[petalas]

Summary

Scoped proof answering Open next question 1 of #4601. For every invertible two-role target over a semilocal central K-algebra R, with Jacobson radical J:

cost_R(M) = max(delta_R(M), cost_(R/J)(Mbar)) <= 3
cost_R(M) <= 2  iff  cost_(R/J)(Mbar) <= 2

Three explicit paid factors have fixed base-field right vectors e1,e2,e1, with no free map needed for the upper bound. The new step admits arbitrary matrix blocks in the semisimple quotient, rather than requiring R to be a product of local algebras. This includes every finite-dimensional K-algebra. No new kappa or selected-frontier change.

Produced by Nick Petalas. Credit Nick Petalas #4601/#4276/#4265/#4243, ToikY #4334, and Wentao Li #4099 following #2513. Stable-range unit completion and ordinary ring-matrix elimination are classical; the new result is their scoped application to the restricted-right-vector cost and radical formula. The proof includes the unit-completion argument and every multiplication order.

Evidence

• Before: Prove radical invariance of right-vector costs over local products #4601 proves the local-product formula and explicitly leaves matrix-block semilocal quotients open.
• After: PROOF.md proves the semilocal theorem and gives the attaining word. Its construction works on arbitrary input data with two data roles and no scratch.
• The verifier constructs words for all 20,160 invertible targets over M2(F2), and all 65,536 radical lifts of one fixed sharp residue target to M2(F2[e]/e^2). Every lift has exact cost 3. Every word is checked by block-ring multiplication and independently through all eight binary basis coordinates of a faithful representation. The residue sharpness is predecessor credit.
• Seven negative controls discriminate scalar-only completion, reversed order, a dropped factor, residue-only substitution, an uncorrected residue inverse, a singular input, and a wrong sign over F3.
• Both warning-strict normal and optimized registered commands pass locally, timed separately at 1.349 and 1.335 seconds using CPython 3.12. The package runner passes. Both modes emit the same generated file-bound receipt. The committed-byte/source-history check passes for eight package/input files at 29e4d3a.

From the repository root, with the Python 3.12 standard library only:

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

Any CPython 3.12 installation can replace the local interpreter path. Expected marker: PASS_RIGHT_VECTOR_SEMILOCAL, with residue targets=20160, radical targets=exact_cost_three=65536, and seven negative controls. The registered runner ends with PASS_REGISTERED_RESEARCH_PACKAGE. Source inputs are pinned to ancestor b58ad9e.

Limitations and unrun obligations

The target is invertible, the role count is two, and the algebra is semilocal. The proof does not classify singular targets or algebras outside that class. Native preparation of algebra-valued left coefficients, conversion, routing, control, precision, restoration, stock and complete supplier cost are unknown. No native supplier or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected operator replay were not run; no registered package obligation remains unrun locally.

Open next questions

• Classify a named singular residue family over a matrix-block radical extension. The unit-pivot proof does not apply there. The split square-zero local case is separately addressed by Prove exact square-zero radical reachability and binary count (no new κ) #4745, so check live overlap first.
• Specify and price algebra-valued left-coefficient preparation for one literal native producer/consumer pair before extending this into a supplier construction.

Related open work

• Prove radical invariance of right-vector costs over local products #4601, head 4b2322f, supplies the local-product theorem and the semilocal question answered here. The shared binary dual-number compatibility control has 96 invertible targets and histogram [6,54,36,0], numerically agreeing with its reported k=1 histogram. This control is not a replay of its engines.
• 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, checks Prove radical invariance of right-vector costs over local products #4601 on four local algebras, including a noncommutative one. Its 96-target binary dual-number control also agrees numerically with [6,54,36,0] here. The new matrix-block theorem is outside its finite check.
• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13, settles radical singular targets over split square-zero local algebras. Those targets and our invertible targets are disjoint, so there is no shared target count to equate. It is complementary to this semilocal invertible-target extension.

Live PR bodies and handoff claims were rechecked before committing, together with distinctive main-source and merged-PR searches. The overlap helper covered 506 open PR bodies and 3,662 Discussion records with reported complete pagination. Its local-note coverage is sparse; Git searches separately inspected relevant main READMEs. This is search coverage, not a proof of novelty.

Claim: #1166 (comment)

Related open work refresh

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22 18:56:10 UTC were listed and their current titles and bodies read. Besides this PR, they are #4745, #4746, #4747 and #4748. The only one studying the same right-vector alphabet is #4745, already listed above; its singular target domain is disjoint from the invertible targets here. #4746 determines Schur ambient dimensions, #4747 counts full-guard headers, and #4748 classifies depth-six chain schedules. None supplies an overlapping right-vector cost or census number. The earlier 96-target compatibility agreement with #4601/#4613 remains recorded above.

CI notes

Both hosted checks passed on 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891 in Registered research packages run 39836998080: plan passed in 2m41s and the registered research/right-vector-semilocal replay passed in 1m18s. The hosted replay certifies the registered package checks at this exact commit within their stated finite scope. It does not supply independent human acceptance or formal proof-kernel verification of the universal theorem. All registered local checks and the committed-byte/source-history check also passed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only the six files in the new research package are added. Existing source inputs and the selected record are unchanged.