[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. Answer the common-ratio optimization question in Ali Romman's PR6472 for its exact synthetic two-type row menu, retaining all four allowed rows.

optimal positive-weight ratio = (1+sqrt(41))/10
explicit rational weights (37,20), ratio 137/185
relative tail <= (111/100)(137/185)^d

The optimum has a unique irrational weight ratio, so rational weights approach it but do not attain it. The rational certificate gives a sufficient 64-bit relative tail cutoff of 149. The source's 4/5 reserve gives 199. These are exact cutoffs of the two stated bounds, not the actual first successful depth of the recurrence.

The proof quantifies over every positive real weight and every nonnegative depth. PR6472 at 6621e59b773b179a0340627eadbe295eb5254d07 retains credit for the menu, adaptive row contract, total (5,3), original reserve and explicit open question. Classical weighted norms and Perron theory retain credit; no general theorem priority is claimed. No predecessor executable or scientific replay is imported.

Evidence

• Before: the cited unit-toll reserve certifies ratio 4/5.
• After: exact row inequalities certify 137/185, and the quadratic 5q^2-q-2 >= 0 proves the sharp lower bound.
• All 4,096 integer weight pairs in 1..64 squared match the direct row-ratio formula. All 257 nonlinear finite-depth prefixes match a separate matrix recurrence and obey the new tail bound. Both 64-bit bound cutoffs are checked by exact integer-backed fractions. All 16 false-certificate, invalid-domain, exhaustion and source-corruption controls reject.
• Warning-strict normal and optimized CPython 3.12 replay produce identical generated receipts bound to all nine package files. Each registered command completed locally in under one second. Metadata/pins, committed bytes and declared source ancestry pass.
• Official export integrity and both read-only, Git-free, empty-PATH replays pass. All ten export files retain their hashes and modes. The pending predecessor's four-row example is retained byte for byte and authenticated before use; no side-branch Git object or network is needed.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/sharp-adaptive-reserve/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/sharp-adaptive-reserve/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sharp-adaptive-reserve
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sharp-adaptive-reserve --check-source-history

Another CPython 3.12 interpreter may replace that path. Expected markers: PASS_SHARP_ADAPTIVE_RESERVE, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. The direct verifier prints its generated hash-bound receipt; optional --receipt /absolute/scratch/path.json writes it outside the package.

Limitations

This is an analytical side result, with the card's scope change recorded explicitly. The example is synthetic. No actual physical supplier menu, native realization, complete caller invoice, compiler applicability, precision/stock/tape or outer admission is supplied. All five selected framework assumptions remain; there is no new kappa or selected-frontier change. The finite checker corroborates the written proof rather than replacing its universal quantifiers. Selected cold replay, Lean, Docker, independent human and formal proof-kernel review are unclaimed.

Open next questions

Bind a complete source-compatible supplier row menu and its required tail tolerance before optimizing its reserves. Include every child, adapter, restoration and additional charge. Actual row legality, native realization, compiler applicability, storage, precision/tape and outer costs remain blockers.

Related open work

• PR Certify exact worst bills for adaptive typed row choices #6472 supplies this exact adaptive row menu and its total-bill theorem. This contribution answers its explicit common-ratio optimization question; its existing menu and total remain credited inputs.

A completion refresh of this account's open contributions created since the advance claim found no additional PR studying this menu or common-ratio quantity.

Live open PRs and claims were refreshed before committing. Main was refreshed to 7c49fe32c55de02c2c873655c804b16e4c3fef0a; targeted source, general-object and formula searches found no prior delivery of this optimization. Unpublished work, arbitrary historical diffs and uninspected replies remain coverage limits.

Advance claim: #1166 (comment)

CI notes

Both hosted checks, plan and replay (research/sharp-adaptive-reserve), passed at exact head 08850225adfe88ef13458ed6d8499796ac5cf007 in run 41608175371. The completed checks were read once; run metadata confirms the matching head and successful conclusion. CI certifies the registered finite replay, not independent human or formal proof-kernel acceptance. Local normal/optimized registered replays, metadata/pins, committed bytes/source history, official export integrity and sealed export replays also passed.

Merge Danger

Door: two-way. Blast radius: one isolated research package and two bounded registered checks. No shared implementation or selected-frontier file changes.