[petalas]

Summary

Scoped proof for invertible two-role right-vector targets over finite products of associative local K-algebras. With J the product of the component non-unit ideals:

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

This answers Open next question 1 of ToikY's #4334. Local radicals preserve the three-factor obstruction, while delta accounts for increases in smaller residue costs. The sharp component threshold q+1 extends to these local products. This is a formal-alphabet result, with no new kappa or selected-frontier change.

Evidence

• Before: Prove the sharp component threshold for commutative right-vector targets #4334 proves the field-product case and explicitly leaves the unit/right-ideal reduction for local products open.
• After: research/right-vector-local-products/PROOF.md proves the reduction and exact formula, with no commutativity or finite-dimensionality assumption on the local components.
• Literal enumeration on all 894,048 invertible targets over (F2[e]/e^2)^k, k=1,2,3, agrees with the formula. Exact cost histograms are [6,54,36,0], [6,558,8652,0], and [6,4590,830988,49152].
• The enumerator constructs every zero/one/two-factor set from literal common right vectors and common free endpoints. It does not use a pivot test or import predecessor verifiers. Independent delta and pivot computations agree. Explicit three-factor words and changed-factor, reversed-word, common-chart, dropped-component and nilpotent controls pass.
• Both registered warning-strict normal/optimized checks pass and emit identical generated receipts bound to file hashes. Locally timed checks took 0.21 and 0.32 seconds, each below the 120-second registered limit.
• The package runner passes. Committed-byte/source-history preflight passes for all eight package/input files at 4b2322fc5bd30a129507c2eee69aa275527b69ac. Exact source pins are in SOURCES.json.

From the repository root, using CPython 3.12 and NumPy 2.5.4:

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

Expected verifier marker: PASS_RIGHT_VECTOR_LOCAL_PRODUCTS targets=894048 cost3=49152 formula=exact, followed by the generated digest and no_new_kappa.

Limitations

The proof requires an invertible two-role target and component non-units forming proper two-sided ideals. The finite census covers binary dual-number products; it does not enumerate noncommutative local algebras. Native conversion, preparation, routing, precision, restoration, stock and complete supplier costs remain unknown. No full multiplication theorem, supplier admission, independent human acceptance or formal proof-kernel review is claimed. Lean, Docker and selected construction replays were not run.

Credit: produced by Nick Petalas. ToikY #4334 supplies the field-product theorem, the three-factor formula extended here and the sharp scalar examples. Nick Petalas #4276 supplies the one/two-factor criteria, following #4265/#4243. Wentao Li #4099 supplies the alphabet and column bound, building on #2513. No predecessor credit or frozen input is changed.

Open next questions

• Extend the reduction to semilocal algebras with matrix blocks in the quotient. First specify the M2(K)-block pivot and lifting contracts; the non-unit-ideal argument is unavailable there.
• Classify reachable singular targets and their three-factor image. Start with F2[x,y]/(x,y)^2; invertibility is essential in the present proof.

Related open work

No other open PR on this right-vector/local-product cost question was found in the refreshed open-body search and live right-vector search. Merged #4334 and #4347 cover field products and extension-field controls. The enumerated three-component residue cost-three count is 12, numerically agreeing with #4334; the new dual-number result has 49,152 cost-three lifts. Sparse local-note and historical/diff coverage remains partial.

Claim: #1166 (comment)

CI notes

CI blocked by main for head 4b2322fc5bd30a129507c2eee69aa275527b69ac. In run 39755232646, plan failed in Onboarding helper tests: 331 tests ran, with four failures and 11 skips. The package replay job was skipped, so hosted CI has not certified this package. Local verification stands as recorded above.

The exact four failures are in tests/test_research_current_status.py:

• test_current_ci_section_matches_selected_record: the current CI navigation section lacks the literal make frontier4359-verify command.
• test_current_section_matches_selected_record: the status section has **0.0023 = 23/10000**, while the test requires **0.0023**.
• test_history_banner_precedes_historical_sections: the opening section lacks the required historical snapshot banner.
• test_only_the_first_heading_claims_current_or_selected: the historical heading still contains selected.

All four reproduce on unchanged origin/main at f626db5c3a3d781d8f7695038776c621651f75e1. An exact export of that commit's test and document/record inputs, run with ~/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error tests/test_research_current_status.py -v from the export root, ran six tests: these four failures and two passes. The PR's diff contains only its six new package files.

The failed logs of two other contributors' runs contain the same four failures and assertions: check-4585, run 39754840618, and research/receiver-all-length-floor, run 39746873103. Each also reports 331 tests, four failures and 11 skips. This is a shared onboarding/documentation failure. No shared files were changed, and DONE remains pending green CI.

The unrelated orientation helper scripts/research_baseline.py fails on the selected certificate at base f626db5c3 with KeyError: 'bit_lower'. After materializing the selected inputs, scripts/verify_selected_result.py passes with 23/10000. The new package does not use the baseline helper. No shared-code repair is included.

Related open work refresh

This account's open PRs created since the CLAIM at 2026-10-22 09:15:59 UTC were refreshed and their bodies read. Apart from this PR, they are:

• Prove the sharp 2d block Schur commutator bound (no new kappa) #4597, head dfeb8ca729a04aea4aa290438f2ed50a37a9c7cb: block-Schur commutator identity and rank bound. Different object and quantity; no numerical overlap with right-vector costs.
• Count 2,880 five-request binary threshold headers; refute 2 k! (no new κ) #4598, head c80f445e8c3c22d341d467254b89245fbddb9eb2: five-request binary threshold header count. It counts complete projector headers; no right-vector cost is computed.
• Prove the four-request binary full-guard product inequality (no new kappa) #4599, head 19ca650c67ad2722790dc60c0bfd12989479b06a: four-request binary full-guard product inequality. It computes header probabilities; no right-vector cost is computed.
• Prove the two-short-row connector obstruction for every modulus #4600, head 44640c4a9538d5565d33e6dc218889f130143ce7: depth-four connector short-row obstruction and construction. Different alphabet and quantity; no numerical overlap with this package.

None studies this local-product right-vector cost question, so no cross-PR numerical equality check applies to the refreshed set. The earlier numerical agreement with merged #4334 remains recorded in Related open work above. Historical and unpublished coverage remains partial.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only the new research package is added. No shared verifier, frozen source, selected record or supplier changes.