[petalas]

Summary

Scoped written proof with exact finite corroboration: signed simple-root calls over F2[UT_d(Z)] have separate positive and negative count floors for every diagonal target, arbitrary finite restored dirty workspace and free binary scalar mixing. For targets admitting sign-consistent words, coordinate calls attain all floors simultaneously, so arbitrary positive sign-dependent prices have an exact optimum.

P_i = sum_j max((h_j)_(i,i+1), 0)
M_i = sum_j max(-(h_j)_(i,i+1), 0)
optimal price = sum_i (alpha_i+ P_i + alpha_i- M_i)

In particular, h=g1^m g2^-n and the two-output endpoint diag([h],[h^-1],1,...) require exactly 2(m+n) calls, even though their aggregate determinant characters all cancel. This closes the sign-consistent target family to call sharing; it does not change kappa.

Produced by Nick Petalas. Cinna Davis #2103/#3316 supplies the UT5 target questions and alphabet; Adam Gall #1911 and ywz65 #2716 supply the all-minor Laurent-degree method. The scalar-idempotent lineage retains eumemic #288 and Ali Romman #468. The new contribution is the integer-unitriangular sign-separated projection, uniform target family and exact heterogeneous price. No predecessor executable was replayed or audited. Exact source provenance is retained in SOURCES.json.

Evidence

Before, the cited UT5 packages left other diagonal targets and a uniform family tariff open. After, the written proof gives the formula for every dimension and every finite restored dirty workspace. The executable corroborates bounded cases rather than extrapolating them to the theorem.

Local CPython 3.12.15 checks pass:

• 155 projectors and 310 signed gates; 13,824 complete mixed three-call words with 55,296 sign-degree checks; 432 free-mixing cases.
• 281 target fixtures, 648 single-call omissions, 1,296 literal arbitrary-dirty basis inputs and 320 dirty samples, plus scope, order and invalid-input controls.
• Every registered normal and optimized command passes. Direct durations were 1.34 and 1.31 seconds, each below its 120-second timeout. Both source-bound JSON receipts agree exactly.
• Three optimized implementation mutants and two receipt-output boundary controls reject.
• Metadata/pins, committed bytes/source history, whitespace and AST/function bounds pass. The official five-file committed export passes integrity checking; an optimized Git-free replay with empty PATH matches the source-bound receipt exactly.

From the repository root:

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

Any CPython 3.12 path can replace the original absolute interpreter path. Expected markers are PASS_UT_SIGNED_MONOTONE_TARIFF, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

Sharpness requires a supplied sign-consistent word for each target. The lower bounds alone hold more broadly but erase genuine central commutators; they do not solve arbitrary-target costs. The alphabet consists only of signed simple-root rank-one idempotent calls, with full formal coefficient equality. Finite quotient actions corroborate the implementation and do not establish the formal lower bound.

Native frames, actual scalar/hosting/precision/tape costs, recursive child prices and complete supplier admission remain unestablished. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, selected cold replay, physical supplier or complete multiplication theorem was run or claimed. There are no locally unrun registered commands. Selected frontier files and frozen sources are unchanged.

Open next questions

• Find a saving, or a stronger obstruction, for one named target with genuine root-sign reversals. Begin with higher-root information and a bounded alphabet; zero character cost does not provide a construction.
• Bind one formal signed simple-root call to a complete native dirty-state/frame contract and charge every conversion and physical traversal before invoking recursive pricing.

Related open work

• Research: exact four-call minimum for an integer-shear target #3905 studies a two-role SL2(Z) integer-shear target; the group and alphabet differ, and no shared numerical quantity is compared.
• Prove exact monotone UT3 rank tariffs (no new κ) #4846 prices arbitrary-rank central UT3(F2) block calls under common monotone rank tariffs. This result concerns signed rank-one simple roots of UT_d(Z) and sign-consistent diagonal outputs; it is not a follow-up to that rank-price problem.
• Solve the exact mixed UT4 coefficient relaxation (no new κ) #4882 solves a mixed UT4(F2) middle-factor coefficient relaxation. This result proves full formal endpoints for a different integer-group target family and does not compute that relaxation quantity.
• Prove optimal square-zero coupling compilation: four calls at rank three #4912 compiles square-zero couplings for an involutory address label. This result keeps idempotent couplings and noninvolutory integer simple roots, with no coupling-compilation claim or shared numerical comparison.
• Prove exact orbit weights for ordered idempotents (no new κ) #4915 counts simultaneous centralizers and orbits of ordered idempotents. This package bounds signed address-label counts; it computes no overlapping centralizer or orbit population.

The completion refresh listed this account's nine other open PRs created since the CLAIM at 2026-10-22T23:39:03Z: #4909, #4910, #4911, #4912, #4914, #4915, #4916, #4917 and #4919. Their scopes were checked. The adjacent coupling and idempotent results are distinguished above; none reports a competing integer-unitriangular signed-monotone tariff. No extra computations were run to audit those results.

Live open PRs, current handoffs, merged searches and main READMEs were refreshed before committing. Sparse local notes and arbitrary unsearched diffs limit novelty coverage. The source base remains canonical main 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Advance claim.

CI notes

Both hosted checks passed at exact contribution head 9f4bf9b3dc7b93d5c5c57c8c4a2d2f2c6b030b0a in run 39874045123: planning in 3m39s and registered package replay in 1m14s. Run metadata confirms the matching head, completed status and success. The checks were read once at completion. The replay executes both warning-strict normal and optimized commands.

All required local checks, source-history preflight, official export integrity and Git-free replay also passed. Hosted CI certifies execution of the bounded checks, not independent mathematical acceptance of the universal theorem. No locally unrun registered command or CI failure is claimed.

Merge danger

Door: two-way. Blast radius: package.

Only five new isolated research files and their opt-in replay manifest. No shared implementation, predecessor credit, pinned input or selected frontier is changed.