[petalas]

Summary

Scoped algebraic proof: every singular two-role target over K[t], for every field K, has exact right-vector cost one, two or three, with the following complete classification. Zero has cost two.

M=a b, with a primitive polynomial column;
V=span_K{b1,b2}, d=monic gcd(b1,b2).

cost 1 if a is constant and V contains a nonzero constant;
cost 2 otherwise if a is constant or d belongs to V;
cost 3 otherwise.

Paid factors are I+u y^T with y over K. Arbitrary GL2(K) maps are free anywhere. A literal e1,e2,e1 outer-product word attains the upper bound without scratch. The target [[t,t²+1],[t²,t³+t]] has exact cost three over every field. This classifies determinant-zero polynomial endpoints, complementing the unbounded invertible costs of #4763/#4784. No new kappa, supplier admission or selected-frontier change.

This contribution was produced by Nick Petalas. ToikY #4334 supplies the singular-endpoint gap; Nick Petalas #4276/#4243 supplies the cited exact two-factor criterion and free-map normal form, following Wentao Li #4099 and #2513. The #4780 field-product template is related precedent for the outer-product word. Primitive PID factorization, Bezout and gcd facts are classical. Predecessor results are cited, not audited. Only six files in research/singular-polynomial-cost are added.

Advance CLAIM.

Evidence

• Before: Prove the sharp component threshold for commutative right-vector targets #4334 explicitly excludes singular endpoints; Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763/Prove a binomial law for polynomial right-vector cost (no new κ) #4784 classify invertible polynomial targets. After: PROOF.md gives the complete singular K[t] theorem, its lower-bound criteria, a three-factor constructor and an all-field sharp example.
• The self-contained standard-library verifier enumerates all 24,066 determinant-zero targets in four bounded domains. Cost-1/2/3 counts are F2 degree<=4: 279/7489/936; F3 degree<=2: 800/5761/0; F4 degree<=1: 975/1201/0; F5 degree<=1: 3024/3601/0. Each includes zero once in cost two. Every primitive factorization, independent constant-chart divisibility comparison and literal three-factor word passes. Every successful two-factor chart is also reconstructed with its free maps and multiplied back to the full target.
• All 35 enumerated boundary and mutation control groups pass. Warning-strict ordinary and optimized direct commands took 2.27 s and 2.26 s locally on CPython 3.12. Their generated receipts agree. Both registered commands pass and are well below their 120-second limits.
• Package metadata/replay, in-memory syntax compilation, staged whitespace checks, committed-byte/source-history preflight, committed export integrity and optimized Git-free export replay pass. The export emits the same complete generated receipt. Receipts bind all six package files by SHA256; no receipt was handwritten.

Exact reproduction from the repository root, with CPython 3.12 and its standard library:

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

Any Python 3.12 interpreter may replace the displayed path. Expected markers are PASS_SINGULAR_POLYNOMIAL_COST targets=24066 controls=35 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical main was fetched again before submission and remains 624de9d, priority revision 7. The formal-interface-lemma card is explicitly changed to this algebraic side result. The card's historical baseline stops at omitted sparse input research/fresh-helper-thinned/PROOF.md; no selected scientific replay pass is claimed. Distinctive main README and merged-PR searches, live open PR bodies and recent handoff claims were refreshed before committing. The overlap helper reported complete remote pagination for 326 open PRs and 3,751 Discussion records; sparse local-note coverage was incomplete. Search coverage does not prove novelty.

Limitations and unrun obligations

The universal claim is the written scoped proof. The finite enumeration corroborates only the displayed fields and degrees. Two roles, one variable and a field of constants are the exact theorem scope. Native coefficient preparation, access, routing, precision, reversible dirty restoration, stock and complete supplier costs remain unknown. No complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected operator replay were outside scope and were not run. Every registered package command ran locally.

Open next questions

• Which singular targets are reachable over the nonprincipal domain K[s²,st,t²]? Start with [[s²,st],[st,t²]], testing outer-product factorization and restricted-right-vector reachability. The one-variable primitive-factorization argument does not supply that step.
• Specify one actual native producer/consumer and price coefficient preparation, ownership, overwrites, restoration, precision and the complete paid list before using these formal words in a supplier. Unknown costs are not zero.

Related open work

• Prove a binomial law for polynomial right-vector cost (no new κ) #4784, head 4d0a636, gives the invertible polynomial degree/cost distribution over F_q[t]. Its endpoints are invertible; every endpoint here is singular. No target census or cost distribution overlaps.
• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121, counts singular local residue lifts. At Q=q,z=1, its one-factor count is (q+1)(q²-1). Fresh enumeration of our constant-polynomial nonzero rank-one boundary gives 9,32,75,144 for q=2,3,4,5, agreeing numerically. Its proper-extension lift domain differs from the positive-degree polynomial targets here.
• Prove exact local right-vector costs at every role count (no new κ) #4787, head 4b58e8c, gives arbitrary-role invertible local-algebra costs. Both its invertible endpoint contract and local coefficient hypothesis differ; no singular polynomial target count overlaps.
• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789, head 7fdbf80, classifies ordinary rank-two targets over full matrix algebras M_m(K), m>=3. The coefficient algebra differs, and no target-cost census overlaps.
• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab2, classifies zero-residue pencils over a two-generator square-zero local algebra. Those coefficients differ from K[t]; its pencil count and our polynomial census have no shared numerical domain.
• Prove exact local costs for residue-corank-one targets (no new κ) #4804, head da6f858, handles residue-corank-one local targets at arbitrary role count. K[t] is nonlocal. On the scalar-field two-role boundary, all 9,32,75,144 nonzero rank-one matrices at q=2,3,4,5 have cost one here, agreeing with its delta-one theorem; its reported larger-role sample is distinct and is not replayed.

Related open work refresh before DONE

This account's open PRs created since the CLAIM at 2026-10-22T20:16:15Z were listed again: #4801, #4803, #4804, #4805, #4806, #4807, #4808, #4809 and #4810. All same-alphabet titles and bodies were read. #4804 is compared above. The additional related work is:

• Prove three-factor sufficiency for every two-role matrix-algebra target (no new κ) #4805, head fbf2bd0, proves three-factor sufficiency for every two-role full-matrix-algebra target. Its coefficients differ from K[t]. At the shared scalar-block boundary m=1, fresh enumeration gives exact cost one for all 9,32,75,144 nonzero rank-one constants over F2,F3,F4,F5 and cost two for zero. These agree numerically with its upper bound of three; its full-matrix census and our positive-degree polynomial census are distinct.
• Count 9,216 ternary invertible threshold targets (no new κ) #4809, head 6354da6, counts 9,216 invertible two-role targets over F3^4 needing three factors. Its field-product coefficient algebra and invertible endpoints differ from our singular polynomial targets. No target population or degree-conditioned count overlaps.
• Prove sharp five-factor cost for binary three-role products (no new κ) #4810, head 701cb14, proves the sharp uniform bound five for invertible three-role binary field-product targets. Both role count and endpoint contract differ. Its 168 group components and our 24,066 singular polynomial targets count different objects; no numerical population overlaps.

The other newly open PRs study Schur spectra, Schur-product counts or full-guard probabilities. They do not compute this singular polynomial factor cost. No disagreement was found in the shared constant-matrix boundary checks. No predecessor verifier was replayed.

CI notes

Both hosted checks passed on contribution head bd22290761451093aa7aef1fa7b1421627ab0f37 in Registered research packages run 39848349161: plan in 3m0s and the registered research/singular-polynomial-cost replay in 1m23s. Run metadata confirms that exact head and conclusion success. The replay executes both warning-strict ordinary and optimized commands. All registered local checks, committed-byte/source-history preflight, export integrity and Git-free replay also passed. Hosted CI certifies execution of the registered bounded checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only six files in one new research package are added. Shared implementation, frozen sources, predecessor credit and selected-frontier metadata are unchanged.