Proof repair: bootstrap cutoffs need only a stock constant (repair (b) of #1956; κ unchanged)

[JosieElliston]

Proof repair for one finite lemma: under a restated cutoff lemma, the selected bootstrap cutoffs reduce to an uncomputed space obligation with a large exact margin. κ, the certificate, the suppliers and the frontier are unchanged.

Follow-up to #1956 (open question 1, "repair (b)"). That PR showed the 14 recorded cutoff_log2 values of the selected #1094 ASSEMBLY.json fail the cutoff lemma as written (PR387 PROOF; five-stage-source527-banks/proof/FINITE_BIT_ACCOUNTING.md) from stage 2 on, because that lemma's C must dominate the carried time constant C_old·2^(L·Δa). This PR proves a restated lemma under hypotheses (H1)–(H3), which are read from the retained sources but not re-proved. Under that lemma the same numbers are valid exactly when each stock coefficient satisfies σ_j ≤ 2^(H_j−3), where H_j is that stage's headroom. This is plausible by a wide margin but is not verified here.

Result

• The time bound needs no cutoff. Each lower-order toll is at most M(δ)·e^(1−a_j) for all e ≥ 1, where M(δ) = max(2, 2/(ℯ·ln2·δ)) ≥ sup_t (1+t)2^(−δt). Hence T_j(e) ≤ C_j·V·e^(1−a_j) with C_j = max(K(4+M), C_(j−1)·2^(L·Δa)). Neither the previous time constant nor the new one enters L.
• The cutoff only makes the borrowed stock fit, k < e/2 (three-stage-cover-rows.tex). With C = 8σ, where σ is the stock coefficient in log_q S(e,w) ≤ σw(1+log2 e), the original 36/δ² and 2(4+ℓ)/δ inequality chain gives k ≤ e/64 + 1. The induction stays sequential: σ → L → C_j.
• Selected receipt, exact: all 14 recorded cutoffs satisfy the fit-only rule with log2(8σ) ≤ 512. The exact headroom H_j (the largest log2(8σ) each cutoff supports) is 6,562,783 bits at stage 1 and about 3·10^23 bits at stage 7. The carried factor alone (log2 C_old ≥ L·Δa) exceeds every stage-2..7 headroom, which reproduces Audit: selected bootstrap cutoffs omit the carried constant (κ unchanged) #1956's finding from the other side. An illustrative time-constant chain, under the extra modeling assumptions that fresh constants are below 2^512 and K_j ≤ 2^512(1+C_(j−1)), gives a 27-digit log2 C_7.

Evidence

python3 -B research/bootstrap-cutoff-time-space-split/verify.py prints PASS_BOOTSTRAP_TIME_SPACE_SPLIT recorded=14 fit_only_valid=14 written_rule_fails=12 time_constant_finite=2 controls_rejected=7. It uses the standard library and exact Fractions, except for a floating-point sanity check of M, and runs in under 0.1 s. It passes on CPython 3.10, 3.11, 3.12, 3.13 and 3.14. research_package.py --check-only and the registered 3.12 replay pass, and the 42 research tests pass. The verifier source-pins the receipt, both lemma texts, three-stage-cover-rows.tex and the assembler formula in upstream1087.tar.gz. It certifies ln2 > 693/1000 and ℯ > 2718/1000 from series with tail bounds and checks the 3√t step. The seven rejected controls are: cutoff −1; stock coefficient one bit over the headroom; M/4; the false bound ln2 > 0.6932; the false bound ℯ > 2.719; the carried time constant in the fit rule; and a tampered gap. A floating-point check at the maximizer is a sanity check only.

Not discharged

• (H3): the stock coefficient σ_j contains no time constant and satisfies log2(8σ_j) ≤ headroom_j. This rests on the retained complete-row contract's logarithmic old-leaf reserve, which has the same bound shape at every stage, so the below-cutoff branch does not multiply it by 2^(LΔa). This PR does not compute σ_j.
• (H2): the retained stopped-leaf estimate holds whenever k < e/2, and its O-constant is independent of the cutoff. The sources state the estimate only asymptotically. Any further "sufficiently large e" condition in the inherited PR185 lineage is another fit condition, with its own coefficient to check against the headroom.
• Other lineages that use the same formula (T37, Conditional κ 0.001676855229071321886206381217337893: verified outer witness refinement, unchanged construction #1424, T74 outer210: verified conditional arithmetic refinement (superseded by #1578) #1586, Conditional κ = 0.0016769381110224387711...: stacked pivot 566 + 554 kernels with 600-digit grid and 600-digit endpoints over PR1692 #1724): the argument applies, but their headrooms are not computed. All other inherited obligations of the selected record are also untouched.

Scope and overlap

Chooser nonce bff97872294a70402bf5bd4940f9436f drew complex-mechanisms. I overrode it, by donor/coordinator preference, to #1956's open question. Card: uniform-transfer-audit, refined. One Codex read-only review was run. Its two must-fixes (narrow the receipt claim, and σ ≤ 2^(H−3) rather than 2^H) and its should-fixes (qualify H2, label the time chain's assumptions) are applied. Claim: #1166 (reply in thread). The overlap helper found nothing beyond #1956. This PR and #1956 are complementary repairs, (b) and (a); maintainers can pick the wording. Base 0b1cccf8.

Credit: rohanarun (PR387 PROOF), eumemic (five-stage bank proof), the retained three-stage-cover-rows.tex, Jeremy Kothe's PR1087 assembler and its lineage, Rohan Arun's PR1094 receipt, and #1956. Prepared by Josie Elliston with Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at c9819a94f9c630f8b42ff5c3027965c4ba451653.

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

Reviewed scope: Conditional time/space separation absorbs finite time coefficients into the induction constant and makes the cutoff a stock-fit obligation. All fourteen sufficient headrooms reproduce. H1-H3 and actual stock coefficients remain assumptions; its phrase valid exactly when refers only to this sufficient guard, as clarified independently by #1990, not a necessary fit condition.

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.