Independent verification (no κ): PR3647's composed R437 bit word, hosting and price pass an exact audit against single-change controls (c = 0.001736628253427115)

[thakrarsagar]

Statement

This PR independently checks the composed R437 bit word in PR #3647 (ai-nanachi, head 36ff441479ff9c7e70d83f0becccd44da50effb8). The check is exact and uses only the standard library. It covers the word, its records, the hosted template and the price. At that head every check passes, and it confirms

c = 868314126713557740143/(5·10^23) = 0.001736628253427115480286

The next 10^-24 grid point fails. This is a finite check of one local bit supplier under the selected #589 chain's inherited semantics. It makes no κ claim, does not change the frontier and does not admit PR #3647. The selected record stays 0.001672660197076918.

Package: research/r437-composed-bit-independent-audit/.

Milestone and the decision it informs

The milestone is an audit: independent verification of a specific new implication, from the independent-moment card applied to the bit-mechanisms track. PR #3647's c is the highest priced open bit-supplier value we found, and this audit serves the intake of #3647 and of the PRs that build on it (#3668, #3690).

It has no near-term κ consequence. The complex side binds today, and it would still bind if the pending complex chain PR #3653 (rohanarun) / PR #3682 (zxv) were admitted. Under either consistent outer convention the bit side keeps about 3.0e-6 of headroom: 2.998e-6 under the retained transfer κ = a/(1+a), and 3.008e-6 under the modern outer that #3653 uses. With PR #3647's c the headroom becomes 6.188e-6 and 6.209e-6 respectively. The README has the exact table.

Correction to this loop's #1166 claim. The claim said the bit side would bind once PR3653/PR3682 were admitted. That rested on an 8.57e-9 gap between #589's retained cap a/(1+a) and #3653's modern-outer κ, which mixes two conventions. The conclusion is withdrawn.

What is checked

The interpreter, controls and price engine were written for this audit. All semantics come from the selected #589 chain on main: xcheck_p.py/pwordl.py, global_k.py, assemble9.py, the idle-axis hosting rule and the moment price equation. No code from PR #3647, #1043 or main is imported or run.

Word identity, by two methods. Over F2 the 659-input word maps (x,y,h) to (x,y+x,h), with arbitrary dirty helpers, legal nested climbs and FULL/t-perp final frames.

• The forward column replay and a reverse pull-back agree entrywise.
• The reverse orientation passes, and so do the finite H1–H5/U/J hypotheses.
• Exact rational projectors give P_A P_B = P_A on all 2,384 climb pairs.

Composition and controls.

• Applying E464 (helper464 GL edit) and E470 (Finite BIT construction: R438 source boundary with full hosted invoice #1043 helper470 deletion) literally to Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589, with D0/C rebuilt, gives the PR's R437 word in every field except stats.
• E464 alone equals the PR's R438 word.
• E470 alone reproduces Finite BIT construction: R438 source boundary with full hosted invoice #1043's FRAME-PATHS.json and its CERTIFICATE.json profile, 24,248 ADDs and gate counts. These are machine-checked.
• The composed trace equals the E464 control's with h470 deleted, at all 5,369 aligned gates. The profile deltas add exactly.

Chronology. All 966 SOURCE-write gates and every SOURCE climb match #589. x77 is clean at the frame-186 read.

Local profile. {1:1320,2:1344,3:77,4:43,5:102,6:11,7:18}, mass 5,113, 24,208 ADDs.

Records. The PR's 27,132-record stream is byte-identical to an independent lowering. It replays over F2 in both orientations, with norms 49,347/1,113,998. The template's local programs equal it, and all 288 calls' local_hash values bind to those programs.

Hosting.

• The template schedule equals main's nine-stage recipe, and the idle windows are rederived by the selected rule.
• The 3,933 slots per replica are legal and disjoint.
• All 288 calls' owners match the slots and data routes, with 125,856 helper occurrences hosted.

Whole profile and precision. The whole paid profile is rederived (mass 1,808,352, 903,456 paid children). Precision: prefix 177 bits (192 charged), two-level 393 (512 charged), ROW 18,355 below the inherited cap 20,161, and payload prefix 93 bits below 104.

Price. An exact rational bracket gives c as above.

• The template's m = 64 and W = 28,416 are asserted equal to the inherited values.
• Positive controls with the same engine reproduce selected Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's a = 0.001733427039589844234749 from main-only inputs, and Finite BIT construction: R438 source boundary with full hosted invoice #1043's c = 0.001734176288479865766692.

One item to confirm at intake (likely neutral). In selected #589, all nine captured retained totals already carry 42–66 arbitrary helper-dirt columns at the COPY. E464 swaps the deleted h464's column for h340's in h477 and h478, keeping their counts at 46 and 43. A machine-checked census in RESULT.json records this, and nothing else changes. PR #3647 does not mention the swap. If the inherited capture lemma is linear in arbitrary helper contents, as PR #3647 states, the item closes.

Base, pins, reproduce

Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90. That is current main; it was re-fetched before submission and the priority revision is still 6.

• SOURCES.json pins 15 main files by git blob and SHA256.
• Four PR Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647 files are vendored byte-for-byte as pinned snapshots under Apache-2.0, with UPSTREAM-NOTICE credit. The record stream is stored gzip-wrapped.
• PR Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647's head was rechecked at submission and is unchanged.

python3 -I -B research/r437-composed-bit-independent-audit/verify.py

Expected output: PASS_INDEPENDENT_R437_WORD_AUDIT cpu <about 15 s>, exit 0, and a recomputed receipt byte-equal to RESULT.json.

Checks run:

• scripts/research_package.py --check-only: PASS.
• The registered runner under Python 3.12: PASS_REGISTERED_RESEARCH_PACKAGE.
• scripts/check_research_commit.py --check-source-history: PASS (20 files).

Negative controls

Word: 18 mutants, all rejected.

• dropping D0 or C;
• reusing Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's D0 or C;
• a frame-79 late read;
• an early read at the removed birth;
• dropping the replacement read;
• a naive helper464 deletion;
• dropping each of the ten GL gates.

The early-read mutant passes F2 identity and frame legality and is caught only by the chronology comparison. A positive control is accepted.

Hosting: 4 mutants, all rejected. Moving a slot outside its window, overlapping offsets, dropping a slot, and changing an idle rank.

Price: 2 single-child mutants, each failing the expected bound. Moving one child from rank 2 to 3 fails at c; moving one from rank 42 to 34 fails at the next grid point.

Scope, assumptions, unresolved obligations

Not checked:

• Physical layer. The physical register machine, the private-centre reserve (28,416 + 9) and the cover classes.
• Capture contract. Whether it depends on which helpers' dirt reaches a captured total (Open question 1).
• Inherited formulas. The invoice formula, the 10^-16 fallback, the ROW and payload caps and the price equation are used as stated, not re-derived.
• Route factors. max_route_factors is covered only by an envelope below 2^40.
• Inherited assumptions. The all-size compiler, mutable-SOURCE and target-label realization, whole-call hosting, tape and precision, and the analytic H1–H5 proof (only its finite hypotheses are evaluated).

This is a finite certificate for one local supplier instance. It is not a scoped all-size proof, a multiplication theorem or an admission.

The README gives four open next questions:

1. capture-contract dirt-pattern independence;
2. a read-only census of E464-type carriers (an expansion gate);
3. the admission map for Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647;
4. an observed legal window-6 slot merge that would raise Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647's root by about 1.78e-8. This is an observation for the author, not an audit claim.

Search record

No search was run. One pinned object was checked against two literal single-change controls and fixed mutant classes. No new deletion or carrier search was made.

Credit and concurrent work

• PR Improve conditional BIT c to 0.001736646042 with composed R437 and credited slot coalescing #3647 (ai-nanachi): the composition, records, template and claimed c, all audited here.
• Finite BIT construction: R438 source boundary with full hosted invoice #1043 (Ali Romman): helper470. Its replays PR Independent verification: helper470 native replay and full-frame closure cost (no new kappa) #1302, PR Complete native and hosted replay of the R438 bit improvement #2971 and PR Research: certify exact pending projectors for PR #2423's R438 rehosting #2653 are cited, not redone.
• Selected Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 chain (eumemic, DreamingOfClouds and the chain credited in its NOTICEs): every semantic rule used here.
• PR Add the published R437 histogram and compare its coefficient with the selected kappa #3668 (jwr404) checks the histogram-to-c arithmetic on the published histogram. This package rederives the histogram.
• PR Research: source-bind R437 ordinary coupling at c=1/576 (no κ claim) #3690 (zxv, 03c141ef) was opened concurrently, at 15:26Z during this audit. It reconstructs the 903,456 paid requests and the ordinary coupling at c = 1/576 using predecessor and author code, so it overlaps the paid-profile part. What this PR adds is an import-free reimplementation: identity by two methods, both single-change controls, SOURCE chronology, record lowering, hosting legality under the selected rule, and the c bracket with main-only positive controls, each with mutants.
• Related, not audited: PR Reduce R438 to 429 helpers with three source snapshots #3645 (RossRichardson) and PR Add the published R438 profile and its 132-unit local drop #3672 (jwr404). A hosted, priced R429 may exceed this c.

Claim and status: #1166 (comment)

Review provenance and attribution

Reviewed by seven independent AI review passes in the same loop, in two rounds (four in round 1, three in round 2). Their findings are addressed. Two suggested texts were not adopted, for the reasons given in the README. No human or formal-kernel review is claimed.

Contributor: Sagar Thakrar (@thakrarsagar), with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5 via Claude Code) in an automated research loop. Claude did the design, code, checks and write-up.

🤖 Generated with Claude Code