[petalas]

Summary

Scoped proof: over every field K, the radical target below has exact right-vector cost three over R=K[x,y]/(x,y)^3, even with arbitrary free GL2(K) maps.

[[x,0],[y,1]] [[1,y],[0,0]] [[x,0],[0,1]] = [[x^2,xy],[xy,y^2]].

All three factors are legal I+u v^T calls with v over K. Two factors are impossible by #4276's exact ideal criterion: every transformed entry is in J^2, so ideal membership reduces to scalar dependence and would force a common bilinear zero. But (ax+by)(cx+dy) is nonzero for every pair of nonzero scalar linear forms. This answers the higher-nilpotency question in #4745 without rechecking its theorem. The same target is unreachable in the smaller square-zero subalgebra K+J^2 by that cited theorem; degree-one coefficients enable the new word.

Produced by Nick Petalas. Credit Nick Petalas #4745/#4276/#4243, ToikY #4334, and Wentao Li #4099 following #2513. Only research/cube-zero-three-factor/ is added. No new kappa or selected-frontier change.

Evidence

• Before: Prove exact square-zero radical reachability and binary count (no new κ) #4745 leaves higher nilpotency open. After: an explicit all-field cost-three target, a written lower bound, and arbitrary-data action with no scratch assumptions.
• Standard-library checks over F2,F3,F4,F5,F7 reject all 794 projective two-factor charts through a six-dimensional ideal-rank calculation. All 3,178 nonzero row/column pairs have nonzero response. The word passes all 60 basis columns and all 4,096 binary R^2 inputs. Eight boundary/mutation controls and positive ideal controls pass.
• Both registered warning-strict commands pass locally on CPython 3.12, timed separately at 0.27 seconds each. The registered runner passes. Committed-byte/source-history checks pass at b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511. Official export integrity and optimized Git-free replay pass, with the identical generated SHA256 receipt and all eight exported files unchanged.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/cube-zero-three-factor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/cube-zero-three-factor/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cube-zero-three-factor
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cube-zero-three-factor --check-source-history

Any CPython 3.12 installation can replace the local interpreter path. Expected markers: PASS_CUBE_ZERO_THREE_FACTOR, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES.

The card's first positive control also passed, reporting pairings=945 max_coverage=31 queries=74 leaves=182; that is orientation, not new research. Base is canonical main b58ad9ef4d4e861b0eaa5dba41742bdc072dca51, priority revision 7. Pre-commit refresh reached 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b; guidance, selected record and both mathematical sources were unchanged. Exact source pins, overlap coverage and the explicit card scope change are in TASK.json/SOURCES.json. Advance claim.

Limitations and unrun obligations

The universal statement is the written algebraic proof, with bounded finite corroboration. This is one target family, not a singular-target classification. Native coefficient preparation, routing, restoration, precision, stock and complete supplier costs remain unknown. No supplier admission or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or selected-construction replay was run; all registered local obligations were run.

Open next questions

• Is every reachable radical target over K[x,y]/(x,y)^3 attainable with three factors? Start with a three-factor normal form; the present word gives no general upper bound.
• Which cube-zero radical multiplication maps admit the same separation? Start with two degree-one elements whose nonzero scalar linear combinations always have nonzero product; degree-two relations may prevent that property.

Related open work

• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13af13b755932852e9306a714b317630dda, supplies the square-zero theorem and this higher-nilpotency question. The targets here use a different ambient algebra. The degree-one quotient of our target is zero, checked numerically; no census total from Prove exact square-zero radical reachability and binary count (no new κ) #4745 is claimed or rerun. Over its three-dimensional-radical subalgebra the same coefficient tuple has no common bilinear zero, agreeing with its cited obstruction, while allowing degree-one factors gives cost three here.
• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891, classifies invertible semilocal targets. Our target has zero scalar residue and is singular, so its endpoint domain is disjoint. Both use three paid factors in their respective examples; there is no overlapping target-cost or census number to equate.

Merged #4276 supplies the two-factor criterion used as a lemma. Merged #4601/#4613 concern invertible local targets, also disjoint from this target. The concurrently claimed all-field square-zero count changes the counting field, whereas this contribution changes the nilpotency and permitted coefficient layers. Searches covered live claims, targeted open/merged PRs and main README terms; sparse local-note and unpublished coverage remain partial.

Related open work refresh before DONE

This account's open PRs created since the CLAIM at 2026-10-22T19:18:23Z were listed before DONE. They are #4761, #4763, #4764, #4766, #4767, #4769, #4770, #4771, #4772 and #4773. The five new PRs studying the same right-vector alphabet are:

• Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763, head d0158a552352c09608295a7ec47dd31327b360ae, proves unbounded costs for invertible targets over F2[t]. Our finite local coefficient algebra and singular endpoint differ, so no target-cost or census number overlaps.
• Count square-zero radical targets over every finite field (no new κ) #4766, head 5af823b1a14b2517235925145e36581bfdfcad41, counts square-zero radical targets over every finite field. Our target also belongs to the subalgebra S=K+J^2, but our factors use the larger cube-zero algebra. At q=2, the target has exactly zero common-zero pairs among all nine projective row/column pairs, checked directly. This agrees numerically with the cited criterion excluding it over S; its cost three over the larger algebra concerns a different allowed alphabet. No census total is recomputed or claimed.
• Count rank-one residue lifts over local algebras (no new κ) #4770, head 39ef149ba3ea8adbda18478ea227d3d0826475ab, counts lifts of nonzero rank-one scalar residues. Our target's scalar residue is the zero matrix, checked directly, so these target domains and census numbers are disjoint.
• Prove exact rank-one matrix-algebra cost at every role count (no new κ) #4772, head 116992ef72c8aa9f8994137b04395584d3b350e8, gives exact costs for ordinary rank-one targets over matrix algebras M_m(K), m>=2. Our coefficient algebra is commutative local and outside that family; no endpoint or numerical census overlaps.
• Prove an all-length radical pencil obstruction for any role count #4773, head f5d0f421efec55c5f4b3dbb142eee1e79a4e21db, excludes radical pencils over square-zero algebras at arbitrary role counts. Our degree-two target and cube-zero ambient algebra are outside its hypotheses; its executed three-role pencil count and our two-role exact cost do not overlap.

#4761 concerns chain retiming, #4767 and #4769 concern full-guard header probabilities, and #4771 concerns scalar-pivot Schur dimensions. They study different objects and quantities. No conflicting overlapping number was found. The earlier #4745/#4749 relationships remain as recorded above.

CI notes

Both hosted checks passed on b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511 in Registered research packages run 39839704596: plan passed in 3m7s and the registered research/cube-zero-three-factor replay passed in 1m9s. Run metadata confirms that exact head and conclusion success. Hosted CI certifies the registered checks within their stated finite scope; it does not provide independent human acceptance or formal proof-kernel verification of the written all-field theorem. All registered local checks, committed-byte/source-history preflight and Git-free export replay also passed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only five files in the new research package are added.