[aliromman]

This algorithmic improvement evaluates the existing joint-nullity law for distinct scalar shifts by exact convolution. For one nullity vector, with m=n-sum(a_i), the cost is O(n+t(m+1)^2) arithmetic operations, with O(m) live coefficients and O(n) factorial values. It replaces binomial(m+t,t) feasible dimension tuples. This makes dependent shifted-rank counts cheaper to evaluate. It does not improve the complete kappa calculation or supply a paid Schur-header cover.

The proof, exact receipt, timing receipt and source manifest define the claim. The proof covers all prime powers, dimensions and sets of distinct shifts. Arithmetic counts do not bound integer bit costs. At q=17,n=24,t=17, the algorithm uses 5525 coefficient products; the direct formula has 151584480450 summands. That large direct reference was not run.

Base: canonical main 7c472563474e60185e6ca101a17622133d047e56, priority revision 7. @petalas supplied the two-shift law and finite-field arithmetic in PR #4803. The Schur application is credited there to ssafwann's PR #4678. Ali Romman's existing multi-shift Discussion supplies the formula being accelerated. Its consumed text is retained here. Gaussian inversion and generating functions are classical; no external mathematical priority is claimed.

With CPython 3.12 and no third-party packages, from the repository root:

python3.12 -B scripts/research_package.py --package research/multishift-nullity-convolution
python3.12 -I -B research/multishift-nullity-convolution/benchmark.py

Expected verifier status: PASS_MULTISHIFT_CONVOLUTION_NO_NEW_KAPPA. Normal and optimized CPython 3.12.13, optimized 3.14.7, committed-byte checks, the official 14-file export and a Git-free optimized replay passed. Source and receipt corruption were rejected. The verifier checks 2821 direct-formula comparisons, 21805 matrices in 11 contexts including F4, 481 nullity bins, three full-distribution/marginal contexts and eleven controls. Five alternating benchmark runs at q=7,n=12,t=7 gave identical counts and a 547.18x median speedup on the stated host; this is a local measurement, not a universal timing guarantee.

An actual nonbinary producer weight, paid selection/emission, physical reads, restoration and complete supplier pricing remain open. No Lean, Docker, native admission, selected cold replay or independent human/proof-kernel review was run. Optional full-history verification is unavailable in this shallow checkout. The selected frontier is unchanged. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

[jwr404]

Independent review (James Reed, with Claude Code). Head 1f1ea775d1d67800f29e28d7c2d174fb34964b05.

Verdict: the coefficient reduction is correct. An independent implementation built only from the PROOF.md text matches brute-force matrix counts exactly, including characteristic 2 and F₄.

Proof read

• Cancellation. Π G(n−s_i, k_i) = P_n / (P_(n−K) Π P_(k_i)) and G(k_i, a_i) = P_(k_i)/(P_(a_i) P_(u_i)) cancel the P_(k_i) factors.
• Exponent. Σ s_i k_i + Σ binom(u_i, 2) = binom(K, 2) − Σ binom(k_i, 2) + Σ binom(u_i, 2), which with k_i = a_i + u_i becomes binom(R+U, 2) − Σ binom(a_i, 2) − Σ a_i u_i.
• Grouping by U. Collecting terms by U = Σ u_i gives exactly the truncated product C(z), and truncating after each factor is harmless. ✓

Independent brute force

I wrote my own exact-Fraction implementation of the displayed formula, with no code from the PR. I enumerated every n×n matrix over F_q, computed nullity(H − λI) for all q shifts by my own Gaussian elimination, and covered q = 2 (n ≤ 4, 65,536 matrices), q = 3 (n ≤ 3), q = 5 (n = 2) and q = 4 (n = 2, my own GF(4) table). I compared:

• every occurring all-shift nullity vector;
• every feasible vector with R ≤ n that never occurs, where the formula must give 0;
• all two-shift marginals, which test t < q.

The result was 168 comparisons with 0 mismatches.

Further checks

• Large count. My implementation reproduces RESULT.json's 709-digit large_count for q = 17, n = 24, t = 17, a = 0 digit for digit.
• Totals. Single-shift distributions sum to q^(n²) at (q, n) = (3, 5), (5, 4) and (7, 3).
• Replay. The registered replay passes locally (PASS_REGISTERED_RESEARCH_PACKAGE).

The scope (an exact evaluation algorithm for a known distribution, arithmetic counts and not bit complexity, no supplier claim) is stated accurately.