[petalas]

Summary

Scoped algebraic proof: every n-role target over a central local K-algebra whose residue has division-algebra rank n-1 attains the paid right-vector column lower bound, including singular endpoints.

E = {v in K^n : Mv belongs to K^n}
cost(M) = delta(M) = n - dim_K M(E) <= n
delta(M) = min_{T in GL_n(K)} dim_K ((MT-I)K^n)

Fix independent base-field images by free charts.
Choose unit pivots through the penultimate column.
Replace the final column without inverting the singular endpoint.

Paid factors are I+u y^T with y over K; arbitrary free GL_n(K) maps are allowed anywhere. The residue division algebra may be larger than K or noncommutative. The proof assumes neither finite dimension nor a nilpotent radical. It supplies an attaining word with no auxiliary roles. Only the six files in research/local-corank-one-cost/ are added. No new kappa, supplier admission or selected-frontier change.

This contribution was produced by Nick Petalas. Credit ToikY #4334 for the singular-target gap; Nick Petalas #4276/#4243/#4787 for predecessor criteria, column bounds and local constructions; Wentao Li #4099 following #2513 for the alphabet and column-bound lineage. Basis extension, Gaussian elimination, Schur complements and local-ring unit lifting retain classical credit. The predecessor results are cited, not audited.

Evidence

• Before: Prove the exact two-factor right-vector criterion at n = 2; the M2(F2) failure lifts to F2[S3] (no new κ) #4276 covers unit-entry local targets on two roles; Prove exact local right-vector costs at every role count (no new κ) #4787 covers invertible local targets at arbitrary role count. After: PROOF.md handles residue-corank-one singular endpoints at every role count. The singular fixed-space formula counts M(E), rather than E: diag(1,1,0) has E=K^3 and exact cost one.
• The self-contained standard-library verifier checks 1,112 three/four-role targets across binary and ternary dual numbers, a binary two-generator square-zero algebra, F4 over F2 and noncommutative F4[e;Frobenius]/e^2 over F2. The binary dual-number domain includes all 512 lifts of diag(1,1,0); the other domains are explicitly bounded samples. Every literal paid word and all 8,272 basis-data executions pass.
• All 5,376 direct chart evaluations pass: every one of the 168 GL3(F2) charts is evaluated on each of 32 selected targets, independently comparing the definition of delta to the constructed cost. Eight domain/word rejection controls pass. Two additional shared two-role targets over F4[e]/e^2 have costs one and two, as detailed below. No predecessor census is rerun.
• Both warning-strict registered commands pass locally. They were timed separately at 0.269 s ordinary and 0.268 s optimized. Complete generated receipts agree and bind all six package files. The registered package runner passes.
• Committed-byte/source-history preflight passes at da6f858f21d407648a711d8fd369c37a9c6df24e. Official committed export integrity and optimized direct replay from a read-only Git-free export with PATH=/nonexistent pass with an identical generated receipt. In-memory compilation and staged whitespace checks pass. No receipt is handwritten.

Exact reproduction from the repository root, using CPython 3.12 and its standard library:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/local-corank-one-cost/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/local-corank-one-cost/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/local-corank-one-cost
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/local-corank-one-cost --check-source-history

Any CPython 3.12 path may replace the displayed local path. Expected markers are PASS_LOCAL_CORANK_ONE_COST targets=1112 controls=8 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Canonical main is pinned at 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, fetched again before committing. The formal-interface card is explicitly changed to this algebraic side question. Its historical baseline command stops with KeyError: 'bit_lower' under the changed selected schema; no baseline pass is claimed. Live claims/open bodies, distinctive main README terms and merged searches were refreshed before committing. The broader overlap helper covered 325 open PRs and 3,742 Discussion records with complete remote pagination; sparse local-note coverage was incomplete. These searches do not certify novelty.

Advance CLAIM.

Limitations and unrun obligations

The universal claim rests on the written scoped proof. The finite checks cover only the specified rings and bounded domains. Residue corank at least two and nonlocal algebras are outside the cost theorem. Physical coefficient preparation, frames, routing, restoration, precision, stock and a complete recursive invoice remain unknown. Independent human and formal proof-kernel review are unclaimed. Lean, Docker and the selected operator replay were not run. Every registered package command ran locally.

Open next questions

• Characterize which residue-corank-two local targets attain delta. Begin with one three-role target over F2[x,y]/(x,y)^2 with residue diag(1,0,0). A nonunit intermediate pivot needs a new completion argument.
• Specify one native dirty-input/output contract for a singular column factor and price preparation, frames, movement and restoration before using it in a supplier.

Related open work

• Prove exact local right-vector costs at every role count (no new κ) #4787, head 4b58e8cd5690bd35cf01b0018eefa2c39a298920, proves arbitrary-role invertible local costs. This theorem requires residue corank exactly one, so endpoint domains and sampled cost counts are disjoint. Its construction and credit are retained as predecessors.
• Count rank-one lifts over residue-field extensions (no new κ) #4781, head b838121c65e1bfd17a65466c95940eb00f12a225, counts two-role rank-one residue lifts over proper field extensions. Its targets satisfy this theorem at n=2. A literal shared check over F4[e]/e^2 over F2 gives cost one for diag(1,e), whose nonzero base-field image space has dimension one, and cost two for [[1,e],[e,0]], whose such image space is zero. These agree with its one-factor line criterion. Its full lift totals are cited, not rechecked or extended here.
• Prove the sharp singular rank-one product threshold (no new κ) #4780, head a4436fab4164eccc498d4d9885eec68211c869de, studies singular rank-one tuples over field products. Multiple components are nonlocal and outside this theorem. At the single-field boundary a separate literal check gives cost one for the shared binary projection diag(1,0), agreeing with its one-component law. No multi-component threshold or population is compared to our samples.
• Classify radical pencil reachability by characteristic-polynomial splitting (no new κ) #4776, head 0764ab259a8ce496a4c5d4debeed4c4e4d9f7ca4, classifies zero-residue square-zero pencils. For n>=2 those have residue corank n, so they are outside this theorem. At n=1 a separate literal check gives exact cost one for both x and x+y over the binary two-generator square-zero algebra, agreeing with both statements. No pencil census overlaps these samples.
• Classify exact rank-two matrix-algebra right-vector costs (no new κ) #4789, head 7fdbf8040ea9a957af0576067df2d9cb6fcf0e51, classifies two-role ordinary rank-two targets over M_m(K), m>=3. Those matrix algebras are nonlocal; no exact-cost census or target domain overlaps.
• Prove a binomial law for polynomial right-vector cost (no new κ) #4784, head 4d0a6363fe4f45dd9a769f5b74d02fc5d9061ae3, counts invertible polynomial-matrix costs over F_q[t]. That coefficient ring is nonlocal and those endpoints are invertible; no cost distribution or target count overlaps.

Related open work refresh before DONE

This account's open PRs created since the CLAIM at 2026-10-22T20:12:38Z were listed before DONE. The four additional same-alphabet titles and bodies were read:

• Prove three-factor sufficiency for every two-role matrix-algebra target (no new κ) #4805, head fbf2bd033f2d3ae1605a3cad690e1fc2fbf477e7, proves a three-factor upper bound for all two-role targets over full matrix algebras. Those algebras are nonlocal for block size at least two. At the shared scalar-block boundary, a fresh literal check gives cost one for the binary projection diag(1,0), compatible with its upper bound three. No matrix-block census overlaps our local-algebra samples.
• Classify exact singular polynomial right-vector costs (no new κ) #4808, head bd22290761451093aa7aef1fa7b1421627ab0f37, classifies singular two-role polynomial targets. K[t] is nonlocal. At the shared constant-field boundary its classification assigns cost one to diag(1,0); the fresh literal binary check above gives one as well. Its positive-degree census and our local-algebra samples are different populations.
• Count 9,216 ternary invertible threshold targets (no new κ) #4809, head 6354da68f7a78b9e4db91c796476ae9ebae89a86, counts invertible targets over F3^4. That nonlocal coefficient algebra and its invertible endpoints differ from this local residue-corank-one theorem; no target count or exact-cost instance overlaps.
• Prove sharp five-factor cost for binary three-role products (no new κ) #4810, head 701cb14b7c461253d7ac8243d0722996918bc6c6, gives the sharp uniform bound for invertible three-role binary field-product targets. Our endpoints have residue corank exactly one, and its sharp coefficient algebra is nonlocal; no target-count or uniform-bound number overlaps.

The other newly opened account PRs #4801/#4803 study Schur spectra/rank pairs, and #4806/#4807 study full-guard probabilities. They do not assert this right-vector factor cost. No discrepancy was found in the stated shared boundary comparison. No predecessor verifier or census was replayed.

CI notes

Both hosted checks passed on contribution head da6f858f21d407648a711d8fd369c37a9c6df24e in Registered research packages run 39848080362: plan in 2m35s and the registered research/local-corank-one-cost replay in 1m25s. Run metadata confirms that exact head and conclusion success. The replay executes both warning-strict ordinary and optimized verification commands. All registered local commands, committed-byte/source-history checks, export integrity and optimized Git-free replay also passed. Hosted CI certifies execution of these bounded package checks; independent human and formal proof-kernel acceptance of the universal proof remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: isolated.

Only one new six-file research package is added. Shared code, frozen evidence, predecessor credit and the selected result are unchanged.