[petalas]

Summary

Scoped algebraic proof with exact finite corroboration, produced by Nick Petalas. This resolves the explicit symmetry-reduction blocker in Josie Elliston's merged PR6137, using JoyboyBrian's merged PR6097 diagonal-pair model. Their scientific results are cited, not audited or replayed.

odd width-two projector tuple
  simultaneous monomial conjugates
  reversed transposes of those conjugates
  least encoded image is the canonical representative

The proof establishes covariance of the universal all-group diagonal-pair problem and its coefficient-class collision relaxation, then gives the exact orbit count over every finite field and every odd positive length. For five calls over F5 there are exactly 1,520,110 representatives under this declared action. No maximal quotient or runtime speedup is claimed.

For n=q(q+1), k=2m+1 and epsilon=0 for even q or 1 for odd q, the count is

[
\frac{n^{2m+1}+2^{2m+1}\left(q-2+\epsilon\frac{q-1}{2}\right)+(q-1)(q+\epsilon)n^m}{4(q-1)}.
]

Only research/projector-word-symmetry/ is added. Advance CLAIM. Canonical base 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The card's changed scope is explicit in TASK.json. Research stopped at this first verified reduction.

Evidence

Before, PR6137 names symmetry soundness as a prerequisite for reducing its field-extension search. After, the written proof establishes the quotient and a standalone checker enumerates all 27,756,608 length-five projector tuples over F2, F3, F4 and F5. Its complete orbit partition, representative hashes and orbit-size histograms agree with an independent fixed-projector calculation.

• Construct and check seven finite fields; compare matrix-entry and outer-product projector inventories; check the formula at lengths 1,3,5.
• Check 612 new formal coefficient covariance fixtures, fifteen rejecting input controls and two semantic controls. Transpose without reversal and a scalar map outside the target normalizer are rejected as substitutes.
• Warning-strict normal and optimized direct commands pass locally, timed at 8.56 and 8.40 seconds. These are deadline observations, not speedup measurements. Final registered replay passes both commands with identical generated receipts bound to all six package files; the combined registered run took 18.26 seconds.
• Exact committed bytes and declared source history pass at cba0d9a40f4815716594d235590b469cfeaa7da0. Official committed export integrity and declared-input coverage pass. Both direct modes also pass from the read-only Git-free export with an empty PATH, identical receipts and all seven exported files unchanged.

Local environment is CPython 3.12.15, macOS 27.0.1, arm64, standard library only. From the repository root, use this exact local interpreter or an equivalent CPython 3.12 path:

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

Expected markers are PASS_PROJECTOR_WORD_SYMMETRY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. No predecessor executable is imported; source pins are historical provenance.

Limitations and unrun obligations

Reversal with transpose transports G to its opposite. It is safe for the universal family; a fixed oriented group/label/target specification must be transported as stated in the proof. The count concerns this declared action, and the executable has explicit finite-domain capacity limits. The universal quantifiers rest on the written proof.

No representative collision solver, new-field six-call obstruction, DRAT certificate, physical implementation, paid native/scalar/frame/movement/restoration bill, supplier admission or kappa improvement is supplied. All five selected framework assumptions remain. Selected cold replay, historical scientific suites, Lean and Docker were not run. Independent human and formal proof-kernel review are unclaimed.

Credit Josie Elliston / PR6137 and PR6048, JoyboyBrian / PR6097, Nick Petalas / PR6038, Wentao Li / PR4089/4093, Cinna Davis / PR2246 and their retained register-model lineage. Projector conjugacy, finite-field orthogonal projections and Burnside's lemma are classical; no global literature-priority claim is made.

Open next questions

1. Transport every coefficient equation and cancellation rule to the F4 or F5 representatives, then produce a checkable refutation or an actual group realization. A smaller search domain alone settles no new call minimum.
2. Prove covariance and an orbit count for a specified Frobenius action before further reducing this domain. No follow-up is reserved.

Related open work

• PR6618 classifies regular rank-one projector triples with active dimension three under full simultaneous GL conjugacy. This package counts width-two projector words under the smaller target-preserving monomial/reversal action. The domains and equivalence relations differ; no common numerical quantity is compared.
• PR6613 prices formal calls from a full monomial reflection alphabet with restored dirty workspace. Here monomial matrices are scalar conjugation symmetries of a search domain; no paid-call tariff is computed.

The completion refresh inspected all 160 open account PRs, including the 15 other contributions created since the advance CLAIM. No other new PR studies this exact width-two quotient or reports a competing same-instance count. No additional computations checked another contribution. PR6097 and PR6137 are merged source contributions. The earlier complete 556-PR overlap snapshot, live claims and main/merged searches remain limited by sparse notes, arbitrary historical diffs and unpublished work.

CI notes

Both hosted checks, plan and replay (research/projector-word-symmetry), passed at exact head cba0d9a40f4815716594d235590b469cfeaa7da0 in run 41634046065. The completed checks were read once with gh pr checks 6609 --repo CrocSwap/integer-mult-bounds; separate run metadata confirms completed success and the matching head.

Hosted replay executes the registered normal and optimized finite checks and package bindings. Local direct/registered replay, committed-byte/source-history checks and read-only Git-free export replay also pass. The universal written proof remains separate from hosted execution; independent human and formal proof-kernel review are unclaimed. No check failed.

Merge danger

Door: two-way. Blast radius: package. Six new isolated files; shared implementations, predecessor evidence and the selected record are unchanged.