[petalas]

Summary

Scoped written proof with exact finite corroboration. Classifies every optimal linear second callee in Jerry Chen / CubeJerry's fixed ordinary-wrapper interface, over every field and dimension. PR2950 supplies the optimum value and Q=P as an attainer; the new result is the complete equality set.

A = P^2-I, R = Q-P, choose A G A=A
Q is optimal iff Q is invertible and
  im R <= im A, ker A <= ker R, R G R=0

Equivalently, the perturbation induces a square-zero endomorphism on V/ker A. For scalar P=lambda I with lambda nonzero and lambda^2 different from 1, every optimal Q is lambda I+R with R^2=0. The exact count is

sum_r [n choose r]_q [n-r choose r]_q |GL_r(F_q)|.

At dimension two it is q^2. Thus P=2I over F5 has exactly 25 optimal second maps, each correct on 25 of 625 pairs. Optimal agreement does not give exact interchange or a new kappa.

Only research/optimal-second-callees/ is added. The proof, verifier and provenance are self-contained. The contribution is produced by Nick Petalas. Credit Jerry Chen / PR2950 for the known optimum; Nick Petalas / PR2808/2786, David Arias / PR2753 and Jose Manuel / PR2697/2899 retain the caller lineage. Square-zero flag counting and block rank elimination are standard techniques.

Evidence

• Before: PR2950 gives the optimal value and one attainer. After: a necessary-and-sufficient classification, including singular A and noncommuting P,Q, plus the scalar finite-field count.
• Literal six-operation execution on every basis input passes for 258,624 ordered map pairs in GL_3(F2) and GL_2(F5).
• Separate finite checks pass for 6,817 arbitrary A,R pairs, all 1,087 inner inverses in the declared small domains, 88,871 matrices for square-zero rank counts, and six control groups. Controls retain both span conditions, the square-zero condition and candidate-Q invertibility.
• Warning-strict direct normal and optimized replays pass in 13.784 and 14.002 seconds on CPython 3.12.15, standard library only. Each registered check has a 300-second timeout. The registered package replay, metadata/pins, syntax inspection and source-history/committed-byte checks pass locally.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/optimal-second-callees
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/optimal-second-callees/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/optimal-second-callees/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/optimal-second-callees --check-source-history

Any CPython 3.12 path may replace the local path. The direct marker is PASS_OPTIMAL_SECOND_CALLEES no_new_kappa. It prints a generated receipt bound to all six package files and writes nothing.

Limitations

This changes the uniform-transfer-audit card to an original algebraic equality question at PR2950's boundary. It does not audit #4100 or replay PR2950's census. The written theorem is all-field; finite evidence is limited to its declared prime-field domains. General finite abelian groups, nonlinear callees and changed wrapper operations are outside scope.

No physical callee, normalization cost, frame/hosting/restoration implementation, tape/precision bound, recursive supplier or multiplication theorem is supplied. Independent human and formal proof-kernel review are unclaimed. The selected cold replay, Lean and Docker are unrun. The selected frontier is untouched.

Open next questions

• Count the invertibility-filtered square-zero quotient maps for nonscalar P from its rational canonical data. Start with one nonsemisimple block and retain singular-Q rejection.
• Pin an actual weighted caller and determine whether a classified second map has a smaller complete invoice on the required correctness domain. No physical caller or cost saving is established here.

Related open work

No open PR studying this fixed-wrapper optimal-second-callee set or its count was found in the live precommit scan. PR2950 is the merged predecessor. Wentao Li's merged PR4053, PR4063 and PR4070 concern paid normalization and reciprocal weighted pairs, rather than this equality classification.

CI notes

CI pending for fcc9079. Local registered normal/optimized replay and publication preflight pass. Hosted CI has not yet certified this commit.

Merge danger

Door: two-way. An isolated research package with no shared-code changes.

Blast radius: research.