']) for L in (12, 24)}, + 'nu2_over_nu1_L': {'L=%d' % L: a2 / (a1 + k * (L ** 3 * lam_inf + H['hole'])) for L in (12, 24)}, + 'note': 'a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)'} + return res + + +def check_run(): + zs = standard_normals(40000, 2026) + ctl = controls(zs) + 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 b in (0.0, 1.0): + require(abs(far_field(b)['lambda'] - ref['far_field'][str(b)]['lambda']) < 1e-12, 'far-field replay b = %r' % b) + require(abs(R3(b) - ref['R_3'][str(b)]) < 1e-9, 'R_3 replay b = %r' % b) + for j in range(3): + r = ctl['far_mc_over_quadrature_b0.0'][j] + require(0.9 < r < 1.1, 'far-point Monte Carlo within 10%% of z_0 E|det|1{j}, j = %d' % j) + # replay one kernel point per (k = 1, b) with the stored sample (same seeds and lengths as the full run) + zfull = [z for i in range(ref['batches']) for z in standard_normals(ref['samples_per_batch'], ref['seeds'][i])] + for key, prof in ref['ray_profiles'].items(): + b = float(key.split('b=')[1]) + z0 = ref['z0'][key] + # the transverse ray is blind to a sign error in the odd derivatives (it is the y -> -y reflection), so the + # axis and the diagonal ray, where the ramp 2 k x1^3 breaks that symmetry, are replayed as well + for name, idx, direction in (('transverse', 3, (0.0, 1.0)), ('diag45', 4, (1 / sqrt(2), 1 / sqrt(2))), ('axis', 5, (1.0, 0.0))): + row = prof[name][idx] + res = lambda_at((direction[0] * row['rho'], direction[1] * row['rho'], 0.0), b, 1.0, zfull, z0) + require(abs(res['lambda'] - row['lambda']) <= REPLAY_TOL * max(abs(row['lambda']), 1e-9), '%s replay %s' % (name, key)) + print(json.dumps({'check': 'ok', 'far_field_b0': ctl['far_field_b0'], 'mutant': MUT})) + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', action='store_true') + ap.add_argument('--fast', action='store_true') + 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 2ecffa826076f8360b18c1830d71e74a235eb1da Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Wed, 30 Sep 2026 15:11:08 +0000 Subject: [PATCH 2/5] d = 3 remote coefficient numerics: mark assembled nu as continuum approximations; trigger on [SC] Takes the two Codex findings of review 5367490653: the assembled coefficients are stored as approx_nu1_L and approx_nu2_over_nu1_L with an annotated note (continuum-kernel approximations of the finite-torus quantities, as the remote_total keys already were), NOTE.md section 3 says so before the values table, and the workflow triggers on the pinned [SC] source path. Numbers unchanged; RESULTS.json re-serialized by the same dumper; manifest refreshed. Scientific effect NONE; certified false. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c6-remote-coefficient-d3-numerics.yml | 1 + .../NOTE.md | 4 ++ .../RESULTS.json | 48 +++++++++---------- .../SOURCE_FILES.json | 18 +++---- .../remote3.py | 6 ++- 5 files changed, 42 insertions(+), 35 deletions(-) diff --git a/.github/workflows/c6-remote-coefficient-d3-numerics.yml b/.github/workflows/c6-remote-coefficient-d3-numerics.yml index 657b344aa..6ccf6f15a 100644 --- a/.github/workflows/c6-remote-coefficient-d3-numerics.yml +++ b/.github/workflows/c6-remote-coefficient-d3-numerics.yml @@ -7,6 +7,7 @@ name: C6 remote coefficient numerics, d = 3 (replay) - '.github/workflows/c6-remote-coefficient-d3-numerics.yml' - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' - 'frontiers/remote_window_20260924/PROOF.md' + - 'frontiers/spectral_cluster_closure_20260929/PROOF.md' - 'frontiers/c6_cluster_law_20260929/PROOF.md' - 'frontiers/remote_collision_20260928/PROOF.md' - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md index ec2fb9ba3..41bb46436 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md @@ -70,6 +70,10 @@ Far field: `Lambda_inf(0) = 0.08574653`, `Lambda_inf(1) = 0.06501408` (index spl `E|det G_3| = 3.3851375013`; level-integrated densities by index `0.0348624089, 0.1065073050, 0.1065073050, 0.0348624089` (closed forms `0.0348624089`, `0.1065073050`). `m_(3,0) = 0.6715728753`, `m_(3,1) = 5.5480602736`; `z_0 = 36 k^2 m_(3,b)`. +The `integral_X Lambda` and `nu` columns are continuum-kernel approximations of the finite-torus quantities (the +parent's periodized covariance is not used; `RESULTS.json` keys `remote_total.approx_L3_lambda_inf_plus_hole`, +`assembled_nu.approx_nu1_L`, `assembled_nu.approx_nu2_over_nu1_L`); see section 5. + | `k` | `b` | `c_hole = integral (Lambda - Lambda_inf)` (± s.e.) | `integral_X Lambda`, `L = 12` | `L = 24` | `a_1^(3)` | `nu(1)` at `L = 24` | `nu(2)/nu(1)` at `L = 24` | |---|---|---|---|---|---|---|---| | `1/2` | `0` | -1.883 ± 0.025 | 146.29 | 1183.48 | 4.71 | 596.4 | 2.45e-04 | diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json index 02cc2bf1d..b94917db7 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/RESULTS.json @@ -7,80 +7,80 @@ "k=0.5,b=0.0": { "a1_d3": 4.7095045254278975, "a2_d3": 0.14633532032248192, - "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", - "nu1_L": { + "approx_nu1_L": { "L=12": 77.85325873143533, "L=24": 596.4483000261176 }, - "nu2_over_nu1_L": { + "approx_nu2_over_nu1_L": { "L=12": 0.0018796299950305758, "L=24": 0.0002453445173975248 - } + }, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)" }, "k=0.5,b=1.0": { "a1_d3": 1.4791546886761118, "a2_d3": 0.04596054323383355, - "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", - "nu1_L": { + "approx_nu1_L": { "L=12": 57.03583271929747, "L=24": 450.2410184528051 }, - "nu2_over_nu1_L": { + "approx_nu2_over_nu1_L": { "L=12": 0.0008058187466119573, "L=24": 0.00010207986689389385 - } + }, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)" }, "k=1.0,b=0.0": { "a1_d3": 5.5975300565772095, "a2_d3": 0.10743529559706387, - "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", - "nu1_L": { + "approx_nu1_L": { "L=12": 150.63442458905413, "L=24": 1187.8245071784188 }, - "nu2_over_nu1_L": { + "approx_nu2_over_nu1_L": { "L=12": 0.0007132187472429238, "L=24": 9.044711146115999e-05 - } + }, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)" }, "k=1.0,b=1.0": { "a1_d3": 1.7580621249054367, "a2_d3": 0.03374291169622627, - "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", - "nu1_L": { + "approx_nu1_L": { "L=12": 111.79376034849392, "L=24": 898.2041318155093 }, - "nu2_over_nu1_L": { + "approx_nu2_over_nu1_L": { "L=12": 0.00030183179804525517, "L=24": 3.7567085811577016e-05 - } + }, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)" }, "k=2.0,b=0.0": { "a1_d3": 9.105999529987802, "a2_d3": 0.071295162408578, - "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", - "nu1_L": { + "approx_nu1_L": { "L=12": 296.29531694755747, "L=24": 2370.6754821262866 }, - "nu2_over_nu1_L": { + "approx_nu2_over_nu1_L": { "L=12": 0.00024062196845722278, "L=24": 3.0073775574138276e-05 - } + }, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)" }, "k=2.0,b=1.0": { "a1_d3": 2.859979352086044, "a2_d3": 0.022392245776304404, - "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)", - "nu1_L": { + "approx_nu1_L": { "L=12": 220.60180425443744, "L=24": 1793.4225471884683 }, - "nu2_over_nu1_L": { + "approx_nu2_over_nu1_L": { "L=12": 0.00010150527033077964, "L=24": 1.2485761267699303e-05 - } + }, + "note": "a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)" } }, "batches": 4, diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json index 9d2165465..0cb803b09 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json @@ -7,9 +7,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 10342, - "sha256": "1f0a69daf45da5c95decef7138c785b25a0ffa8af855417c85670ad75fc48b53", - "git_blob": "ec2fb9ba304bbec54b5e3829df4ce455cc34d56a" + "bytes": 10650, + "sha256": "2e544c70701b844a963c9f07d70748b39957ac2e74c81e8eb91c71a328124115", + "git_blob": "41bb46436096747e0190c141a62b59ea7f2dd0bd" }, { "path": "README.md", @@ -19,9 +19,9 @@ }, { "path": "RESULTS.json", - "bytes": 26653, - "sha256": "8a953cce275c5de86712c2de7ed2ae33f539c60685129b43b95907f4c076bc31", - "git_blob": "02cc2bf1d8b56a44c777e1cdb92223628f7f78c9" + "bytes": 27403, + "sha256": "2d3690d336460b6b06c080fa4b51e01a1fdfaa2e2cb24593e504bc5c9bb73a50", + "git_blob": "b94917db7479e8074d8d881e7de2e4ea28b12e57" }, { "path": "SOURCE_MAP.json", @@ -31,9 +31,9 @@ }, { "path": "remote3.py", - "bytes": 26354, - "sha256": "9058ca89fbecbbd72e46ca70bf5d5d5cc739aa23f31724701dd3d0ec24b7a2c5", - "git_blob": "834b732f85224535889a55227bac8d015061595f" + "bytes": 26608, + "sha256": "0d0d78364b643690c02c09f3d5247cd589626ae288f72000416047e89e17f7d6", + "git_blob": "1c2eb8033ff481e6db8bd3817cb9a1b8d996304f" } ], "nonauthor_reads": [] diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py index 834b732f8..1c2eb8033 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py @@ -533,8 +533,10 @@ def full_run(fast=False): a1 = alphas['%s,b=%d' % (kstr, int(b))]['alpha1_gh60'] * res['R_3'][str(b)] a2 = alphas['%s,b=%d' % (kstr, int(b))]['alpha2_gh60'] * res['R_3'][str(b)] res['assembled_nu'][key] = {'a1_d3': a1, 'a2_d3': a2, - 'nu1_L': {'L=%d' % L: a1 + k * (L ** 3 * lam_inf + H['hole']) for L in (12, 24)}, - 'nu2_over_nu1_L': {'L=%d' % L: a2 / (a1 + k * (L ** 3 * lam_inf + H['hole'])) for L in (12, 24)}, + # continuum-kernel approximations of the finite-torus coefficients (the parent's + # periodized covariance is not used; see NOTE.md section 5), like remote_total + 'approx_nu1_L': {'L=%d' % L: a1 + k * (L ** 3 * lam_inf + H['hole']) for L in (12, 24)}, + 'approx_nu2_over_nu1_L': {'L=%d' % L: a2 / (a1 + k * (L ** 3 * lam_inf + H['hole'])) for L in (12, 24)}, 'note': 'a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)'} return res From 5925b3fc623c4f16824327e71b9ccb06041e600c Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Wed, 30 Sep 2026 16:44:45 +0000 Subject: [PATCH 3/5] d = 3 remote coefficient numerics: producer note synchronized with RESULTS.json; reporting-consistency check Takes the Codex reconciliation finding 5915572509: the assembled-row note is now a module constant used by full_run, identical to the committed RESULTS.json text, and --check verifies for every (k, b) row the key set, the note, a_j^(3) = R_3(b) alpha_j^(2) from the stored R_3 and the pinned planar values, the remote totals L^3 Lambda_inf + hole, and the assembled approx_nu1_L and approx_nu2_over_nu1_L from the stored components. RESULTS.json unchanged; manifest refreshed. Scientific effect NONE; certified false. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../SOURCE_FILES.json | 6 ++--- .../remote3.py | 25 ++++++++++++++++++- 2 files changed, 27 insertions(+), 4 deletions(-) diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json index 0cb803b09..4d7adcb7a 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json @@ -31,9 +31,9 @@ }, { "path": "remote3.py", - "bytes": 26608, - "sha256": "0d0d78364b643690c02c09f3d5247cd589626ae288f72000416047e89e17f7d6", - "git_blob": "1c2eb8033ff481e6db8bd3817cb9a1b8d996304f" + "bytes": 28268, + "sha256": "b1f8a70c7ce874f7d310286be17f587058004c6a1ca88e52630117a9d787ebff", + "git_blob": "b43e6bdd644c50e9a94f5ed949aa69c6073cbf36" } ], "nonauthor_reads": [] diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py index 1c2eb8033..b43e6bdd6 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py @@ -32,6 +32,9 @@ RULES = (20, 16, 16) # nested Gauss-Legendre nodes for the A_0 cone quadrature (5120 nodes) MUT = None REPLAY_TOL = 1e-9 +ASSEMBLED_NOTE = ('a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating); ' + 'continuum-kernel approximation of the finite-torus coefficients (nonperiodic kernel; not the parent quantity)') +ASSEMBLED_KEYS = ('a1_d3', 'a2_d3', 'approx_nu1_L', 'approx_nu2_over_nu1_L', 'note') D = 3 Z3 = (0, 0, 0) @@ -537,7 +540,7 @@ def full_run(fast=False): # periodized covariance is not used; see NOTE.md section 5), like remote_total 'approx_nu1_L': {'L=%d' % L: a1 + k * (L ** 3 * lam_inf + H['hole']) for L in (12, 24)}, 'approx_nu2_over_nu1_L': {'L=%d' % L: a2 / (a1 + k * (L ** 3 * lam_inf + H['hole'])) for L in (12, 24)}, - 'note': 'a_j^(3) = R_3(b) alpha_j^(2) (Math-#184, unmerged, conditional); alpha from Math-#168 GH60 (floating)'} + 'note': ASSEMBLED_NOTE} return res @@ -551,6 +554,26 @@ def check_run(): for b in (0.0, 1.0): require(abs(far_field(b)['lambda'] - ref['far_field'][str(b)]['lambda']) < 1e-12, 'far-field replay b = %r' % b) require(abs(R3(b) - ref['R_3'][str(b)]) < 1e-9, 'R_3 replay b = %r' % b) + # reporting consistency of the assembled rows with their stored components (keys, note, arithmetic) + alphas = planar_alphas() + for k in (0.5, 1.0, 2.0): + for b in (0.0, 1.0): + key = 'k=%s,b=%s' % (k, b) + row = ref['assembled_nu'][key] + require(tuple(sorted(row)) == tuple(sorted(ASSEMBLED_KEYS)) and row['note'] == ASSEMBLED_NOTE, 'assembled row keys and note %s' % key) + kstr = {0.5: 'k=1/2', 1.0: 'k=1', 2.0: 'k=2'}[k] + R = ref['R_3'][str(b)] + a1 = alphas['%s,b=%d' % (kstr, int(b))]['alpha1_gh60'] * R + a2 = alphas['%s,b=%d' % (kstr, int(b))]['alpha2_gh60'] * R + require(abs(row['a1_d3'] - a1) < 1e-12 * a1 and abs(row['a2_d3'] - a2) < 1e-12 * a2, 'assembled a_j %s' % key) + lam_inf = ref['far_field'][str(b)]['lambda'] + hole = ref['hole'][key]['hole'] + for L in (12, 24): + tot = L ** 3 * lam_inf + hole + require(abs(ref['remote_total'][key]['L=%d' % L]['approx_L3_lambda_inf_plus_hole'] - tot) < 1e-9 * tot, 'remote total %s L=%d' % (key, L)) + nu1 = a1 + k * tot + require(abs(row['approx_nu1_L']['L=%d' % L] - nu1) < 1e-9 * nu1, 'assembled nu1 %s L=%d' % (key, L)) + require(abs(row['approx_nu2_over_nu1_L']['L=%d' % L] - a2 / nu1) < 1e-9 * (a2 / nu1), 'assembled nu2/nu1 %s L=%d' % (key, L)) for j in range(3): r = ctl['far_mc_over_quadrature_b0.0'][j] require(0.9 < r < 1.1, 'far-point Monte Carlo within 10%% of z_0 E|det|1{j}, j = %d' % j) From 4c5b8e18b7eb44caba80cea31e961990e5848718 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Sat, 3 Oct 2026 23:23:34 +0000 Subject: [PATCH 4/5] C6 remote coefficient d = 3: apply R1/R2/R3 nonauthor reads Applies the Math-#192 Grok-support reads routed by the Chief of Staff: - R1 Slice A (review 5403289484) F1-F6: full_run writes the far-point controls from the first batch (the sample --check uses), so RESULTS.json regenerates; BBKS normalization, attribution and 6e-11 precision stated exactly; continuum-kernel scope; --check adds a 5 % gate for j = 1, 2. - R2 Slice B (review 5403404967) A1-A5: node allocation overshoot and reuse described; cone-rule accuracy near the pin; far-point decoupling; the Rc = 5 difference is midpoint-grid alignment (re-measured: the shell contributes 1e-5, a half-cell shift reproduces -0.019 / -0.021). - R3 Slice C (5974291103) findings 2-9: observed far-point ratios with their sample; Reading paragraph corrected (the d = 2 / d = 3 share factor carries R_3(b)); same-seed grid differences; ray-profile precision; conditional status of the assembled columns; continuum title. SOURCE_MAP records the nonauthor reads. RESULTS.json unchanged. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../NOTE.md | 92 +++++++++++++------ .../SOURCE_FILES.json | 18 ++-- .../SOURCE_MAP.json | 9 +- .../remote3.py | 18 ++-- 4 files changed, 95 insertions(+), 42 deletions(-) diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md index 41bb46436..444837ee0 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md @@ -1,4 +1,4 @@ -# Numerical values of the remote singleton coefficient k ∫_X Λ of ν(1) in dimension d = 3 (parent kernel; not certified) +# Numerical values of the remote singleton coefficient k ∫_X Λ of ν(1) in dimension d = 3 (continuum kernel `exp(-|x|^2/2)`, not periodized; not certified) Object `CL-C6-REMOTE-COEFF-D3-NUMERICS-20260930-v1`. Author: Anthropic Claude (Claude Code `session_015wNj8LPTKXsaT68G3DgPPh`). **Numerical evaluation only** (exact quadrature for the far field and the @@ -16,7 +16,8 @@ with `U_0 = (f, f_x, f_xx, f_xxx, f_y, f_z, f_xy, f_xz)(0)`, `v_0 = (b, 0, 0, 12 ([RM] (11)), `Y_x = (grad f(x), f(x))`, `H_x = D^2 f(x)` and `F_j(H) = |det H| 1{index H = j}`, `j = 0, ..., 3`. `Lambda = sum_j Lambda_j` is the density in [SC] (22), `beta_far = k integral_(X \ {0}) Lambda dx`, so that the cluster-law coefficients are `nu_1 = a_1 + beta_far`, `nu_2 = a_2` ([SC] (20)–(22); [CL] (1.4)). Math-#174 evaluated -this in `d = 2`; this note does `d = 3` for the continuum kernel `K(x) = exp(-|x|^2/2)`. +this in `d = 2`; this note evaluates the formula (13) in `d = 3` with the continuum covariance `K(x) = exp(-|x|^2/2)` in +place of [RM]'s `K_L`; no [RM] theorem is asserted for it. Exact structure used (all checked in `remote3.py --check`): @@ -24,7 +25,8 @@ Exact structure used (all checked in `remote3.py --check`): `2` and off-diagonal variance `1` (exact rational regression; Math-#184 Lemma 1), so that `z_0 = 36 k^2 m_(3,b)`, `m_(3,b) = E[(det A_0)^2 1{A_0 < 0}]`, `m_(3,0) = (7 - 4 sqrt2)/2` (Math-#184 Lemma 2, companion, conditional; the finite-`r` normalizer `Z_r / r^2` was corroborated there by Monte Carlo). -- **Far field.** At separation `|x| -> infinity` the two sites decouple: `p_(Y_x | U_0)(0, b) -> phi(b) (2 pi)^(-3/2)`, +- **Far field.** For the continuum kernel `K` only (on the torus `|x| <= sqrt3 L/2` and `K_L` does not decorrelate), + at separation `|x| -> infinity` the two sites decouple: `p_(Y_x | U_0)(0, b) -> phi(b) (2 pi)^(-3/2)`, `E[w_0 F_j(H_x)] -> z_0 E[F_j(H_b)]` with `H_b = -b I_3 + G_3` (Hessian at a level-`b` critical point of the unconditioned field), hence @@ -33,9 +35,13 @@ Exact structure used (all checked in `remote3.py --check`): evaluated by the ordered-eigenvalue quadrature (Vandermonde density with `c_3` of [SC] (8)); at `b = 0` the split is symmetric under `j <-> 3 - j`. Integrated over the level, `integral Lambda_inf_j(b) db` is the density of index-`j` critical points of the unconditioned field, whose classical closed forms in this normalization (unit - kernel: `sigma_0 = 1`, per-component `sigma_1 = 1`, `sigma_2^2 = 15`) are `(29 sqrt3 - 18 sqrt2)/(72 pi^2)` for - maxima and for minima and `(29 sqrt3 + 18 sqrt2)/(72 pi^2)` for each saddle index (Bardeen–Bond–Kaiser–Szalay - 1986); the quadrature reproduces all four to `1e-12`, an exact control of the far-field machinery (the `d = 2` + kernel: `Var f = 1`; `Var f_i = 1` per component, i.e. BBKS `sigma_1^2 = E|grad f|^2 = 3`; `Var f_ii = 3`, + `Cov(f_ii, f_jj) = Var f_ij = 1`, i.e. BBKS `sigma_2^2 = E(Lap f)^2 = 15`; `R_* = sqrt3 sigma_1/sigma_2 = 3/sqrt15`) are `(29 sqrt3 - 18 sqrt2)/(72 pi^2)` for + maxima and for minima and `(29 sqrt3 + 18 sqrt2)/(72 pi^2)` for each saddle index (maxima/minima: Bardeen–Bond–Kaiser–Szalay + 1986, `n_pk`; index 1 and 2: the same expression with the opposite sign, as in Azaïs–Delmas, arXiv:1911.02300, + Prop. 3.7, an attribution taken from the Slice A review 5403289484 and not checked here against the source; the + values themselves are checked numerically); the quadrature reproduces all four to `6e-11` (absolute; check + tolerance `1e-9`), an exact control of the far-field machinery (the `d = 2` analogue in Math-#174 was the Longuet-Higgins total `2/(pi sqrt3)`). - **Symmetry.** `Lambda` depends on `(x_1, r)`, `r = |x_perp|`, only (rotations about the pin axis leave `U_0`, `v_0` and the kernel invariant); the spatial integrals are cylindrical midpoint rules with weight `2 pi r`. @@ -51,16 +57,23 @@ conditional Gaussian density. Then the `A_0` integral is a nested Gauss–Legendre cone quadrature in `(a, c, e) = ((a_11 + a_22)/2, (a_11 - a_22)/2, a_12)` (cone `a < 0`, `c^2 + e^2 < a^2`; `20 x 16 x 16` nodes; the integrand vanishes to second order on the cone -boundary, and against the exact `m_(3,b)` the rule is accurate to `1e-8`), and `E[F_j(H_x) | A_0]`, a six-dimensional +boundary; against the exact `m_(3,b)` at the decoupled point the rule is accurate to `1e-8`; in the cells nearest the +pin (`r = 0.2` at `h = 0.4`, `r = 0.15` at `h = 0.3`) the conditional `A_0` law is narrow and the rule is `1`–`3 %` +off in the cone mass, which changes the hole by at most about `1e-4` at `h = 0.4`, as measured in the Slice B review +5403404967), and `E[F_j(H_x) | A_0]`, a six-dimensional Gaussian expectation of `|det H| 1{index}`, is Monte Carlo: `H_x | A_0` is Gaussian with mean affine in `A_0`, the budget of `4 x 40000` standard normals (four independent seeds) is allocated to the cone nodes in proportion to their -weights, and the same normals are used at every `x` (common random numbers). The correlation hole +weights (rounded, with a floor of one draw per node; at `40000` draws per batch the floor overshoots the budget by +about `11 %`, and the overshoot, about `2450` nodes carrying `4`–`5 %` of the cone weight, reuses the first draws of +the batch; each node's draws are still standard normal, so the estimator is unbiased, and the batch spread contains +the induced correlation), and the same normals are used at every `x` (common random numbers). The correlation hole c_hole_j = integral (Lambda_j(x) - Lambda_j(x_far)) d^3x, x_far = (10, 0, 0), is estimated by the difference of the two common-random-number estimates, cell by cell, on the cylinder `|x_1| <= 6`, `r <= 6` (`h = 0.4`); the standard error is the spread of the four batch means. `Lambda(x_far) = Lambda_inf` to -`1e-16`. The torus integral is then `integral_X Lambda = L^3 Lambda_inf + c_hole` for `L >= 12` (the depletion radius +`1e-16` (every cross-covariance between the two sites is below `2e-17` at `|x| = 10`). The torus integral is then approximated by `L^3 Lambda_inf + c_hole` for `L >= 12` (continuum kernel; not checked +against a periodized integral in `d = 3`; the depletion radius is about `3`, section 3); no direct periodized integral is computed (at `L = 6` the additive form is not valid, as in `d = 2`, and no value is given). @@ -72,7 +85,9 @@ Far field: `Lambda_inf(0) = 0.08574653`, `Lambda_inf(1) = 0.06501408` (index spl The `integral_X Lambda` and `nu` columns are continuum-kernel approximations of the finite-torus quantities (the parent's periodized covariance is not used; `RESULTS.json` keys `remote_total.approx_L3_lambda_inf_plus_hole`, -`assembled_nu.approx_nu1_L`, `assembled_nu.approx_nu2_over_nu1_L`); see section 5. +`assembled_nu.approx_nu1_L`, `assembled_nu.approx_nu2_over_nu1_L`); see section 5. The `a_1^(3)`, `nu(1)` and +`nu(2)/nu(1)` columns are moreover conditional on Math-#184's `R_3(b)` (unmerged) and on Math-#168's uncertified GH60 +`alpha_j`. | `k` | `b` | `c_hole = integral (Lambda - Lambda_inf)` (± s.e.) | `integral_X Lambda`, `L = 12` | `L = 24` | `a_1^(3)` | `nu(1)` at `L = 24` | `nu(2)/nu(1)` at `L = 24` | |---|---|---|---|---|---|---|---| @@ -83,32 +98,45 @@ parent's periodized covariance is not used; `RESULTS.json` keys `remote_total.ap | `2` | `0` | -4.575 ± 0.035 | 143.59 | 1180.78 | 9.106 | 2370.7 | 3.01e-05 | | `2` | `1` | -3.473 ± 0.009 | 108.87 | 895.28 | 2.86 | 1793.4 | 1.25e-05 | -Index decomposition of the hole at `k = 1`: `b = 0`: `-0.132 ± 0.006, -2.936 ± 0.017, -0.263 ± 0.023, 0.198 ± 0.006`; `b = 1`: `-0.001 ± 0.000, -0.282 ± 0.006, -1.823 ± 0.008, -0.202 ± 0.001`. +Index decomposition of the hole at `k = 1`: `b = 0`: `-0.132 ± 0.006, -2.936 ± 0.017, -0.263 ± 0.023, 0.198 ± 0.006`; `b = 1`: `-0.0008 ± 0.0004, -0.282 ± 0.006, -1.823 ± 0.008, -0.202 ± 0.001`. The pin depletes a region of radius about `3` (as in `d = 2`): along the axis `Lambda/Lambda_inf` is below `1e-3` at `|x_1| <= 2` and `0.0329` at `2.5`, `0.62` at `3`, `1.01` at `4`, `1` at `5`; transversally `0.182` at `0.5`, `0.237` at `1`, `0.644` at `2`, `0.998` at `3`, `1` at `4`; along the `45` degree ray `0.00917` at `1`, `0.777` at `2`, `1.07` at `2.5`, `1.03` at `3`, `1` at `4` (values at -`k = 1, b = 0`, `160000` samples per point, Monte Carlo error about `0.7 %` per point). The three profiles are within -Monte Carlo noise of `1` from `|x| = 4` on. +`k = 1, b = 0`, `160000` samples per point, Monte Carlo error about `0.2`–`0.7 %` per point). The three profiles are within +Monte Carlo noise of `1` from `|x| = 5` on (on the axis `1.013` at `4`, a residual overshoot of about `1 %`). **Reading.** `a_j^(3) = R_3(b) alpha_j^(2)` (Math-#184; `R_3(0) = (32 + 28 sqrt2)/17 = 4.2116458674`, `R_3(1) = 4.6201205312`) with -Math-#168's planar `alpha_j`. On the parent torus sizes the remote singletons dominate `nu(1)` by a factor of order -`10^2`–`10^3` (`L^3 Lambda_inf = 13824 Lambda_inf` at `L = 24`), so the two-point share of a nonempty window is -`nu(2)/(nu(1) + nu(2))` of order `1e-4` at `L = 12` and `1e-5` at `L = 24`, smaller than in `d = 2` (Math-#174) by -the factor `L Lambda_inf^(3) / Lambda_inf^(2)` up to the near terms. +Math-#168's planar `alpha_j` (conditional, as above). On the parent torus sizes the remote singletons dominate +`nu(1)`: `k integral_X Lambda / a_1` is about `15.5`–`76` at `L = 12` and `126`–`626` at `L = 24` +(`L^3 Lambda_inf = 13824 Lambda_inf` at `L = 24`), so the two-point share of a nonempty window +`nu(2)/(nu(1) + nu(2))` is between `1.0e-4` and `1.9e-3` at `L = 12` and between `1.25e-5` and `2.45e-4` at `L = 24` +(largest at `k = 1/2, b = 0`), smaller than in `d = 2` (Math-#174, additive values) by about +`L Lambda_inf^(3) / (R_3(b) Lambda_inf^(2))`, i.e. about `2` at `L = 12` and `4` at `L = 24` (observed `2.04`–`2.25` +and `4.00`–`4.28`). ## 4. Precision (empirical, not bounds) - Far field: quadrature orders `30`, `40` agree to `1e-12`; the Monte Carlo at the decoupled point `x_far` reproduces - `z_0 E[|det| 1{j}]` within its statistical error (`1.052, 0.985, 1.001, 1.137` at `b = 0`, indices `j = 0..3`; the maxima/minima indices - carry `1.4 %` of the mass and have correspondingly larger errors). + `z_0 E[|det| 1{j}]` to within 1.5 % for the saddle indices and 14 % for the extremal indices (observed ratios, no + error computed; `1.052, 0.985, 1.001, 1.137` at `b = 0`, indices `j = 0..3`, from `40000` samples, seed `2026`, the + sample `--check` uses; the maxima/minima indices carry `1.4 %` of the mass and have correspondingly larger errors: + over the four seed batches `j = 3` is `1.113 ± 0.020` at `b = 0` and `1.027 ± 0.006` at `b = 1`). - Cone quadrature for the pin weight: `1e-8` relative against the exact `m_(3,b)` at the far point; at the near point `(0, 1, 0)` the `16 x 12 x 12` and `20 x 16 x 16` rules differ by `7e-5` relative. -- Monte Carlo: per-point relative error about `1.28 / sqrt(160000) = 0.3 %` on `Lambda`; the hole standard errors are - in the table (batch means of four independent seeds; common random numbers make the far cells contribute almost no +- Monte Carlo: per-point relative error nominally `1.28 / sqrt(160000) = 0.3 %` on `Lambda` (observed four-batch + spread `0.2`–`0.7 %` on the ray profiles); the hole standard errors are in the table (batch means of four + independent seeds, 3 degrees of freedom; the grid rows use two seeds, 1 degree of freedom; common random numbers make the far cells contribute almost no noise). -- Grid (`k = 1, b = 0`, two batches): hole `-3.155 ± 0.055` (`h = 0.6`), `-3.133 ± 0.029` (`h = 0.4`, tabulated), `-3.138 ± 0.057` (`h = 0.3`); - cutoff `Rc = 5` instead of `6`: `-3.164 ± 0.056`. These differences are measured, not bounds; the grid error of the - tabulated values is of the order of the difference between `h = 0.4` and `h = 0.3`. +- Grid (`k = 1, b = 0`, seeds 2026–2027 only; same-seed `h = 0.4` value `-3.143 ± 0.056`): `h = 0.6`: `-3.155`, + `h = 0.3`: `-3.138`, `Rc = 5`: `-3.164`. Same-seed differences from `h = 0.4, Rc = 6` (common random numbers): + `-0.011`, `+0.005`, `-0.021`. The last is systematic (identical in both batches) and is a midpoint-grid alignment + effect, not the cutoff: the `Rc = 5` run also shifts the `x_1` cell centres by `h/2` (`hr = 5/12`); shifting the + centres at `Rc = 6` gives `-0.019`, `-0.021` on the same two seeds, while the cells of the `Rc = 6` grid outside the + `Rc = 5` box contribute `1e-5` (Slice B review 5403404967; on nested grids that review finds the shell + `5 < |x| <= 6` at most `7e-5` over the six rows and `6 < |x| <= 8` below `1e-6`). Against `h = 0.1`, that review's + paired runs give a grid error of the tabulated `h = 0.4` hole of about `-0.013` at `k = 1, b = 0` (`O(h^2)`: + `-0.007` at `h = 0.31`, `-0.003` at `h = 0.2`), systematic and about `0.45` of the reported s.e. These differences + are measured, not bounds. - Near-pin points: the conditioning never lost positivity on the grid (`r >= 0.2`); `Lambda` is below `1e-20 Lambda_inf` on the axis for `|x_1| <= 1.5`. @@ -116,7 +144,9 @@ the factor `L Lambda_inf^(3) / Lambda_inf^(2)` up to the near terms. Not certified. Continuum kernel only: no periodized (finite-`L`) jets, so the values at `L = 12, 24` are the additive approximation `L^3 Lambda_inf + c_hole` and no `L = 6` value is given. `d = 3` only; the pin direction is immaterial by -isotropy. The assembled `nu(1)`, `nu(2)` inherit Math-#168's empirical precision on `alpha_j` and are conditional on +isotropy. The assembled `nu(1)`, `nu(2)` inherit Math-#168's empirical precision on `alpha_j` (at `k = 2` Math-#168 finds GH60 `J_2` 2.6 % (3.2 s.e.) below +MC and does not establish the third digit of `alpha_2(2)`; the `k = 2` `nu(2)/nu(1)` entries inherit this) and are +conditional on Math-#184's factor `R_3(b)` (unmerged). No rate, no finite-`r` statement. Values of the formulas [RM] (13) and [SC] (22); not a re-review of either. @@ -125,7 +155,8 @@ Math-#184's factor `R_3(b)` (unmerged). No rate, no finite-`r` statement. Values `python -B -S remote3.py --check` and `-B -O -S`: exact rational contact law (`A_0 | U_0 = -b I + GOE(2, 1)`), `c_2 = pi`, `m_(3,0) = (7 - 4 sqrt2)/2`, eigenvalue-density normalization, far-field index symmetry at `b = 0`, the decoupled density `phi(b) (2 pi)^(-3/2)` at `x_far`, the far-point Monte Carlo against `z_0 E|det| 1{j}` within -`10 %` for the three largest indices, replay of the far field and of `R_3(b)` against `RESULTS.json`, and replay of +`10 %` for `j = 0, 1, 2` and within `5 %` for `j = 1, 2` at `b = 0` with the fixed seed `2026` (across seeds the s.d. at +`40000` samples is about `8 %, 1.4 %, 1.3 %` for `j = 0, 1, 2`, so the `j = 0` gate is weak), replay of the far field and of `R_3(b)` against `RESULTS.json`, and replay of one kernel point per `b` with the stored seeds to `1e-9`. Mutants `hermite-sign`, `det-abs`, `weight-indicator` exit `1`. The full run regenerates `RESULTS.json` deterministically (seeded) in about an hour. @@ -134,3 +165,12 @@ one kernel point per `b` with the stored seeds to `1e-9`. Mutants `hermite-sign` Pins (`SOURCE_MAP.json`, all on `main`): [RM] `b383bfcc`, [SC] `16c56821`, [CL] `ba492c8e`, [LP] `dfed3b8d`, [NUM] `2648314f` (Math-#168), [NUM2] `3e4a7ddb` (Math-#174, merged). Unmerged companion: Math-#184 `PROOF.md` `91d22fce` (Lemma 2 and `R_3(b)`, recomputed here from the displayed formulas). + +Revision (Slice A review 5403289484, Slice B review 5403404967 and Slice C review 5974291103 on Math-#192; wording, one regeneration fix, one +added check gate): `full_run` writes the far-point controls from the first batch (the sample `--check` uses), so the +stored `controls` block regenerates; the far-point ratios are stated as observed values with their sample; the +BBKS normalization, attribution and `6e-11` control precision are stated exactly; the continuum-kernel scope is +stated in section 1; the Reading paragraph is corrected (the `d = 2` / `d = 3` share factor carries `R_3(b)`); the +conditional status of the assembled columns, the grid rows' same-seed differences (the `Rc = 5` effect is grid +alignment, per the Slice B review 5403404967, re-measured here), the cone-rule accuracy near the pin, the node +allocation of the Monte Carlo budget and the ray-profile precision are stated; `--check` gains a `5 %` gate for `j = 1, 2`. `RESULTS.json` unchanged. diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json index 4d7adcb7a..e8488abca 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json @@ -7,9 +7,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 10650, - "sha256": "2e544c70701b844a963c9f07d70748b39957ac2e74c81e8eb91c71a328124115", - "git_blob": "41bb46436096747e0190c141a62b59ea7f2dd0bd" + "bytes": 15042, + "sha256": "cd6549409a474e1e981239821a9edcc5d99106d84fc691ad42cba3c3b79ecd9d", + "git_blob": "444837ee05471f5e0b4ad9fcaaf142d7c7e65653" }, { "path": "README.md", @@ -25,15 +25,15 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 4295, - "sha256": "0ace734fa8900a27da3e0cf987a5cd06d351c6b3210e9a0793c52ac01bba1667", - "git_blob": "1d65daa6fd28a5673ea73326ed55e9405c652ae2" + "bytes": 5713, + "sha256": "cbf39af4c64ef8355e70cf72c9ad396b89d6c3737326831e11b7ffef7db10e49", + "git_blob": "c20c831030cc194b8e453fbe75b45321185a494e" }, { "path": "remote3.py", - "bytes": 28268, - "sha256": "b1f8a70c7ce874f7d310286be17f587058004c6a1ca88e52630117a9d787ebff", - "git_blob": "b43e6bdd644c50e9a94f5ed949aa69c6073cbf36" + "bytes": 28824, + "sha256": "e282c93af1e1126b5e8e7556b3189ac03b4d89c630e0556ada9e1d8e0dedea10", + "git_blob": "8f29909f1f0037950a91fe26f03e43d8d1f377a3" } ], "nonauthor_reads": [] diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json index 1d65daa6f..c20c83103 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json @@ -8,7 +8,14 @@ "review_status": "AUTHOR_SIDE_NUMERICAL_NOTE", "nonauthor_acceptance": false, "governing_status_changed": false, - "nonauthor_reads": [], + "nonauthor_reads": [ + {"comment": "5367490653", "kind": "pull_request_review", "pr": "Math-#192", "lane": "OpenAI / Codex (automated Codex review)", "scope": "two P2 findings, applied in 2ecffa8"}, + {"comment": "5915572509", "kind": "issue_comment", "pr": "Math-#192", "lane": "OpenAI / Codex", "scope": "reconciliation finding (producer note vs RESULTS.json), applied in 5925b3f"}, + {"comment": "5915786179", "kind": "issue_comment", "pr": "Math-#192", "lane": "OpenAI / Codex", "scope": "reconciliation recorded as resolved at 5925b3f"}, + {"comment": "5403289484", "kind": "pull_request_review", "pr": "Math-#192", "lane": "Grok Bot agent 1 (Cursor-hosted, Grok support routed by the Chief of Staff 5974102560)", "scope": "R1 Slice A (exact structure and far field) at 5925b3f: ACCEPT_WITH_FINDINGS; F1-F6 applied"}, + {"comment": "5974291103", "kind": "issue_comment", "pr": "Math-#192", "lane": "Grok Bot agent 4 (Grok support routed by the Chief of Staff 5974102560)", "scope": "R3 Slice C (reporting) at 5925b3f: ACCEPT_WITH_FINDINGS; findings 2-9 applied (finding 9 as NOTE wording; finding 4 attribution superseded by R2 A4)"}, + {"comment": "5403404967", "kind": "pull_request_review", "pr": "Math-#192", "lane": "Grok Bot agent 3 (Grok support routed by the Chief of Staff 5974102560)", "scope": "R2 Slice B (method) at 5925b3f: ACCEPT_WITH_FINDINGS; A1-A5 applied (A3 PR body; A4 re-measured by the author)"} + ], "sources": [ {"id": "RM", "path": "frontiers/remote_window_20260924/PROOF.md", "bytes": 18355, "sha256": "a332bae9bdc0106ce17047f7e0409cc3d94eb610a0c7b74ba5ba2d01a1620cb7", "git_blob": "b383bfcc88ec4ad497dff01fb6640e429ba24a84", "commit": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "sections 2, 4, 5: the contact frame U_0 (4), the weight w_0 = (6k)^2 (det A_0)^2 1{A_0 < 0} (10), the full normalizer (11) and the contact kernel Lambda_j (13) in every fixed d >= 2; the object evaluated", "current_required": true, "on_main": true}, {"id": "SC", "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", "bytes": 25006, "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", "commit": "de54d1da2f6cdde59df3c34bb50ecd85c25ca333", "role": "(8) c_m; (20) the near coefficients a_1, a_2; (22) beta_far = k integral_X Lambda in every fixed d; nu_1 = a_1 + beta_far, nu_2 = a_2", "current_required": true, "on_main": true}, diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py index b43e6bdd6..8f29909f1 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/remote3.py @@ -248,7 +248,8 @@ def m3(b, n=40): def far_field(b, n=40): - """Lambda_inf_j(b) = phi(b) (2 pi)^(-3/2) E[|det(-b I + G_3)| 1{index j}]: the decoupled far-field kernel.""" + """Lambda_inf_j(b) = phi(b) (2 pi)^(-3/2) E[|det(-b I + G_3)| 1{index j}]: the decoupled far-field kernel. + Continuum kernel K on R^3 only; not a statement about K_L or the torus.""" E = goe_functional(3, b, lambda l: abs(l[0] * l[1] * l[2]), n) pref = phi(b) * (2 * pi) ** (-1.5) return {'E_absdet_index': E, 'lambda_j': [pref * e for e in E], 'lambda': pref * sum(E)} @@ -354,7 +355,7 @@ def weighted(self, zs, rules=None): second order on the cone boundary). E[F_j(H_x) | A_0] is Monte Carlo on the supplied 6-dim standard normals zs (common random numbers across x), H_x | A_0 being Gaussian with mean affine in A_0 and fixed covariance; the sample budget len(zs) is allocated to the nodes in proportion to their weights (at least one sample per node), - so the effective sample size is the whole budget.""" + approximately proportional; the floor of one draw per node overshoots the budget, see the reuse branch below.""" if rules is None: if getattr(self, '_nodes', None) is None: self._nodes = self.cone_nodes() @@ -373,7 +374,8 @@ def weighted(self, zs, rules=None): block = zs[pos:pos + n] pos += n if not block: - # budget exhausted by rounding: reuse the first samples (deterministic, tiny weights only) + # budget exhausted by rounding: reuse the first samples (deterministic; about 4-5 % of the cone weight at 40000 + # draws, 1 % at 160000) block = zs[:n] part = [0.0] * 4 for z in block: @@ -483,8 +485,9 @@ def controls(zs): out['far_mc_over_quadrature_b%s' % b] = [Emc[j] / (z0 * Eq[j]) for j in range(4)] # E|det G_3| at b = 0 (the level-0 factor) and the level-integrated far field: integral_b Lambda_inf_j(b) db is the # density of index-j critical points of the unconditioned field, whose classical closed forms in this normalization - # (unit Gaussian kernel: sigma_0 = 1, per-component sigma_1 = 1, sigma_2^2 = 15) are (29 sqrt3 - 18 sqrt2)/(72 pi^2) - # for maxima and minima and (29 sqrt3 + 18 sqrt2)/(72 pi^2) for each saddle index (Bardeen-Bond-Kaiser-Szalay 1986) + # (unit Gaussian kernel: Var f = 1, Var f_i = 1 per component, i.e. BBKS sigma_1^2 = E|grad f|^2 = 3, and BBKS + # sigma_2^2 = E(Lap f)^2 = 15) are (29 sqrt3 - 18 sqrt2)/(72 pi^2) for maxima and minima (Bardeen-Bond-Kaiser-Szalay + # 1986, n_pk) and (29 sqrt3 + 18 sqrt2)/(72 pi^2) for each saddle index (opposite sign; cf. Azais-Delmas 2019) out['E_absdet_G3_quadrature'] = sum(goe_functional(3, 0.0, lambda l: abs(l[0] * l[1] * l[2]))) rule = gauss_legendre(60) dens = [gl(lambda bb: far_field(bb, n=24)['lambda_j'][j], -8.0, 8.0, rule) for j in range(4)] @@ -506,7 +509,8 @@ def full_run(fast=False): res = {'schema': 1, 'object': 'CL-C6-REMOTE-COEFF-D3-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, 'kernel': 'exp(-|x|^2/2), continuum, d = 3', 'mutant': MUT, 'python_requirement': '>= 3.11', 'samples_per_batch': per_batch, 'batches': nb, 'rules': list(RULES), 'seeds': [2026 + i for i in range(nb)]} - res['controls'] = controls(zs) + # the far-point Monte Carlo controls use the first batch (40000 samples, seed 2026), the sample --check uses + res['controls'] = controls(zbatches[0]) res['far_field'] = {str(b): far_field(b) for b in (0.0, 1.0)} res['z0'] = {} res['hole'] = {} @@ -577,6 +581,8 @@ def check_run(): for j in range(3): r = ctl['far_mc_over_quadrature_b0.0'][j] require(0.9 < r < 1.1, 'far-point Monte Carlo within 10%% of z_0 E|det|1{j}, j = %d' % j) + if j in (1, 2): + require(0.95 < r < 1.05, 'far-point Monte Carlo within 5%% of z_0 E|det|1{j}, j = %d' % j) # replay one kernel point per (k = 1, b) with the stored sample (same seeds and lengths as the full run) zfull = [z for i in range(ref['batches']) for z in standard_normals(ref['samples_per_batch'], ref['seeds'][i])] for key, prof in ref['ray_profiles'].items(): From 50737069b3f9b177df29ecb0ddcc65ace57d8890 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Sat, 3 Oct 2026 23:45:58 +0000 Subject: [PATCH 5/5] C6 remote coefficient d = 3: apply readback AMEND (far-point ratios at b = 0 only) MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Applies Math-#192 readback 5974705557 (Grok Bot agent 3): - NOTE §4: the 1.5 % / 14 % far-point agreement is stated at b = 0 only, with the stored b = 1 ratios (3.8 % for j = 1, 23 % for j = 0). - Advisory: the re-measured 1e-5 outside the Rc = 5 box is attributed to this note, not to the Slice B review. - Advisory: SOURCE_FILES.json nonauthor_reads synchronized with SOURCE_MAP. Wording and metadata only; remote3.py and RESULTS.json unchanged. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../NOTE.md | 7 +- .../SOURCE_FILES.json | 64 +++++++++++++++++-- .../SOURCE_MAP.json | 3 +- 3 files changed, 63 insertions(+), 11 deletions(-) diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md index 444837ee0..98a14c82c 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/NOTE.md @@ -117,7 +117,7 @@ and `4.00`–`4.28`). ## 4. Precision (empirical, not bounds) - Far field: quadrature orders `30`, `40` agree to `1e-12`; the Monte Carlo at the decoupled point `x_far` reproduces - `z_0 E[|det| 1{j}]` to within 1.5 % for the saddle indices and 14 % for the extremal indices (observed ratios, no + `z_0 E[|det| 1{j}]` at `b = 0` to within 1.5 % for the saddle indices and 14 % for the extremal indices (at `b = 1`: `1.234, 0.962, 1.003, 1.025`, i.e. 3.8 % for `j = 1` and 23 % for `j = 0`, which carries `3e-4` of the mass; observed ratios, no error computed; `1.052, 0.985, 1.001, 1.137` at `b = 0`, indices `j = 0..3`, from `40000` samples, seed `2026`, the sample `--check` uses; the maxima/minima indices carry `1.4 %` of the mass and have correspondingly larger errors: over the four seed batches `j = 3` is `1.113 ± 0.020` at `b = 0` and `1.027 ± 0.006` at `b = 1`). @@ -132,7 +132,7 @@ and `4.00`–`4.28`). `-0.011`, `+0.005`, `-0.021`. The last is systematic (identical in both batches) and is a midpoint-grid alignment effect, not the cutoff: the `Rc = 5` run also shifts the `x_1` cell centres by `h/2` (`hr = 5/12`); shifting the centres at `Rc = 6` gives `-0.019`, `-0.021` on the same two seeds, while the cells of the `Rc = 6` grid outside the - `Rc = 5` box contribute `1e-5` (Slice B review 5403404967; on nested grids that review finds the shell + `Rc = 5` box contribute `1e-5` (measured here; on nested grids the Slice B review 5403404967 finds the shell `5 < |x| <= 6` at most `7e-5` over the six rows and `6 < |x| <= 8` below `1e-6`). Against `h = 0.1`, that review's paired runs give a grid error of the tabulated `h = 0.4` hole of about `-0.013` at `k = 1, b = 0` (`O(h^2)`: `-0.007` at `h = 0.31`, `-0.003` at `h = 0.2`), systematic and about `0.45` of the reported s.e. These differences @@ -173,4 +173,5 @@ BBKS normalization, attribution and `6e-11` control precision are stated exactly stated in section 1; the Reading paragraph is corrected (the `d = 2` / `d = 3` share factor carries `R_3(b)`); the conditional status of the assembled columns, the grid rows' same-seed differences (the `Rc = 5` effect is grid alignment, per the Slice B review 5403404967, re-measured here), the cone-rule accuracy near the pin, the node -allocation of the Monte Carlo budget and the ray-profile precision are stated; `--check` gains a `5 %` gate for `j = 1, 2`. `RESULTS.json` unchanged. +allocation of the Monte Carlo budget and the ray-profile precision are stated; `--check` gains a `5 %` gate for `j = 1, 2`. The readback 5974705557 restricted the far-point ratio sentence to +`b = 0` and added the `b = 1` ratios. `RESULTS.json` unchanged. diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json index e8488abca..b595bcf6b 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_FILES.json @@ -7,9 +7,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 15042, - "sha256": "cd6549409a474e1e981239821a9edcc5d99106d84fc691ad42cba3c3b79ecd9d", - "git_blob": "444837ee05471f5e0b4ad9fcaaf142d7c7e65653" + "bytes": 15285, + "sha256": "7d9ac0247e95c5723074dab786215d8dab3499dc4bfeb1f29905ad3ef61c63ed", + "git_blob": "98a14c82c08ba7c6dff420c70b6d573e521c6960" }, { "path": "README.md", @@ -25,9 +25,9 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 5713, - "sha256": "cbf39af4c64ef8355e70cf72c9ad396b89d6c3737326831e11b7ffef7db10e49", - "git_blob": "c20c831030cc194b8e453fbe75b45321185a494e" + "bytes": 5973, + "sha256": "9aedb5508b8996ad739fa86c6142bce110be06ac56d1489cbed24119d8f48bca", + "git_blob": "46bc42db4a0395e87f9199922388324233e11ba7" }, { "path": "remote3.py", @@ -36,5 +36,55 @@ "git_blob": "8f29909f1f0037950a91fe26f03e43d8d1f377a3" } ], - "nonauthor_reads": [] + "nonauthor_reads": [ + { + "comment": "5367490653", + "kind": "pull_request_review", + "pr": "Math-#192", + "lane": "OpenAI / Codex (automated Codex review)", + "scope": "two P2 findings, applied in 2ecffa8" + }, + { + "comment": "5915572509", + "kind": "issue_comment", + "pr": "Math-#192", + "lane": "OpenAI / Codex", + "scope": "reconciliation finding (producer note vs RESULTS.json), applied in 5925b3f" + }, + { + "comment": "5915786179", + "kind": "issue_comment", + "pr": "Math-#192", + "lane": "OpenAI / Codex", + "scope": "reconciliation recorded as resolved at 5925b3f" + }, + { + "comment": "5403289484", + "kind": "pull_request_review", + "pr": "Math-#192", + "lane": "Grok Bot agent 1 (Cursor-hosted, Grok support routed by the Chief of Staff 5974102560)", + "scope": "R1 Slice A (exact structure and far field) at 5925b3f: ACCEPT_WITH_FINDINGS; F1-F6 applied" + }, + { + "comment": "5974291103", + "kind": "issue_comment", + "pr": "Math-#192", + "lane": "Grok Bot agent 4 (Grok support routed by the Chief of Staff 5974102560)", + "scope": "R3 Slice C (reporting) at 5925b3f: ACCEPT_WITH_FINDINGS; findings 2-9 applied (finding 9 as NOTE wording; finding 4 attribution superseded by R2 A4)" + }, + { + "comment": "5403404967", + "kind": "pull_request_review", + "pr": "Math-#192", + "lane": "Grok Bot agent 3 (Grok support routed by the Chief of Staff 5974102560)", + "scope": "R2 Slice B (method) at 5925b3f: ACCEPT_WITH_FINDINGS; A1-A5 applied (A3 PR body; A4 re-measured by the author)" + }, + { + "comment": "5974705557", + "kind": "issue_comment", + "pr": "Math-#192", + "lane": "Grok Bot agent 3 (Grok support; author of R2, disclosed)", + "scope": "readback of 5925b3f..4c5b8e1: 28/29 resolved; AMEND on the far-point ratio sentence (b = 0 only) applied" + } + ] } diff --git a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json index c20c83103..46bc42db4 100644 --- a/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json +++ b/frontiers/c6_remote_coefficient_d3_numerics_20260930/SOURCE_MAP.json @@ -14,7 +14,8 @@ {"comment": "5915786179", "kind": "issue_comment", "pr": "Math-#192", "lane": "OpenAI / Codex", "scope": "reconciliation recorded as resolved at 5925b3f"}, {"comment": "5403289484", "kind": "pull_request_review", "pr": "Math-#192", "lane": "Grok Bot agent 1 (Cursor-hosted, Grok support routed by the Chief of Staff 5974102560)", "scope": "R1 Slice A (exact structure and far field) at 5925b3f: ACCEPT_WITH_FINDINGS; 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