[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. Equal-count, equal-rank-mass integer child profiles can have any prescribed rational improvement threshold t in (0,1): the new power bill is smaller exactly for 0<a<t, equal at t, and larger for t<a<1.

P_qs(x) = (x-1)(x-2)(x^q-2^s),  x=2^(1-a),  t=1-s/q

first exchange:  4 rank-1 + 3 rank-8 -> 6 rank-2 + 1 rank-16
paid example:   {1:508,8:381} -> {1:1,2:762,16:127}

The first exchange improves exactly on 0<a<1/2. On 0.003<=a<=0.0031, 127 copies fund one added rank-one child with a uniform raw-bill margin of at least 1646873/2000000000. This is a sufficient allowance, not a minimum batch size. It answers Dugongue's #3868 Open next question 2 about interval-specific stencils. #3868/#3863 retain credit for the all-exponent comparison and paid-overhead framework; their computations are cited rather than replayed.

The independent-moment card is explicitly changed to this original arithmetic side result. No source circuit, supplier admission, selected-frontier change or new kappa is claimed. The proof and exact scope are in research/rational-threshold-rank-exchange/README.md.

Evidence

Before, the cited binomial-stencil family supplied an all-exponent sufficient comparison. The new construction has a proved exact crossing, so it also supplies explicit exchanges that an all-exponent improvement requirement excludes.

Local CPython 3.12.15 checks pass at b9d6e4adc8ec01cb4e16a1d844f832ee86cf22a9:

• All 66 constructed threshold pairs through q=12; 264 rational polynomial evaluations; 1,994 independent direct-cost power comparisons; 52 crossing cases; exact count/mass and algebraic crossing checks.
• Both endpoint integer inequalities, the fully charged overhead profiles and all 12 controls.
• Warning-strict normal and optimized direct checks, about 0.09 seconds each; both registered checks, about 0.43 seconds for the complete runner. Each registered timeout is 120 seconds.
• Exact committed-byte/source-history preflight for all five package files; whitespace check; committed Git archive and optimized Git-free replay with complete receipt/hash equality.
• Selected source-binding baseline passed as orientation only.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/rational-threshold-rank-exchange/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/rational-threshold-rank-exchange/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rational-threshold-rank-exchange
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rational-threshold-rank-exchange --check-source-history

Any CPython 3.12 installation may replace the displayed interpreter. Expected marker: PASS_RATIONAL_THRESHOLD_RANK_EXCHANGE, followed by PASS_REGISTERED_RESEARCH_PACKAGE for the runner. Generated stdout receipts bind all five package files by SHA256. No predecessor executable or receipt is a runtime dependency.

Limitations

The written proof establishes the universal threshold family; the finite checks corroborate only their declared domains. The first exchange requires ambient rank m>16 for strict child contraction. Actual inventory, operand compatibility, legal recursion, native implementation, dirty restoration, frames, hosting, stock, precision, tape and the complete compiler/consumer bill remain unknown. Arithmetic allowance alone does not establish a physical construction or complete multiplication theorem. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific replays and selected operator replay were outside scope and were not run.

Open next questions

Can an actual producer/consumer implement the seven-child exchange with a favorable complete bill? First name four rank-one and three rank-eight operations in one source word, prove the operator replacement and endpoint compatibility, and charge every added operation. No such circuit is supplied here. Optimizing the arithmetic batch size does not resolve that obligation.

Related open work

No open PR found in the refreshed searches studies this same rational-threshold stencil family. Nearby paid-profile work remains separate:

• Prove bounded native corrective ADD chains cannot improve paid profiles #5170 proves a source-specific corrective-ADD screen for concave tariffs; this abstract exchange supplies no corrective ADD realization or counterexample to that screen.
• Price native helper reuse: even a free third H17 loses (no new κ) #5147 prices lost native helper rewards across all power prices; this package assumes no native inventory and does not alter its bill.
• Screen paid nonmonotone owned contacts with shared late native maps (no new κ) #5144 screens actual nonmonotone owned contacts with complete paid native boundaries; this package supplies neither those contacts nor an improving circuit candidate.
• Classify every intact-region minimizer: exact 43-bit cube (no new κ) #5181 classifies every minimizing chart assignment in a fixed intact-region moment model. This package changes abstract child profiles and provides no chart assignment or source-level exchange. No directly comparable numeric quantity is computed.

Live open PRs, recent account contributions, handoff claims, active Discussions, cached body searches, distinctive all-main README searches and merged-PR searches were checked before the claim and commit. The local-note cache was sparse; these checks do not prove novelty or cover unpublished work.

CI notes

Both hosted checks passed for exact head b9d6e4adc8ec01cb4e16a1d844f832ee86cf22a9 in Registered research packages run 39948610997: plan in 4m37s and the focused package replay in 1m9s. The replay runs both registered normal and optimized exact checks. Run metadata confirms that head and success. All local checks above also passed. Automated replay is separate from independent human or formal proof-kernel acceptance.

The completion refresh examined this account's nine open PRs created since the advance CLAIM at 2026-10-23 11:41:13 UTC, including this PR. #5181 is added above as nearby work on the recursive moment. The other seven concern different objects or cost quantities; none supplies a directly comparable computed number for this stencil.

Merge Danger

Door: two-way. Blast radius: isolated research package. Five added files; existing scientific sources, shared code and the selected frontier are unchanged.