Audit selected bootstrap stock fit: conditional factor-16 coefficient extension

research/selected-bootstrap-stock-audit/README.md

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+# Selected bootstrap stock: independent audit and a factor-16 fit extension
+
+This package independently corroborates the fourteen selected bootstrap cutoffs
+and stock headrooms from [Josie Elliston's PR1965](https://github.com/CrocSwap/integer-mult-bounds/pull/1965),
+then proves a small extension of its **conditional stock-fit bound**. Under the
+same stock-envelope hypothesis, every recorded cutoff supports a coefficient
+16 times the coefficient admitted by that proof's sufficient guard. The actual
+selected coefficients remain unknown. There is no new κ, supplier adoption,
+frontier change, or proof of the full multiplication theorem.
+
+The subject is the seven native and seven conservative stages in
+`research/community-frontier1094-audit/ASSEMBLY.json`, bound by
+`certificates/selected-result.json` on main commit
+`4d3248d6d52f8ddfe612e8508b74a36acb67fbe3`. PR1965 is pinned at
+`c9819a94f9c630f8b42ff5c3027965c4ba451653`; its README and result are retained
+byte for byte under `sources/pr1965/`. Its checker is neither imported nor run.
+
+## Exact arithmetic scope
+
+For each branch, reconstruct the selected producer's seven-step chain:
+
+```
+beta = 1/10^24
+s = min(bit_lower, (1-beta)*complex_lower) - 1/10^27
+a_0 = 384599/10^10
+a_j = (1-s)*s + s*a_(j-1)
+delta_j = min(s-a_j, 1-a_j-s, 1-a_j-s*(1-a_(j-1)), 1-s)
+L_j = max(ceil(36/delta_j^2), ceil(1032/delta_j))
+H_j = floor(L_j*delta_j/2) - 4
+```
+
+Use `conservative_complex_lower` for the conservative branch. The formula is
+read from frozen PR1087's `code/assemble.py`, lines 205–211, inside the pinned
+`upstream1087.tar.gz`; source is inspected, not executed. Integer rational
+pairs, Euclidean reduction and cross multiplication reconstruct every stage.
+The tests use a separately implemented `Fraction` oracle. Neither arithmetic
+path calls the retained assembler or PR1965's functions. The inequalities
+`2(H_j+4) <= L_j*delta_j < 2(H_j+5)` certify that H is the **largest integer
+log bound admitted by this guard**. They do not characterize all possible fit
+proofs.
+
+Both branches have `H_1 = 6,562,783`. Their original guard coefficient cap is
+`2^6,562,780`; the extension below admits `2^6,562,784`. At stage seven, H is
+`296674203132883528250615` (native) and `296674207198239164557594`
+(conservative). `RESULT.json` retains all fourteen exact values and symbolic
+coefficient exponents; it never materializes these enormous powers of two.
+
+## Conditional factor-16 fit lemma
+
+Let `0<s<1`, `0<delta<=1-s`, `L>=3`, `L*delta^2>=36`, and
+`H=floor(L*delta/2)-4`. For every width `e>=2^L`, set
+`t=log2(e)`, `w=ceil(e^s)` and `k=ceil(log_q S(e,w))`, where `q>1`.
+Assume, uniformly at those widths,
+
+```
+log_q S(e,w) <= sigma*w*(1+log2 e),   0 < sigma <= 2^(H+1).
+```
+
+Then `k<e/2`.
+
+**Proof.** Put `sigma_0=2^(H-3)`. For `t>=L`, we have
+`H+4<=delta*t/2` and `3*sqrt(t)<=delta*t/2`. The elementary inequality
+`log2(1+t)<=3*sqrt(t)` for `t>=1` therefore gives
+
+```
+8*sigma_0*(1+t)*2^(-delta*t) <= 1/16.
+```
+
+For completeness, `ln(2)>=1/2` follows by integrating `1/x` on `[1,2]`.
+Also `sqrt(t)-ln(1+t)` is positive at one and nondecreasing for `t>=1`,
+because `1+t>=2*sqrt(t)`. Thus the stronger inequality
+`log2(1+t)<=2*sqrt(t)` suffices for the displayed elementary bound.
+Since `w<=2*e^s`, `s+delta<=1`, and `sigma<=16*sigma_0`,
+
+```
+k <= sigma*w*(1+t)+1
+  <= 16*e^(s+delta)/64+1
+  <= e/4+1 < e/2                 (e>=8).
+```
+
+All fourteen selected cutoffs meet the lemma's arithmetic premises. This proof
+changes only a sufficient fit coefficient and leaves the recorded L values
+fixed. It does not measure or derive sigma.
+
+At `sigma=2^(H+1)`, `log2(8*sigma)=H+4` exceeds the original guard's maximum
+H. Nevertheless the displayed proof still establishes fit under the envelope
+hypothesis. Consequently PR1965's phrase "valid exactly when" should be read
+as passing that particular sufficient guard, rather than as a necessary
+condition for `k<e/2`. The restated repair lemma itself remains a sufficient
+statement; this package finds no discrepancy in its fourteen headrooms.
+The factor 16 is a convenient proved extension, not an optimal coefficient.
+
+## Source boundary: the coefficient is still uncomputed
+
+The explicit new contribution in
+`five-stage-source527-banks/proof/FINITE_BIT_ACCOUNTING.md`, lines 90–95,
+is `57,604*w*ceil(log2(max(2,n)))`. That source fixes its own candidate
+`m=120`, `W=23,683` and uses a four-level halving bound. The selected
+complex assembler prices an `m=73` supplier. A numerical increment from the
+former source cannot become a selected-stage coefficient merely by citation.
+The earlier `three-stage-cover-rows.tex` has a different depth statement,
+`47*ceil(log2(max(2,n)))+O(1)`, and leaves the inherited reserve logarithmic.
+These are distinct source statements, not a contradiction to resolve by
+silently replacing a depth factor.
+
+The reviewed PR185 proof, retained in PR387's `bootstrap-sources/`, states
+that completed ordinary leaves need no **external** extra stock: their own
+borrowing uses the chunks passed to them, preserves outer spectators and
+restores the row coordinate. It also says the parent stock is `O(w log e)`
+and explicitly declines to identify the external 252 allowance with the
+leaf's actual internal reserve. Neither that text nor PR387 supplies a
+numerical coefficient for the selected fourteen stages.
+
+In particular, neither `sigma_j=sigma_0+j*57604` nor use of 252 as a selected
+sigma is discharged here. An additive reserve recurrence needs an explicit
+same-domain reserve bound and a selected-producer binding; completed inner
+borrowing is not an external reserve multiplication. A very large arithmetic
+headroom cannot replace that missing source obligation.
+
+## Reproduction and evidence limits
+
+```
+python3 -I -B research/selected-bootstrap-stock-audit/verify.py
+python3 -I -O -B research/selected-bootstrap-stock-audit/test_verify.py
+python3 -B scripts/research_package.py --package research/selected-bootstrap-stock-audit
+```
+
+Use Python 3.12 for registered replay; no dependencies or network are required.
+The checker binds all consumed source bytes, reproduces the fourteen stage
+records and PR1965 headrooms, certifies guard maximality, and checks the
+factor-16 arithmetic premises. Tests cover independent positive cases and
+semantic/source/receipt corruptions, including incorrect acceptance above
+the original guard and substitution of an unproved selected stock claim.
+Assertions are not the acceptance gate. The stored receipt is required;
+explicit regeneration writes a new external file without overwriting one.
+See `VERIFY.json` for the exact registered commands. This is a finite
+arithmetic and conditional proof audit, not a full cold supplier replay.
+The originating task catalogue is pinned as historical provenance by the
+registered metadata check; later priority edits do not alter this arithmetic.
+
+## Open next questions
+
+1. What explicit uniform stock envelope does the selected stopped producer
+   supply at each stage? Start with its per-level divisor and depth constant,
+   then account for outer borrowing and completed inner leaves at their actual
+   atom widths. The missing numerical source binding blocks comparison with
+   either the original or extended caps.
+2. Does every other "sufficiently large width" condition in the inherited
+   producer fit the same recorded cutoffs? PR1965's H2 uniform time estimate
+   and inherited hosting/compiler/precision/tape assumptions remain open.
+   Inventory such conditions before treating the fit lemma as adoption.
+3. Should the retained proof use PR1956's carried-constant cutoffs or PR1965's
+   time/space restatement? Maintainer review must choose the proof wording;
+   this package changes neither source proof nor selected result.
+
+## Credit
+
+Josie Elliston, with Claude Code, supplied PR1956's carried-constant audit and
+PR1965's time/space separation, guard, and fourteen headrooms. This package
+corroborates those headrooms and derives the factor-16 extension from the slack
+in that proof. Rohan Arun supplied PR387 and the selected PR1094 assembly;
+Jeremy Kothe supplied its PR1087 predecessor. The retained PR185/PR384
+reviewed lineage and eumemic's five-stage accounting provide the source
+contracts examined here. Tekalign and Wronskode supplied earlier fixed-allowance
+checks. No predecessor mathematical result is claimed as new.
+
+Prepared for ToikY with substantial OpenAI Codex assistance, including separate
+source-contract and arithmetic review. No independent human review is claimed.
+Apache-2.0, following the repository license; retained predecessor files preserve
+their original attribution.