[petalas]

Summary

Prove the exact count and determinant distribution of optimal second callees for a regular first map P=J_3(lambda) over every finite field. This settles the regular three-dimensional case explicitly left open by Nicholas Stone's merged PR5556. Claim type: scoped written proof with exact finite corroboration. No new kappa is claimed.

For lambda nonzero and lambda^2!=1:

all optimal Q = P+(P^2-I)S, S^2=0, det(Q)!=0     [credited PR5228]
  rank-one determinant filter                  [new J3 calculation]
    q^4-q                  if lambda^2=-1
    q^4-q^2-q              otherwise

The full determinant distribution is also proved, including characteristic two. Only six files in research/jordan-three-callee-count are added.

Evidence

Before, PR5556 computes J2 and leaves J3 open. After, PROOF.md counts every rank-one square-zero fiber and verify.py compares the formula with literal matrices:

• 62,512 distinct square-zero parameters in fields of sizes 2,3,4,5,7,8,9,11,13.
• 519,494 second-map determinants in all 38 regular field/lambda contexts.
• Independent complete enumeration of all 262,144 matrices over F4 and 1,953,125 over F5, recovering exactly 316 and 745 square-zero parameters.
• All six input columns of the literal wrapper for 1,712 accepted J3 maps over F4/F5, with error rank three.
• Thirteen rejection controls and extension-field, non-square-zero and invertible-but-suboptimal discriminators.

Local warning-strict normal and optimized checks pass on CPython 3.12.15, macOS arm64, in 4.59 and 4.73 seconds. Both registered commands pass through the package launcher. Each has a 180-second timeout. Metadata/pins, AST compilation, exact committed bytes/modes and full source-history ancestry pass. Official export integrity and declared-input coverage pass. Both direct modes also pass from a read-only Git-free export with an empty executable PATH; all seven exported files retain their bytes and modes.

From the worktree root, with the standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/jordan-three-callee-count/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/jordan-three-callee-count/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/jordan-three-callee-count
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/jordan-three-callee-count --check-source-history --check-modes

Another CPython 3.12 path may replace the interpreter prefix. Expected markers are PASS_JORDAN_THREE_CALLEE_COUNT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The verifier emits a generated package-hash receipt and writes nothing. Predecessor sources are historical provenance, with no imported predecessor executable or runtime dependency.

Limitations and unrun obligations

This is an algebraic side result, explicitly changing the uniform-transfer-audit card rather than claiming to discharge a selected assumption. The general wrapper characterization and its optimum are cited premises. Resonant lambda^2=1, larger or multiple Jordan blocks, arbitrary first maps, physical/native implementation, restoration, movement, precision, stock, compiler and complete supplier/outer costs remain outside scope. No selected cold replay, predecessor scientific suite, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. No registered check is deferred to CI alone.

Open next questions

• Count the resonant J3 branch using PR5228's noninvertible-A characterization, separating characteristic two.
• Choose two explicitly specified Jordan blocks and count their coupled square-zero parameters. Off-diagonal couplings prevent a naive product rule.

Related open work

No open PR studying this regular J3 optimal-second-callee count or determinant distribution was identified in the live targeted searches. The completion refresh read the titles and bodies of all 13 other open account PRs created since the advance CLAIM. None computes this J3 count, so no numerical agreement or disagreement is asserted. PR5556 and PR5228 are merged predecessors.

• PR6402 studies complex Euclidean error of square-zero tensor powers. This result counts finite-field square-zero parameters through a determinant filter; the quantities differ.
• PR6407 gives pure-shear swap tariffs. Its stage alphabet and tariff differ from the fixed ordinary wrapper and optimal-callee count here.

No additional computation was run to check those contributions.

Credit and source scope

This contribution was produced by Nick Petalas. Credit Nicholas Stone / PR5556 at f87b6ac for the question, filtered Jordan method and rank-one suggestion; Nick Petalas / PR5228 for the general characterization; Jerry Chen / CubeJerry / PR2950 for the wrapper optimum and attainer; and the retained ordinary/native lineage. Rank-nullity, rank-one determinants and finite-field fiber counting are classical. Apache-2.0; no general literature-priority claim.

Base and refreshed canonical main are 7c49fe3, priority revision 8. Advance CLAIM. The delivered scope matches the claim. Live open/merged searches, current main README searches, recent account work, local reservations and Discussion claims were checked. Unpublished work and arbitrary historical diffs remain coverage limits.

CI notes

Both hosted checks passed at exact head 1eeff4a in run 41527963342: plan in 4m43s and replay (research/jordan-three-callee-count) in 1m55s. The completed checks were read once; separate run metadata confirms completed/success and the matching head. Hosted replay certifies the registered bounded checks, including normal and optimized execution. Local checks listed above also pass. Independent human and formal proof-kernel acceptance of the all-field written proof remain unclaimed. No CI failure remains.

Merge danger

Door: two-way. Blast radius: isolated. Six new files in an opt-in research package. No shared implementation, selected-frontier record, frozen source or predecessor credit is changed.