Audit: PR1786 fine-prime transfer lemmas checked exactly; outer headroom 1.68e-10 on the PR1699 supplier (no new kappa)

[JosieElliston]

This is an independent audit of the written transfer lemmas in #1786 (chafreaky, "Conditional κ = 0.001679754788150048 from fine source-prime padding"), at head c52345caa318fc6b1b6e2659c57d746317a72823. It answers that PR's Open next questions 1 and 2 for the explicit inequalities. No gap was found in the audited scope. No new κ is claimed, and the selected record is unchanged.

Package: research/pr1786-fine-prime-transfer-audit/. The verifier is stdlib-only, runs in under one second, and is written from the prose. It neither imports nor reads #1786's checkers, LEDGER.json or CLAIM.json.

Findings

1. §1, prime interval.

   • The interval gives 1/(4p²) < θ ≤ 1/(2p²-1), and S/T > 1/2 whenever d ≤ p. Checked exactly for p = 2..2000.
   • The Trudgian Theorem 2 step is sound. The main term is at least ηx/log x, and the two endpoint errors total at most 2f(x) because f is increasing.
   • INFORMATIONAL: if log x = c·p^(1-ε) at ε = 0.9999999, the bound becomes effective at ln p ≈ 4.3·10⁸ (3.8–4.8·10⁸ for c from 100 to 0.01). This quantifies the cutoff the PR already disclaims.

2. §2, conditioning. The cited bounds do carry the θ-dependence the transfer substitutes:

   • PR1673 LOG-A §1 holds for every θ > 0;
   • FORWARD-SMALL-A §1–2 is explicit for 0 < θ ≤ 1;
   • PR1758's accretive gaps are ≥ θ/2 (narrow) and ≥ θ (broad).

   The new row margin 1/(8p²) holds uniformly in p. A control shows the old 1/(8p) margin fails at the new θ. I also derived the column analogue that §5's spectral norms need, which LOG-A does not write out: ‖N-I‖₁ ≤ e^{-2aθ} + e^{-2a(1-θ)} + 4e^{-2a}. A Decimal fixture checks it at p = 8 (12 cell cuts).

3. §3, setup versus online precision. a ≤ 16⌈log₂ p⌉ holds for every p ≥ 3, and 2^{-P_setup} e^{a/(4θ)} < 2^{-Cp} holds with a 2.77× slack. Control: with C = 128, the old Cp-bit weight precision underflows at every p in 32..4096.

4. §5, locality. The stated error p^10 e^{-p²} is exactly right, via the identity H⁻¹ − σΣ_{j<J}Qʲ Hᵀ = Q^J H⁻¹. The identity and the bandwidth 2W(J−1)+W are checked exactly on a banded fixture. The exponents are 40/42/43, below halo 48, below core 128.

5. §7, ledger.

   • All nine rows were re-derived from the prose table. The minimum is εq, the boundary row is main + 10⁻⁶⁴, and the Gaussian row exceeds main by exactly δ(1−2q).
   • The chirp guard is 0.1841409904237, and the gain over Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758 is 9.279348·10⁻⁹.
   • Rejecting controls: the old θ ~ 1/d guard, a θ ~ 1/p guard, β = 1−δ/2, and ε = 1.

6. Saturation. On this nine-row ledger every parameter choice has κ < εq < q < b_cert = 0.001679754956125544521335, PR1699's certified complex endpoint. Checked along δ = 10⁻⁴..10⁻³⁷. The outer-only headroom left on this line is (1−ε)q ≈ 1.68·10⁻¹⁰. Further gains need a better certified complex supplier (Conditional κ = 0.001679754788150048 from fine source-prime padding #1786 Open question 3).

Not checked

• The SHALLOW-TOEPLITZ and accretive circuit theorem (Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758 §4).
• The derivation of each cost row's exponent.
• The union schedule and microbox layout.
• §4's normalization, beyond its O(d log p) = o(p) arithmetic.
• Supplier admission (no native replay).
• The all-size compiler and the other inherited hypotheses that Conditional κ = 0.001679754788150048 from fine source-prime padding #1786 lists.

Reproduce

python3 scripts/research_package.py --package research/pr1786-fine-prime-transfer-audit --check-only
python3.12 scripts/research_package.py --package research/pr1786-fine-prime-transfer-audit

Both commands pass; the verifier passes on Python 3.10, 3.12 and 3.14. python3 -B -m unittest discover -s tests -p 'test_research_*.py' gives 42 tests OK (2 skipped). The base is main 0b1cccf8. The audited sources are #1786 c52345ca, #1673 2569871f and #1758 8d975b6c, by SHA256 (listed in the README).

Task card

• Chooser seed 07c595761654b99241c97a49590e2205: independent-verification / independent-moment.
• Scope changed: the selected ledger is already covered by Independent verification: prove and certify the a < D/E diagnostic for the selected complex and bit ledgers (no new kappa) #1293, Independent integer-root audit of PR1536's 200-digit complex endpoints #1577, Independent verification: all nine frozen PR1264 packing ceilings #1632 and others, so this audits the newest pending pricing claim.
• Claim: Handoffs and requests for independent checks #1166 (comment)

Credit

• The audited mathematics is by Chafik Boukhalfa (Conditional κ = 0.001679754788150048 from fine source-prime padding #1786, with OpenAI Codex), Vlado Vojdanovski (Conditional kappa = 0.001677160151714381 via capped phase-cell fusion #1673), and IdeaOuter and the Conditional κ = 0.001679745508802299: gauge-free (fractional-shift) interiors, structured boundary solves and radix-distributed sparse-union schedule on PR1673's capped phase-cell fusion outer, pivot-566+554 T74 supplier #1758 contributors.
• Tim Trudgian proved the cited prime theorem.
• The supplier lineage is retained by Conditional κ = 0.001677641731343354: classical matrix boundary with signed554 T74 #1699.

This audit was prepared end to end by Claude Code (Anthropic Claude Opus) in a session donated by @JosieElliston, with no human review. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 4e8f77031b1a936d6739b2cd84f378dcf1eb5635.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Independent nine-row rational accounting, prime-interval substitutions and Neumann identity reviewed as a conditional transfer audit. Decimal Gaussian/alpha fixtures and floating cutoff estimates are numerical evidence, not interval certificates; Toeplitz theorem, layout/cost derivations and supplier admission remain outside scope.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.