[petalas]

Summary

Scoped written algebraic proof with exact finite corroboration, produced by Nick Petalas. For an ordered rank-one idempotent triple over any field, put S=P1+P2+P3. Under rank(S)=rank(S²)=3 and nonzero pair traces, the ambient dimension and four traces completely determine simultaneous GL conjugacy.

p = tr(P1P2), q = tr(P1P3), r = tr(P2P3), u = tr(P1P2P3)
regular class iff p,q,r,u != 0 and 1-p-q-r+u+pqr/u != 0
finite-field class count = Q^4 - 5Q^3 + 9Q^2 - 9Q + 6

The proof gives an explicit representative and centralizer. Two F5 triples have identical pair traces and different oriented triple traces, so pair-trace data alone is insufficient. This is a new state-classification side result, with changed bit-new-register-model scope. No native supplier, paid-call saving, kappa or selected-frontier change is claimed. Research stopped at this first verified result.

Only six new files in research/regular-projector-triple-orbits/ change. Written proof, verifier, advance CLAIM.

Evidence

Before, the explicit F5 ambiguity has one pair-trace descriptor for two distinct simultaneous-conjugacy classes. After, the four-trace key separates them and classifies the entire stated regular class.

• Complete six-entry Gram domains over F2,F3,F4,F5,F7 give exactly 0, 6, 50, 186 and 1,070 classes. The direct Gram enumeration and full expansion of the four-parameter diagonal orbits agree as sets. Six-term determinants, row reduction and actual projector matrix traces provide separate checks of the required identities.
• All 11,232 GL3(F3) basis changes of each ternary representative pass: 67,392 conjugated triples and 33,696 distinct endpoints. Zero spectator dimensions one and two, ten domain rejections and an oriented-trace ambiguity fixture pass. F4 uses t²+t+1, not arithmetic modulo four.
• Warning-strict normal and optimized direct checks and the registered package replay pass on CPython 3.12.15, standard library only. Registered commands were timed individually at 3.071 s and 3.191 s, rounded upward. These are check durations, not a speedup claim. Receipts are generated and bind all six files; normal and optimized outputs agree byte for byte.
• Both deliberate source corruptions reject under optimized Python for their expected mathematical reasons: a false count and a reversed oriented trace. Source compilation, function-size inspection and staged whitespace checks pass.
• All six committed bytes/modes and declared source-history ancestry pass at 2c69372760776c6f522ba31fef3357e2aa9d309d. Official committed export integrity and declared-input coverage pass. Both direct modes pass from a read-only Git-free export with empty PATH, identical receipts and all seven exported files unchanged, including the export manifest.

From the repository root, reproduce with the exact local interpreter or another CPython 3.12 executable:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/regular-projector-triple-orbits/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/regular-projector-triple-orbits/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/regular-projector-triple-orbits
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/regular-projector-triple-orbits --check-source-history --check-modes

Expected markers: PASS_REGULAR_PROJECTOR_TRIPLE_ORBITS, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. No predecessor scientific checker is imported or run. The verifier has no scientific runtime inputs outside its six files.

Limitations and unrun obligations

The universal result rests on the written proof, with fixed finite corroboration. It requires rank(S)=rank(S²)=3 and nonzero pair traces. Singular or partly uncoupled triples are excluded. The equivalence preserves projector order, field and ambient dimension; transpose, reversal, field automorphisms and projector permutations are not included.

This is not a complete word-state invariant or a drop-in physical search quotient. A consumer must restrict conjugation to its target stabilizer, preserve other state, and price basis transport, actual reads, native maps, arbitrary dirty restoration, ownership, precision/tape/stock and the full recursive invoice. Those obligations are unpriced. All five selected framework assumptions remain.

Selected scientific cold replay, predecessor scientific suites, Lean, Docker, independent human review and formal proof-kernel acceptance are unrun or unclaimed. All registered checks ran locally. Python managed-allocation bounds and available diagnostics are documented in the README.

Credit and overlap

Credit MatShep / PR4762 at 3b4e0449262b0b1e07b80e009ef71cc0ca5bf816 for the related two-idempotent factor descriptor, codyw912 / PR4735 at 4afbea22e79e19d1d65702eb4090d454ee39d967 for its inclusion/observation framework, and the eumemic/woklerx scalar-projector and retained native/frame/helper/compiler/outer lineage. The new proof derives its three-factor reduction directly; those packages are context, not proof premises or audit targets. Rank-one factorizations, diagonal similarity and cycle products are classical; no global literature-priority claim is made.

Canonical main was refreshed before publication and remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The source-binding baseline passed. Live claims, local reservations, recent account PRs, main README/general-form searches and merged/all-state PR searches were checked before claiming and committing. The broader snapshot covered 556 open PRs and 6,605 Discussion records with complete remote pagination; sparse local notes, arbitrary historical diffs and unpublished work remain limits.

Open next questions

1. Classify one singular or partially uncoupled triple. Start with a fixed singular Gram and its factor kernel and observation image; the regular direct-sum normalization no longer applies.
2. Bind the descriptor to one target-preserving native search and charge every basis change. Begin with a concrete alphabet and target stabilizer; an actual caller and full paid invoice are missing.

Related open work

• PR6609 counts width-two projector words under a monomial/reversal action. This result classifies regular triples with active dimension three under full simultaneous GL conjugacy. The domains and equivalence relations differ; no common numerical quantity is compared.
• PR6565 gives Schur continuation/header counts for ordered rank-two binary idempotent pairs. This result concerns rank-one triples and a four-trace conjugacy key, and computes no header probability or continuation table.

No open PR in the refreshed candidate-object searches supplied this regular triple class count. No extra computation audited another contribution.

Before DONE, the completion refresh inspected all fourteen other open PRs created by this account since the advance CLAIM at 2026-11-03T12:11:52Z. None reports this regular fully coupled triple class count or a competing value computed here. PR6609 studies the distinct equivalence relation described above. No extra scientific computations checked another contribution.

CI notes

Both hosted checks passed at exact head 2c69372760776c6f522ba31fef3357e2aa9d309d in run 41634962887: plan in 4m38s and registered package replay in 1m35s. The completed checks were read once. Separate run metadata confirms completed status, successful conclusion and the matching head.

Hosted replay certifies execution of both registered warning-strict normal and optimized bounded checks. Local mathematical, package, committed-byte/source-history, corruption and standalone export checks also passed. The all-field theorem remains a written proof; independent human and formal proof-kernel acceptance remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated package. Shared code, predecessor files and credit, frozen evidence and the selected record retain their bytes.