[petalas]

Summary

Scoped algebraic counting proof. For the two-role right-vector alphabet over F_q[t], with base-field right vectors and arbitrary free GL2(F_q) maps, a uniformly chosen invertible target of exact entry degree D>=1 satisfies

cost(M)-1 ~ Binomial(D-1, (q-1)/q).

# targets of degree D and cost r
= |GL2(F_q)| (q+1) (q-1)^r q^(D-1) binom(D-1,r-1),  1<=r<=D.

The mean is 1+(D-1)(q-1)/q and the variance is (D-1)(q-1)/q^2. The written proof extends the binary constant-free reduction of #4763 to arbitrary fields, then counts its unique expressions. This gives the exact all-field, all-degree distribution rather than another finite binary census. No new kappa, native supplier or selected-frontier change is claimed.

Produced by Nick Petalas. Nick Petalas #4763/#4276/#4243 supplies the binary cost question and free-map lineage. Hirosi Nagao's 1959 polynomial-matrix presentation and degree method retain their classical credit; no new group-presentation theorem is claimed. ToikY #4334, Nick Petalas #4601/#4749 and Wentao Li #4099 following #2513 retain the local/semilocal and alphabet credit. Primary sources and exact pins are in the package.

Evidence

• Before: Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763 gives the binary cost theorem and degree-at-most-four finite histogram. After: PROOF.md gives the joint cost count and binomial law for every finite field and every exact degree.
• An independent determinant-unit enumeration matches all 63,504 literal polynomial-matrix targets: 1,536 over F2 through degree four; 3,888 over F3 through degree two; 46,080 over F4 through degree two; 12,000 over F5 through degree one. The enumeration performs 135,570 product-tail comparisons. It is not presented as a census of every rejected matrix.
• All 924 invertible degree-one paid parameter tuples pass constant splitting. All free conjugations retain the required scalar. Exact probability normalization, mean and variance pass for 480 integer pairs; non-prime-power q in those arithmetic checks does not denote a finite field. Eight control groups cover the stated boundary and mutation cases.
• Both registered warning-strict ordinary and optimized commands passed locally, timed separately at 0.61 s and 0.59 s. Package metadata and replay passed. In-memory compilation passed; optional external Python analyzers were unavailable. The largest function has 29 physical lines. Finite bounds and the managed-allocation deviation are documented in README.md.
• Committed-byte/source-history preflight passed for seven files at 4d0a6363fe4f45dd9a769f5b74d02fc5d9061ae3. Official committed export and integrity checks passed. An optimized Git-free export replay with PATH=/nonexistent matches the complete generated final receipt. Receipts are generated on stdout and bind all six package files; the predecessor proof hash is checked against SOURCES.json.

From the repository root, using CPython 3.12 and the standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/polynomial-cost-distribution/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/polynomial-cost-distribution/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/polynomial-cost-distribution
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/polynomial-cost-distribution --check-source-history

Another CPython 3.12 installation can replace the local interpreter path. Expected verifier marker:

PASS_POLYNOMIAL_COST_DISTRIBUTION targets=63504 fields=4 probability_cases=480 controls=8 no_new_kappa

The package runner ends with PASS_REGISTERED_RESEARCH_PACKAGE. The original card's 945-pairing positive control and source-bound selected metadata orientation also passed. The selected operator replay was not run. Canonical main, relevant guidance and the predecessor proof were unchanged at the pre-submission fetch, 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Live overlap refresh covered 305 open PR bodies and 3,723 Discussion records with complete remote pagination; sparse local notes remain incomplete. Main README and targeted merged searches were checked separately.

Advance claim.

Limitations and unrun obligations

The universal claim is a written scoped proof with bounded finite corroboration. Two roles, one polynomial variable, base-field right vectors and free base-field maps are essential to this classification. Singular endpoints, multivariate rings and restored auxiliary roles remain unclassified. Uniform exact-degree targets are the sampling measure; uniform words and degree-bounded targets have different distributions.

Native preparation, conversion, routing, control, precision, restoration, stock and a complete physical consumer remain unpriced. No complete multiplication theorem, supplier admission, independent human acceptance or formal proof-kernel review is supplied. Lean, Docker and the selected operator replay were outside scope and were not run. Every registered local command ran successfully.

Open next questions

• Derive closed-form moments under a uniform degree-bounded target measure. The proved joint counts give a finite weighted mixture; first fix that measure and sum the mean.
• Identify one literal native polynomial-coefficient producer and price left-coefficient preparation, conversion and the complete consumer before using these counts for a supplier.

Related open work

• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab259a8ce496a4c5d4debeed4c4e4d9f7ca4, classifies singular square-zero radical pencils in the same formal alphabet. Its coefficient algebra and endpoints differ from our invertible F_q[t] targets. No target count or cost domain overlaps.

• Prove the sharp singular rank-one product threshold (no new κ) #4780, head a4436fab4164eccc498d4d9885eec68211c869de, studies singular rank-one tuples over finite field products and the sharp component threshold. The finite product algebra differs from F_q[t]; no target census number overlaps.

• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121c65e1bfd17a65466c95940eb00f12a225, counts lifts of rank-one residues over finite local algebras and proper residue-field extensions. Those endpoints and algebras differ from the present invertible polynomial targets; no target count overlaps.

• Prove exact local right-vector costs at every role count (no new κ) #4787, head 4b58e8cd5690bd35cf01b0018eefa2c39a298920, proves exact arbitrary-role costs over local algebras. F_q[t] is nonlocal. On the common constant-field two-role endpoints both formulas assign cost zero; our enumerated F2/F3 constant totals are 6 and 48, and their basis images give delta zero. The positive-degree polynomial distribution and its target totals are outside that local theorem.

• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789, head 7fdbf8040ea9a957af0576067df2d9cb6fcf0e51, classifies singular ordinary rank-two targets over matrix algebras. Our invertible scalar-polynomial endpoints have different coefficients and ranks, so no target count or exact-cost census overlaps.

Before DONE, this account's open PRs created since the CLAIM were listed again. All newly relevant titles and bodies were read. The two additional right-vector results are recorded above; the other additions study Schur dimensions or full-guard probabilities. No conflicting shared numerical claim was found.

Merged #4763 is the direct predecessor. At binary exact degree four its joint cost vector agrees numerically with the present [144,432,432,144]; summing degrees zero through four reproduces [6,270,612,504,144]. The six constant binary free maps also agree. The new result is the quantified all-q distribution. No other open PR studying this polynomial degree-cost quantity was found in the refreshed snapshots. The account's open PRs created since the advance claim were listed again during publication preparation.

CI notes

Both hosted checks passed on 4d0a6363fe4f45dd9a769f5b74d02fc5d9061ae3 in Registered research packages run 39844489882: plan in 3m51s and the registered polynomial-cost-distribution replay in 1m12s. Run metadata confirms this exact head and success. All registered local checks, committed-byte/source-history preflight, export integrity and Git-free replay also passed. Hosted replay certifies the registered finite checks within their stated scope; it does not provide independent human acceptance or formal proof-kernel verification of the written universal theorem.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only six files in the new research/polynomial-cost-distribution package are added. Shared code, frozen sources, predecessor credit and the selected record are unchanged.