[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. For d repeated A2 targets st, independent nonnegative nondecreasing rank-price tables and separate finite caps have exact complete formal cost H_s(d)+H_t(d), where

H_u(0) = 0
H_u(m) = min_{1 <= k <= min(b_u,d,m)} [tau_u(k) + H_u(m-k)]

This removes the common-price and concavity restrictions from ToikY's #6821. Its separate-letter rank demands are cited inputs. Each rank list can be reduced to exact demand using monotonicity; independent minimizing s and t partitions then attain the full group-algebra endpoint. The theorem holds over every field with arbitrary rank-capped scalar couplings, free scalar GL mixing and arbitrary finite restored dirty spectators.

The first example has d=7, cap four, tau_s(k)=3+k^2 for k>0 and tau_t(k)=k/(k+1). The s-ranks are (2,2,2,1), costing 25; the t-ranks are (4,3), costing 31/20. The exact bill is 531/20. Applying parts (4,3) to both letters costs 651/20, six more under this declared abstract tariff. This comparison evaluates a legal word; it does not apply the predecessor's concave formula outside its hypotheses.

Only research/a2-st-monotone-tariff/ is added. Advance CLAIM. The delivered scope matches the claim and permits separate caps explicitly.

Credit ToikY #6821 for the formal endpoint and selective-cut application; Adam Gall #1941 and Ross Richardson #1966/#1989 for separate-letter cuts; Benny Zhao / zhaobenny #6817 and David Arias #2738 for the retained framework; and the ywz65, eumemic, Ali Romman and Gareth Davidson register-model lineage. Classical partition dynamic programming and monotone clipping are not claimed as new methods. Nick Petalas's older #4846 is a related monotone-tariff precedent for the distinct UT3 central target. No predecessor scientific program was replayed.

Evidence

Local CPython 3.12.15 on macOS arm64, standard library only, at head fc55af2:

• 1,125 exact partition optima agree with an independent count-vector oracle, including 846 nonconcave cases and zero prices; 17,772 excess-rank truncation cases satisfy the new price bound.
• 212 complete signed formal endpoints over Q/F2/F3/F5 include 64 unequal-cap cases. All 1,180 paid couplings have their asserted ranks; 424 couple an arbitrary dirty role through scalar gauges with both boundary maps included.
• A separate faithful regular-action implementation checks 1,344 full basis columns. All 1,180 paid-call omissions, 212 missing scalar boundary maps and seven invalid optimizer inputs reject.
• Warning-strict normal and optimized direct commands print PASS_A2_ST_MONOTONE_TARIFF and compare the entire freshly generated source-hash receipt. Each took 1.44 seconds in a single local timeout check. The registered runner executed both commands and printed PASS_REGISTERED_RESEARCH_PACKAGE in 3.12 seconds. These are timeout observations, not a performance comparison.
• Metadata, exact source pins, committed bytes, modes and source-history ancestry pass. The official seven-file export passes integrity checks and normal/optimized Git-free direct replays. No local mathematical check is deferred solely to CI.

Reproduce from the repository root; another installation may substitute its own CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/a2-st-monotone-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/a2-st-monotone-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/a2-st-monotone-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/a2-st-monotone-tariff --check-source-history --check-modes

Direct replay writes no files and imports no predecessor code. RESULT.json is generated by --emit-receipt, never handwritten. Universal coverage comes from PROOF.md using the cited rank-demand premise; finite checks are corroboration rather than enumeration of all circuits.

Limitations

This is a formal tariff, not a native implementation, supplier admission, new kappa or multiplication theorem. Scalar coupling rank is not an admitted recursive child rank. Native realization, scalar-map costs, frames, movement, ownership, storage, stock, precision, all-size compilation and the complete supplier/consumer invoice remain unknown. All five selected framework assumptions remain. The selected frontier and frozen sources are unchanged. Decreasing prices, changed targets and additional paid/free address labels are outside the theorem. Independent human and formal proof-kernel review are unclaimed; Lean, Docker, the selected cold replay and predecessor scientific suites were not run.

Open next questions

• Can an actual native rank-capped call support a complete nonconcave or letter-dependent price table that benefits from independent partitions? First specify its arbitrary-input entry/exit boundary and charge every physical and recursive obligation. No such native call or rank conversion is supplied here.
• For one explicitly decreasing price table, can a complete dirty-restored word beat the exact-mass partition relaxation, or can an exact replacement be proved? Monotone truncation no longer applies. No continuation is reserved.

Related open work

• #7179 studies five longest-element A2 targets with rank cap three and a common concave price. This contribution treats repeated st with independent monotone tables and computes no common numerical optimum.
• #3893 studies the mixed three-role S3 endpoint diag([s],[t],[st]) with rank-one unit-priced calls. The endpoint and tariff differ; no common numerical optimum is computed here.

CI notes

CI did not certify this package. At head fc55af2406dd0be798d1a1fa241098f117fafa4a, run 42993110077, plan job 130430507159, failed in Onboarding helper tests. The log reports 760 tests, exactly one failure and 24 skips. The failed test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError that `make frontier7000-verify` was not found in the CI verification document. The package replay job was skipped.

The unchanged main files at e54523517d0f03e58352e647c403a5af3e525a57 disagree: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. The log's checkout records this PR head merged into that exact main base. The same assertion was independently reproduced locally before publication, with one failure among six navigation tests.

The complete failed-job logs were also checked for run 42905094314 and run 42903313790. Each reports the same exact test and missing make frontier7000-verify, 760 tests, one failure and 24 skips. The earlier verified #7176 plan job has the same assertion and counts.

All package evidence above remains local at the unchanged head: both registered commands, both direct modes, generated-receipt comparisons, committed bytes/modes/source history and normal/optimized Git-free exported replays passed. No push, retrigger commit, workflow rerun or shared-file change was made in response to this repository-level failure. Maintainer repair of main remains separate from this contribution.

Merge Danger

Door: two-way. Seven isolated research-package files are added.

Blast Radius: research. Shared code, frozen sources and selected certificates are unchanged.