[petalas]

Summary

Claim type: scoped algebraic proof, direct construction and exhaustive finite corroboration. No new kappa.

The normalized labelled characteristic-two Desargues interface satisfies

exact thirty-incidence interface <=> rank_F2(1,u,v,w) = 4
normalized labelled count over Fq = (q-2)(q-4)(q-8)

Here B0=(u,1,0), B1=(v,0,1), B2=(1+w,1,1), after fixing the ordered projective frame. A binary basis beginning with 1 directly constructs the parameters. F32 works without an F16 subfield.

This answers Open next question 4 of Sagar Thakrar / PR2274. The new repository contribution is the explicit parameter criterion, constructor and complete polynomial certificate. The predecessor's scientific suites were not replayed. Classical embedding and enumeration priority is unclaimed. Bruen, Bruen and McQuillan's primary paper gives the simplex-section context and distinguishes extra incidences; it also credits Puystjens and Thas for earlier strict enumeration.

Produced by Nick Petalas. Credit thakrarsagar for the question and normalization, DreamingOfClouds for the retained three-total/CR2 context, and the classical projective geometry and finite-field lineage. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before, PR2274 left the all-field explanation and direct parameter criterion open. The added proof gives the normalization argument and all 100 factored point-line products.

Local CPython 3.12.15 / macOS arm64 checks passed:

• 30 symbolic incidences, 70 nonincidences and all 14 required affine forms.
• Independent cross-product geometry on all 32,768 F32 triples: 20,160 exact and 12,608 rejected.
• All 14 omitted-condition controls, one degree-four constructor and three rejecting mutations.
• Normal and optimized commands, registered replay, metadata/pins, unrelated-directory execution, CLI rejection, syntax/function bounds, committed bytes and declared source-history preflight.

Normal and optimized registered checks were timed locally at 0.83 s and 0.78 s. These are check durations, not a runtime improvement claim. Each registered timeout is 120 s.

From the repository root:

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

Expected markers are PASS_DESARGUES_BINARY_PARAMETER_LAW, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Each verifier run generates a receipt bound to all six package files.

Limitations

The universal statement is a written algebraic proof, with exact symbolic and finite corroboration. It is not proof-kernel acceptance. Counts concern normalized labelled realizations, not unlabelled orbits. Support-restricted frame transport, native realization, dirty restoration, physical caller costs and supplier admission remain separate. No new kappa or selected-frontier change. Lean, Docker, predecessor scientific suites and the selected cold replay were not run.

Open next questions

1. Transport the constructor through one actual support-restricted frame and prove its required zeros and nonzeros. General projectivities need not preserve the prescribed supports.
2. Derive the normalized factors for another rich configuration before assuming a similar binary-independence criterion.

Related open work

No open PR in the refreshed live list studied this normalized Desargues interface or its parameter count. Searches covered current main, merged Desargues PRs, open PR bodies, active claims and the cached card corpus; arbitrary historical diffs and unpublished work remain coverage limits. PR2274 and Aaron Wang / PR2427 are merged predecessors with different retained contributions.

The completion refresh examined this account's 14 open PRs created since the advance CLAIM, including this PR. None of the 13 others studies the same Desargues interface or parameter count. No shared numerical disagreement was found; no extra computation checked another contribution.

CI notes

Both hosted checks, plan and replay (research/desargues-binary-parameter-law), passed at head 29cf4fcef263b56b850e95b56628cc2912541629 in run 41544387750. The completed checks were read once; run metadata independently confirms the matching head and successful conclusion. Both registered checks and publication preflight also passed locally. CI certifies the registered replay, not proof-kernel acceptance or supplier admission.

Merge Danger

Door: two-way. Blast radius: isolated. Only the six-file research package is added; no shared implementation or selected record changes.