[petalas]

Summary

Scoped algebraic proof with finite corroboration. In the paid alphabet I+u v^T on n roles, with u over R=K[x,y]/(x,y)^2, v over K and arbitrary free GL_n(K) maps:

A has no K-valued left eigenvector
    => xI+yA is unreachable at every finite paid length.

The proof follows the first zero scalar prefix and shows that an invertible radical coefficient forces a common row for the two coefficient matrices. Later factors preserve that restriction. This extends the two-role square-zero direction of #4745 to a necessary condition at every role count. It does not recheck that predecessor or claim a converse.

The binary three-role companion of t^3+t+1 gives an explicit excluded target. Exactly 48 of the 512 binary three-by-three matrices have no scalar left eigenvector; all 48 corresponding pencils are excluded by the theorem. This is not the complete unreachable-pencil count. No new kappa, supplier admission or selected-frontier change.

Only research/multirole-radical-pencil/ is added. This package was produced by Nick Petalas. Credit Nick Petalas #4745 for the two-role square-zero theorem and first-zero idea, #4243 for the free-map normal form, Wentao Li #4099 following #2513 for the alphabet, and ToikY #4334 and Nick Petalas #4276/#4601/#4749 for the related two-role successors. No predecessor code is imported; the proof and verifier are self-contained.

Evidence

• Before: Prove exact square-zero radical reachability and binary count (no new κ) #4745 classifies two-role square-zero radical targets and explicitly excludes more roles from its scope. After: the written proof gives the displayed arbitrary-field, arbitrary-length necessary condition at every role count.
• All 196 compatible binary first-zero scalar pairs pass all 802,816 single-coefficient transition checks. Exactly 150,528 have an invertible new coefficient, and every one has a nonzero common-row multiplier.
• Direct enumeration of all 512 binary three-by-three A gives 48 excluded pencils. All 168 free conjugations of the displayed companion preserve the obstruction and give 24 distinct conjugates.
• Three legal reachable controls pass 1,536 chronological comparisons on all dirty ring inputs. They distinguish a rootful triangular pencil, a singular first coefficient, and identified radical variables. Two further controls detect reversed chronology and an omitted radical contribution.
• Both warning-strict registered commands passed locally on CPython 3.12, timed separately at 0.318 s and 0.319 s. The registered package runner, five-file committed-byte/source-history preflight, official committed export and export-integrity check pass at f5d0f421efec55c5f4b3dbb142eee1e79a4e21db. Optimized Git-free replay passes in 0.330 s. All three generated receipts agree completely and bind all five package files by SHA256.

Reproduce from the repository root with CPython 3.12 and the standard library:

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

Any CPython 3.12 installation may replace the displayed interpreter. Expected markers are PASS_MULTIROLE_RADICAL_PENCIL transitions=802816 full_rank=150528 excluded_pencils=48 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

The originating card and selected record are pinned at b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. The branch starts from refreshed canonical main 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b; relevant guidance and the selected record are unchanged. The connector card's 945-pairing positive control passed, and TASK.json explicitly changes scope to this algebraic side question. Live PR bodies, recent claims, local reservations, distinctive main terms and merged PR searches were refreshed before the commit. The overlap helper's remote pagination was complete at approximately 19:22 UTC; sparse local-note coverage was incomplete. This describes search coverage, not proof of novelty.

Limitations and unrun obligations

The universal claim is the written scoped proof. Executed corroboration is binary and three-role. A scalar eigenvalue is not proved sufficient. Two independent square-zero radical generators, base-field right vectors and base-field free maps are essential hypotheses. Restored auxiliary roles, ring-valued free maps and higher nilpotency are not covered. No native preparation, routing, restoration, precision, stock or complete supplier cost is supplied. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and selected operator replay were not run; every registered package command ran locally.

Open next questions

• Is an eigenvalue sufficient for reachability when n>=3? Start with the binary direct sum of a scalar block and the companion of t^2+t+1. The missing step is a recursive condition on the common row's quotient.
• Can a restored auxiliary role make the excluded cubic pencil reachable? Start with diag(xI_3+yA,I_1). Its nonzero scalar endpoint is outside the first-zero proof.

Related open work

• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13af13b755932852e9306a714b317630dda, supplies the two-role square-zero theorem and the first-zero idea. Our executed three-role census and its two-role census are disjoint; the reported 48 excluded pencils and its 12 unreachable radical targets are different quantities. No shared numerical total is claimed or rechecked.
• Count square-zero radical targets over every finite field (no new κ) #4766, head 5af823b1a14b2517235925145e36581bfdfcad41, counts all two-role radical targets over every finite field. We give a multi-role necessary condition and execute only three-role pencils, so no numeric census overlaps.
• Prove a cube-zero radical target has exact cost three (no new κ) #4764, head b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511, has a cost-three two-role target in a cube-zero ambient algebra. Our ring kills all radical products and our executed targets have three roles. Its exact cost and our unreachable-pencil count do not overlap.
• Count rank-one residue lifts over local algebras (no new κ) #4770, head 39ef149ba3ea8adbda18478ea227d3d0826475ab, counts two-role lifts of nonzero rank-one residue matrices. Our targets have zero scalar residue, so the target domains and numerical totals are disjoint.
• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891, studies invertible semilocal targets. These are disjoint from our zero-residue radical targets; no cost histogram or census total overlaps.
• Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763 studies invertible two-role targets over F2[t]. Its nonsemilocal coefficient ring and endpoints differ from the square-zero radical pencils here; no numerical claim overlaps.

Claim: #1166 (comment)

CI notes

CI pending on f5d0f421efec55c5f4b3dbb142eee1e79a4e21db. No hosted certification is claimed yet. All registered local checks, committed-byte/source-history checks and the exported replay pass as recorded above.

Merge Danger

Door: two-way.

Blast Radius: isolated. Five new package files; no shared code or selected record changes.