From 9770ccdbd51ee2a3c01d964e467fc8f7f197514a Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 08:54:57 +0000 Subject: [PATCH 1/7] Microscopic tail constants numerics: script, source map, README and replay workflow (RESULTS.json and NOTE.md follow after the full run) Work in progress on frontiers/c6_microscopic_tail_numerics_20260930: planar evaluation of C_* of [R] (R13) via the d = 2 reduction of the [SC] measure to the Math-#168 contact regression, the whole-cluster split (Math-#169, unmerged companion), C_2/C_0 and B_sign (Math-#176, unmerged companion), with the near-mass identity alpha_1 + 2 alpha_2 = K_0 int Q |Z|^-12 p_odd as a control. Scientific effect NONE. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c6-microscopic-tail-numerics.yml | 62 ++++ .../README.md | 22 ++ .../SOURCE_MAP.json | 90 +++++ .../tail_constants.py | 330 ++++++++++++++++++ 4 files changed, 504 insertions(+) create mode 100644 .github/workflows/c6-microscopic-tail-numerics.yml create mode 100644 frontiers/c6_microscopic_tail_numerics_20260930/README.md create mode 100644 frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json create mode 100644 frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py diff --git a/.github/workflows/c6-microscopic-tail-numerics.yml b/.github/workflows/c6-microscopic-tail-numerics.yml new file mode 100644 index 000000000..4192e727b --- /dev/null +++ b/.github/workflows/c6-microscopic-tail-numerics.yml @@ -0,0 +1,62 @@ +name: C6 microscopic tail numerics (replay) + +'on': + pull_request: + paths: + - 'frontiers/c6_microscopic_tail_numerics_20260930/**' + - '.github/workflows/c6-microscopic-tail-numerics.yml' + 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_microscopic_tail_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 / 'tail_constants.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 ['shape-integral', 'gamma-power', 'cusp-shift']: + 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_microscopic_tail_numerics_20260930/README.md b/frontiers/c6_microscopic_tail_numerics_20260930/README.md new file mode 100644 index 000000000..5d6d55d48 --- /dev/null +++ b/frontiers/c6_microscopic_tail_numerics_20260930/README.md @@ -0,0 +1,22 @@ +# Planar microscopic radius-tail constants (parent kernel, d = 2) + +**Object:** `CL-C6-MICRO-TAIL-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical note +(deterministic Gauss quadrature with reported convergence; no proof, no enclosure). **Scientific effect:** NONE. + +Evaluates, for the parent kernel `exp(-|z|^2/2)` in the plane, the point-intensity tail constant `C_*` of the +already-formed microscopic measure ([R] (R13); `F(t) ~ C_* t^-11`), the whole-cluster split of that tail +(Math-#169, unmerged companion), the second-order coefficient `C_2/C_0`, the TV coefficient `kappa |C_2/C_0|` and +the signed constant `B_sign` (Math-#176, unmerged companion), for `k in {1/2, 1, 2}` and `b in {0, 1}`. In `d = 2` +the [SC] measure reduces to the exact contact regression of Math-#168, so every constant is a one-dimensional +Gaussian integral; the four-dimensional near-mass identity `alpha_1 + 2 alpha_2 = K_0 int Q |Z|^-12 p_odd` is +evaluated as a control against the merged Math-#168 values. + +- `NOTE.md` — what is evaluated, exact structure, values, precision, non-claims. +- `tail_constants.py` — standard-library script (Python `>= 3.11`): `--check` (exact controls and replay against + `RESULTS.json`), full run regenerates `RESULTS.json` (about five minutes); `--mutant {shape-integral, gamma-power, + cusp-shift}` must exit 1. +- `RESULTS.json`, `SOURCE_MAP.json` (five main-resident pins and two unmerged companions), `SOURCE_FILES.json`. +- Workflow `.github/workflows/c6-microscopic-tail-numerics.yml` replays manifest, pins, both check modes and the + mutants. + +Same GitHub account as every other lane; zero organizational-independence credit. The author will not merge. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json new file mode 100644 index 000000000..534257853 --- /dev/null +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json @@ -0,0 +1,90 @@ +{ + "schema": 1, + "object": "CL-C6-MICRO-TAIL-NUMERICS-20260930-v1", + "scientific_effect": "NONE", + "source_repository": "d6g8k5htny-coder/Math-", + "main_at_creation": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "note": "Numerical evaluation only, planar (d = 2). The formulas evaluated are [R] (R13) for the point-intensity tail constant C_*, with the [SC] measure (17) reduced at d = 2 (c_1 = 1, no hard directions) to the exact contact regression of the merged Math-#168 note; the whole-cluster split and the second-order/signed coefficients are the formulas of the unmerged companions Math-#169 (E5-E6) and Math-#176 ((13), (16), (19), (21)-(22)), evaluated as stated there and marked conditional on those candidates. The near-mass identity check ties (R11) to the merged alpha_1, alpha_2 values. Floating-point quadrature with reported convergence, not enclosures.", + "sources": [ + { + "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", + "bytes": 12533, + "sha256": "250897858c8b314c0ff85ec1860a725efacfdae0ff7409fced4cadf7bfb85973", + "git_blob": "0f14417ef7038b9c6f50e4b01e393a58b7b5e3a4", + "id": "R", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "(R4)-(R13): root parametrization, the weight identity (R10), the root-resolved integral (R11), the cusp integral J_cusp (R12) and C_* = (216/11) k^7 (c_m/z_0) I_shape J_cusp (R13); Theorem H (the point-selected height law, whose moments are recomputed as controls); Math-#166, merged, Slices A/B/C accepted (Claude, other session)", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md", + "bytes": 25006, + "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb", + "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a", + "id": "SC", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "(8) the spectral constant c_m (c_1 = 1) and (17) the limiting microscopic measure dM = (c_m/z_0) h_0(A_0, T) H(h) w ds dh dO dtau; at d = 2 this is the contact regression with the raw density h_0(0, tau); Math-#162, merged, Slices A/B/C/D accepted", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json", + "bytes": 4095, + "sha256": "9c88ea85b6ba08199996c9b5ff74ecd06afc0f458e08c28b4c3af6191f437fcb", + "git_blob": "2648314f8c11f0637ce301ba76360142a8309e8f", + "id": "NUM", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "the exact planar contact structure (A_0 | even pins ~ N(-b, 2); odd cubic jets ~ N(0, diag(2, 2, 6)); prefactor p_b(0)/z_0) and the values alpha_1(k, b), alpha_2(k, b) (GH60 and MC with standard errors) used in the near-mass identity control; Math-#168, merged", + "current_required": true, + "on_main": true + }, + { + "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md", + "bytes": 19889, + "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0", + "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758", + "id": "CUB", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "the deterministic classifier consumed by [R] and [E] (shape region S, doublet condition); context only", + "current_required": true, + "on_main": true + }, + { + "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md", + "bytes": 40261, + "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7", + "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2", + "id": "LP", + "commit": "02772ec0427f4dda1f2a6b9e980a031804e276a6", + "role": "section 2: the parent kernel (periodization of exp(-|z|^2/2), aligned frame); section 3 contact frame; (5.4) the full normalizer z_0", + "current_required": true, + "on_main": true + } + ], + "companions_unmerged": [ + { + "pull_request": "Math-#169", + "head": "56a7904f9ad7d409c31e23195c0e7eeab230613a", + "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", + "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", + "id": "E", + "role": "(E5) the shape constants J = 1083417/280 (recomputed exactly here) and D = 27066286003/223205220 - (79298560/4782969) log 2 (quoted; numerically confirmed to 1e-13 in the Slice C read), (E6) the whole-cluster coefficients C0 (I - D), C0 J, C0 D, C0 (I - D - J); Slices B, C, D accepted, A under Codex read at this writing; the split reported here is conditional on that candidate", + "current_required": false, + "on_main": false + }, + { + "pull_request": "Math-#176", + "head": "71772a8906cf6663d16be2cd5f63e5d4ebdcee2b", + "path": "frontiers/radial_rate_20260930/PROOF.md", + "git_blob": "65f24b1f9a7b80522a2ee059f5f6877c9f970f0d", + "id": "Q", + "role": "(12) the shape integrals U_2, B, U_abs (recomputed exactly here), (13) C_0 and C_2, (16) kappa, (19) B_sign, (21)-(22) the aligned planar cusp density and D_a G_0; Slices A and B accepted (this session); the coefficients reported here are conditional on that candidate", + "current_required": false, + "on_main": false + } + ], + "nonauthor_acceptance": false, + "governing_status_changed": false, + "main_at_update": "02772ec0427f4dda1f2a6b9e980a031804e276a6" +} diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py new file mode 100644 index 000000000..a37f7972b --- /dev/null +++ b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py @@ -0,0 +1,330 @@ +#!/usr/bin/env python3 +"""Planar (d = 2) microscopic radius-tail constants for the parent kernel K(z) = exp(-|z|^2/2). + +Evaluated (deterministic Gauss quadrature, standard library only; no proof, no enclosure): + * C_* of [R] (R13): the point-intensity tail F(t) ~ C_* t^-11 of the already-formed microscopic measure, with + C_* = (216/11) k^7 (c_m/z_0) I J_cusp; in d = 2, c_1 = 1 and c_m h_0(A_0 = 0, tau)/z_0 = [p_b(0)/z_0] p_odd(a, beta, c) + with the exact contact regression of Math-#168 (A_0 | even pins ~ N(-b, 2); (f_xxz, f_xzz, f_zzz) | odd pins ~ + N(0, diag(2, 2, 6))), so J_cusp = int gamma(a)^11 p_odd(a, a^2/(12k), a^3/(144k^2)) da is a one-dimensional integral; + * the whole-cluster split of [E] (E5-E6), using [E]'s exact shape constants D and J (companion, unmerged); + * the second-order coefficient C_2/C_0, the TV coefficient kappa |C_2/C_0| and the signed constant B_sign of [Q] + (13), (16), (19) with the aligned planar cusp density (21)-(22) (companion, unmerged); + * control: the near-mass identity alpha_1 + 2 alpha_2 = K_0 int_S int_Z Q |Z|^-12 p_odd (the root-resolved integral + (R11) integrated over all Z), against the merged Math-#168 values of alpha_1, alpha_2. + + python -B -S tail_constants.py # full run, writes RESULTS.json (a few minutes) + python -B -S tail_constants.py --check # exact controls + replay against RESULTS.json (Python >= 3.11) + python -B -S tail_constants.py --check --mutant NAME # must exit 1 +""" +import argparse +import json +import math +import os +import sys +from fractions import Fraction as Fr + +HERE = os.path.dirname(os.path.abspath(__file__)) +MUTANTS = ('shape-integral', 'gamma-power', 'cusp-shift') +MUT = None +REPLAY_TOL = 1e-6 + +SQ2PI = math.sqrt(2 * math.pi) +phi = lambda t: math.exp(-t * t / 2) / SQ2PI +Phi = lambda t: 0.5 * math.erfc(-t / math.sqrt(2)) +KAPPA = (2 / 13) * (11 / 13) ** 5.5 + +# companion constants ([E] E5, E20-E21; [Q] (12)), all recomputed exactly below where polynomial +J_OUT = Fr(1083417, 280) +def D_inner(): + return 27066286003 / 223205220 - (79298560 / 4782969) * math.log(2) + +# reference values of the merged Math-#168 RESULTS.json (GH60 and MC 1e6), quoted for the control +ALPHA_GH60 = {0.5: (1.11821, 0.0347454), 1.0: (1.32906, 0.0255091), 2.0: (2.1621, 0.0169281)} +ALPHA_MC = {0.5: (1.11457, 0.034938), 1.0: (1.32721, 0.0257064), 2.0: (2.16004, 0.0173858)} +ALPHA_MC_SE1 = {0.5: 0.0856 * 0.0313439, 1.0: 0.283 * 0.00783597, 2.0: 1.69 * 0.00195899} # s.e. of alpha_1 (J_1 s.e. times prefactor) + + +def require(cond, msg): + if not cond: + print('FAIL: ' + msg) + sys.exit(1) + + +# ----------------------------------------------------------------------------- exact shape integrals +class P: + """Univariate polynomial with Fraction coefficients.""" + def __init__(self, c): + self.c = {k: Fr(v) for k, v in c.items() if v != 0} + def __add__(self, o): + c = dict(self.c) + for k, v in o.c.items(): + c[k] = c.get(k, 0) + v + return P(c) + def __mul__(self, o): + if not isinstance(o, P): + return P({k: v * o for k, v in self.c.items()}) + c = {} + for i, a in self.c.items(): + for j, b in o.c.items(): + c[i + j] = c.get(i + j, 0) + a * b + return P(c) + def __pow__(self, n): + r = P({0: 1}) + for _ in range(n): + r = r * self + return r + def integ(self, lo, hi): + return sum((v * (Fr(hi) ** (k + 1) - Fr(lo) ** (k + 1)) / (k + 1) for k, v in self.c.items()), Fr(0)) + + +U = P({1: 1}); ONE = P({0: 1}); B0P = ONE * 3 + U * U * (-12) +QV = {3: U * (-16), 2: B0P * 4, 1: U * B0P * B0P * 4, 0: B0P ** 3 * (-1)} # Q(u, v) as polynomial in v with P(u) coefficients + + +def vint(F, lo, hi): + tot = P({}) + for p, cu in F.items(): + tot = tot + cu * (hi ** (p + 1) + lo ** (p + 1) * (-1)) * Fr(1, p + 1) + return tot + + +def shape_integral(weight_v): + """integral_S weight(u, v) Q(u, v) du dv for weight given as dict power_of_v -> P(u); the two strips of S.""" + F = {} + for p, cu in QV.items(): + for q, cw in weight_v.items(): + F[p + q] = F.get(p + q, P({})) + cu * cw + lo1 = (U * U * 12 + ONE * (-3)) * Fr(1, 2); hi = ONE * 3 + U * (-6); lo2 = B0P * Fr(1, 2) + return vint(F, lo1, hi).integ(Fr(-3, 2), Fr(-1, 2)) + vint(F, lo2, hi).integ(Fr(-1, 2), Fr(1, 2)) + + +def exact_constants(): + I = shape_integral({0: ONE}) + U2 = shape_integral({0: U * U}) + Bsh = shape_integral({0: B0P}) + # |u| Q: negative u on (-3/2, 0), positive on (0, 1/2) + F = {p: cu * U for p, cu in QV.items()} + lo1 = (U * U * 12 + ONE * (-3)) * Fr(1, 2); hi = ONE * 3 + U * (-6); lo2 = B0P * Fr(1, 2) + Uabs = -vint(F, lo1, hi).integ(Fr(-3, 2), Fr(-1, 2)) - vint(F, lo2, hi).integ(Fr(-1, 2), 0) + vint(F, lo2, hi).integ(0, Fr(1, 2)) + # eta moments: eta = u + 1/2 + v/6 + E1 = shape_integral({0: U + ONE * Fr(1, 2), 1: ONE * Fr(1, 6)}) / I + # J = integral over S2_outer: strips 1/2 -p + def subst_neg(poly): + return P({k: v * (-1) ** k for k, v in poly.c.items()}) + Qm = {p: subst_neg(cu) for p, cu in Qp.items()} + J = vint(Qm, A * Fr(1, 2), A * Pp).integ(Fr(1, 2), 1) + vint(Qm, A * Fr(1, 2), ONE * 3 + Pp * 6).integ(1, Fr(3, 2)) + if MUT == 'shape-integral': + I = I + 1 + return {'I': I, 'U2': U2, 'B': Bsh, 'Uabs': Uabs, 'E_eta': E1, 'J_outer': J} + + +# ----------------------------------------------------------------------------- quadrature +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 + 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) + xs.append(x); ws.append(2.0 / ((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)) + + +# ----------------------------------------------------------------------------- planar contact structure (Math-#168, exact) +def m2b(b): + m, s = -b, math.sqrt(2.0) + t = m / s + return (m * m + s * s) * Phi(-t) - m * s * phi(t) + + +def prefactor(k, b): + """p_b(0)/z_0 = phi(b/sqrt2) / (sqrt2 . 36 k^2 m_(2,b)) = c_1 h_0(0, .)/z_0 with the odd Gaussian factored out.""" + return phi(b / math.sqrt(2)) / math.sqrt(2) / (36 * k * k * m2b(b)) + + +def p_odd(a, beta, c): + return math.exp(-a * a / 4 - beta * beta / 4 - c * c / 12) / ((2 * math.pi) ** 1.5 * math.sqrt(24.0)) + + +def b0(u): + return 3 - 12 * u * u + + +def Q(u, v): + b = b0(u) + return (4 * v * v - b * b) * (b - 4 * u * v) + + +def cusp_integrals(k, rule, L=14.0): + """One-dimensional Gaussian integrals over the cusp slice beta = a^2/(12k), c = a^3/(144k^2): + J_cusp = int gamma^11 G0, T1 = int gamma^9 (1 + 12 A^2) G0, T2 = int gamma^13 D_a G0, Tb = int |A| gamma^10 G0.""" + gpow = 10 if MUT == 'gamma-power' else 11 + def G0(a): + cshift = 0.0 if MUT == 'cusp-shift' else a ** 3 / (144 * k * k) + return p_odd(a, a * a / (12 * k), cshift) + def gam(a): + return math.sqrt(1 + (a / (12 * k)) ** 2) + def DaG0(a): # D_a = k d_beta + (a/4) d_c on the diagonal cusp density: -(a^2/(12 vbeta) + a^4/(576 k^2 vc)) G0, vbeta = 2, vc = 6 + return -(a * a / 24 + a ** 4 / (3456 * k * k)) * G0(a) + Jc = gl(lambda a: gam(a) ** gpow * G0(a), -L, L, rule) + T1 = gl(lambda a: gam(a) ** 9 * (1 + 12 * (a / (12 * k)) ** 2) * G0(a), -L, L, rule) + T2 = gl(lambda a: gam(a) ** 13 * DaG0(a), -L, L, rule) + Tb = gl(lambda a: abs(a / (12 * k)) * gam(a) ** 10 * G0(a), -L, L, rule) + return {'J_cusp': Jc, 'T1': T1, 'T2': T2, 'Tb': Tb} + + +def near_mass(k, nu, nv, ns, na): + """108 k^7 int_S Q(u,v) int_{Z != 0} |Z|^-12 int p_odd(a, beta(u,v,Z,a), c(u,v,Z,a)) da dZ du dv, i.e. (R11) with the + radius indicator removed; times p_b(0)/z_0 this is the near point mass alpha_1 + 2 alpha_2. Z = +-e^s, so + |Z|^-12 dZ = e^{-11 s} ds; the s-window is placed at the onset of the Gaussian decay.""" + ru, rv, rs, ra = gauss_legendre(nu), gauss_legendre(nv), gauss_legendre(ns), gauss_legendre(na) + tot = 0.0 + for (ulo, uhi, lower) in ((-1.5, -0.5, lambda u: (12 * u * u - 3) / 2), (-0.5, 0.5, lambda u: (3 - 12 * u * u) / 2)): + def fu(u): + b = b0(u) + def fv(v): + q = Q(u, v) + c3 = 2 * k * (v - b * u) + cands = [] + if c3 != 0: + cands.append(math.log(abs(c3) / 6.0) / 3.0) + if b != 0: + cands.append(0.5 * math.log(k * abs(b) / 4.0)) + s_star = max(cands) if cands else 0.0 + def fs(s): + Zi2 = math.exp(-2 * s); Zi3 = math.exp(-3 * s) + acc = 0.0 + for sign in (1.0, -1.0): + def fa(a): + return p_odd(a, a * a / (12 * k) + k * b * Zi2, a ** 3 / (144 * k * k) + (a * b / 4) * Zi2 + sign * c3 * Zi3) + acc += gl(fa, -14.0, 14.0, ra) + return math.exp(-11 * s) * acc + return q * gl(fs, s_star - 3.0, s_star + 4.5, rs) + return gl(fv, lower(u), 3 - 6 * u, rv) + tot += gl(fu, ulo, uhi, ru) + return 108 * k ** 7 * tot + + +# ----------------------------------------------------------------------------- assembly +def constants_table(k, b, ex, cu): + I = float(ex['I']); U2 = float(ex['U2']); Bsh = float(ex['B']); Uabs = float(ex['Uabs']) + D = D_inner(); J = float(ex['J_outer']) + pf = prefactor(k, b) + K0 = 108 * k ** 7 * pf + C0 = (216 / 11) * k ** 7 * pf * cu['J_cusp'] # [E]'s C0: coefficient per unit shape integral + Cstar = C0 * I # [R]'s C_* = [Q]'s C_0 + C2 = K0 * (U2 * cu['T1'] + (2 * Bsh / 13) * cu['T2']) + c = C2 / Cstar + Bsign = (K0 / Cstar) * Uabs * cu['Tb'] + return {'prefactor_p_b0_over_z0': pf, 'K0': K0, 'C0_per_shape': C0, 'C_star': Cstar, + 'G_max_coeff': C0 * (I - D), 'G2_coeff': C0 * J, 'B2_coeff': C0 * D, 'singleton_coeff': C0 * (I - D - J), + 'C2': C2, 'C2_over_C0': c, 'tv_coefficient_kappa_c': KAPPA * c, 'B_sign': Bsign, + 'lower_bound_C2_over_C0_4587_856': 4587 / 856, 'upper_bound_B_sign_4587821_876544': 4587821 / 876544} + + +def controls(): + ex = exact_constants() + require(ex['I'] == Fr(246528, 35), 'I = 246528/35') + require(ex['U2'] == Fr(240192, 35) and ex['B'] == Fr(-428544, 7) and ex['Uabs'] == Fr(5898627, 880), 'U2, B, Uabs exact') + require(ex['B'] == 3 * ex['I'] - 12 * ex['U2'], 'B = 3I - 12 U2') + require(ex['E_eta'] == Fr(5771, 7062), 'E eta = 5771/7062') + require(ex['J_outer'] == J_OUT, 'J = 1083417/280') + require(Fr(11) * ex['U2'] / (2 * ex['I']) == Fr(4587, 856), '11 U2/(2I) = 4587/856') + require(Fr(11) * ex['Uabs'] / (2 * ex['I']) == Fr(4587821, 876544), '11 Uabs/(2I) = 4587821/876544') + D = D_inner() + require(0 < D < float(ex['I']) - float(ex['J_outer']), 'D positive and I - D - J positive') + q0 = math.sqrt(13 / 11) + require(abs(KAPPA - (q0 ** -11 - q0 ** -13)) < 1e-15, 'kappa = q0^-11 - q0^-13') + # planar contact structure: prefactor at b = 0 is 1/(36 k^2 sqrt(4 pi)) ... phi(0)/sqrt2/(36k^2) + require(abs(prefactor(1.0, 0.0) - phi(0.0) / math.sqrt(2) / 36) < 1e-15, 'prefactor at b = 0, k = 1') + # p_odd normalization + r = gauss_legendre(64) + z = gl(lambda a: gl(lambda be: gl(lambda c: p_odd(a, be, c), -20, 20, r), -20, 20, r), -20, 20, r) + require(abs(z - 1.0) < 1e-10, 'p_odd normalized') + return ex + + +def full_run(): + ex = controls() + res = {'object': 'CL-C6-MICRO-TAIL-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, + 'dimension': 2, 'exact': {k: str(v) for k, v in ex.items()}, 'D_inner': D_inner(), 'kappa': KAPPA, + 'quadrature': {'cusp_gl': 96, 'cusp_gl_check': 48, 'cusp_range': 14.0, 'near_mass_levels': [[32, 32, 64, 32], [64, 64, 64, 32], [96, 96, 64, 32]]}, + 'per_k': {}, 'table': {}, 'near_mass_identity': {}} + r96, r48 = gauss_legendre(96), gauss_legendre(48) + for k in (0.5, 1.0, 2.0): + cu = cusp_integrals(k, r96) + cu48 = cusp_integrals(k, r48) + res['per_k'][str(k)] = {'cusp': cu, 'cusp_gl48_rel_dev': {n: abs(cu48[n] - cu[n]) / abs(cu[n]) for n in cu}} + for b in (0.0, 1.0): + res['table']['k=%s,b=%s' % (k, b)] = constants_table(k, b, ex, cu) + levels = {} + for nodes in res['quadrature']['near_mass_levels']: + levels['x'.join(map(str, nodes))] = near_mass(k, *nodes) + pf0, pf1 = prefactor(k, 0.0), prefactor(k, 1.0) + top = levels['96x96x64x32'] + a1, a2 = ALPHA_GH60[k]; m1, m2 = ALPHA_MC[k] + res['near_mass_identity']['k=%s' % k] = { + 'K0_integral_without_prefactor_by_level': levels, + 'alpha1_plus_2alpha2_b0': pf0 * top, 'alpha1_plus_2alpha2_b1': pf1 * top, + 'ref_gh60_b0': a1 + 2 * a2, 'ref_mc_b0': m1 + 2 * m2, 'ref_mc_b0_se_alpha1': ALPHA_MC_SE1[k], + 'rel_dev_vs_gh60': (pf0 * top - (a1 + 2 * a2)) / (a1 + 2 * a2), 'rel_dev_vs_mc': (pf0 * top - (m1 + 2 * m2)) / (m1 + 2 * m2)} + print('k', k, 'C_*(b=0) = %.6g' % res['table']['k=%s,b=0.0' % k]['C_star'], 'near mass', pf0 * top, 'ref', a1 + 2 * a2, m1 + 2 * m2, flush=True) + return res + + +def check_run(): + ex = controls() + path = os.path.join(HERE, 'RESULTS.json') + require(os.path.exists(path), 'RESULTS.json present') + with open(path) as fh: + ref = json.load(fh) + r96 = gauss_legendre(96) + for k in (0.5, 1.0, 2.0): + cu = cusp_integrals(k, r96) + for n in cu: + require(abs(cu[n] - ref['per_k'][str(k)]['cusp'][n]) <= REPLAY_TOL * abs(ref['per_k'][str(k)]['cusp'][n]), 'cusp integral replay %s k=%s' % (n, k)) + t = constants_table(k, 0.0, ex, cu) + require(abs(t['C_star'] - ref['table']['k=%s,b=0.0' % k]['C_star']) <= REPLAY_TOL * ref['table']['k=%s,b=0.0' % k]['C_star'], 'C_* replay k=%s' % k) + nm = near_mass(1.0, 32, 32, 64, 32) + require(abs(nm - ref['near_mass_identity']['k=1.0']['K0_integral_without_prefactor_by_level']['32x32x64x32']) <= REPLAY_TOL * nm, 'near-mass replay (low level)') + top = ref['near_mass_identity']['k=1.0']['alpha1_plus_2alpha2_b0'] + require(abs(top - ref['near_mass_identity']['k=1.0']['ref_mc_b0']) < 3 * ref['near_mass_identity']['k=1.0']['ref_mc_b0_se_alpha1'] + 2e-3, 'near-mass identity within tolerance of the Math-#168 MC value') + print(json.dumps({'check': 'ok', 'mutant': MUT})) + + +def main(): + global MUT + ap = argparse.ArgumentParser() + ap.add_argument('--check', 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() + 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 1c51e675398b51f8055e4951eb106b0b7671e5bf Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 08:58:39 +0000 Subject: [PATCH 2/7] Microscopic tail constants numerics: half-line quadrature for the even cusp integrands; note template (interim) Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../NOTE_template.md | 123 ++++++++++++++++++ .../tail_constants.py | 10 +- 2 files changed, 129 insertions(+), 4 deletions(-) create mode 100644 frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md new file mode 100644 index 000000000..8d9bac3bb --- /dev/null +++ b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md @@ -0,0 +1,123 @@ +# Numerical values of the planar microscopic radius-tail constants for the parent kernel + +**Object:** `CL-C6-MICRO-TAIL-NUMERICS-20260930-v1`. **Author lane:** Anthropic / Claude. **Kind:** numerical note. +**Scientific effect:** NONE. **Certified:** no (deterministic Gauss quadrature with reported convergence; not an +enclosure). Third of the planar numerics notes after Math-#168 (near coefficients `alpha_1, alpha_2`) and +Math-#174 (remote coefficient `k integral_X Lambda`). + +## 1. What is evaluated + +[R] (Math-#166, merged) proves for the already-formed limiting microscopic measure `M` (the `r -> 0` limit taken +first) the point-intensity radius tail + + F(t) = integral sum_i 1{R_i > t} dM ~ C_* t^(-11), + C_* = (216/11) k^7 (c_m/z_0) I J_cusp, I = 246528/35, (R13) + +with `J_cusp = integral H(h) gamma(a)^11 h_0(A_0, Rot_O tau_cusp) da dxi dh dO` (R12) and +`gamma(a) = sqrt(1 + (a/(12k))^2)`, and lists "no numerical value of its Gaussian integral" as a non-claim. The +unmerged companions Math-#169 ([E]) and Math-#176 ([Q]) express further constants of the same measure through the +same cusp integrals: the whole-cluster split of the tail (E6) and the second-order coefficient `C_2`, the +total-variation coefficient `kappa |C_2/C_0|` of the radial ratio and the signed constant `B_sign` ((13), (16), +(19)). This note evaluates all of them in the plane for the parent kernel `K(z) = exp(-|z|^2/2)` ([LP] §2). + +**Planar reduction.** In `d = 2` the spectral measure (17) of [SC] has `m = d - 1 = 1`, `c_1 = 1`, no hard +curvatures and no Haar variable; `H(h) = 1`. Its density `h_0(A_0, tau)` at `A_0 = 0` is the raw contact density of +`(f_zz; f_xxz, f_xzz, f_zzz)` at `(0; a, beta, c)` given the contact pins `(f, f_x, f_z, f_xx, f_xz, f_xxx) = +(b, 0, 0, 0, 0, 12k)`, which by the exact regression of Math-#168 factors as `p_b(0) . p_odd(a, beta, c)` with +`p_b(0) = phi(b/sqrt2)/sqrt2` (the density of `f_zz | (f, f_xx, f_xz) = (b, 0, 0) ~ N(-b, 2)` at `0`) and +`p_odd = N(0, diag(2, 2, 6))`, for every `k, b`. Hence + + c_1 h_0(0, tau)/z_0 = [p_b(0)/z_0] p_odd(a, beta, c), p_b(0)/z_0 = phi(b/sqrt2) / (sqrt2 . 36 k^2 m_(2,b)), + +the prefactor of Math-#168, and `J_cusp(k) = integral gamma(a)^11 p_odd(a, a^2/(12k), a^3/(144k^2)) da` is a +one-dimensional Gaussian integral, `b`-free; all `b`-dependence of the constants sits in the prefactor. + +**Near-mass identity (control).** The same root-resolved integral (R11) without the radius indicator is the near +point mass `integral n dM = alpha_1 + 2 alpha_2` of Math-#168 ((1.6) of [CL], `Lambda_1 = nu_near(1) + 2 nu_near(2)`): + + alpha_1 + 2 alpha_2 = 108 k^7 [p_b(0)/z_0] integral_S Q(u, v) integral_(Z != 0) |Z|^(-12) + integral p_odd(a, beta(u, v, Z, a), c(u, v, Z, a)) da dZ du dv, (N) + +with `beta = a^2/(12k) + k b_0(u)/Z^2`, `c = a^3/(144k^2) + a b_0(u)/(4Z^2) + 2k(v - b_0(u) u)/Z^3`. This +four-dimensional integral is evaluated independently here and compared with the merged Math-#168 values; agreement +ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight `9`, multiplicity by the map) to the +[CL]/[CUB] parametrization of Math-#168, which integrates the classifier over the jets directly. + +## 2. Exact structure + +- Shape integrals over `S = {-3/2 < u < 1/2, |b_0|/2 < v < 3 - 6u}` with `Q = (4v^2 - b_0^2)(b_0 - 4uv)`, in rational + arithmetic: `I = 246528/35`, `U_2 = integral u^2 Q = 240192/35`, `B = integral b_0 Q = -428544/7 = 3I - 12 U_2`, + `U_abs = integral |u| Q = 5898627/880` ([Q] (12)); `E eta = 5771/7062` under `Q/I` ([R] H3); the outer doublet + integral `J = 1083417/280` ([E] E20); `11 U_2/(2I) = 4587/856`, `11 U_abs/(2I) = 4587821/876544` ([Q] + corollary). The inner doublet integral `D = 27066286003/223205220 - (79298560/4782969) log 2 = 109.7699503...` + ([E] E21) is quoted, not recomputed exactly (it was confirmed to `1e-13` by quadrature in the Slice C read). +- `kappa = (2/13)(11/13)^(11/2) = q_0^(-11) - q_0^(-13)` at `q_0 = sqrt(13/11)` ([Q] (16)). +- With the aligned planar cusp density ([Q] (21)–(22), `v_a = v_beta = 2`, `v_c = 6` for the continuum kernel): + `D_a G_0 = -[a^2/24 + a^4/(3456 k^2)] G_0 <= 0`, so `C_2 > 0`. +- Assembly (`K_0 = 108 k^7 p_b(0)/z_0`; `C_0` below is [E]'s per-shape amplitude, `C_* = C_0 I` is [Q]'s `C_0`): + + C_0 = (216/11) k^7 [p_b(0)/z_0] J_cusp, C_* = C_0 I, + G_max ~ C_0 (I - D), G_2 ~ C_0 J, B_2 ~ C_0 D, singleton ~ C_0 (I - D - J), + C_2 = K_0 [ U_2 T_1 + (2B/13) T_2 ], T_1 = integral gamma^9 (1 + 12A^2) G_0 da, T_2 = integral gamma^13 D_a G_0 da, + c = C_2/C_*, d_TV(R/t, Pareto_11) = kappa c t^(-2) + O(t^(-4)), + B_sign = (K_0/C_*) U_abs integral |A| gamma^10 G_0 da, A = a/(12k). + + Every integrand in `a` is even, so the integrals are taken on `[0, 14]` and doubled (this also removes the kink of + `|A|` at `0`); 96-point Gauss–Legendre, checked against 48 points. + +## 3. Values + +**Cusp integrals** (`b`-free): + +| `k` | `J_cusp` | `T_1` | `T_2` | `integral |A| gamma^10 G_0` | max rel. dev. 48 vs 96 points | +|---|---|---|---|---|---| +{{cusp_rows}} + +**Tail constants** (`F(t) ~ C_* t^(-11)` is the `r^(-3)`-scaled expected number of near points at scaled radius +`R > t`; the split rows are conditional on [E]; `c`, `kappa c`, `B_sign` on [Q]): + +| `k` | `b` | `C_*` | `G_max` | `G_2` | `B_2` | singleton | `c = C_2/C_*` | `kappa c` | `B_sign` | +|---|---|---|---|---|---|---|---|---|---| +{{table_rows}} + +`c` and `B_sign` are `b`-free (the prefactor cancels); the ratios `G_max/C_* = (I-D)/I = 0.98442`, `G_2/C_* = J/I += 0.54934`, `B_2/C_* = D/I = 0.015584` are universal. The bounds of [Q]'s corollary hold: `c > 4587/856 = 5.3586` +and `0 < B_sign < 4587821/876544 = 5.2340`. Reading: at `k = 1`, `b = 0` the near point mass is `1.378` and `C_* += 52.6`, so the asymptotic law puts `C_* 2^(-11) = 0.026` of it beyond twice the pin distance and `3e-4` beyond +three; the radial ratio's total-variation distance from Pareto(11) at threshold `t` is `0.42/t^2` to leading order +(`4.7%` at `t = 3`), and the signed-mark distance decays only like `0.51/t`. + +**Near-mass identity** (`alpha_1 + 2 alpha_2` from (N) at three shape-grid levels, against Math-#168): + +| `k` | (N) at 32/64/96 shape points | (N), `b = 0` | Math-#168 GH60 | Math-#168 MC (s.e. of `alpha_1`) | rel. dev. vs GH60 / MC | +|---|---|---|---|---|---| +{{nm_rows}} + +The four-dimensional integral converges from below (observed order about two in the shape grid; the integrand has +an integrable boundary singularity where the `|Z|`-scale of the Gaussian cutoff diverges) and at the finest level +lies within one Monte Carlo standard error of the Math-#168 value at every `k`. This is the check that the two +parametrizations agree; it is not a certification of either. + +## 4. Precision (empirical, not bounds) + +- The one-dimensional cusp integrals agree between 48 and 96 Gauss–Legendre points to the relative deviations in + the table (all below `{{cusp_dev_max}}`); the constants inherit that precision and the exact prefactor. +- The near-mass integral moves by `{{nm_move}}` relative between the 64- and 96-point shape levels; its remaining + error is of that order, below the Math-#168 Monte Carlo standard error. + +## 5. What the numbers are not + +Not a proof and not an enclosure. `C_*` is a value of the reviewed formula (R13) of the merged [R]; the split and the +second-order/signed constants are values of formulas in the unmerged candidates [E] and [Q] (their Slices B, C, D and +A, B respectively carry nonauthor reads at this writing) and are conditional on them. Planar only (`d = 2`), parent +kernel only, aligned frame; no rate for `r -> 0`, no finite-`r` statement, no statement about the remote singleton +population. No register, catalog or STATUS change. Same GitHub account as every lane; zero +organizational-independence credit. The author will not merge. + +## 6. Provenance + +`tail_constants.py` (standard library; `--check` replays the exact controls, the cusp integrals and `C_*` for each +`k`, the low-level near-mass integral, and the identity tolerance against `RESULTS.json`; mutants `shape-integral`, +`gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about {{minutes}} minutes), `SOURCE_MAP.json` +(pins [R], [SC], [NUM], [CUB], [LP] on `main` `02772ec`; companions [E], [Q] unmerged, not workflow-checked), +`SOURCE_FILES.json`, workflow `c6-microscopic-tail-numerics.yml`. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py index a37f7972b..d47c4599a 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py +++ b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py @@ -183,10 +183,12 @@ def gam(a): return math.sqrt(1 + (a / (12 * k)) ** 2) def DaG0(a): # D_a = k d_beta + (a/4) d_c on the diagonal cusp density: -(a^2/(12 vbeta) + a^4/(576 k^2 vc)) G0, vbeta = 2, vc = 6 return -(a * a / 24 + a ** 4 / (3456 * k * k)) * G0(a) - Jc = gl(lambda a: gam(a) ** gpow * G0(a), -L, L, rule) - T1 = gl(lambda a: gam(a) ** 9 * (1 + 12 * (a / (12 * k)) ** 2) * G0(a), -L, L, rule) - T2 = gl(lambda a: gam(a) ** 13 * DaG0(a), -L, L, rule) - Tb = gl(lambda a: abs(a / (12 * k)) * gam(a) ** 10 * G0(a), -L, L, rule) + # every integrand is even in a (G0 depends on a through a^2, beta_0^2 and c_0^2), so integrate on [0, L] and double; + # this also removes the kink of |A| at a = 0 from the quadrature + Jc = 2 * gl(lambda a: gam(a) ** gpow * G0(a), 0.0, L, rule) + T1 = 2 * gl(lambda a: gam(a) ** 9 * (1 + 12 * (a / (12 * k)) ** 2) * G0(a), 0.0, L, rule) + T2 = 2 * gl(lambda a: gam(a) ** 13 * DaG0(a), 0.0, L, rule) + Tb = 2 * gl(lambda a: (a / (12 * k)) * gam(a) ** 10 * G0(a), 0.0, L, rule) return {'J_cusp': Jc, 'T1': T1, 'T2': T2, 'Tb': Tb} From f60e209189ed8b9f6d1683da1177174c8e94d3ca Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 09:02:11 +0000 Subject: [PATCH 3/7] Microscopic tail constants numerics: RESULTS.json, NOTE.md and manifest Planar values of C_* ([R] R13), the whole-cluster split (Math-#169, unmerged), C_2/C_0, kappa c and B_sign (Math-#176, unmerged) for k in {1/2, 1, 2}, b in {0, 1}; near-mass identity alpha_1 + 2 alpha_2 checked against the merged Math-#168 values (within 1.5 Monte Carlo standard errors). Scientific effect NONE; not certified. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../{NOTE_template.md => NOTE.md} | 28 ++- .../RESULTS.json | 227 ++++++++++++++++++ .../SOURCE_FILES.json | 39 +++ 3 files changed, 284 insertions(+), 10 deletions(-) rename frontiers/c6_microscopic_tail_numerics_20260930/{NOTE_template.md => NOTE.md} (83%) create mode 100644 frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json create mode 100644 frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md similarity index 83% rename from frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md rename to frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md index 8d9bac3bb..79dda17c3 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE_template.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md @@ -71,38 +71,46 @@ ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight ` | `k` | `J_cusp` | `T_1` | `T_2` | `integral |A| gamma^10 G_0` | max rel. dev. 48 vs 96 points | |---|---|---|---|---|---| -{{cusp_rows}} +| `0.5` | 0.0569758 | 0.0933384 | -0.0077782 | 0.0115205 | `1e-14` | +| `1.0` | 0.0485522 | 0.0561351 | -0.00467793 | 0.00468021 | `1e-15` | +| `2.0` | 0.0465858 | 0.0483768 | -0.0040314 | 0.0022051 | `2e-15` | **Tail constants** (`F(t) ~ C_* t^(-11)` is the `r^(-3)`-scaled expected number of near points at scaled radius `R > t`; the split rows are conditional on [E]; `c`, `kappa c`, `B_sign` on [Q]): | `k` | `b` | `C_*` | `G_max` | `G_2` | `B_2` | singleton | `c = C_2/C_*` | `kappa c` | `B_sign` | |---|---|---|---|---|---|---|---|---|---| -{{table_rows}} +| `0.5` | `0.0` | 1.9297 | 1.8996 | 1.0601 | 0.030073 | 0.83957 | 9.7826 | 0.6 | 1.058 | +| `1.0` | `0.0` | 52.621 | 51.801 | 28.907 | 0.82006 | 22.894 | 6.9042 | 0.424 | 0.5045 | +| `2.0` | `0.0` | 1615.7 | 1590.5 | 887.55 | 25.179 | 702.95 | 6.2011 | 0.381 | 0.2477 | +| `0.5` | `1.0` | 0.55249 | 0.54388 | 0.30351 | 0.0086102 | 0.24038 | 9.7826 | 0.6 | 1.058 | +| `1.0` | `1.0` | 15.066 | 14.831 | 8.2763 | 0.23479 | 6.5549 | 6.9042 | 0.424 | 0.5045 | +| `2.0` | `1.0` | 462.58 | 455.37 | 254.11 | 7.209 | 201.26 | 6.2011 | 0.381 | 0.2477 | `c` and `B_sign` are `b`-free (the prefactor cancels); the ratios `G_max/C_* = (I-D)/I = 0.98442`, `G_2/C_* = J/I = 0.54934`, `B_2/C_* = D/I = 0.015584` are universal. The bounds of [Q]'s corollary hold: `c > 4587/856 = 5.3586` -and `0 < B_sign < 4587821/876544 = 5.2340`. Reading: at `k = 1`, `b = 0` the near point mass is `1.378` and `C_* -= 52.6`, so the asymptotic law puts `C_* 2^(-11) = 0.026` of it beyond twice the pin distance and `3e-4` beyond +and `0 < B_sign < 4587821/876544 = 5.2340`. Reading: at `k = 1`, `b = 0` the near point mass is `1.378` and `C_* = 52.6`, so the asymptotic law puts `C_* 2^(-11) = 0.026` of it beyond twice the pin distance and `3e-4` beyond three; the radial ratio's total-variation distance from Pareto(11) at threshold `t` is `0.42/t^2` to leading order -(`4.7%` at `t = 3`), and the signed-mark distance decays only like `0.51/t`. +(`4.7%` at `t = 3`), and the signed-mark distance decays only like `0.50/t`. **Near-mass identity** (`alpha_1 + 2 alpha_2` from (N) at three shape-grid levels, against Math-#168): | `k` | (N) at 32/64/96 shape points | (N), `b = 0` | Math-#168 GH60 | Math-#168 MC (s.e. of `alpha_1`) | rel. dev. vs GH60 / MC | |---|---|---|---|---|---| -{{nm_rows}} +| `0.5` | 37.9798 / 37.8556 / 37.867 | 1.1869 | 1.1877 | 1.1844 (0.0027) | -6.76e-04 / +2.07e-03 | +| `1.0` | 175.615 / 175.762 / 175.773 | 1.3774 | 1.3801 | 1.3786 (0.0022) | -1.97e-03 / -9.20e-04 | +| `2.0` | 1116.83 / 1117.77 / 1117.93 | 2.19 | 2.196 | 2.1948 (0.0033) | -2.70e-03 / -2.18e-03 | The four-dimensional integral converges from below (observed order about two in the shape grid; the integrand has an integrable boundary singularity where the `|Z|`-scale of the Gaussian cutoff diverges) and at the finest level -lies within one Monte Carlo standard error of the Math-#168 value at every `k`. This is the check that the two +lies within `0.9`, `0.6` and `1.5` Monte Carlo standard errors of the Math-#168 values at `k = 1/2, 1, 2`. This is the check that the two parametrizations agree; it is not a certification of either. ## 4. Precision (empirical, not bounds) - The one-dimensional cusp integrals agree between 48 and 96 Gauss–Legendre points to the relative deviations in - the table (all below `{{cusp_dev_max}}`); the constants inherit that precision and the exact prefactor. -- The near-mass integral moves by `{{nm_move}}` relative between the 64- and 96-point shape levels; its remaining + the table (all below `1e-14`); the constants inherit that precision and the exact prefactor. +- The near-mass integral moves by `at most `3e-04`` relative between the 64- and 96-point shape levels; its remaining error is of that order, below the Math-#168 Monte Carlo standard error. ## 5. What the numbers are not @@ -118,6 +126,6 @@ organizational-independence credit. The author will not merge. `tail_constants.py` (standard library; `--check` replays the exact controls, the cusp integrals and `C_*` for each `k`, the low-level near-mass integral, and the identity tolerance against `RESULTS.json`; mutants `shape-integral`, -`gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about {{minutes}} minutes), `SOURCE_MAP.json` +`gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about 5 minutes), `SOURCE_MAP.json` (pins [R], [SC], [NUM], [CUB], [LP] on `main` `02772ec`; companions [E], [Q] unmerged, not workflow-checked), `SOURCE_FILES.json`, workflow `c6-microscopic-tail-numerics.yml`. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json b/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json new file mode 100644 index 000000000..b4ef8048e --- /dev/null +++ b/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json @@ -0,0 +1,227 @@ +{ + "D_inner": 109.76995034568799, + "certified": false, + "dimension": 2, + "exact": { + "B": "-428544/7", + "E_eta": "5771/7062", + "I": "246528/35", + "J_outer": "1083417/280", + "U2": "240192/35", + "Uabs": "5898627/880" + }, + "kappa": 0.06138439418216763, + "near_mass_identity": { + "k=0.5": { + "K0_integral_without_prefactor_by_level": { + "32x32x64x32": 37.97976778641956, + "64x64x64x32": 37.85556130368196, + "96x96x64x32": 37.866974622954146 + }, + "alpha1_plus_2alpha2_b0": 1.1868973690409972, + "alpha1_plus_2alpha2_b1": 0.33981935655174555, + "ref_gh60_b0": 1.1877008, + "ref_mc_b0": 1.184446, + "ref_mc_b0_se_alpha1": 0.0026830378399999998, + "rel_dev_vs_gh60": -0.0006764590534946305, + "rel_dev_vs_mc": 0.0020696334328429286 + }, + "k=1.0": { + "K0_integral_without_prefactor_by_level": { + "32x32x64x32": 175.6152363636971, + "64x64x64x32": 175.76241702393807, + "96x96x64x32": 175.77336837738886 + }, + "alpha1_plus_2alpha2_b0": 1.37735421532813, + "alpha1_plus_2alpha2_b1": 0.39434885897069694, + "ref_gh60_b0": 1.3800781999999998, + "ref_mc_b0": 1.3786228, + "ref_mc_b0_se_alpha1": 0.0022175795099999996, + "rel_dev_vs_gh60": -0.0019737900880325843, + "rel_dev_vs_mc": -0.0009201825705117285 + }, + "k=2.0": { + "K0_integral_without_prefactor_by_level": { + "32x32x64x32": 1116.8334428953553, + "64x64x64x32": 1117.7663737273087, + "96x96x64x32": 1117.9311635466386 + }, + "alpha1_plus_2alpha2_b0": 2.190017769432071, + "alpha1_plus_2alpha2_b1": 0.6270217195330132, + "ref_gh60_b0": 2.1959562000000004, + "ref_mc_b0": 2.1948116, + "ref_mc_b0_se_alpha1": 0.0033106930999999996, + "rel_dev_vs_gh60": -0.0027042572925312937, + "rel_dev_vs_mc": -0.0021841649497063554 + } + }, + "object": "CL-C6-MICRO-TAIL-NUMERICS-20260930-v1", + "per_k": { + "0.5": { + "cusp": { + "J_cusp": 0.056975765507371816, + "T1": 0.09333844205882272, + "T2": -0.007778203504901892, + "Tb": 0.011520519255772783 + }, + "cusp_gl48_rel_dev": { + "J_cusp": 1.948588166808413e-15, + "T1": 5.9472978128640934e-15, + "T2": 1.2935372742979406e-14, + "Tb": 4.969035981477091e-15 + } + }, + "1.0": { + "cusp": { + "J_cusp": 0.04855218515605281, + "T1": 0.05613513535587141, + "T2": -0.0046779279463226185, + "Tb": 0.004680210947657693 + }, + "cusp_gl48_rel_dev": { + "J_cusp": 1.1433296164289626e-15, + "T1": 1.1124947813746234e-15, + "T2": 5.562473906873115e-16, + "Tb": 7.413014051620834e-16 + } + }, + "2.0": { + "cusp": { + "J_cusp": 0.046585754487253814, + "T1": 0.04837676216650269, + "T2": -0.00403139684720856, + "Tb": 0.002205099388800259 + }, + "cusp_gl48_rel_dev": { + "J_cusp": 1.489488359752928e-15, + "T1": 1.5777788658173336e-15, + "T2": 6.454549905616361e-16, + "Tb": 7.866872054781294e-16 + } + } + }, + "quadrature": { + "cusp_gl": 96, + "cusp_gl_check": 48, + "cusp_range": 14.0, + "near_mass_levels": [ + [ + 32, + 32, + 64, + 32 + ], + [ + 64, + 64, + 64, + 32 + ], + [ + 96, + 96, + 64, + 32 + ] + ] + }, + "scientific_effect": "NONE", + "table": { + "k=0.5,b=0.0": { + "B2_coeff": 0.030073037204841252, + "B_sign": 1.0583142135298427, + "C0_per_shape": 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"upper_bound_B_sign_4587821_876544": 5.233988253869743 + }, + "k=1.0,b=0.0": { + "B2_coeff": 0.82006047733282, + "B_sign": 0.50453278357996, + "C0_per_shape": 0.007470719215507359, + "C2": 363.3047025936172, + "C2_over_C0": 6.904152846784693, + "C_star": 52.62118476458853, + "G2_coeff": 28.906800715383344, + "G_max_coeff": 51.8011242872557, + "K0": 0.8462843753216343, + "lower_bound_C2_over_C0_4587_856": 5.358644859813084, + "prefactor_p_b0_over_z0": 0.00783596643816328, + "singleton_coeff": 22.89432357187236, + "tv_coefficient_kappa_c": 0.4238072398409664, + "upper_bound_B_sign_4587821_876544": 5.233988253869743 + }, + "k=1.0,b=1.0": { + "B2_coeff": 0.23479066599155163, + "B_sign": 0.50453278357996, + "C0_per_shape": 0.002138933881742206, + "C2": 104.01739315282099, + "C2_over_C0": 6.904152846784693, + "C_star": 15.065916914232647, + "G2_coeff": 8.276276176269628, + "G_max_coeff": 14.831126248241095, + "K0": 0.2422988030666534, + "lower_bound_C2_over_C0_4587_856": 5.358644859813084, + 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"G2_coeff": 254.1144173196259, + "G_max_coeff": 455.3742437416232, + "K0": 7.753561698132909, + "lower_bound_C2_over_C0_4587_856": 5.358644859813084, + "prefactor_p_b0_over_z0": 0.0005608768589505866, + "singleton_coeff": 201.25982642199736, + "tv_coefficient_kappa_c": 0.38065018241423904, + "upper_bound_B_sign_4587821_876544": 5.233988253869743 + } + } +} diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json new file mode 100644 index 000000000..43d659d2d --- /dev/null +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json @@ -0,0 +1,39 @@ +{ + "schema": 1, + "scientific_effect": "NONE", + "certified": false, + "review_status": "AUTHOR_SIDE_NUMERICAL_NOTE", + "object": "CL-C6-MICRO-TAIL-NUMERICS-20260930-v1", + "files": [ + { + "path": "NOTE.md", + "bytes": 9480, + "sha256": "a535aafbb52d3d93e77e66253219431f2f153c2fdb080afe75694000431f7eeb", + "git_blob": 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numerics: converged log-Z quadrature (ns = 160), full-table replay in --check, workflow trigger on consumed sources; regenerated RESULTS.json and note Answers the three Codex findings (review 5364178009): the near-mass integral was under-resolved in the log-Z order by 1.8e-3 relative; with ns = 160 (and the recorded ns and a-order studies) the identity now lies within 1.1, 0.5 and 0.2 Monte Carlo standard errors of the Math-#168 values and within 3e-4 of its GH60 values; --check compares every table entry for both birth marks and the near-mass entries for every k; the workflow paths include the five pinned sources. Cusp integrals and tail constants unchanged. Scientific effect NONE. Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../c6-microscopic-tail-numerics.yml | 5 + .../NOTE.md | 34 +++--- .../RESULTS.json | 102 +++++++++++++----- .../SOURCE_FILES.json | 18 ++-- .../tail_constants.py | 33 ++++-- 5 files changed, 136 insertions(+), 56 deletions(-) diff --git a/.github/workflows/c6-microscopic-tail-numerics.yml b/.github/workflows/c6-microscopic-tail-numerics.yml index 4192e727b..e5ffa2e93 100644 --- a/.github/workflows/c6-microscopic-tail-numerics.yml +++ b/.github/workflows/c6-microscopic-tail-numerics.yml @@ -5,6 +5,11 @@ name: C6 microscopic tail numerics (replay) paths: - 'frontiers/c6_microscopic_tail_numerics_20260930/**' - '.github/workflows/c6-microscopic-tail-numerics.yml' + - 'frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md' + - 'frontiers/spectral_cluster_closure_20260929/PROOF.md' + - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' + - 'frontiers/planar_cubic_cluster_20260929/PROOF.md' + - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' workflow_dispatch: permissions: diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md index 79dda17c3..674cb9772 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md @@ -95,23 +95,28 @@ three; the radial ratio's total-variation distance from Pareto(11) at threshold **Near-mass identity** (`alpha_1 + 2 alpha_2` from (N) at three shape-grid levels, against Math-#168): -| `k` | (N) at 32/64/96 shape points | (N), `b = 0` | Math-#168 GH60 | Math-#168 MC (s.e. of `alpha_1`) | rel. dev. vs GH60 / MC | +| `k` | (N) at 32/64/96 shape points (`ns = 160`) | (N), `b = 0` | Math-#168 GH60 | Math-#168 MC (s.e. of `alpha_1`) | rel. dev. vs GH60 / MC | |---|---|---|---|---|---| -| `0.5` | 37.9798 / 37.8556 / 37.867 | 1.1869 | 1.1877 | 1.1844 (0.0027) | -6.76e-04 / +2.07e-03 | -| `1.0` | 175.615 / 175.762 / 175.773 | 1.3774 | 1.3801 | 1.3786 (0.0022) | -1.97e-03 / -9.20e-04 | -| `2.0` | 1116.83 / 1117.77 / 1117.93 | 2.19 | 2.196 | 2.1948 (0.0033) | -2.70e-03 / -2.18e-03 | - -The four-dimensional integral converges from below (observed order about two in the shape grid; the integrand has -an integrable boundary singularity where the `|Z|`-scale of the Gaussian cutoff diverges) and at the finest level -lies within `0.9`, `0.6` and `1.5` Monte Carlo standard errors of the Math-#168 values at `k = 1/2, 1, 2`. This is the check that the two -parametrizations agree; it is not a certification of either. +| `0.5` | 37.9952 / 37.871 / 37.8825 | 1.1874 | 1.1877 | 1.1844 (0.0027) | -2.67e-04 / +2.48e-03 | +| `1.0` | 175.923 / 176.071 / 176.082 | 1.3798 | 1.3801 | 1.3786 (0.0022) | -2.22e-04 / +8.33e-04 | +| `2.0` | 1119.62 / 1120.53 / 1120.7 | 2.1954 | 2.196 | 2.1948 (0.0033) | -2.34e-04 / +2.87e-04 | + +The shape-grid refinement at fixed `ns = 160` moves the integral by at most `3e-04` relative between the 64- and +96-point levels (observed order about two; the integrand has an integrable boundary singularity where the +`|Z|`-scale of the Gaussian cutoff diverges), and the log-`Z` order is converged: at `32 x 32` shape points, +`ns = 64, 96, 160, 256` give `175.6152`, `175.9278`, `175.9235`, `175.9235` at `k = 1` (an under-resolved `ns = 64`, the order of the +first version of this note, biased the identity low by `1.8e-3`, exactly the deviation then reported; the `a`-order +`32` and `48` agree to all digits). At the finest level the identity lies within `1.1`, `0.5` and `0.2` Monte Carlo +standard errors of the Math-#168 values at `k = 1/2, 1, 2`, and within `2.7e-04`, `2.2e-04`, `2.3e-04` relative of the GH60 +values. This is the check that the two parametrizations agree; it is not a certification of either. ## 4. Precision (empirical, not bounds) - The one-dimensional cusp integrals agree between 48 and 96 Gauss–Legendre points to the relative deviations in the table (all below `1e-14`); the constants inherit that precision and the exact prefactor. -- The near-mass integral moves by `at most `3e-04`` relative between the 64- and 96-point shape levels; its remaining - error is of that order, below the Math-#168 Monte Carlo standard error. +- The near-mass integral moves by at most `3e-04` relative between the 64- and 96-point shape levels at `ns = 160`, + and by `4e-09` between `ns = 160` and `256`; its remaining error is of the former order, below the Math-#168 Monte + Carlo standard error. ## 5. What the numbers are not @@ -124,8 +129,9 @@ organizational-independence credit. The author will not merge. ## 6. Provenance -`tail_constants.py` (standard library; `--check` replays the exact controls, the cusp integrals and `C_*` for each -`k`, the low-level near-mass integral, and the identity tolerance against `RESULTS.json`; mutants `shape-integral`, -`gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about 5 minutes), `SOURCE_MAP.json` +`tail_constants.py` (standard library; `--check` replays the exact controls, the cusp integrals and the complete +constants table for each `k` and both `b`, the low-level near-mass integral for each `k` with its log-`Z` study +entry, the assembly of the identity from the stored numbers, and the identity tolerance against `RESULTS.json`; mutants `shape-integral`, +`gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about fifteen minutes), `SOURCE_MAP.json` (pins [R], [SC], [NUM], [CUB], [LP] on `main` `02772ec`; companions [E], [Q] unmerged, not workflow-checked), `SOURCE_FILES.json`, workflow `c6-microscopic-tail-numerics.yml`. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json b/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json index b4ef8048e..0b5ce76f5 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/RESULTS.json @@ -14,45 +14,75 @@ "near_mass_identity": { "k=0.5": { "K0_integral_without_prefactor_by_level": { - "32x32x64x32": 37.97976778641956, - "64x64x64x32": 37.85556130368196, - "96x96x64x32": 37.866974622954146 + "32x32x160x32": 37.99524712549494, + "64x64x160x32": 37.871016097916645, + "96x96x160x32": 37.88250534791175 + }, + "a_study_32x32_ns160": { + "32": 37.99524712549494, + "48": 37.99524693012451 + }, + "alpha1_plus_2alpha2_b0": 1.18738416199911, + "alpha1_plus_2alpha2_b1": 0.3399587297394487, + "logZ_study_32x32": { + "160": 37.99524712549494, + "256": 37.99524722503159, + "64": 37.97976778641956, + "96": 37.99599034986138 }, - "alpha1_plus_2alpha2_b0": 1.1868973690409972, - "alpha1_plus_2alpha2_b1": 0.33981935655174555, "ref_gh60_b0": 1.1877008, "ref_mc_b0": 1.184446, "ref_mc_b0_se_alpha1": 0.0026830378399999998, - "rel_dev_vs_gh60": -0.0006764590534946305, - "rel_dev_vs_mc": 0.0020696334328429286 + "rel_dev_vs_gh60": -0.00026659744684023244, + "rel_dev_vs_mc": 0.0024806213192579392 }, "k=1.0": { "K0_integral_without_prefactor_by_level": { - "32x32x64x32": 175.6152363636971, - "64x64x64x32": 175.76241702393807, - "96x96x64x32": 175.77336837738886 + "32x32x160x32": 175.92347251130104, + "64x64x160x32": 176.070695835646, + "96x96x160x32": 176.08183192252775 + }, + "a_study_32x32_ns160": { + "32": 175.92347251130104, + "48": 175.92347251342343 + }, + "alpha1_plus_2alpha2_b0": 1.379771325315235, + "alpha1_plus_2alpha2_b1": 0.39504089922789, + "logZ_study_32x32": { + "160": 175.92347251130104, + "256": 175.92347306968713, + "64": 175.6152363636971, + "96": 175.92782107142853 }, - "alpha1_plus_2alpha2_b0": 1.37735421532813, - "alpha1_plus_2alpha2_b1": 0.39434885897069694, "ref_gh60_b0": 1.3800781999999998, "ref_mc_b0": 1.3786228, "ref_mc_b0_se_alpha1": 0.0022175795099999996, - "rel_dev_vs_gh60": -0.0019737900880325843, - "rel_dev_vs_mc": -0.0009201825705117285 + "rel_dev_vs_gh60": -0.00022236035955408722, + "rel_dev_vs_mc": 0.0008330961269717983 }, "k=2.0": { "K0_integral_without_prefactor_by_level": { - "32x32x64x32": 1116.8334428953553, - "64x64x64x32": 1117.7663737273087, - "96x96x64x32": 1117.9311635466386 + "32x32x160x32": 1119.6168729751753, + "64x64x160x32": 1120.5321068435828, + "96x96x160x32": 1120.70035333438 + }, + "a_study_32x32_ns160": { + "32": 1119.6168729751753, + "48": 1119.6168737328314 + }, + "alpha1_plus_2alpha2_b0": 2.1954425889914826, + "alpha1_plus_2alpha2_b1": 0.6285748940029996, + "logZ_study_32x32": { + "160": 1119.6168729751753, + "256": 1119.6168775975489, + "64": 1116.8334428953553, + "96": 1119.6578677719954 }, - "alpha1_plus_2alpha2_b0": 2.190017769432071, - "alpha1_plus_2alpha2_b1": 0.6270217195330132, "ref_gh60_b0": 2.1959562000000004, "ref_mc_b0": 2.1948116, "ref_mc_b0_se_alpha1": 0.0033106930999999996, - "rel_dev_vs_gh60": -0.0027042572925312937, - "rel_dev_vs_mc": -0.0021841649497063554 + "rel_dev_vs_gh60": -0.0002338894594153412, + "rel_dev_vs_mc": 0.0002874911867071494 } }, "object": "CL-C6-MICRO-TAIL-NUMERICS-20260930-v1", @@ -104,26 +134,50 @@ "cusp_gl": 96, "cusp_gl_check": 48, "cusp_range": 14.0, + "near_mass_a_study": { + "na": [ + 32, + 48 + ], + "ns": 160, + "shape": [ + 32, + 32 + ] + }, "near_mass_levels": [ [ 32, 32, - 64, + 160, 32 ], [ 64, 64, - 64, + 160, 32 ], [ 96, 96, + 160, + 32 + ] + ], + "near_mass_logZ_study": { + "a_nodes": 32, + "ns": [ 64, + 96, + 160, + 256 + ], + "shape": [ + 32, 32 ] - ] + } }, "scientific_effect": "NONE", "table": { diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json index 43d659d2d..0ba359496 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json @@ -7,9 +7,9 @@ "files": [ { "path": "NOTE.md", - "bytes": 9480, - "sha256": "a535aafbb52d3d93e77e66253219431f2f153c2fdb080afe75694000431f7eeb", - "git_blob": "79dda17c38af80fa1e78fe8d1d0f6c56ef29d2fd" + "bytes": 10188, + "sha256": "6ed848a1a30824930441c461097ef5fb852a0b43063f160c0bbb878e53428ecb", + "git_blob": "674cb9772dde7fe5ddbf35c6fe11c1d462795c6a" }, { "path": "README.md", @@ -19,9 +19,9 @@ }, { "path": "RESULTS.json", - "bytes": 6609, - "sha256": "66e0294e50b7391e2a7daedf0cbc545ae6445a37bcf1d62775c29f49dfa02872", - "git_blob": "b4ef8048e82f5756fb9ce8d0ed676b6c976e26b8" + "bytes": 7587, + "sha256": "771ead07d7693b23fdceff0f73a2f930b8b929e58be0f8f3f42ebe27f9cd9604", + "git_blob": "0b5ce76f5f950f3ef09fc0e26dafa884a43d8f63" }, { "path": "SOURCE_MAP.json", @@ -31,9 +31,9 @@ }, { "path": "tail_constants.py", - "bytes": 15893, - "sha256": "5e2e059ced3377a115c6ed93ae3543b667c81af159135eb35e4f4cb8ba452b7c", - "git_blob": "d47c4599a6dc7525e43f4bcbbcdd169e260bb227" + "bytes": 17123, + "sha256": "12d150bdd6bf78cdced6cc905c7e1774b6f78e428bddd462f5d7b48d4abe2f40", + "git_blob": "ff1f5d4b61859367ffcbebe9e1b5bf7990b2ef0b" } ] } diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py index d47c4599a..ff1f5d4b6 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py +++ b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py @@ -12,7 +12,7 @@ * control: the near-mass identity alpha_1 + 2 alpha_2 = K_0 int_S int_Z Q |Z|^-12 p_odd (the root-resolved integral (R11) integrated over all Z), against the merged Math-#168 values of alpha_1, alpha_2. - python -B -S tail_constants.py # full run, writes RESULTS.json (a few minutes) + python -B -S tail_constants.py # full run, writes RESULTS.json (about fifteen minutes) python -B -S tail_constants.py --check # exact controls + replay against RESULTS.json (Python >= 3.11) python -B -S tail_constants.py --check --mutant NAME # must exit 1 """ @@ -267,7 +267,10 @@ def full_run(): ex = controls() res = {'object': 'CL-C6-MICRO-TAIL-NUMERICS-20260930-v1', 'scientific_effect': 'NONE', 'certified': False, 'dimension': 2, 'exact': {k: str(v) for k, v in ex.items()}, 'D_inner': D_inner(), 'kappa': KAPPA, - 'quadrature': {'cusp_gl': 96, 'cusp_gl_check': 48, 'cusp_range': 14.0, 'near_mass_levels': [[32, 32, 64, 32], [64, 64, 64, 32], [96, 96, 64, 32]]}, + 'quadrature': {'cusp_gl': 96, 'cusp_gl_check': 48, 'cusp_range': 14.0, + 'near_mass_levels': [[32, 32, 160, 32], [64, 64, 160, 32], [96, 96, 160, 32]], + 'near_mass_logZ_study': {'shape': [32, 32], 'a_nodes': 32, 'ns': [64, 96, 160, 256]}, + 'near_mass_a_study': {'shape': [32, 32], 'ns': 160, 'na': [32, 48]}}, 'per_k': {}, 'table': {}, 'near_mass_identity': {}} r96, r48 = gauss_legendre(96), gauss_legendre(48) for k in (0.5, 1.0, 2.0): @@ -279,11 +282,14 @@ def full_run(): levels = {} for nodes in res['quadrature']['near_mass_levels']: levels['x'.join(map(str, nodes))] = near_mass(k, *nodes) + logz = {str(ns): near_mass(k, 32, 32, ns, 32) for ns in res['quadrature']['near_mass_logZ_study']['ns']} + astudy = {str(na): near_mass(k, 32, 32, 160, na) for na in res['quadrature']['near_mass_a_study']['na']} pf0, pf1 = prefactor(k, 0.0), prefactor(k, 1.0) - top = levels['96x96x64x32'] + top = levels['96x96x160x32'] a1, a2 = ALPHA_GH60[k]; m1, m2 = ALPHA_MC[k] res['near_mass_identity']['k=%s' % k] = { 'K0_integral_without_prefactor_by_level': levels, + 'logZ_study_32x32': logz, 'a_study_32x32_ns160': astudy, 'alpha1_plus_2alpha2_b0': pf0 * top, 'alpha1_plus_2alpha2_b1': pf1 * top, 'ref_gh60_b0': a1 + 2 * a2, 'ref_mc_b0': m1 + 2 * m2, 'ref_mc_b0_se_alpha1': ALPHA_MC_SE1[k], 'rel_dev_vs_gh60': (pf0 * top - (a1 + 2 * a2)) / (a1 + 2 * a2), 'rel_dev_vs_mc': (pf0 * top - (m1 + 2 * m2)) / (m1 + 2 * m2)} @@ -302,12 +308,21 @@ def check_run(): cu = cusp_integrals(k, r96) for n in cu: require(abs(cu[n] - ref['per_k'][str(k)]['cusp'][n]) <= REPLAY_TOL * abs(ref['per_k'][str(k)]['cusp'][n]), 'cusp integral replay %s k=%s' % (n, k)) - t = constants_table(k, 0.0, ex, cu) - require(abs(t['C_star'] - ref['table']['k=%s,b=0.0' % k]['C_star']) <= REPLAY_TOL * ref['table']['k=%s,b=0.0' % k]['C_star'], 'C_* replay k=%s' % k) - nm = near_mass(1.0, 32, 32, 64, 32) - require(abs(nm - ref['near_mass_identity']['k=1.0']['K0_integral_without_prefactor_by_level']['32x32x64x32']) <= REPLAY_TOL * nm, 'near-mass replay (low level)') - top = ref['near_mass_identity']['k=1.0']['alpha1_plus_2alpha2_b0'] - require(abs(top - ref['near_mass_identity']['k=1.0']['ref_mc_b0']) < 3 * ref['near_mass_identity']['k=1.0']['ref_mc_b0_se_alpha1'] + 2e-3, 'near-mass identity within tolerance of the Math-#168 MC value') + for b in (0.0, 1.0): + t = constants_table(k, b, ex, cu) + rt = ref['table']['k=%s,b=%s' % (k, b)] + require(sorted(t) == sorted(rt), 'table keys k=%s b=%s' % (k, b)) + for name in t: + require(abs(t[name] - rt[name]) <= REPLAY_TOL * max(abs(rt[name]), 1e-300), 'table replay %s k=%s b=%s' % (name, k, b)) + for k in (0.5, 1.0, 2.0): + nm = near_mass(k, 32, 32, 160, 32) + require(abs(nm - ref['near_mass_identity']['k=%s' % k]['K0_integral_without_prefactor_by_level']['32x32x160x32']) <= REPLAY_TOL * nm, 'near-mass replay (low level) k=%s' % k) + require(abs(nm - ref['near_mass_identity']['k=%s' % k]['logZ_study_32x32']['160']) <= REPLAY_TOL * nm, 'near-mass logZ-study consistency k=%s' % k) + pf0 = prefactor(k, 0.0) + require(abs(pf0 * ref['near_mass_identity']['k=%s' % k]['K0_integral_without_prefactor_by_level']['96x96x160x32'] - ref['near_mass_identity']['k=%s' % k]['alpha1_plus_2alpha2_b0']) < 1e-12, 'near-mass assembly k=%s' % k) + for k in (0.5, 1.0, 2.0): + top = ref['near_mass_identity']['k=%s' % k]['alpha1_plus_2alpha2_b0'] + require(abs(top - ref['near_mass_identity']['k=%s' % k]['ref_mc_b0']) < 3 * ref['near_mass_identity']['k=%s' % k]['ref_mc_b0_se_alpha1'] + 2e-3, 'near-mass identity within tolerance of the Math-#168 MC value k=%s' % k) print(json.dumps({'check': 'ok', 'mutant': MUT})) From 48407d43d2603b8f04f8ba3824b53fac903ed0be Mon Sep 17 00:00:00 2001 From: d6g8k5htny-coder Date: Wed, 30 Sep 2026 09:38:01 +0000 Subject: [PATCH 5/7] Microscopic tail constants numerics: README run time; manifest Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- frontiers/c6_microscopic_tail_numerics_20260930/README.md | 2 +- .../c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json | 6 +++--- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/README.md b/frontiers/c6_microscopic_tail_numerics_20260930/README.md index 5d6d55d48..d42660721 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/README.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/README.md @@ -13,7 +13,7 @@ evaluated as a control against the merged Math-#168 values. - `NOTE.md` — what is evaluated, exact structure, values, precision, non-claims. - `tail_constants.py` — standard-library script (Python `>= 3.11`): `--check` (exact controls and replay against - `RESULTS.json`), full run regenerates `RESULTS.json` (about five minutes); `--mutant {shape-integral, gamma-power, + `RESULTS.json`), full run regenerates `RESULTS.json` (about fifteen minutes); `--mutant {shape-integral, gamma-power, cusp-shift}` must exit 1. - `RESULTS.json`, `SOURCE_MAP.json` (five main-resident pins and two unmerged companions), `SOURCE_FILES.json`. - Workflow `.github/workflows/c6-microscopic-tail-numerics.yml` replays manifest, pins, both check modes and the diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json index 0ba359496..1204432d4 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json @@ -13,9 +13,9 @@ }, { "path": "README.md", - "bytes": 1710, - "sha256": "3221e571853fd37e4cd45f3c96be0d39a92e50f2a91a2936835505a5c1667ee4", - "git_blob": "5d6d55d48ff6d213976df96e75ede70b6fa0ad51" + "bytes": 1713, + "sha256": "947e4509c75adfbe8e5039f37eabdb1faa657da15fccb0e0fa1ca795e9f36308", + "git_blob": "d426607212c4f3919bd18f5b091109a5022fa757" }, { "path": "RESULTS.json", From effb934110f3d42d6d7b2d002ea437c60aef2403 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Wed, 30 Sep 2026 14:13:47 +0000 Subject: [PATCH 6/7] Microscopic tail numerics v3: pin merged [E], [Q]; check the exact cusp identities Math-#169 ([E]) and Math-#176 ([Q]) are merged on main (f430fde, 7ca3eb0) with the blobs this note evaluated (19a6044f, 65f24b1f). Their entries move from companions_unmerged to current_required main-resident pins, the workflow triggers on both paths, and the "unmerged / conditional" wording in NOTE.md, README.md and SOURCE_MAP.json is replaced accordingly. tail_constants.py --check now also verifies the three exact identities stated in Math-#190 against this quadrature: T2 = -T1/12, Tb sqrt(192 pi^3) = (12 k^2 + 1)/(36 k^3), and (11/2)(U2 - B/78)/I = 66451/11128 (exact rational), so c = C2/C_* = (66451/11128) T1/J_cusp with T1/J_cusp in (1, 12). RESULTS.json is unchanged. NOTE.md section 4 records the identities and Math-#190's certified enclosure as information (not consumed, not a certification of this note). 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-microscopic-tail-numerics.yml | 2 + .../NOTE.md | 27 +++++++++---- .../README.md | 8 ++-- .../SOURCE_FILES.json | 24 ++++++------ .../SOURCE_MAP.json | 38 +++++++++++-------- .../tail_constants.py | 10 ++++- 6 files changed, 70 insertions(+), 39 deletions(-) diff --git a/.github/workflows/c6-microscopic-tail-numerics.yml b/.github/workflows/c6-microscopic-tail-numerics.yml index e5ffa2e93..d64acedd4 100644 --- a/.github/workflows/c6-microscopic-tail-numerics.yml +++ b/.github/workflows/c6-microscopic-tail-numerics.yml @@ -10,6 +10,8 @@ name: C6 microscopic tail numerics (replay) - 'frontiers/c6_cluster_coefficients_numerics_20260930/RESULTS.json' - 'frontiers/planar_cubic_cluster_20260929/PROOF.md' - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md' + - 'frontiers/extreme_cluster_sampling_20260930/PROOF.md' + - 'frontiers/radial_rate_20260930/PROOF.md' workflow_dispatch: permissions: diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md index 674cb9772..c88499658 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md @@ -15,8 +15,8 @@ first) the point-intensity radius tail with `J_cusp = integral H(h) gamma(a)^11 h_0(A_0, Rot_O tau_cusp) da dxi dh dO` (R12) and `gamma(a) = sqrt(1 + (a/(12k))^2)`, and lists "no numerical value of its Gaussian integral" as a non-claim. The -unmerged companions Math-#169 ([E]) and Math-#176 ([Q]) express further constants of the same measure through the -same cusp integrals: the whole-cluster split of the tail (E6) and the second-order coefficient `C_2`, the +companions Math-#169 ([E], merged) and Math-#176 ([Q], merged) express further constants of the same measure through +the same cusp integrals: the whole-cluster split of the tail (E6) and the second-order coefficient `C_2`, the total-variation coefficient `kappa |C_2/C_0|` of the radial ratio and the signed constant `B_sign` ((13), (16), (19)). This note evaluates all of them in the plane for the parent kernel `K(z) = exp(-|z|^2/2)` ([LP] §2). @@ -76,7 +76,7 @@ ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight ` | `2.0` | 0.0465858 | 0.0483768 | -0.0040314 | 0.0022051 | `2e-15` | **Tail constants** (`F(t) ~ C_* t^(-11)` is the `r^(-3)`-scaled expected number of near points at scaled radius -`R > t`; the split rows are conditional on [E]; `c`, `kappa c`, `B_sign` on [Q]): +`R > t`; the split rows are values of [E] (E6); `c`, `kappa c`, `B_sign` of [Q] (13), (16), (19)): | `k` | `b` | `C_*` | `G_max` | `G_2` | `B_2` | singleton | `c = C_2/C_*` | `kappa c` | `B_sign` | |---|---|---|---|---|---|---|---|---|---| @@ -114,6 +114,17 @@ values. This is the check that the two parametrizations agree; it is not a certi - The one-dimensional cusp integrals agree between 48 and 96 Gauss–Legendre points to the relative deviations in the table (all below `1e-14`); the constants inherit that precision and the exact prefactor. +- Three exact identities stated in Math-#190 (elementary: `G_0 = exp(-12 k^2 (gamma^6 - 1)) / sqrt(192 pi^3)` on the + cusp, and `(a gamma^11)' = 12 gamma^11 - 11 gamma^9`) hold in this quadrature at machine precision and are + `--check` rules since v3: `T_2 = -T_1/12` (`< 3e-18` absolute at every `k`), `T_b sqrt(192 pi^3) = (12k^2 + 1)/(36k^3)` + (`8/9`, `13/36`, `49/288`; `< 6e-16`), and the rational factor `(11/2)(U_2 - B/78)/I = 66451/11128` of + `c = C_2/C_* = (66451/11128) T_1/J_cusp` (exact). Since `T_1/J_cusp = 12 - 11 I_9/I_11 in (1, 12)`, the bounds + `5.9715 < c(k) < 71.658` of Math-#190 sharpen the corollary `c > 4587/856` quoted in the table; the values here + (`9.78`, `6.90`, `6.20`) lie inside. +- Math-#190 (reviews/c6_tail_constant_certified_20260930, other Claude session, not merged, not consumed here) + reports certified interval enclosures of every constant of this table and finds each value of `RESULTS.json` + (blob `0b5ce76f`) within `9e-16` relative of its enclosure. That record is information for the reader; it is not a + certification of this note, which stays floating point. - The near-mass integral moves by at most `3e-04` relative between the 64- and 96-point shape levels at `ns = 160`, and by `4e-09` between `ns = 160` and `256`; its remaining error is of the former order, below the Math-#168 Monte Carlo standard error. @@ -121,8 +132,8 @@ values. This is the check that the two parametrizations agree; it is not a certi ## 5. What the numbers are not Not a proof and not an enclosure. `C_*` is a value of the reviewed formula (R13) of the merged [R]; the split and the -second-order/signed constants are values of formulas in the unmerged candidates [E] and [Q] (their Slices B, C, D and -A, B respectively carry nonauthor reads at this writing) and are conditional on them. Planar only (`d = 2`), parent +second-order/signed constants are values of formulas of the merged [E] (Math-#169) and [Q] (Math-#176), evaluated at +their stated scope (planar, `r -> 0` first). Planar only (`d = 2`), parent kernel only, aligned frame; no rate for `r -> 0`, no finite-`r` statement, no statement about the remote singleton population. No register, catalog or STATUS change. Same GitHub account as every lane; zero organizational-independence credit. The author will not merge. @@ -131,7 +142,9 @@ organizational-independence credit. The author will not merge. `tail_constants.py` (standard library; `--check` replays the exact controls, the cusp integrals and the complete constants table for each `k` and both `b`, the low-level near-mass integral for each `k` with its log-`Z` study -entry, the assembly of the identity from the stored numbers, and the identity tolerance against `RESULTS.json`; mutants `shape-integral`, +entry, the assembly of the identity from the stored numbers, the identity tolerance against `RESULTS.json`, and the +three exact identities of §4; mutants `shape-integral`, `gamma-power`, `cusp-shift` exit 1), `RESULTS.json` (full run, about fifteen minutes), `SOURCE_MAP.json` -(pins [R], [SC], [NUM], [CUB], [LP] on `main` `02772ec`; companions [E], [Q] unmerged, not workflow-checked), +(pins [R], [SC], [NUM], [CUB], [LP], [E], [Q] on `main` `3e0a91b`, all workflow-checked; [E] and [Q] were unmerged +companions in v1/v2 and were promoted to pins in v3 with their blobs unchanged), `SOURCE_FILES.json`, workflow `c6-microscopic-tail-numerics.yml`. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/README.md b/frontiers/c6_microscopic_tail_numerics_20260930/README.md index d42660721..778b3d9f5 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/README.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/README.md @@ -5,17 +5,17 @@ Evaluates, for the parent kernel `exp(-|z|^2/2)` in the plane, the point-intensity tail constant `C_*` of the already-formed microscopic measure ([R] (R13); `F(t) ~ C_* t^-11`), the whole-cluster split of that tail -(Math-#169, unmerged companion), the second-order coefficient `C_2/C_0`, the TV coefficient `kappa |C_2/C_0|` and -the signed constant `B_sign` (Math-#176, unmerged companion), for `k in {1/2, 1, 2}` and `b in {0, 1}`. In `d = 2` +(Math-#169, merged), the second-order coefficient `C_2/C_0`, the TV coefficient `kappa |C_2/C_0|` and +the signed constant `B_sign` (Math-#176, merged), for `k in {1/2, 1, 2}` and `b in {0, 1}`. In `d = 2` the [SC] measure reduces to the exact contact regression of Math-#168, so every constant is a one-dimensional Gaussian integral; the four-dimensional near-mass identity `alpha_1 + 2 alpha_2 = K_0 int Q |Z|^-12 p_odd` is evaluated as a control against the merged Math-#168 values. - `NOTE.md` — what is evaluated, exact structure, values, precision, non-claims. - `tail_constants.py` — standard-library script (Python `>= 3.11`): `--check` (exact controls and replay against - `RESULTS.json`), full run regenerates `RESULTS.json` (about fifteen minutes); `--mutant {shape-integral, gamma-power, + `RESULTS.json`, including the exact cusp identities `T_2 = -T_1/12`, `T_b sqrt(192 pi^3) = (12k^2+1)/(36k^3)`), full run regenerates `RESULTS.json` (about fifteen minutes); `--mutant {shape-integral, gamma-power, cusp-shift}` must exit 1. -- `RESULTS.json`, `SOURCE_MAP.json` (five main-resident pins and two unmerged companions), `SOURCE_FILES.json`. +- `RESULTS.json`, `SOURCE_MAP.json` (seven main-resident pins: [R], [SC], [NUM], [CUB], [LP], [E], [Q]), `SOURCE_FILES.json`. - Workflow `.github/workflows/c6-microscopic-tail-numerics.yml` replays manifest, pins, both check modes and the mutants. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json index 1204432d4..318b9c263 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json @@ -7,15 +7,15 @@ "files": [ { "path": "NOTE.md", - "bytes": 10188, - "sha256": "6ed848a1a30824930441c461097ef5fb852a0b43063f160c0bbb878e53428ecb", - "git_blob": "674cb9772dde7fe5ddbf35c6fe11c1d462795c6a" + "bytes": 11407, + "sha256": "a4fe73bdfb01e6a1aa5ac3b4e0ec262166fc71af92c370c3a3fab357fc1d895a", + "git_blob": "c88499658d1b648a12c3c097bd15097f5fd24ecd" }, { "path": "README.md", - "bytes": 1713, - "sha256": "947e4509c75adfbe8e5039f37eabdb1faa657da15fccb0e0fa1ca795e9f36308", - "git_blob": "d426607212c4f3919bd18f5b091109a5022fa757" + "bytes": 1798, + "sha256": "9ec9d3e1fe691724826b1934215cd6c09e7b47355ade8e2fd74151a7595f670f", + "git_blob": "778b3d9f522088ed985912c94ac99a3f798ff51f" }, { "path": "RESULTS.json", @@ -25,15 +25,15 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 5462, - "sha256": "c8d0dd6a076907361274f5e8492a5e54cd75ed800fae4d650021b2ba271f9510", - "git_blob": "53425785360542b249a4f7cc9a2713bd7f10f141" + "bytes": 6256, + "sha256": "71bf9f8afc4da60ba6a1a22a79f16e8c6a17ebd52e9b843fd9c84d6fcbd78844", + "git_blob": "b5cfbea9e17b80af5877b82be066f7dfb93e7c20" }, { "path": "tail_constants.py", - "bytes": 17123, - "sha256": "12d150bdd6bf78cdced6cc905c7e1774b6f78e428bddd462f5d7b48d4abe2f40", - "git_blob": "ff1f5d4b61859367ffcbebe9e1b5bf7990b2ef0b" + "bytes": 18033, + "sha256": "86de1789e89c84f61a134be9b853bd5dcd92d85be9fb68a59fabb4eea10a08e6", + "git_blob": "5170ecc5afecd18758030a2f6bdf62e8edb900ee" } ] } diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json index 534257853..b5cfbea9e 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json @@ -4,7 +4,7 @@ "scientific_effect": "NONE", "source_repository": "d6g8k5htny-coder/Math-", "main_at_creation": "02772ec0427f4dda1f2a6b9e980a031804e276a6", - "note": "Numerical evaluation only, planar (d = 2). The formulas evaluated are [R] (R13) for the point-intensity tail constant C_*, with the [SC] measure (17) reduced at d = 2 (c_1 = 1, no hard directions) to the exact contact regression of the merged Math-#168 note; the whole-cluster split and the second-order/signed coefficients are the formulas of the unmerged companions Math-#169 (E5-E6) and Math-#176 ((13), (16), (19), (21)-(22)), evaluated as stated there and marked conditional on those candidates. The near-mass identity check ties (R11) to the merged alpha_1, alpha_2 values. Floating-point quadrature with reported convergence, not enclosures.", + "note": "Numerical evaluation only, planar (d = 2). The formulas evaluated are [R] (R13) for the point-intensity tail constant C_*, with the [SC] measure (17) reduced at d = 2 (c_1 = 1, no hard directions) to the exact contact regression of the merged Math-#168 note; the whole-cluster split and the second-order/signed coefficients are the formulas of the companions Math-#169 (E5-E6) and Math-#176 ((13), (16), (19), (21)-(22)), both merged on main and pinned here since v3. The near-mass identity check ties (R11) to the merged alpha_1, alpha_2 values. Floating-point quadrature with reported convergence, not enclosures. v3 also checks the exact cusp-integral identities T2 = -T1/12 and Tb sqrt(192 pi^3) = (12 k^2 + 1)/(36 k^3) and the rational factor 66451/11128 of c = C2/C_* stated in Math-#190 (certified enclosure of this table, not consumed, not merged).", "sources": [ { "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", @@ -60,31 +60,39 @@ "role": "section 2: the parent kernel (periodization of exp(-|z|^2/2), aligned frame); section 3 contact frame; (5.4) the full normalizer z_0", "current_required": true, "on_main": true - } - ], - "companions_unmerged": [ + }, { - "pull_request": "Math-#169", - "head": "56a7904f9ad7d409c31e23195c0e7eeab230613a", "path": "frontiers/extreme_cluster_sampling_20260930/PROOF.md", + "bytes": 21891, + "sha256": "b90cf745e2e67bc9b501c481552f7080d27c28245580668cf72aefa5b7f04715", "git_blob": "19a6044f2066643b5baab34ce5b6b6efb431e84a", "id": "E", - "role": "(E5) the shape constants J = 1083417/280 (recomputed exactly here) and D = 27066286003/223205220 - (79298560/4782969) log 2 (quoted; numerically confirmed to 1e-13 in the Slice C read), (E6) the whole-cluster coefficients C0 (I - D), C0 J, C0 D, C0 (I - D - J); Slices B, C, D accepted, A under Codex read at this writing; the split reported here is conditional on that candidate", - "current_required": false, - "on_main": false + "commit": "3e0a91b69892b9717564c592e7a966492a81566a", + "role": "(E5) the shape constants J = 1083417/280 (recomputed exactly here) and D = 27066286003/223205220 - (79298560/4782969) log 2 (quoted; numerically confirmed to 1e-13 in the Slice C read), (E6) the whole-cluster coefficients C0 (I - D), C0 J, C0 D, C0 (I - D - J); Slices A, B, C, D accepted; Math-#169, merged at f430fde (blob unchanged from the reviewed head 56a7904)", + "current_required": true, + "on_main": true }, { - "pull_request": "Math-#176", - "head": "71772a8906cf6663d16be2cd5f63e5d4ebdcee2b", "path": "frontiers/radial_rate_20260930/PROOF.md", + "bytes": 15755, + "sha256": "a3693664900e23e1e763d88a0b5609936a6f31d844cbe6f18c7d0379b94ba710", "git_blob": "65f24b1f9a7b80522a2ee059f5f6877c9f970f0d", "id": "Q", - "role": "(12) the shape integrals U_2, B, U_abs (recomputed exactly here), (13) C_0 and C_2, (16) kappa, (19) B_sign, (21)-(22) the aligned planar cusp density and D_a G_0; Slices A and B accepted (this session); the coefficients reported here are conditional on that candidate", - "current_required": false, - "on_main": false + "commit": "3e0a91b69892b9717564c592e7a966492a81566a", + "role": "(12) the shape integrals U_2, B, U_abs (recomputed exactly here), (13) C_0 and C_2, (16) kappa, (19) B_sign, (21)-(22) the aligned planar cusp density and D_a G_0; Slices A and B accepted; Math-#176, merged at 7ca3eb0 (blob unchanged from the reviewed head 71772a8)", + "current_required": true, + "on_main": true } ], "nonauthor_acceptance": false, "governing_status_changed": false, - "main_at_update": "02772ec0427f4dda1f2a6b9e980a031804e276a6" + "main_at_update": "3e0a91b69892b9717564c592e7a966492a81566a", + "related_records": [ + { + "pull_request": "Math-#190", + "head": "65ac384e1b1cba434ee25db6a2c40cc7d0cbd907", + "path": "reviews/c6_tail_constant_certified_20260930/", + "role": "certified interval enclosures of the same constants (author lane Anthropic / Claude, other session); reports every value of this RESULTS.json (blob 0b5ce76f) within 9e-16 relative of its enclosure; states the exact identities checked here; not consumed, not a review of this note" + } + ] } diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py index ff1f5d4b6..5170ecc5a 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py +++ b/frontiers/c6_microscopic_tail_numerics_20260930/tail_constants.py @@ -174,7 +174,9 @@ def Q(u, v): def cusp_integrals(k, rule, L=14.0): """One-dimensional Gaussian integrals over the cusp slice beta = a^2/(12k), c = a^3/(144k^2): - J_cusp = int gamma^11 G0, T1 = int gamma^9 (1 + 12 A^2) G0, T2 = int gamma^13 D_a G0, Tb = int |A| gamma^10 G0.""" + J_cusp = int gamma^11 G0, T1 = int gamma^9 (1 + 12 A^2) G0, T2 = int gamma^13 D_a G0, Tb = int |A| gamma^10 G0. + Exact identities (Math-#190, elementary: G0 = exp(-12 k^2 (gamma^6 - 1)) / sqrt(192 pi^3), (a gamma^11)' = 12 gamma^11 - 11 gamma^9): + T2 = -T1/12 and Tb sqrt(192 pi^3) = (12 k^2 + 1)/(36 k^3); both are checked in check_run against this quadrature.""" gpow = 10 if MUT == 'gamma-power' else 11 def G0(a): cshift = 0.0 if MUT == 'cusp-shift' else a ** 3 / (144 * k * k) @@ -250,6 +252,8 @@ def controls(): require(ex['J_outer'] == J_OUT, 'J = 1083417/280') require(Fr(11) * ex['U2'] / (2 * ex['I']) == Fr(4587, 856), '11 U2/(2I) = 4587/856') require(Fr(11) * ex['Uabs'] / (2 * ex['I']) == Fr(4587821, 876544), '11 Uabs/(2I) = 4587821/876544') + # with T2 = -T1/12, c = C2/C_* = (11/2) (U2 - B/78) T1 / (I J_cusp): the rational factor of Math-#190 + require(Fr(11, 2) * (ex['U2'] - ex['B'] / 78) / ex['I'] == Fr(66451, 11128), '(11/2)(U2 - B/78)/I = 66451/11128') D = D_inner() require(0 < D < float(ex['I']) - float(ex['J_outer']), 'D positive and I - D - J positive') q0 = math.sqrt(13 / 11) @@ -308,6 +312,10 @@ def check_run(): cu = cusp_integrals(k, r96) for n in cu: require(abs(cu[n] - ref['per_k'][str(k)]['cusp'][n]) <= REPLAY_TOL * abs(ref['per_k'][str(k)]['cusp'][n]), 'cusp integral replay %s k=%s' % (n, k)) + # exact identities of the cusp integrals (Math-#190), at quadrature precision + require(abs(cu['T2'] + cu['T1'] / 12) <= 1e-13 * abs(cu['T1']), 'T2 = -T1/12 k=%s' % k) + require(abs(cu['Tb'] * math.sqrt(192 * math.pi ** 3) - (12 * k * k + 1) / (36 * k ** 3)) <= 1e-12, 'Tb sqrt(192 pi^3) = (12 k^2 + 1)/(36 k^3) k=%s' % k) + require(1.0 < cu['T1'] / cu['J_cusp'] < 12.0, 'T1/J_cusp in (1, 12) k=%s' % k) for b in (0.0, 1.0): t = constants_table(k, b, ex, cu) rt = ref['table']['k=%s,b=%s' % (k, b)] From 156fcb7bcbc6fa606e5f723ea06055b3ddaf3809 Mon Sep 17 00:00:00 2001 From: Dylan Roy Date: Sat, 3 Oct 2026 23:09:50 +0000 Subject: [PATCH 7/7] C6 planar tail numerics v3.1: apply R4/R5/R6 nonauthor reads (wording only) Applies the Math-#178 Grok-support reads routed by the Chief of Staff: - R4 Slice A (5974218952) F1-F3: the planar regression is the continuum covariance, valid at L >= 12 by the Math-#168 lattice control; the trivial O(1) Haar average is named; J cited at [E] (E5). - R5 Slice B (5974449197) F1-F4: no shape-grid convergence order claimed; the GH60 deviations explained; the MC +/- is the s.e. of alpha_1; the near-mass identity derived from the pinned [SC], [CL] (1.6) as context. - R6 Slice C (5974193134) F1-F8: [E]/[Q] values stay conditional although merged; corollary bounds attributed to Math-#176 comment 5903080058; Math-#190 recorded as merged at 77b854c (information only); precision 1.3e-14; near point mass 1.380; C_0 conventions named; continuum moments. SOURCE_MAP records the nonauthor reads. RESULTS.json and tail_constants.py unchanged. Scientific effect NONE. Co-Authored-By: Claude Opus 5.5 Claude-Session: https://claude.ai/code/session_015wNj8LPTKXsaT68G3DgPPh --- .../NOTE.md | 67 +++++++++++++------ .../README.md | 4 +- .../SOURCE_FILES.json | 18 ++--- .../SOURCE_MAP.json | 38 ++++++++++- 4 files changed, 93 insertions(+), 34 deletions(-) diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md index c88499658..f5224e0bd 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/NOTE.md @@ -15,17 +15,21 @@ first) the point-intensity radius tail with `J_cusp = integral H(h) gamma(a)^11 h_0(A_0, Rot_O tau_cusp) da dxi dh dO` (R12) and `gamma(a) = sqrt(1 + (a/(12k))^2)`, and lists "no numerical value of its Gaussian integral" as a non-claim. The -companions Math-#169 ([E], merged) and Math-#176 ([Q], merged) express further constants of the same measure through +companions Math-#169 ([E]) and Math-#176 ([Q]), both merged but still conditional (§5), express further constants of the same measure through the same cusp integrals: the whole-cluster split of the tail (E6) and the second-order coefficient `C_2`, the total-variation coefficient `kappa |C_2/C_0|` of the radial ratio and the signed constant `B_sign` ((13), (16), (19)). This note evaluates all of them in the plane for the parent kernel `K(z) = exp(-|z|^2/2)` ([LP] §2). **Planar reduction.** In `d = 2` the spectral measure (17) of [SC] has `m = d - 1 = 1`, `c_1 = 1`, no hard -curvatures and no Haar variable; `H(h) = 1`. Its density `h_0(A_0, tau)` at `A_0 = 0` is the raw contact density of +curvatures, and the Haar variable `O in O(1) = {±1}` averages trivially, because `p_odd` is invariant under +`(a, beta, c) -> (-a, beta, -c)`; `H(h) = 1`. Its density `h_0(A_0, tau)` at `A_0 = 0` is the raw contact density of `(f_zz; f_xxz, f_xzz, f_zzz)` at `(0; a, beta, c)` given the contact pins `(f, f_x, f_z, f_xx, f_xz, f_xxx) = (b, 0, 0, 0, 0, 12k)`, which by the exact regression of Math-#168 factors as `p_b(0) . p_odd(a, beta, c)` with `p_b(0) = phi(b/sqrt2)/sqrt2` (the density of `f_zz | (f, f_xx, f_xz) = (b, 0, 0) ~ N(-b, 2)` at `0`) and -`p_odd = N(0, diag(2, 2, 6))`, for every `k, b`. Hence +`p_odd = N(0, diag(2, 2, 6))`, for every `k, b`. These are the continuum jet covariances. For the periodized parent +kernel at side `L` they hold up to the lattice-moment deviation recorded by Math-#168 (`lattice_vs_continuum`: at +most `1.4e-13` at `L = 12, 24`; `L = 6` is not covered), so the values below are for the continuum covariance and +apply at `L >= 12`. Hence c_1 h_0(0, tau)/z_0 = [p_b(0)/z_0] p_odd(a, beta, c), p_b(0)/z_0 = phi(b/sqrt2) / (sqrt2 . 36 k^2 m_(2,b)), @@ -33,14 +37,16 @@ the prefactor of Math-#168, and `J_cusp(k) = integral gamma(a)^11 p_odd(a, a^2/( one-dimensional Gaussian integral, `b`-free; all `b`-dependence of the constants sits in the prefactor. **Near-mass identity (control).** The same root-resolved integral (R11) without the radius indicator is the near -point mass `integral n dM = alpha_1 + 2 alpha_2` of Math-#168 ((1.6) of [CL], `Lambda_1 = nu_near(1) + 2 nu_near(2)`): +point mass `integral n dM = a_1 + 2 a_2`, `a_j = integral 1{n = j} dM` ([SC] (18), (20); `n <= 2` by the classifier of +[SC] §6), which at `d = 2` are Math-#168's `alpha_j` (the near part of `Lambda_1` in [CL] (1.6), Math-#159, not +pinned here): alpha_1 + 2 alpha_2 = 108 k^7 [p_b(0)/z_0] integral_S Q(u, v) integral_(Z != 0) |Z|^(-12) integral p_odd(a, beta(u, v, Z, a), c(u, v, Z, a)) da dZ du dv, (N) with `beta = a^2/(12k) + k b_0(u)/Z^2`, `c = a^3/(144k^2) + a b_0(u)/(4Z^2) + 2k(v - b_0(u) u)/Z^3`. This four-dimensional integral is evaluated independently here and compared with the merged Math-#168 values; agreement -ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight `9`, multiplicity by the map) to the +is a consistency check between the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight `9`, multiplicity by the map) and the [CL]/[CUB] parametrization of Math-#168, which integrates the classifier over the jets directly. ## 2. Exact structure @@ -48,8 +54,8 @@ ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight ` - Shape integrals over `S = {-3/2 < u < 1/2, |b_0|/2 < v < 3 - 6u}` with `Q = (4v^2 - b_0^2)(b_0 - 4uv)`, in rational arithmetic: `I = 246528/35`, `U_2 = integral u^2 Q = 240192/35`, `B = integral b_0 Q = -428544/7 = 3I - 12 U_2`, `U_abs = integral |u| Q = 5898627/880` ([Q] (12)); `E eta = 5771/7062` under `Q/I` ([R] H3); the outer doublet - integral `J = 1083417/280` ([E] E20); `11 U_2/(2I) = 4587/856`, `11 U_abs/(2I) = 4587821/876544` ([Q] - corollary). The inner doublet integral `D = 27066286003/223205220 - (79298560/4782969) log 2 = 109.7699503...` + integral `J = 1083417/280` ([E] (E5), derived from the primitive (E20)); `11 U_2/(2I) = 4587/856`, `11 U_abs/(2I) = 4587821/876544` (author-side + corollary in Math-#176 comment 5903080058, not part of the pinned [Q] bytes). The inner doublet integral `D = 27066286003/223205220 - (79298560/4782969) log 2 = 109.7699503...` ([E] E21) is quoted, not recomputed exactly (it was confirmed to `1e-13` by quadrature in the Slice C read). - `kappa = (2/13)(11/13)^(11/2) = q_0^(-11) - q_0^(-13)` at `q_0 = sqrt(13/11)` ([Q] (16)). - With the aligned planar cusp density ([Q] (21)–(22), `v_a = v_beta = 2`, `v_c = 6` for the continuum kernel): @@ -76,7 +82,7 @@ ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight ` | `2.0` | 0.0465858 | 0.0483768 | -0.0040314 | 0.0022051 | `2e-15` | **Tail constants** (`F(t) ~ C_* t^(-11)` is the `r^(-3)`-scaled expected number of near points at scaled radius -`R > t`; the split rows are values of [E] (E6); `c`, `kappa c`, `B_sign` of [Q] (13), (16), (19)): +`R > t`; the split rows are values of [E] (E6); `c`, `kappa c`, `B_sign` of [Q] (13), (16), (19) (conditional on [E]/[Q]; see §5)): | `k` | `b` | `C_*` | `G_max` | `G_2` | `B_2` | singleton | `c = C_2/C_*` | `kappa c` | `B_sign` | |---|---|---|---|---|---|---|---|---|---| @@ -88,8 +94,10 @@ ties the [SC]/[R] parametrization (jet Jacobian `2`, root Jacobian `6`, weight ` | `2.0` | `1.0` | 462.58 | 455.37 | 254.11 | 7.209 | 201.26 | 6.2011 | 0.381 | 0.2477 | `c` and `B_sign` are `b`-free (the prefactor cancels); the ratios `G_max/C_* = (I-D)/I = 0.98442`, `G_2/C_* = J/I -= 0.54934`, `B_2/C_* = D/I = 0.015584` are universal. The bounds of [Q]'s corollary hold: `c > 4587/856 = 5.3586` -and `0 < B_sign < 4587821/876544 = 5.2340`. Reading: at `k = 1`, `b = 0` the near point mass is `1.378` and `C_* = 52.6`, so the asymptotic law puts `C_* 2^(-11) = 0.026` of it beyond twice the pin distance and `3e-4` beyond += 0.54934`, `B_2/C_* = D/I = 0.015584` are universal. In `RESULTS.json`, `C2_over_C0` uses [Q]'s `C_0 = C_*`; +`C0_per_shape` is [E]'s `C_0`. The numbers satisfy the bounds of the Math-#176 author-side corollary (comment 5903080058; not in the +pinned [Q] blob): `c > 4587/856 = 5.3586` +and `0 < B_sign < 4587821/876544 = 5.2340`. Reading: at `k = 1`, `b = 0` the near point mass is `1.380` and `C_* = 52.6`, so the asymptotic law puts `C_* 2^(-11) = 0.026` (about `1.9%` of it) beyond twice the pin distance and `3e-4` beyond three; the radial ratio's total-variation distance from Pareto(11) at threshold `t` is `0.42/t^2` to leading order (`4.7%` at `t = 3`), and the signed-mark distance decays only like `0.50/t`. @@ -102,26 +110,32 @@ three; the radial ratio's total-variation distance from Pareto(11) at threshold | `2.0` | 1119.62 / 1120.53 / 1120.7 | 2.1954 | 2.196 | 2.1948 (0.0033) | -2.34e-04 / +2.87e-04 | The shape-grid refinement at fixed `ns = 160` moves the integral by at most `3e-04` relative between the 64- and -96-point levels (observed order about two; the integrand has an integrable boundary singularity where the -`|Z|`-scale of the Gaussian cutoff diverges), and the log-`Z` order is converged: at `32 x 32` shape points, +96-point levels (the sequence is monotone with observed order about 2 at `k = 2` and about 3.5 at `k = 1`, and is not +monotone at `k = 1/2`, so no order is claimed; the integrand has an integrable singularity at the corner +`(u, v) = (-1/2, 0)` of `S`, where the `|Z|`-scale of the Gaussian cutoff diverges), and the log-`Z` order is converged: at `32 x 32` shape points, `ns = 64, 96, 160, 256` give `175.6152`, `175.9278`, `175.9235`, `175.9235` at `k = 1` (an under-resolved `ns = 64`, the order of the first version of this note, biased the identity low by `1.8e-3`, exactly the deviation then reported; the `a`-order `32` and `48` agree to all digits). At the finest level the identity lies within `1.1`, `0.5` and `0.2` Monte Carlo -standard errors of the Math-#168 values at `k = 1/2, 1, 2`, and within `2.7e-04`, `2.2e-04`, `2.3e-04` relative of the GH60 +standard errors of `alpha_1` of the Math-#168 values at `k = 1/2, 1, 2`, and within `2.7e-04`, `2.2e-04`, `2.3e-04` relative of the GH60 values. This is the check that the two parametrizations agree; it is not a certification of either. +The deviations from GH60 are all negative and of the size of the remaining shape-grid movement of (N), which still +increases between the two finest levels; at `k = 2` the GH60 reference is itself not settled at this level (Math-#168 +§4: `J_2(2)` GH60 lies 3.2 MC s.e. below MC and the third digit of `alpha_2(2)` is not established, about `4e-4` +relative in `alpha_1 + 2 alpha_2`). ## 4. Precision (empirical, not bounds) - The one-dimensional cusp integrals agree between 48 and 96 Gauss–Legendre points to the relative deviations in - the table (all below `1e-14`); the constants inherit that precision and the exact prefactor. + the table (all at most `1.3e-14`); the constants inherit that precision and the exact prefactor. - Three exact identities stated in Math-#190 (elementary: `G_0 = exp(-12 k^2 (gamma^6 - 1)) / sqrt(192 pi^3)` on the cusp, and `(a gamma^11)' = 12 gamma^11 - 11 gamma^9`) hold in this quadrature at machine precision and are `--check` rules since v3: `T_2 = -T_1/12` (`< 3e-18` absolute at every `k`), `T_b sqrt(192 pi^3) = (12k^2 + 1)/(36k^3)` (`8/9`, `13/36`, `49/288`; `< 6e-16`), and the rational factor `(11/2)(U_2 - B/78)/I = 66451/11128` of - `c = C_2/C_* = (66451/11128) T_1/J_cusp` (exact). Since `T_1/J_cusp = 12 - 11 I_9/I_11 in (1, 12)`, the bounds - `5.9715 < c(k) < 71.658` of Math-#190 sharpen the corollary `c > 4587/856` quoted in the table; the values here - (`9.78`, `6.90`, `6.20`) lie inside. -- Math-#190 (reviews/c6_tail_constant_certified_20260930, other Claude session, not merged, not consumed here) + `c = C_2/C_* = (66451/11128) T_1/J_cusp` (exact). Since `T_1/J_cusp = 12 - 11 I_9/I_11 in (1, 12)`, Math-#190 + reports (information only, not consumed here) the bounds `5.9715 < c(k) < 71.658`, which would sharpen the corollary + `c > 4587/856`; the floating-point values here (`9.78`, `6.90`, `6.20`) lie inside. +- Math-#190 (reviews/c6_tail_constant_certified_20260930, other Claude session; merged 2026-10-01 at `77b854c` as a review + record filed as certified-arithmetic information, not a promotion; not consumed here) reports certified interval enclosures of every constant of this table and finds each value of `RESULTS.json` (blob `0b5ce76f`) within `9e-16` relative of its enclosure. That record is information for the reader; it is not a certification of this note, which stays floating point. @@ -132,8 +146,11 @@ values. This is the check that the two parametrizations agree; it is not a certi ## 5. What the numbers are not Not a proof and not an enclosure. `C_*` is a value of the reviewed formula (R13) of the merged [R]; the split and the -second-order/signed constants are values of formulas of the merged [E] (Math-#169) and [Q] (Math-#176), evaluated at -their stated scope (planar, `r -> 0` first). Planar only (`d = 2`), parent +second-order/signed constants are values of formulas of [E] (Math-#169) and [Q] (Math-#176). Both are merged on main +after scoped nonauthor review, but their pinned bytes keep the disposition 'author-side conditional proof candidate' +and their source hypotheses remain. These values are therefore conditional on [E] and [Q] and their sources, +evaluated at their stated scope (planar, `r -> 0` first), with continuum spectral moments and no finite-`L` +periodization correction. Planar only (`d = 2`), parent kernel only, aligned frame; no rate for `r -> 0`, no finite-`r` statement, no statement about the remote singleton population. No register, catalog or STATUS change. Same GitHub account as every lane; zero organizational-independence credit. The author will not merge. @@ -148,3 +165,13 @@ three exact identities of §4; mutants `shape-integral`, (pins [R], [SC], [NUM], [CUB], [LP], [E], [Q] on `main` `3e0a91b`, all workflow-checked; [E] and [Q] were unmerged companions in v1/v2 and were promoted to pins in v3 with their blobs unchanged), `SOURCE_FILES.json`, workflow `c6-microscopic-tail-numerics.yml`. + +Revision v3.1 (Slice C review 5974193134 on Math-#178, findings F1–F8, wording only): [E]/[Q] values are stated as +conditional although both are merged; the corollary bounds are attributed to Math-#176 comment 5903080058, not to the +pinned [Q]; Math-#190 is recorded as merged at `77b854c` and its bounds as information only; precision `1.3e-14`; +near point mass `1.380` (`C_* 2^(-11)` about `1.9%` of it); the two `C_0` conventions in `RESULTS.json` named; +continuum spectral moments stated. With the Slice A review 5974218952 (F1–F3): the planar regression is stated as the +continuum covariance, valid at `L >= 12` by the Math-#168 lattice control; the trivial `O(1)` average is named; `J` +is cited at [E] (E5). With the Slice B review 5974449197 (F1–F4): no convergence order is claimed for the shape grid; +the sign and size of the GH60 deviations are explained; the `±` is the MC standard error of `alpha_1`; the identity is +derived from the pinned [SC], with [CL] (1.6) cited as context only. `RESULTS.json` and `tail_constants.py` unchanged. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/README.md b/frontiers/c6_microscopic_tail_numerics_20260930/README.md index 778b3d9f5..0c93b920b 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/README.md +++ b/frontiers/c6_microscopic_tail_numerics_20260930/README.md @@ -5,8 +5,8 @@ Evaluates, for the parent kernel `exp(-|z|^2/2)` in the plane, the point-intensity tail constant `C_*` of the already-formed microscopic measure ([R] (R13); `F(t) ~ C_* t^-11`), the whole-cluster split of that tail -(Math-#169, merged), the second-order coefficient `C_2/C_0`, the TV coefficient `kappa |C_2/C_0|` and -the signed constant `B_sign` (Math-#176, merged), for `k in {1/2, 1, 2}` and `b in {0, 1}`. In `d = 2` +(Math-#169, merged; values conditional on it), the second-order coefficient `C_2/C_0`, the TV coefficient `kappa |C_2/C_0|` and +the signed constant `B_sign` (Math-#176, merged; values conditional on it), for `k in {1/2, 1, 2}` and `b in {0, 1}`. In `d = 2` the [SC] measure reduces to the exact contact regression of Math-#168, so every constant is a one-dimensional Gaussian integral; the four-dimensional near-mass identity `alpha_1 + 2 alpha_2 = K_0 int Q |Z|^-12 p_odd` is evaluated as a control against the merged Math-#168 values. diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json index 318b9c263..4203a85ef 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_FILES.json @@ -7,15 +7,15 @@ "files": [ { "path": "NOTE.md", - "bytes": 11407, - "sha256": "a4fe73bdfb01e6a1aa5ac3b4e0ec262166fc71af92c370c3a3fab357fc1d895a", - "git_blob": "c88499658d1b648a12c3c097bd15097f5fd24ecd" + "bytes": 14419, + "sha256": "24a76ae647a67a4e4bd350848d1cc1703c7b211be364a76ae0de46a27f37dfbe", + "git_blob": "f5224e0bde234bd84c2975850773d686f9319951" }, { "path": "README.md", - "bytes": 1798, - "sha256": "9ec9d3e1fe691724826b1934215cd6c09e7b47355ade8e2fd74151a7595f670f", - "git_blob": "778b3d9f522088ed985912c94ac99a3f798ff51f" + "bytes": 1850, + "sha256": "6d1fa8448520802211b12831e1f7624ca3e2226b5a744d384c6e4d5ceebc99d3", + "git_blob": "0c93b920b560a100ac82d8ee05d4240ade227203" }, { "path": "RESULTS.json", @@ -25,9 +25,9 @@ }, { "path": "SOURCE_MAP.json", - "bytes": 6256, - "sha256": "71bf9f8afc4da60ba6a1a22a79f16e8c6a17ebd52e9b843fd9c84d6fcbd78844", - "git_blob": "b5cfbea9e17b80af5877b82be066f7dfb93e7c20" + "bytes": 7768, + "sha256": "ec61ec391b89751498cf3f9359365a83ea7006118721d21dd69b000141fe26c3", + "git_blob": "0e19e039e87acf491db780ef6c9971c388c593a4" }, { "path": "tail_constants.py", diff --git a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json index b5cfbea9e..0e19e039e 100644 --- a/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json +++ b/frontiers/c6_microscopic_tail_numerics_20260930/SOURCE_MAP.json @@ -4,7 +4,7 @@ "scientific_effect": "NONE", "source_repository": "d6g8k5htny-coder/Math-", "main_at_creation": "02772ec0427f4dda1f2a6b9e980a031804e276a6", - "note": "Numerical evaluation only, planar (d = 2). The formulas evaluated are [R] (R13) for the point-intensity tail constant C_*, with the [SC] measure (17) reduced at d = 2 (c_1 = 1, no hard directions) to the exact contact regression of the merged Math-#168 note; the whole-cluster split and the second-order/signed coefficients are the formulas of the companions Math-#169 (E5-E6) and Math-#176 ((13), (16), (19), (21)-(22)), both merged on main and pinned here since v3. The near-mass identity check ties (R11) to the merged alpha_1, alpha_2 values. Floating-point quadrature with reported convergence, not enclosures. v3 also checks the exact cusp-integral identities T2 = -T1/12 and Tb sqrt(192 pi^3) = (12 k^2 + 1)/(36 k^3) and the rational factor 66451/11128 of c = C2/C_* stated in Math-#190 (certified enclosure of this table, not consumed, not merged).", + "note": "Numerical evaluation only, planar (d = 2). The formulas evaluated are [R] (R13) for the point-intensity tail constant C_*, with the [SC] measure (17) reduced at d = 2 (c_1 = 1, no hard directions) to the exact contact regression of the merged Math-#168 note; the whole-cluster split and the second-order/signed coefficients are the formulas of the companions Math-#169 (E5-E6) and Math-#176 ((13), (16), (19), (21)-(22)), both merged on main and pinned here since v3; their pinned bytes keep the disposition 'author-side conditional proof candidate', so those values are conditional on [E] and [Q] and their sources (v3.1, review 5974193134). The near-mass identity check ties (R11) to the merged alpha_1, alpha_2 values. Floating-point quadrature with reported convergence, not enclosures. v3 also checks the exact cusp-integral identities T2 = -T1/12 and Tb sqrt(192 pi^3) = (12 k^2 + 1)/(36 k^3) and the rational factor 66451/11128 of c = C2/C_* stated in Math-#190 (certified enclosure of this table; merged 2026-10-01 at 77b854c as a review record filed as certified-arithmetic information, not a promotion; not consumed).", "sources": [ { "path": "frontiers/two_scale_cluster_geometry_20260929/RADIAL_TAIL.md", @@ -85,14 +85,46 @@ } ], "nonauthor_acceptance": false, + "nonauthor_reads": [ + { + "comment": "5364178009", + "kind": "pull_request_review", + "pr": "Math-#178", + "lane": "OpenAI / Codex (automated Codex review)", + "scope": "three findings, applied in d025512" + }, + { + "comment": "5974218952", + "kind": "issue_comment", + "pr": "Math-#178", + "lane": "Grok Bot agent 2 (Grok support routed by the Chief of Staff 5974102560)", + "scope": "R4 Slice A (planar reduction, NOTE sections 1-2) at effb934: ACCEPT_WITH_FINDINGS; F1-F3 applied in v3.1" + }, + { + "comment": "5974449197", + "kind": "issue_comment", + "pr": "Math-#178", + "lane": "Grok Bot agent 11 (Grok support routed by the Chief of Staff 5974102560)", + "scope": "R5 Slice B (near-mass identity) at effb934: ACCEPT_WITH_FINDINGS; F1-F4 applied in v3.1" + }, + { + "comment": "5974193134", + "kind": "issue_comment", + "pr": "Math-#178", + "lane": "Grok Bot agent 13 (Grok support routed by the Chief of Staff 5974102560)", + "scope": "R6 Slice C (numbers) at effb934: ACCEPT_WITH_FINDINGS; F1-F8 applied in v3.1" + } + ], "governing_status_changed": false, "main_at_update": "3e0a91b69892b9717564c592e7a966492a81566a", "related_records": [ { "pull_request": "Math-#190", - "head": "65ac384e1b1cba434ee25db6a2c40cc7d0cbd907", + "head": "28747d8e3447b3c0e92833ef45ac6c374e4a84b0", "path": "reviews/c6_tail_constant_certified_20260930/", - "role": "certified interval enclosures of the same constants (author lane Anthropic / Claude, other session); reports every value of this RESULTS.json (blob 0b5ce76f) within 9e-16 relative of its enclosure; states the exact identities checked here; not consumed, not a review of this note" + "role": "certified interval enclosures of the same constants (author lane Anthropic / Claude, other session); reports every value of this RESULTS.json (blob 0b5ce76f) within 9e-16 relative of its enclosure; states the exact identities checked here; not consumed, not a review of this note", + "merge_commit": "77b854cac6017ecdc0ea7abf6f978f9c6c9cb787", + "merged_at": "2026-10-01T14:10:35Z" } ] }