Research audit: tensor E8 rate and finite-depth center-copy correction

[swbiggart]

Research note / mathematical audit; no new multiplication κ.

PR647's retained provider proof states a tensor-depth moment limit, while its packaged verifier focuses on depth one. This contribution independently reconstructs the paid profiles at every depth 1–66 and derives explicit finite-depth bounds. It corroborates the predecessor's leading coefficient and adds the correction

9^n k_n = 28/(2739 log 3) * (1 - (3/2)(3/5)^n) + O(n/9^n),

where k_n is the root of the paid profile's moment equation. The remaining center-copy cost determines the leading shortfall. At depth 66 the scaled root is below its limit by less than 4e-15 relatively. This audits the numerical profile and its analytic consequence; it does not newly certify the physical construction, RAM compiler, Fourier transfer or integer multiplication theorem.

The package contains a proof with explicit error envelopes, independent lifetime and level-sum implementations, exact rational root bounds without root finding, source hashes, a deterministic receipt, attribution and a path-scoped workflow. Nine negative controls include a mutation preserving both child count and rank mass while changing the moment. No existing file, upstream source or selected-frontier pointer changes.

Reproduce (Python 3.9+, standard library):

python3 -B research/tensor-e8-rate-audit/verify.py --check research/tensor-e8-rate-audit/RESULTS.json

Expected: PASS_TENSOR_E8_RATE_AUDIT: 66 depths; 9 negative controls; exact rational bounds.

Validation: exact receipt and five tests pass on Python 3.9.6, 3.11.16 and 3.13.16, including optimized execution. A separate Decimal implementation checks nonlinear moments at 250 digits; these diagnostics supplement the rational proof. PR647's full original checker was freshly replayed from its immutable source and passed. Source hashes, selected-record integrity, workflow syntax and whitespace pass. The unchanged full repository suite was not repeated: the prior attempt stopped at the selected bit supplier's missing local Boost headers. Scoped Linux CI passed on Python 3.11 and 3.13 at submitted head cfc7130e0e371395f4cf2f96c122c240cdbeab97, including the optimized receipt, all five tests and the clean-tree check. This is separate from broader repository CI.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.
Audited PR647: 956bb581e7f97328e1b53baa5aca3d2490f2f607.
Assignment seed 66, mathematical audit. No pinned coordination issue existed; the local claim records the scope, stopping rule and result. Overlap refreshed through #681 (501 open PRs); PR647's head is unchanged.

Credits: Artus / @artuskg's PR647, its OpenAI Pro provider proof, its Codex verification work and all preserved E8/community predecessors. The construction and original limit belong to that contribution. Prepared by Scott Woolsey Biggart (@swbiggart) with substantial OpenAI Codex assistance, under Apache-2.0. No human-only or proof-assistant review is claimed.

[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Independent tensor E8 profile audit at depths 1–66 and center-copy asymptotic correction. Root enclosure and derivative-bound argument reviewed; original physical/analytic supplier claims not newly certified.

The scoped package check passed at cfc7130e0e371395f4cf2f96c122c240cdbeab97. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.