[petalas]

Summary

Scoped algebraic proof with finite corroboration. Over every commutative K-domain, a singular two-role target in the constant-right-vector alphabet is reachable exactly when it factors as a column times a row over that domain.

paid: I+u v^T, u in R^2, v in K^2; free: GL_2(K) anywhere
singular M is reachable <=> M=a b over R <=> cost <= 3
R=K[s²,st,t²]: [[s²,st],[st,t²]] is unreachable at every finite length

The new direction is the necessary factorization invariant. Any singular word contains a determinant-zero paid factor, whose image lies in the kernel of a nonzero constant row. This gives an R-valued outer-product factorization of the complete endpoint. A polynomial gcd argument excludes the explicit Veronese target over every field. The sufficient three-factor constructor is cited from #4808 and is not claimed as new.

This contribution was produced by Nick Petalas. Credit Nick Petalas #4808 for the explicit open target and constructor; #4276/#4243 for the alphabet/free-map context, following Wentao Li #4099 and #2513; and ToikY #4334 for the commutative field-product theorem and singular-endpoint gap. Classical UFD, gcd and factorization facts retain their classical status. No predecessor scientific verifier is imported, audited or replayed. Six files are added in research/veronese-singular-obstruction. No new kappa or supplier admission is claimed.

Advance CLAIM.

Evidence

Before, #4808 explicitly leaves the K[s²,st,t²] target open. After, PROOF.md proves the all-domain reachability criterion and an all-field obstruction for that target.

• Exact polynomial enumeration tests 768 paid-factor parameter choices over F2 and 52,488 over F3, with coefficients in span{1,s²,st,t²}. The 48 and 648 singular choices all yield checked factorizations and complete wrapped-word endpoints. Parameter choices may give duplicate matrices; no distinct-matrix census is claimed.
• Complete factor searches over F2,F3,F5 test 256, 6,561 and 390,625 first-entry pairs. Exactly 2,4,8 multiply to s²; none extends to the complete target. Total degree additivity bounds every possible factor entry by degree two, so this finite factor search omits no higher-degree factorization in those three domains. The all-field obstruction rests on the written proof.
• Ten exact constructor words over Z,F2,F3,F5,F7, fifteen single-factor omissions, five determinant mutations, five larger-ring controls, 3,123 scalar input comparisons, three malformed-input controls, one capacity boundary and two capacity/singular-premise rejection controls pass. Scalar specializations are reachable and cannot establish the polynomial obstruction.
• Both registered warning-strict commands passed locally on CPython 3.12.15 / macOS arm64 in 1.013 s and 1.009 s. Normal and optimized modes produce identical generated hash-bound receipts. Each command has a 120-second limit. These are check durations, not speed comparisons.
• Registered metadata/replay, in-memory compilation, staged whitespace checks and committed-byte/source-history preflight pass at 913a75e. Official committed export and declared-input integrity pass. Optimized replay in a read-only Git-free export with empty PATH passes in 1.013 s, reproduces the complete receipt and preserves all seven exported files. No receipt is handwritten.

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

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

Expected markers are PASS_VERONESE_SINGULAR_OBSTRUCTION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 interpreter may replace the local path. The source-history check requires full Git history; direct verification also works without Git or predecessor files.

Canonical main was refreshed before submission and remains de5ec2e, priority revision 8. Selected source-binding orientation passed; it is metadata orientation only. TASK.json explicitly changes the formal-interface card to this algebraic side result. Live PRs, recent claims, local reservations, main README terms and merged-PR searches were checked before claiming and committing. Cached remote pagination covers 416 open PRs and 5,312 Discussion records; sparse local-note coverage is incomplete. The explicit target was located only as #4808's open question. Merged #5454 studies factorable targets over cube-zero algebras. These checks do not cover unpublished work, arbitrary historical diffs or global literature priority.

Limitations and unrun obligations

The universal result is a written scoped proof. Finite execution corroborates the stated domains and identities. The theorem uses exactly two roles, a commutative integral domain, base-field right vectors and base-field free maps. It does not classify exact minimum one versus two versus three, rings with zero divisors or extra roles.

Native coefficient preparation, access, routing, precision, reversible restoration, stock and the complete supplier bill remain unknown. No complete multiplication theorem or selected-frontier update follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected scientific replay were outside scope and unrun. Every registered local command ran successfully. The bounded managed-Python allocation deviation is recorded in the README; no external analyzer was installed or claimed.

Open next questions

• Can one extra restored role realize diag(M,1) for the obstructed target? Start with the same alphabet while preserving the third input/output exactly. A three-role obstruction or word is missing.
• Which factorable singular targets over a specified non-PID domain have exact cost one, two or three? Start with the cited constant-chart two-factor divisibility criterion. The reachability theorem supplies only the uniform upper bound three.

Related open work

• Prove exact all-role right-vector costs for small local products #5861 studies the same paid right-vector alphabet for invertible targets over small products of local algebras at arbitrary role count. This result concerns singular targets over coefficient domains. No shared numerical cost census is asserted.
• Prove split determinants do not ensure radical reachability (no κ) #5854 gives an all-length obstruction for a singular three-role target over a square-zero radical algebra. Both the coefficient ring and role count differ from this two-role domain criterion. No shared numerical count or minimum is asserted.

Related open work refresh before DONE

The account's open PRs created since the CLAIM at 2026-10-27T10:04:08Z were refreshed: #5865, #5866, #5867, #5868, #5869, #5870, #5871, #5872, #5873 and #5874. Their titles and result descriptions were read. Apart from this PR, they concern cyclotomic column completion, binary phase defects, finite-field connector chart ranks, Clebsch query restrictions, sunflower inertia, matrix-tensor determinant characters, dirty-word error bounds, C5 fitting classes and rank-capped A5 calls. None studies this singular two-role coefficient-domain target or supplies an overlapping numerical factorization census. The related same-alphabet work remains #5861 and #5854, described above. No new computation was run to check another contribution or compare its numbers.

CI notes

On contribution head 913a75e5735702cc39559af1be8a8fd510c42c02, run 40486873507 failed in plan, at Onboarding helper tests. The log identifies all ten failed/error entries in unchanged tests/test_research_current_assignment.py and ends Ran 511 tests / FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures expect sixteen frozen-card transformations, missing-reference exceptions and nonempty sparse-reference omission lists. Actual messages include 0 != 16, ValueError not raised, and empty lists where selected-baseline or historical-replay paths were expected. All ten failure headers and exception/assertion messages match PR #5854's run 40478863245 exactly. This is the shared onboarding failure on main base de5ec2e7a6074068452f45b1a191b5c72832df4a; maintainers will repair it. No shared tests or implementation files were changed.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific checks have run. Normal/optimized registered checks, committed-byte/source-history preflight and read-only Git-free export replay all passed locally as recorded above. No mathematical result or package file was changed in response to the shared failure.

Merge danger

Door: two-way.

Blast radius: isolated.

The change adds one self-contained research package. Shared code, frozen sources, predecessor credit and selected-frontier metadata are unchanged.