[petalas]

Summary

Scoped all-size proof with exact finite corroboration. For the central UT3(F2) block-call model, every common nonnegative nondecreasing rank tariff has exact optimum 2 F(2w), where F is the integer-partition recurrence with parts at most min(k,w). The new monotone truncation lemma removes the predecessor's concavity requirement, including arbitrary restored dirty workspace and words spending excess rank mass.

w=5, rank cap=4, complete price tau(d)=5+d^2
inherited balanced six-call word:       98
inherited concentrated six-call word:  102
new optimal ten-call rank-two word:     90

Only research/ut3-monotone-rank-tariff/ is added. Credit Arham Tahir #1919 for the rank-cap model, coefficient bounds and nonlinear-pricing gap; Nick Petalas #2129 for the concave special case and loopless-incidence attainment; steph4n-gh #1573, Arham Tahir #1589, Hanson Sun #1607 and Josie Elliston #1792 for the retained model/rank lineage. The new result is the monotone reduction and its exact optimizer. Predecessor executables were not replayed.

Produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed. Advance claim.

Evidence

Before, concavity determined the concentrated rank list. After, the exact recurrence covers every common monotone table and detects a cheaper longer word for the displayed nonconcave tariff.

Local CPython 3.12.15 checks pass:

• Both registered warning-strict normal and optimized replays: 736 formal constructions, 6,368 faithful regular-action basis columns forward/inverse, 766 monotone integer price tables, 2,010 relaxed rank lists, and 18 rejection controls.
• Every registered command was timed directly: 0.70 seconds normally and 0.69 seconds optimized, below its 120-second limit.
• Metadata/source bindings, committed-byte/source-history preflight and whitespace checks pass on 4a940386c07eab4dd4aebcb6ed65462abeb1bda7.
• The official five-file committed export passes integrity checking. An isolated optimized Git-free replay with empty PATH matches the local receipt exactly. Receipts are generated and bind all package files by SHA256.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/ut3-monotone-rank-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/ut3-monotone-rank-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/ut3-monotone-rank-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/ut3-monotone-rank-tariff --check-source-history

Expected markers: PASS_UT3_MONOTONE_RANK_TARIFF no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The written proof covers the stated formal alphabet; finite checks corroborate it. Both letters use the same complete rank price. No native realization, scalar/frame/hosting/routing/precision/tape bill, recursive-child identification, supplier admission, new kappa or complete multiplication theorem is supplied. The selected frontier is unchanged. No Lean, Docker, selected scientific cold replay or physical implementation was run. There are no locally unrun registered checks.

Open next questions

• Can different nonproportional letter prices attain both independent partition minima? First choose tables with incompatible minimizing rank lists and supply a full alternating-incidence construction or obstruction.
• Can a native complete interface realize the 90-cost boundary more cheaply than the six-call comparator after all physical costs? First price both entire dirty-input boundaries and justify the recursive-rank interpretation.

Related open work

• Prove the UT3(F4) central floor survives nilpotent calls (no new κ) #4849, opened after the CLAIM, studies rank-one nilpotent couplings and variable F4 axis coefficients for the UT3(F4) central triple. This PR studies common monotone rank tariffs for UT3(F2) block calls. The results are independent, and no shared numerical value is compared or recomputed.
• Research: exact four-call minimum for an integer-shear target #3905 concerns an SL2(Z) integer-shear target and a different alphabet, with no shared numerical quantity.

Before DONE, this account's open PRs created since the CLAIM at 21:45:24 UTC on 2026-10-22 were refreshed. No other submission in that snapshot studies this exact monotone block-call tariff or reports a competing value for the 90-cost fixture. Sparse local notes, arbitrary diffs and unpublished work limit overlap coverage. The scientific base is canonical main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7.

CI notes

Both hosted checks passed on publication head 4a940386c07eab4dd4aebcb6ed65462abeb1bda7 in run 39860659616: planning in 3m30s and registered package replay in 1m17s. Run metadata confirms the exact head, completed status and success. The package job executes both warning-strict normal and optimized commands. These automated checks certify the bounded replay; independent human and formal proof-kernel review remain unclaimed.

Merge danger

Door: two-way. Blast radius: package.

Five isolated research files and an opt-in replay manifest. No shared implementation, frozen predecessor, selected source or frontier changes.