[pinterbanget]

Summary

This adds research/pr4532-proof-check/, an independent proof check of Dillon's #4532 §2.2–2.3. That is the funded critical-root majorant L(n) ≤ A n^(t*)(c + log n)^p, on which hexapode's #4535 (conditional κ = 0.003020165) rests. My #4537 checked only #4532's finite figures. Every step checks out, and I found no gap. No new κ.

Checks, step by step in the form each is used

• S0, §1.1. Both mirror-slack integral identities, ∫(S−E)dz/z = PT' − TP' and the z^(t−2) rank-one form, hold as exact finite sums over the actual p16 windows at seven exponents. Strict slack gives a positive mean gap.
• S8, §3. In the tail-gate feasibility lemma, the convexity bound on H(λ) and (13) at the crossing hold. The sufficiency construction (a = b = 1, M = 512) passes all gates, and necessity holds on 149 random passing samples.
• S1. (ln f)' = τ/x and the f''/f identity hold exactly at 2,000 random rational points (rational-function identity with dy/dx = 1/x).
• S2. c ≥ p/(1−t*) gives τ ≤ 1. Then (f/x)' ≤ 0 and f(n0) = n0, so L(n) = n ≤ f(n) on the base range.
• S3, S4. (11) f(e)/f(n) ≤ (e/n)^τ and (12) f(n) − f(n−s) ≥ η(τ)f(n)(s/n)^τ (including s = n) hold on 60-digit grids.
• S5. The recurrence closure from (10) holds.
• S6. The K_ℓ monotonicity identity holds, and n < m·M^(ℓ+1) holds in 17,118 exhaustive cases.
• S7. On the actual p16 profile over each cover interval, the monotonicity directions that critical_guards' endpoint evaluation relies on all hold: forward ratios, tilted means, every mirror cell sum, η, and the root-tail and moment-ratio endpoints. So does (6), log q ≥ d(t − t*).

S2–S4 and S7 run for both #4532's p16 witness and my #4565 retuned witness. One remark, not a defect: the proof's +log 2 in the rank bound is slack, because n < m·M^(ℓ+1) already holds.

python3.12 -B scripts/research_package.py --package research/pr4532-proof-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4532-proof-check outcome=completed   (~70 s, stdlib only)

Limitations:

• Remove tensor padding (199.32× smaller uniform-rank exponent) #4483's imported identities are not re-proved: the cancellation, the Chernoff bounds and the mirror identity. My Independently certify #4483's finite gates (p16 2^2800000000, all four witnesses) and its arbitrary-height flow identities (no new κ) #4529 and Independently check #4544's sharp cancellation audit: exact identities, necessity of #4483's (10), controls, source binding (no new κ) #4551 cover parts of them.
• Native, physical and outer premises are out of scope.
• The inequalities are checked on finite grids, alongside their one-line written proofs.

Head: c4607e578e09a7cc7fe60ee0b7685b941822135d. Base: canonical main f626db5c.

Credit: Dillon (#4532), hexapode (#4535), zxv and Jacob Alan Sussman. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code