[1337isnot1337]

Change and mathematical scope

Constructs one fixed e1,e2,e1 three-factor word for every two-role target over a finite product of M_m(K[t]/t^d), for every field K and positive m,d. Singular and zero targets are included; no scratch roles or free endpoint maps are used in the upper construction. A singular binary rank-two example has exact cost three at every nilpotency depth. A single scalar chain-ring component has cost at most two with a base-field column permutation, and the proof gives its exact zero/one/two classification.

This answers the named singular radical-extension question in #5156 for matrix chain rings. Rectangular valuation elimination completes the possibly singular top block row before the inherited unit-completion/three-factor absorption argument is applied. The target itself is never inverted. The known multigenerator radical obstructions remain valid outside this domain.

Scoped algebraic proof with exact finite corroboration; no new kappa or supplier admission. Arbitrary coefficient preparation, overwritten-input recovery, physical routing, ownership, precision, stock and the complete recursive tariff remain unknown. The selected conditional kappa remains 0.00302015. Only seven files in research/chain-ring-common-words/ are added.

Task and overlap

Base: canonical main a5df48c55ea806d7952b65acc9c76d878ff09b13, priority revision7; canonical main was fetched again before submission and remains at that commit. Task bit-new-register-model, explicitly changed to the coefficient-algebra question in the canonical claim. This is a side result on an interface that could support a changed native model, with no native budget credited.

Nearest predecessor is JoyboyBrian #5156 at 410240294708410635cdbc3d885cf999f8f0c33a: it handles products of full matrix algebras and leaves singular radical extensions open. This package retains its common-word absorption skeleton and supplies the specified arbitrary-depth chain-ring extension. That pending written proof is an immutable bibliographic reference in SOURCES.json, not a runtime input or publication ancestor. Nick Petalas #4749 at 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891 already handles invertible targets over arbitrary semilocal algebras; singular endpoints are the difference here. The exact two-factor boundary is inherited from #4276. The square-zero obstructions of #4745/#4776 are not weakened.

The initial overlap snapshot had complete remote pagination and all 6,918 local notes at the base. A submission-time live PR scan was also performed. Adjacent #5163 studies destination-priced dirty fanout over local rings, not factorization of arbitrary matrix-chain-ring endpoints. This coverage is not a novelty certificate.

Validation

Exact contribution head: 987ef2db3ea27e80fd6e37213cf5f247648cfae2.

With CPython3.12.15 and the standard library:

python3.12 -B scripts/research_package.py --package research/chain-ring-common-words
python3.12 -B scripts/check_research_commit.py --package research/chain-ring-common-words
python3.12 -B scripts/export_research_package.py export --package research/chain-ring-common-words --output /tmp/chain-ring.tar.gz
python3.12 -B scripts/export_research_package.py check /tmp/chain-ring.tar.gz

Registered normal and optimized replay pass and emit identical file-bound receipts. A read-only Git-free optimized replay, with PATH=/nonexistent and the actual interpreter path, emits the identical receipt; all seven exported files remain unchanged. Metadata/pins, committed-byte preflight, official export integrity and whitespace checks pass.

The bounded verifier checks 79,572 scalar targets, 582 three-factor target constructions, forty arbitrary dirty-input executions, a common three-component word, the complete one-factor/free-map orbit in two small rings, nine rejecting two-factor sharpness charts and eleven failure controls. The multiplication oracle separately convolves polynomial coefficients; it does not use the constructor's ring-operation tables. Odd characteristic, torsion top rows, zero targets and higher nilpotency are included. The written proof, rather than extrapolation from finite tests, supplies universal coverage.

The optional --check-source-history preflight is unavailable in this shallow checkout and was not counted as passing. Ordinary committed-byte preflight passed. Hosted CI and independent human/proof-kernel acceptance are pending. No native selected-operator suite, historical exhaustive suite, Lean or Docker replay was run.

Credit

Prepared for Cinna Davis / 1337isnot1337 with substantial OpenAI Codex assistance; no subagents. Retain JoyboyBrian #5156, Nick Petalas #4805/#4749/#4276/#4776, ToikY #4334, Wentao Li #4099 and the retained #2513 lineage. Smith elimination, basis completion and stable range one are classical. Apache-2.0.

Open next questions

See the package README, PROOF.md and TASK.json. The next K-oriented decision is a fully priced actual native producer/consumer for one emitted coefficient tuple, including recovery of overwritten inputs. Three formal updates alone are insufficient. Algebraic family expansion is stopped until that real consumer budget is favorable. Multigenerator radical extensions remain a separately scoped mathematical question, with the existing quotient obstructions retained.