Prove sharp borrowing-fit cutoff for PR2234's prescribed stock

[aakashthakkar]

Change and mathematical scope

PR2234 leaves its borrowing condition ceil(log_q S(e)) < e/2 at “eventually.” This proof establishes 311,311 as the least uniform integer cutoff for its prescribed reserve S(e)=K(ceil(e^(17/10000)))^D(e), over every prime q>2^80. A certified prime makes borrowing exactly half the width at 311,310, so sharpness concerns the actual stock, not just a sufficient guard.

The exact depth staircase covers the finite boundary, and a dyadic tail bound covers every later atom-width ceiling jump. A modular prime certificate and integer power comparisons prove sharpness without floating point or a probable-prime assumption.

This is a scoped stock-fit theorem. It does not certify the complete GL64 payload, physical time/locality/restoration contracts, a practical runtime threshold, or adoption of PR2234. It does not optimize the reserve itself or claim a new multiplication exponent. The selected frontier and frozen predecessors remain unchanged.

Task and overlap

Revision-4 card selected-stock-envelope, explicitly changed from the retained selected-stage reserve to the actual reserve of the proposed replacement caller. Live overlap already covers conditional coefficients; the remaining unnamed fit threshold supplies a distinct, bounded proof obligation.

Base and refreshed canonical main: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2. Exact predecessor: Jose Manuel's PR2234 at d047ef86fbc5b1614d3d8e2b8d3441413beab73f, proof sections 1, 6 and 8. Contribution commit: 0ec8a26572069b6109decadd488b9770944a4296. Unchanged predecessor proof and notice are authenticated by Git blob and SHA256; no predecessor code executes.

Prior #2084/#2120/#2105 address conditional coefficients/depth, #2097 fresh fields, #2289/#2295 recurrence/moment, and #2310/Discussion2319 control/dispatch. They are credited rather than repeated. This result does not identify the replacement reserve with the fourteen retained selected-stage coefficients.

Final overlap covered 539 open PRs, 1,036 Discussion/comment/reply records and 1,538 local records, with complete pagination and no warnings. The targeted cutoff/borrowing-threshold search matched only this package and its own claim; broad card/predecessor searches were also inspected. Text coverage excludes arbitrary code diffs and does not certify novelty. Canonical coordination claim.

Validation

macOS arm64 / CPython 3.12.13:

python3.12 -Werror -B scripts/research_package.py --package research/bit2234-stock-threshold
python3.12 -B -m unittest discover -s tests -p 'test_research_*.py'
python3 -B scripts/research_baseline.py

The exact certificate and eight package tests pass normally and with -O. Independent checks include all child ranks 1–42, a depth oracle through 2^21, distant exact atom-width boundaries, and normal/optimized public-CLI source/receipt corruption controls, including JSON type drift. The 65 workflow tests report OK with three skips; the selected source-binding baseline passes. These are local results; hosted CI is separate.

The universal claim rests on the written proof, not finite-search extrapolation. No complete selected cold replay, full physical caller execution, formal-kernel proof or independent human review was performed.

Credit

Prepared for Aakash (@aakashthakkar) with substantial OpenAI Codex assistance, including separate mathematical and implementation review. Jose Manuel supplied PR2234's construction, reserve and depth recurrence; its retained notice preserves the selected PR589/PR555, PR1627, GL, source527 and routing/parking lineage. The package also credits the stock/cutoff predecessors. New work is Apache-2.0.

Open next questions

Structured next questions and proof.

A smaller reserve based on the post-borrow body could fit earlier, but first needs a new complete-row divisibility proof. Separately, joined caller/primitive implementation and review must establish fresh fields, restoration and uniform time; start with PR2234 sections 5–7 and the PR2310/Discussion2319 work. Neither follows from stock fit alone.