[petalas]

Summary

Prove the exact sign-dependent query tariff for every one of the 19,683 signed 3x3 dirty-connector targets, over every nonzero unital coefficient ring. This answers #4241 open question 1 and combines its canonical envelope with #944's support-attaining class. Claim type: scoped proof with finite corroboration. No new κ.

S(M) = active source/output operators
sigma_F = p_F^+ + p_F^-; mu_F = min(p_F^+, p_F^-)
c = 0:  b_F = sigma_F,             u_F = 2 sigma_F
c >= 2: b_F = min(sigma_F,c mu_F),  u_F = min(2 sigma_F,c mu_F)
minimum(M) = sum_S b_F + delta(M) min_S(u_F-b_F)
delta(M) = 1 on the 288-member signed/permuted obstruction orbit
minimum(0) = 0

The new proof transports all valid words and their actual sign prices through signed row/column permutations. The two price statistics survive sign swaps, so the canonical formula extends to the entire orbit. Outside the orbit, independent cheaper-sign conversions of the support-attaining words attain sum b. The constructor emits the word at its actual fixed ports; no physical permutation or frame change is treated as free.

Only research/signed-matrix-price-classification/ is added. Produced by Nick Petalas. Predecessor credit: Ross Richardson's #944/#943/#934 and their PR889/PR628 lineage, martinxxx9's #1088 repairs, Sagar Thakrar's #1349, and #4233/#4241. Their results are used as lemmas, not independently audited here.

Evidence

Before, #4241 covered only the canonical target. After, the written proof and constructor cover the full target domain. The registered normal and optimized scientific checks each pass with:

• 472,392 priced words: all 19,683 targets, six characteristics and four price vectors;
• 27,648 transported anchors, covering every orbit target and operator, eight characteristics and both global signs;
• 6,912 rational-price orbit words, 768 count-profile DPs, 2,304 noncommuting dense dirty-array executions and 744 query-deletion controls.

Reproduce from the repository root:

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

Expected markers: PASS_SIGNED_MATRIX_PRICE_CLASSIFICATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Any CPython 3.12 interpreter can replace the local interpreter path. Standard library only. Timed scientific runs took 18.45 and 20.18 seconds locally on CPython 3.12.15/macOS arm64, below their individual 180-second limits. These are check durations, not performance comparisons. Committed-byte/source-history preflight and official export integrity pass. Optimized Git-free replay passes on a read-only committed export; all seven payload files remain unchanged. A modified predecessor input is rejected at its SHA256 pin. Normal, optimized and exported source-bound receipt digests agree: abe049c995ebac53d97a670ad31f420e04548b13fba6c26508e83d6d44b6a8f9. Each scientific run generates a receipt bound to all package and live-input hashes; use --receipt /path/outside to save its JSON outside the checkout.

Limitations and unrun obligations

The all-ring theorem is a written proof using #944's finite integer classification and the unmerged canonical theorem #4241 at ec573175d36cb016661f37293dd9ee2aefb737bb. This does not independently establish those lemmas. Finite price/characteristic checks corroborate the proof, rather than enumerate every ring or price. Both live certificate inputs are pinned to main; no pending executable or Git object is required by replay. Independent human and formal proof-kernel review are unclaimed. No physical caller, instantiated complete prices, routing/admission, precision, supplier integration or outer multiplication theorem is supplied. No selected cold replay or formal build was run.

Open next questions

1. Bind one real caller, its demand matrix and actual forward/inverse complete prices, then compare routing, admission and total schedules.
2. Find a signed 3x4 support obstruction with both an arbitrary-operator integer repair and finite-characteristic unary anchors. Without those attainment properties the same lower bound need not be sharp.

Related open work

• Prove the exact signed connector tariff with sign-dependent prices (no new κ) #4241, head ec573175d36cb016661f37293dd9ee2aefb737bb, supplies the canonical sign-dependent theorem and anchors; this extends its target domain. At forward/inverse prices (1,3), the canonical bills 28,18,24,25,28 in characteristics 0,3,4,5,8 agree, checked numerically.
• Prove the exact signed connector tariff over every coefficient ring #4233, head d56d43c726dddc37a97a5dabf9d13ef861394133, supplies the canonical equal-sign characteristic law through Prove the exact signed connector tariff with sign-dependent prices (no new κ) #4241. Unit-price bills 14,12,13,14,14,14 in characteristics 0,2,3,4,5,8 agree, checked numerically.

Live claims and open PRs, main README searches and merged-PR searches were checked before claiming and before committing. The overlap helper's complete snapshot covered 364 open PRs and 3,135 Discussion records; this is evidence of search coverage, not a novelty certificate. Main remained 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7.

CI notes

Hosted CI is green on head 6dfe100bc622da9615db77653195360d5fadf2be: run 39504785884. Planning passed in 2m55s and the package replay in 1m40s. Both hosted scientific modes printed the local counts and receipt digest abe049c995ebac53d97a670ad31f420e04548b13fba6c26508e83d6d44b6a8f9; PASS_REGISTERED_RESEARCH_PACKAGE and the unchanged-input step passed. No check failed. All registered commands and source checks passed locally; there is no unrun registered local obligation assigned to CI.

Merge danger

Door: two-way. Blast radius: isolated. The change adds one research package and its explicit replay manifest.