[petalas]

Summary

Scoped proof answering the singular-target question of #4601. For any field K and split square-zero algebra R = K ⊕ J, a radical two-role target has the exact right-vector cost

cost(M) = 2  iff some nonzero base-field row φ and column v have φMv = 0;
cost(M) = ∞  otherwise.

A first-zero-residue invariant excludes arbitrary longer words, and a triangular two-factor word attains every allowed target. Over F₂, with z = 2^dim(J), the reachable count is exactly 9z³ − 30z² + 42z − 20. No new κ or selected-frontier change.

Produced by Nick Petalas. Credit Nick Petalas's #4276/#4243, ToikY's #4334, and Wentao Li's #4099 following #2513. The predecessor inputs and original card are pinned at main b58ad9e. The #4601 invertible-target theorem is not a dependency of the proof.

Evidence

• Before: Prove radical invariance of right-vector costs over local products #4601 leaves singular reachability open, starting with F₂[x,y]/(x,y)²; Prove the exact two-factor right-vector criterion at n = 2; the M2(F2) failure lifts to F2[S3] (no new κ) #4276 records 12 unreachable targets in that example.
• After: the written proof classifies every radical target over every split square-zero K-algebra, at arbitrary finite word length. The new binary all-dimension count gives reachable/unreachable pairs (1,0), (16,0), (244,12), (3004,1092) for dimensions 0..3.
• The verifier compares the entire literal two-factor/free-endpoint image against the criterion in those four dimensions and reconstructs every accepted target. It independently recovers the polynomial from all 511 nonempty subsets of the nine rank-one forms. All 1,152 first-zero transitions at dim(J)=1 and six rejecting controls pass.
• Both warning-strict registered commands passed locally, ordinary and optimized, each timed at 0.07 seconds on CPython 3.12.15. The package runner passed. Committed bytes and declared source history passed at 3edea13. Official export integrity and optimized direct replay from a Git-free export passed with an identical complete generated receipt.

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

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

Any CPython 3.12 installation can replace the local interpreter path. Expected marker: PASS_SQUARE_ZERO_RADICAL_REACHABILITY, followed by the declared counts and no_new_kappa. A generated hash-bound receipt precedes it.

Limitations

Two roles, residue exactly K, split algebra and J²=0 are essential hypotheses. The theorem classifies targets in M₂(J); it does not classify other singular targets, higher nilpotency or more roles. The finite corroboration is binary and dim(J)≤3; arbitrary fields and dimensions are covered by the written argument. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker or selected-construction replay was run. Native preparation, routing, restoration, precision, stock and complete supplier cost remain unknown; no multiplication theorem or supplier admission is supplied.

Open next questions

• Determine the second-order obstruction over F₂[x,y]/(x,y)³, where products of radical entries survive.
• Derive the corresponding finite-field counting law for F_q with q>2 by classifying rank-one-free subspaces of M₂(F_q).

These follow-ups are not part of this contribution.

Related open work

• Prove radical invariance of right-vector costs over local products #4601, head 4b2322f, proves the invertible local-product formula and poses this singular-target question. Its invertible targets and this package's radical targets are disjoint, so there is no overlapping count to equate.
• Literally check #4601's radical-invariance cost formula on a noncommutative local algebra (F4[e;Frob]/e^2) and three further local rings: holds on every target (no new κ) #4613, head a59c2bf, literally checks Prove radical invariance of right-vector costs over local products #4601 on invertible targets over four local algebras. Even on its binary dual-number algebra, those invertible targets are disjoint from our 16 radical targets, so no numerical equality is claimed.
• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a, extends Prove radical invariance of right-vector costs over local products #4601 to invertible semilocal targets and their radical lifts. Our radical-entry targets have zero scalar residue and are disjoint from its invertible targets, including its 65,536 invertible lifts; its cost-three count and our 1,092 unreachable radical targets concern different endpoints, so no overlapping numerical count is asserted.
• Merged Prove the exact two-factor right-vector criterion at n = 2; the M2(F2) failure lifts to F2[S3] (no new κ) #4276 supplies the shared alphabet and triangular construction. Its 12 unreachable radical targets at dim(J)=2 agree with our independently enumerated 12; the new contribution is the quantified reachability theorem and all-dimension count.

Live overlap refreshed through 19:00 UTC covered 506 open PR bodies and 3,669 Discussion records with complete reported pagination. Main-source searches checked distinctive terms, the formula and counts across tracked research READMEs. The helper's local-note coverage remains sparse; arbitrary diffs, closed remote bodies and unpublished work are not exhaustively covered. The previously concurrent invertible semilocal claim is now PR #4749, listed above. Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 18:55:11 UTC were refreshed: #4745, #4746, #4747, #4748 and #4749. The Schur dimension, corank-one header and depth-six schedule results in #4746–#4748 study different objects and quantities. Only #4749 studies the same right-vector alphabet; its target domain is disjoint as stated above.

Claim: #1166 (comment)

CI notes

Both hosted checks passed at 3edea13 in run 39836662678: plan passed in 2m30s and the registered package replay passed in 1m30s. The replay log reports PASS_SQUARE_ZERO_RADICAL_REACHABILITY in both ordinary and optimized modes and PASS_REGISTERED_RESEARCH_PACKAGE. Local checks and their limits are recorded above. These checks certify the registered replay, not independent mathematical acceptance or formal proof-kernel verification.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only five files in the new research package are added.