Cubature audit

Every complete radial Class-I grid is a spherical five-design. The limiting equal-weight sampling measure differs from uniform spherical area first at harmonic degree six. Paper IV also gives finite-frequency computations and an asymptotic expansion for the degree-six moment. The displayed exact invariant evaluation is a paper theorem, not a site-wide certificate.

Paper IV DOI

Paper VI v1.3 proves a strictly positive $H_3$-orbit-constant rule of degree $t_f$ on each frozen grid $X_f$ for $1\le f\le 9$. Target degrees are

Exact weights: orbit_masses_f1_f9.csv. Frequency-9 affine family: f9_affine_family.csv. Frozen contract: CLASS_I_CUBATURE/1.0.

Replay of those frozen files is an implementation property of G6_RELEASE/1.1. It does not enlarge Paper VI beyond f = 9.

A positive-weight rule that integrates every product from a finite-dimensional space of continuous functions has injective samples on that space. Degree $2d$ is sufficient for polynomial restrictions of degree at most $d$. That sufficient condition, and its use on the separate twelve-node boundary decoder, is the exact field interface. It is not a new cubature theorem and it does not enlarge Paper VI.

Reproduce these certificates

A post-publication computation gives a strictly positive target-degree rule on each fixed Class-I grid with $10\le f\le 21$. Maximality is not claimed. This is not Paper VI v1.3, and it is not the f = 22 obstruction.

Recovered exact certificate package for 10 ≤ f ≤ 21 is hosted under /cubature/. Combined file classI_exact_certificates_f2_f21.json (1,815,267 bytes, SHA-256 7b3945ad…). Historical verifier and r2 revision are hosted. Per-frequency records at /cubature/frequencies/f10.json through f21.json. Python 3.11.2 r2 PASS on this archive: 383 identities, 404 positive masses. Post-publication computation; does not enlarge Paper VI. Maximality is not claimed. Not peer review. Not independent third-party reproduction.

Combined certificate: classI_exact_certificates_f2_f21.json · 1,815,267 bytes · SHA-256 7b3945ad58b76341bbc22e7812a359f64b61a68d413b2668012dc155803edb5eCopy. Historical verifier: verify_classI_f2_f21.py. Recovery revision: verify_classI_f2_f21.r2.py. Generators are homogeneous harmonics $H_6,H_{10}$, not the f=23,24 raw even vertex power sums.

This-archive Python 3.11.2 r2 ALL PASS: 20 frequencies, 383 exact identities, 404 strictly positive masses. Author-supplied evidence_* transcripts in the Word recovery package are a different run. Independent third-party reproduction is currently recorded as none. Frozen CLASS_I_CUBATURE/1.0 and G6_RELEASE/1.1 bytes are unchanged.

On the frozen node set $X_{22}$ a strictly positive orbit-constant rule exists through degree 69, and no nonnegative weights on those same nodes reach degree 70. See the dedicated audit.

Audit this result

A locally hosted existence theorem states that $X_{23}$ admits strictly positive $H_3$-invariant cubature weights exact through degree 71, and $X_{24}$ through degree 75. No maximality assertion is included. The argument is exact linear algebra over $\mathbb{Q}(\sqrt{5})$. It does not enlarge Paper VI v1.3.

Original certificate: ClassI_Cubature_f23_f24_exact_certificate_v1.1.json · 1,407,835 bytes · SHA-256 e886510ec03afe014ad143d6d0b45677fc2bb58a7df495374b5780cf79d5b5afCopy. Verifier: verify_ClassI_f23_f24_exact_v1.1.py · SHA-256 41d83727473850ebfbe0bbea0f57f38a64c08d9849170a880b23163769509ce1Copy. Ledger: ClassI_Cubature_f23_f24_verification_ledger_v1.1.txt. Launcher: replay_recovered.py. The printed audit extract and the LaTeX-emitted JSON (SHA-256 3ecbf665…) remain distinct files.

The three original f=23,24 files are hosted under /cubature/ with their recorded hashes: ClassI_Cubature_f23_f24_exact_certificate_v1.1.json (1,407,835 bytes, SHA-256 e886510e…), verify_ClassI_f23_f24_exact_v1.1.py (6,818 bytes, SHA-256 41d83727…), ClassI_Cubature_f23_f24_verification_ledger_v1.1.txt (10,905 bytes, SHA-256 4d56dc2c…). The guarded launcher replay_recovered.py (1,621 bytes, SHA-256 a3bf98e7…) hash-pins those three files and refuses python3 -O. This-archive Python 3.11.2 replay PASS; that is implementation conformance of recovered source, not independent third-party reproduction and not peer review. The printed finite-audit extract remains a distinct artifact. The LaTeX-emitted JSON (SHA-256 3ecbf665…) has identical mathematics and different bytes; do not substitute it for the original. Does not enlarge Paper VI. No maximality. Neither argument proves failure or impossibility at any higher degree.

Read the existence theorem · Typed network dossier · X23_X24_positive_existence.md · f23.json · f24.json