[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. Separate monotone letter-price minima need not compose into one legal UT3(F2) block-call word, even with arbitrary restored dirty workspace.

three central targets; scalar rank cap 3
complete prices: a(d)=5+d^2, b(d)=10+d
separate rank-partition minima: 27+26 = 53
exact full chronological minimum:       54

Equality in the separate bound forces three rank-two a calls and two rank-three b calls. The interval between the two b calls and its complement each require a-rank at least three, which excludes that rank list. Four rank-three calls attain 54. This settles the nonproportional-price question left open by #4846 with a sharp counterexample.

Only research/ut3-asymmetric-monotone-gap/ is added. Produced by Nick Petalas. Credit Arham Tahir #1919 for the block alphabet and rank-volume lemma, Nick Petalas #2129/#4846 for the concave/common-monotone predecessors and open question, and steph4n-gh #1573, Arham Tahir #1589, Hanson Sun #1607, Josie Elliston #1792 and the retained model lineage. Predecessor files and credit lines are unchanged. No predecessor executable was replayed.

Advance claim. Scientific base: canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8.

Evidence

Before, two independent partition optimizers leave simultaneous attainability unresolved. After, the chronological cut excludes their 53-cost equality case and a complete formal endpoint attains the sharp 54-cost bill.

Local CPython 3.12.15 checks pass:

• Registered warning-strict normal and optimized replay, including all 3,840 two-role cut identities, seven rank partitions, ten equality-case letter orders, 36 full endpoint fixtures and 1,296 separately implemented faithful regular-action basis columns.
• Thirty-two deleted-call and three semantic controls reject. A read-only Git-free optimized replay passes; wrong receipt, changed source and missing source copies fail.
• Each registered command was timed directly, at 0.210 seconds normal and 0.208 seconds optimized, below its 120-second limit. These are local check durations, not a performance improvement claim.
• Metadata/source bindings, exact committed bytes/source-history preflight, AST/function-length inspection and whitespace checks pass. The generated receipt binds all five source/metadata files. Publication head: c33d673cb7e913f0e013c55111558297473d0ce8.

From the repository root:

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

Expected markers: PASS_UT3_ASYMMETRIC_MONOTONE_GAP separate=53 exact=54 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The written proof covers the stated formal alphabet for every finite register count. Finite checks corroborate its equations and attaining endpoint. Native implementation, scalar/frame/hosting/routing/precision/tape costs, recursive-child rank interpretation, compiler admission and analytic transfer remain outside scope. No new kappa, physical supplier or complete multiplication theorem is claimed. No Lean, Docker or selected scientific cold replay was run. Independent human and formal proof-kernel review are unclaimed. All registered local checks ran.

Open next questions

• Characterize compatibility of separate monotone rank minima. Start with exactly two b calls and rank partitions that satisfy the inside/outside cut. The cut alone need not suffice for arbitrary noncommuting words.
• Specify and price one physical UT3 producer/consumer interface, including scalar mixing, frames, preparation, restoration, hosting and precision. Physical realization and recursive rank identification are missing.

Related open work

• Research: exact four-call minimum for an integer-shear target #3905 studies an integer-shear target in a different address group; no shared numerical quantity is compared. The live paginated open-PR corpus contains no other identified open PR on this asymmetric UT3(F2) block-price question. The directly relevant Research: sharp invocation tariff for rank-bounded UT3 block calls #1919/Prove sharp concave tariffs for rank-capped UT3 block calls #2129/Prove exact monotone UT3 rank tariffs (no new κ) #4846 predecessors are merged.
• Cut rank-two clean compilation from 16 to 10 CNOTs (no κ) #5857, opened since the CLAIM, studies scalar conjugator sharing in a clean fixed-involution compiler. Its paid operations and endpoint differ from this dirty central UT3 block-price model; no common minimum is compared.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-27 09:05:23 UTC were refreshed: #5854 through #5862. No other PR in that snapshot studies this asymmetric UT3(F2) pricing question or computes its 53/54 fixture. No new numerical disagreement was found in that scope.

Entry overlap covered 405 open PRs and 5,286 Discussion records, plus distinctive main-branch README and merged-PR searches. Sparse local notes, arbitrary diffs and unpublished work limit coverage. The live open lists and handoff claims were rechecked before committing; no overlapping claim was found.

CI notes

Hosted run 40479202777 finished on publication commit c33d673cb7e913f0e013c55111558297473d0ce8: plan failed and replay was skipped. Hosted replay has not certified this package; all package verification reported above ran locally.

The sole failed step is Onboarding helper tests, running python -B -m unittest discover -s tests -p 'test_research_*.py' -v. All ten failure/error entries are in unchanged tests/test_research_current_assignment.py: FAILED (failures=8, errors=2, skipped=22) after 511 tests. The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failures concern frozen-card counts, missing historical/current helper rejection and sparse first-step metadata.

The same ten failure/error headers and aggregate result were checked in PR5854's run and PR5855's run, also based on main de5ec2e7a6074068452f45b1a191b5c72832df4a. This PR changes only its six package files. Maintainers will address the shared onboarding failure; no shared tests or other repository files were changed. No replacement hosted replay or scientific certification is claimed.

Merge danger

Door: two-way. Blast radius: package.

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