[petalas]

Summary

Scoped counting proof for the two-role right-vector alphabet. Let R be a finite-dimensional local F_q-algebra with central base field and canonical residue R/J=F_q. Put z=|J|. Above each fixed rank-one residue matrix, the exact distribution is

one paid call:   q*z^2-q+1
two paid calls:  z^4-q*z^2+q-1

The formula is independent of radical multiplication and includes noncommutative algebras and higher nilpotency. The new contribution is this uniform count. The one-factor criterion and unit-pivot two-factor upper bound are credited to Nick Petalas's #4276, following #4243 and Wentao Li's #4099/#2513. #4601 posed the singular-target direction, following ToikY's #4334. Produced by Nick Petalas. No new kappa or selected-frontier change.

Evidence

• Before: the two-factor upper bound was known, but no uniform count above each rank-one residue was supplied by the cited packages. After: PROOF.md counts the q eligible projective kernel lines and their intersections, giving the displayed law for every q and radical dimension.
• The independent standard-library verifier enumerates 63,417 lifts over all 152 rank-one residues in seven local algebras, including F4, two distinct dimension-two binary radicals and a noncommutative local algebra. It compares each lift with the complete literal one-factor image, constructs an attaining two-factor word, and checks its action on every basis input. Controls discriminate zero scalar images, reversed or omitted factors, odd-field signs and noncommutative multiplication order; invalid field orders are rejected.
• Both registered warning-strict normal and optimized commands pass locally on CPython 3.12.15, timed separately at 0.39 s and 0.41 s. Their generated receipts are identical and bind all six package files by SHA256. The registered runner, six-file committed-byte/source-history preflight, committed export integrity and optimized read-only Git-free export replay pass at 39ef149ba3ea8adbda18478ea227d3d0826475ab.

Exact reproduction from the repository root:

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

Any CPython 3.12 installation may replace the displayed interpreter. Expected markers are PASS_RIGHT_VECTOR_RANKONE_RESIDUE targets=63417 residues=152, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

The scientific baseline is b58ad9ef4d4e861b0eaa5dba41742bdc072dca51. Pre-publication fetch reached 1f87c3578ae546c8f903f7f4d83dc2e2fb99650b; relevant guidance and both predecessor proof inputs were unchanged. The original card's 945-pairing positive control passed before the explicitly recorded change to this counting side question. Live open PR bodies, current claims, account PRs since CLAIM, distinctive main-source terms and merged PRs were checked. The overlap helper reported complete remote pagination with 426 open PR records and 3,682 Discussion records; local-note coverage was sparse. This records search coverage, not a novelty certificate.

Limitations and unrun obligations

The universal claim is the written scoped counting proof. Finite corroboration covers only the seven stated algebras. Two roles and the canonical scalar residue field are essential. Zero-residue targets, proper residue-field extensions, matrix-block residues and products of local algebras are outside scope. Native coefficient preparation, conversion, routing, control, restoration, precision, stock and complete supplier cost remain unknown. No supplier admission or complete multiplication theorem follows. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected operator replay were not run; all registered package checks ran locally.

Open next questions

• Determine the lift distribution over a proper local residue-field extension L/K. Start with one rank-one residue in F4[e]/e^2 viewed over F2; the common right vectors are still F2-valued.
• Specify one literal native producer/consumer and its complete preparation, access, restoration and precision bill before using the formal count in a supplier.

Related open work

• Prove the semilocal right-vector cost formula (no new κ) #4749, head 29e4d3a25225a219f97c3a2815ed3a5a8fd9b891, treats invertible semilocal targets, disjoint from our singular rank-one-residue targets. Our separate binary dual-number invertible compatibility control is [6,54,36], agreeing numerically with its reported control and merged Prove radical invariance of right-vector costs over local products #4601/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.
• Prove exact square-zero radical reachability and binary count (no new κ) #4745, head 3edea13af13b755932852e9306a714b317630dda, treats radical targets with zero residue. Those endpoints are excluded here, so no target count overlaps.
• Count square-zero radical targets over every finite field (no new κ) #4766, head 5af823b1a14b2517235925145e36581bfdfcad41, extends that zero-residue count to all finite fields. Its zero-residue census and our nonzero rank-one-residue census are disjoint; no numerical total is shared.
• Prove a cube-zero radical target has exact cost three (no new κ) #4764, head b1b163cd5553c75c9bd5ca3ac31a2eab4e2f3511, gives a zero-residue cost-three target in a cube-zero local algebra. Its endpoint has no unit entry, whereas every endpoint here does. No target-cost or census number overlaps.
• Prove unbounded two-role right-vector cost over F2[t] (no new κ) #4763, head d0158a552352c09608295a7ec47dd31327b360ae, proves unbounded invertible-target costs over F2[t]. That infinite nonlocal algebra is outside the finite-dimensional local hypothesis here; no target-cost or census number overlaps.

Advance CLAIM.

CI notes

CI pending for 39ef149ba3ea8adbda18478ea227d3d0826475ab. All registered commands passed locally. No hosted result is claimed yet.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only the six files in the new research package are added.