Audit: selected bootstrap cutoffs omit the carried constant (κ unchanged)

[JosieElliston]

Scoped audit of one finite lemma instantiation in the selected record. κ, the certificate, the suppliers and the selected frontier are unchanged.

Package: research/bootstrap-cutoff-carry/. Base main 0b1cccf8. Card uniform-transfer-audit (chooser nonce 9c18a425e1fae055c7b797453ed1df50, priority revision 3), scope refined. Claim: #1166 (comment)

Question

The selected research/community-frontier1094-audit/ASSEMBLY.json records 14 finite bootstrap cutoffs, seven per branch. Frozen PR1087 code/assemble.py computes them as max(ceil(36/gap^2), ceil(2*(4+512)/gap)), which takes ceil(log2 C) = 512 at every stage.

The lemma they instantiate appears in PR387's PROOF and in five-stage-source527-banks/proof/FINITE_BIT_ACCOUNTING.md. There, C dominates "previous ordinary constants", and the next constant is C_large + C_old*2^(L*(a_(j+1)-a_j)). Do the recorded cutoffs satisfy that rule once the carried factor enters the next stage's C?

Result (exact rationals, stdlib)

• All 14 recorded cutoffs reproduce from the receipt and the formula. Stage 1 is valid in both branches.
• Stages 2–7 fail in both branches (12 of 14), even with the most favorable carry (C_old = 1, no C_large, the recorded smaller previous cutoff). The shortfall ratio is 1.996649 at every failing stage.
• Closed form. The gaps satisfy δ_(i+1) = sδ_i and the steps satisfy Δa = (1−s)δ/s, so the quadratic cutoff ⌈36/δ²⌉ can never pay the carry when s < 1/2. The shortfall tends to 2(1−s). Every lineage that uses this formula with a fixed allowance should therefore fail the same way, including T37 (1024) and the deeper chains in Conditional κ 0.001676855229071321886206381217337893: verified outer witness refinement, unchanged construction #1424, T74 outer210: verified conditional arithmetic refinement (superseded by #1578) #1586 and Conditional κ = 0.0016769381110224387711...: stacked pivot 566 + 554 kernels with 600-digit grid and 600-digit endpoints over PR1692 #1724. Only the selected record is checked here.
• Corrected chain. The exact minimal cutoffs that carry the constant are about 2^(i−1) times the recorded ones; stage 7 has 48 digits instead of 46. This assumes fresh per-stage constants below 2^512, as the receipt does.
• The numerical κ calculation is unaffected. The pinned outer engine takes no cutoff input, and κ reproduces exactly from the chain end. The asymptotic statement needs only some finite cutoff per stage, but it still rests on the bootstrap lemma under one of the repairs below.

This is a receipt/proof-text inconsistency, conditional on the lemma's wording. It does not establish that the bound fails. There are two repairs:

• (a) carry C_old into the cutoff, which gives the corrected chain above;
• (b) restate the lemma so that previous-algorithm tolls are absorbed against the new constant. This is the usual "large induction constant" argument, and we expect it to work. It would leave the recorded numbers valid, but it is not written here.

Evidence

• python3.12 scripts/research_package.py --package research/bootstrap-cutoff-carry prints PASS_REGISTERED_RESEARCH_PACKAGE, and --check-only passes.
• verify.py passes on CPython 3.10–3.14.
• python3 -B -m unittest discover -s tests -p 'test_research_*.py': 42 tests OK.
• Six controls are rejected: a tampered recorded cutoff, the next-grid κ, a shifted starting saving, a corrected cutoff minus one (which shows minimality), the recorded cutoffs under the carry rule, and an outer signature with a cutoff input. As a positive check, dropping the carry from the corrected chain reproduces the receipt's 14 cutoffs exactly.
• scan_receipts.py (supplementary, unregistered) checks stages 3 onward of every in-repo research/*/ASSEMBLY.json that records cutoffs: 12 receipts and 23 branches, including the Conditional κ = 0.001661202643272239: eight B-frame coalescences after SW4 #741–Conditional κ = 0.001672188119205421: improved T16 owner packing on the seven-slot k8 word #1042 frontier audits and T37's 1024-bit receipt. It uses recorded exponents only. All 114 such stages fail the minimal carry.
• One independent read-only review by OpenAI Codex confirmed the lemma reading and the index (L_j(a_(j+1)−a_j)) and the closed form. Its must-fix and should-fix items are applied: the finding is stated as conditional on the lemma's wording, the closed-form scope is narrowed, "least" is qualified, the κ claim is narrowed to the numerical calculation, and a vacuous control was replaced by an in-memory mutation of the outer engine source.

Not checked

• Whether repair (b) holds.
• The actual fresh per-stage constants.
• Open PRs' receipts. The in-repo scan covers stages 3 onward only and is unpinned.
• Every inherited all-size, precision, hosting and analytic obligation of the selected record.

Overlap

complex-t37-validation/check_bootstrap_reserve.py (Tekalign) and frontier1042-invoice-audit-seed81 (Wronskode) check the same cutoffs against a fixed allowance; those checks are correct as stated. bootstrap-depth-audit and t37-bootstrap-depth-seed60 study depth, not cutoffs. A fresh overlap refresh found no work on the carried constant.

Credit

The cutoff lemma and the next-constant rule are from rohanarun's PR387 and eumemic's five-stage bank proof. The chain and the 512 allowance come from the frozen PR1087 assembler (Jeremy Kothe's package and its lineage) and Rohan Arun's PR1094 receipt.

Prepared by Josie Elliston with Claude Code (Claude Opus 5.5) assistance.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at f8003953e1719923f769df46c478aa0fa6d3ed9a.

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

Reviewed scope: Exact reconstruction of both seven-stage bootstrap chains confirms twelve recorded cutoffs fail the carried-time-constant rule as written. Corrected finite cutoff chains pass under the stated fresh-constant allowance; numerical kappa is independent of the cutoff input. This is a scoped proof/receipt mismatch, not a refutation of the asymptotic bound.

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.