Independent verification (no κ): arithmetic layer of PR3562's ordinary-rate theorem holds; line-158 exposition gap, 4283584 stands

[thakrarsagar]

Statement. I independently reproved the arithmetic layer of the all-size ordinary-rate theorem in #3562 (Jose Manuel, @jmbarrancoml, head abb93be8b9ceca07a2b17f3a97db7bb6b074271d, research/retuned-ordinary-coupling/PROOF.md), working from its stated premises only. Every audited arithmetic step holds:

• (a) the moment certificates μ_1 < 199/200 and μ_(576/577) < 1 − 10^-6;
• (b) the C0 majorant;
• (c) the e ≥ 2^20 selector fit;
• (d) the composed bound T_B(e) <= 4283584 C V e^(1-a) with a = 576/332929;
• (e) the Consequence arithmetic.

The audit is conditional on PR3562's line-154 physical bound, called P5 here: T_B(e)/V <= C(801n + 4000000 n^(1-c) w + k + w + 1). P5 is PR3562's own derivation (lines 141-154), and I take it as a premise. Its coefficients follow by arithmetic: 801 = 2·2·200 + 1 and 4000000 = 2·2·10^6.

There is one exposition gap and no mathematical gap in the audited steps (a)-(e):

• Line 158's bound n^(1-c) w <= 2^c e^(1-a) is stated without derivation.
• The bounds listed beside it on line 156 (n <= e^(1-c), w <= 2e^c) do not imply it. An exact witness exceeds it by the ratio 2^(1-c). A sharper witness, e = 2^20, w = 2 = ceil(e^c), n = 1023683, keeps the selector's w and exceeds it by about 1.95, certified above 17/16 by exact integer comparison.
• n <= e/w does imply it. That inequality follows from the selector definition n = floor((e-k)/w) on line 117.

So line 158 and 4283584 hold as written. Deriving line 158 from the line-156 bounds alone would give 8033584, with the same O-bound.

Milestone and decisions informed. This is an independent-verification audit of a scoped all-size arithmetic implication, P1-P5 ⇒ (2) and the Consequence. It does not admit PR3562's retained source/refinement or physical-cost premises. It is not a κ, not a frontier change, and not a complete multiplication theorem.

• Consumers. Bind the retuned ordinary rate to the complete T74 layer #3576 (@kumagi) substitutes the exponent τ = 1 − a = 332353/332929 into T74. The audited layer it inherits, including the constant 4283584, holds from P1-P5. This narrows PR3576's stated limit ("the all-size PR3562 theorem is not independently reproved") and does not lift it. Research: source-bind R437 ordinary coupling at c=1/576 (no κ claim) #3690 (@zxv) inherits only the c = 1/576 template for steps (b)-(d), and only if it keeps the same D, K, P5 composition and A = 200. Its R437 moments (step (a)) are not checked here.
• Proof-shape ceiling. This compares primitive against primitive under the modern outer. Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979, Combine T37, T8 and T16: conditional κ > 0.00173009636352 #3633, Improve conditional T530 κ through two commuting release retimings #3642 and Compose retimed T488 suppliers: conditional kappa 0.00173041889976 #3653 state κ = X − 10^-46 − 10^-70.
  • On the selected bit program, the c(1−c) shape has supremum b(1−b) = 0.0017304222702882630233…, which is not attained.
  • The reciprocal a = 576/332929 compared with each X: PR2979 +1.0646e-6, PR3633 +2.246e-9, PR3642 −2.958e-7, PR3653 −3.203e-7.
  • b(1−b) exceeds PR3653's X by 3.371e-9.
  • Pending PR3690 reports a = 575/331776, which exceeds every X here by at least +2.678e-6. So this ceiling matters only if R437 is not admitted.
  • This is a statement about the proof shape, not a barrier or an endpoint. It avoids the mixed-convention comparison that Independent verification (no κ): PR3647's composed R437 bit word, hosting and price pass an exact audit against single-change controls (c = 0.001736628253427115) #3700 withdrew.

Base and pins. The base is 20ecf2891d80b5acf18d980b2164fb44ea7faa90. The package adds 6 new files under research/pr3562-ordinary-coupling-reproof-audit/. It does not touch upstream/, certificates/ or any frozen packages.

• SOURCES.json pins the selected bit supplier (research/e8-bit-p111-tuned/tune/EXPECTED.json, "nine" profile), the Selected.lean moment literal, ASSEMBLY.json, certificates/selected-result.json and PR2979's archived comparator.
• PR3562's PROOF.md is not vendored. It is checked optionally by sha256 3e554832c38b4b801f55b20868679f3283bc90cd5a9d8ce1960f828ed2655f22 together with its audited literals.

Reproduce (stdlib, exact Fraction arithmetic; about 5 s and under 40 MB RSS):

python3 -I -B research/pr3562-ordinary-coupling-reproof-audit/verify.py
# optional: --pr3562-proof <git show abb93be8...:research/retuned-ordinary-coupling/PROOF.md>

Expected last line:

PASS_PR3562_REPROOF_AUDIT: audited arithmetic steps (a)-(e) hold from P1-P5; line-156 bounds alone give factor 2, n <= e/w from line 117 gives 2^c; coefficient 4283584 justified

Certificates.

• Two exact moment engines, each implemented independently using the methods PR3562 states (lines 64-72):
  • E1: integer-root bisection at Q = 2^128;
  • E2: rational interval atanh/exp series with outward dyadic rounding.
• Results:
  • μ_1 = 2874298095721977/2890625000000000;
  • μ_(576/577) = 0.999998938581682162866772413602788…, with E1 width 9.358e-38 and E2 width 1.695e-77, both inside PR3562's stated 10^-30 enclosure;
  • 10^6(1−μ) = 1.0614.
• Source binding: H, W and εF/W match the selected bit supplier on main. The selected bit_saving 0.001733427039589844234749 is the fallback-inclusive root truncated to 24 digits (E2). E1 brackets the root in (10/5769, 9/5192).
• Steps (b)-(d) are proved in writing. Lemma B is parametric in (c, histogram). The selector fit has an exhaustive exact check of D(e) ≤ 2 log2 e on [2, 2^20]. At e = 2^20 the boundary values are D = 22, k ≤ 180246, n ≥ 434165.

Negative controls. Seven controls run through the same audit() function, and all are rejected. The code was frozen before the single sweep (verify.py sha256 e91884740bd07b208d6b735f5c46b6b024f3e857be9ba1f0a239eadbcbca7e15, recorded in VERIFY.json).

control	rejected at
c = 9/5192 (just above b)	(a)
c = 10/5769 with M = 10^6	(a)/(b)
H_7 + 1	source binding
fallback term dropped	(a), outside PR3562's enclosure only
reciprocal j = 576	(e)
reciprocal j = 578	(e)
factor 2 instead of 2^c	(d), 8033584 ≠ 4283584

The gap classification rests on the written argument in PROOF.md. The verifier checks the exponent arithmetic only.

Scope, assumptions, unresolved obligations.

• P5's physical inputs are assumed, not audited: exactly two reversed C0 calls, padded volume below 2V per call, cost proportional to padded volume at the same n, and a complete wrapper bill. So are the retained PR2753/PR2809/PR2899/PR2808 inventory, history and wrapper arguments.
• Not audited:
  • PR3562 lines 84-104 (history, preparation, reserve);
  • lines 172-174 (binary transfer);
  • lines 20-25 (space bound, restoration, ownership).
• I read the elementary branch as costing at most C·V·e (lines 142 and 165-166).
• PR3562's code, EVIDENCE.md and factory were deliberately not read, to keep the reproof independent.
• PR3576's T74 step and PR3690's R437 moments and source binding are not re-audited.
• E1 confirms the root only to about 1e-8. The 1e-24 root binding rests on E2.
• Open next questions are in the README and TASK.json:
  • a one-line derivation fix in PR3562 (citing line 117);
  • an audit of P5's physical inputs;
  • a decoupled width exponent γ < c;
  • an independent recheck of PR3690's R437 moments.

Search record. No search was run. The only finite enumeration is the exhaustive D(e) check over all integers e in [2, 2^20]. The E1 bracket fractions are the best rational approximations to b with denominator ≤ 6000.

Credit, concurrent work and what this adds.

• Object: Prove and bind the stronger ordinary rate at c=1/577 #3562 by Jose Manuel (@jmbarrancoml). It builds on Complete all-size logical BIT caller and recursive storage bound #2753 (David Arias, @DavidAriasC), Implement fixed-role recursive control for the selected BIT caller #2809 and Complete the physical ordinary-interchange wrapper #2899 (@jmbarrancoml) and Implement the logical BIT outer borrowing wrapper #2808 (Nick Petalas, @petalas).
• Consumers: Bind the retuned ordinary rate to the complete T74 layer #3576 (Hiroki KUMAZAKI, @kumagi) and Research: source-bind R437 ordinary coupling at c=1/576 (no κ claim) #3690 (@zxv).
• Endpoints cited: Conditional κ ≈ 0.0017290339826465: joint rank-7 end frames, five paired extractions and #2957's rewrites on #2942's words (+5.00e-6 over #2969) #2979 (@mdumitrean, merged archive), Combine T37, T8 and T16: conditional κ > 0.00173009636352 #3633 (Pierre-Loic Doulcet, @hexapode), Improve conditional T530 κ through two commuting release retimings #3642 (@roger-sato), Compose retimed T488 suppliers: conditional kappa 0.00173041889976 #3653 (@rohanarun).
• Convention note: Independent verification (no κ): PR3647's composed R437 bit word, hosting and price pass an exact audit against single-change controls (c = 0.001736628253427115) #3700 (this account's sibling loop).
• What this adds: at submission, PR3562 had no comments or reviews, and Selected BIT first-node caller and recursive-return handoff #2319 and Handoffs and requests for independent checks #1166 contained no other audit. This is the first independent check of the arithmetic layer that PR3576 and PR3690 inherit.

Claim. #2319 (reply in thread)

Review provenance. The work was reviewed by seven independent AI review passes in the same loop, over two rounds. Round 1 covered reproducibility, adversarial mathematics, provenance and referee; round 2 covered reproducibility, adversarial mathematics and referee. No human review is claimed.

Attribution. Sagar Thakrar (@thakrarsagar), with substantial AI assistance from Claude Opus 5.5 via Claude Code in an automated research loop. Apache-2.0.

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