[petalas]

Summary

Finite classification certificate: positive total-seven binary C5 fitting matrices have exactly 1, 51 and 6 rank-optimal local-similarity classes in the three width profiles. Only 1, 5 and 2 classes have integral edge/private-channel form. All 58 canonical representatives are supplied; 50 are nonchannel classes.

widths         local classes    edge/private classes
(3,1,1,1,1)          1                   1
(2,2,1,1,1)         51                   5
(2,1,2,1,1)          6                   2

Equivalence is block-diagonal similarity with labeled cycle vertices. Rotation/reflection reduces the width patterns to these three profiles; vertex automorphisms are not additionally quotiented. Arbitrary directed rectangular neighbor blocks are allowed, with identity diagonal blocks and both ordered nonedge blocks zero.

This answers the binary part of JoyboyBrian #5215's explicit total-seven question. #5215, JoyboyBrian #5023 and Nick Petalas #5180/#3248 are cited predecessors, not replay targets. Credit Tekalign, Sagar Thakrar, ywz65, Cinna Davis, Adam Gall, ctoth, Philippe Van Velzen and the retained P112/query-mask lineage. This contribution is produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed.

Evidence

Before: the all-field half-total theorem and a rational total-seven counterexample left the odd-total classification open. After: every one of 1,572,864 binary fitting matrices in the three profiles is enumerated, giving 1,716 rank-four optima and their complete local-group orbit partition.

An independent row-product enumeration computes rank from all 128 possible kernel vectors and enumerates every local invertible matrix by its determinant. It agrees on every rank histogram, canonical orbit, orbit size, stabilizer and channel label. Eight corrupted certificates reject; 7,200 comparisons check literal conjugation and involution. The completeness argument, all representatives and exact source pins are in research/binary-c5-total-seven-orbits/.

Local CPython 3.12.15, macOS arm64, standard library only: registered package and metadata/pin checks pass. Each registered check was timed separately: primary 1.424 s, independent 0.448 s, controls 1.907 s. Optimized direct replay passes in 1.326/0.445/1.926 s. These are local environment timings, not a speedup claim. Committed-byte/source-history preflight passes. A separate read-only internal audit found no mathematical or coverage flags. No external human or proof-kernel review is claimed.

Exact reproduction commands from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/binary-c5-total-seven-orbits
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/binary-c5-total-seven-orbits/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/binary-c5-total-seven-orbits/independent.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/binary-c5-total-seven-orbits/controls.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/binary-c5-total-seven-orbits --check-source-history

Any CPython 3.12 installation can replace the local interpreter path. Add -O to each direct scientific command for optimized replay. Expected markers are PASS_BINARY_C5_TOTAL_SEVEN_ORBITS, PASS_BINARY_C5_INDEPENDENT_KERNEL_AND_FULL_GROUP, PASS_BINARY_C5_CONTROLS and PASS_REGISTERED_RESEARCH_PACKAGE. The canonical classification digest is 304003782bdccd32be10e12706a5a433b7a53d3b9fd58008213ca3fff14de5c5.

Limitations and unrun obligations

This is a complete finite binary certificate for the declared profiles, not an all-field classification or formal-kernel proof. Physical basis changes, source acquisition, queries/copies, dirty restoration, ownership/hosting, coefficient preparation, stock/precision, tapes, compiler and complete caller costs remain unprovided. No native supplier, new kappa, selected-frontier change or complete multiplication theorem is claimed. No Lean, Docker or selected cold replay was run.

Open next questions

• Classify the total-seven profiles over larger finite fields or symbolically. Start with F3 and (3,1,1,1,1); the binary certificate gives no universal parameter classification.
• Bind one nonchannel representative to an actual producer/consumer and price its full physical basis/query/restoration bill. Rank four alone does not supply a useful decoder.

Related open work

• Certify the sharp 13-vertex Clebsch query threshold (no new κ) #5868 studies induced Clebsch masks and query dimension versus edge sharing. This contribution classifies every optimal fitting matrix on three C5 payload profiles; it does not study or compare the Clebsch threshold.
• Count every finite-field connector chart repair rank (no new κ) #5867 counts residual repair ranks under unrestricted finite-field connector charts. This package counts local-similarity classes of C5 fitting matrices; the action and statistic differ, with no common numerical result asserted.
• Prove sharp integer inertia for every sunflower downset (no new κ) #5870 constructs sharp integer inertia certificates for sunflower-downset fitting matrices. Its symmetric disjointness-supported matrices and inertia objective differ from this directed binary C5 orbit classification.
• Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim) #3045 studies binary fitting-rank floors for square root-code decoders on geometric masks. The total-seven C5 orbit domain here is different; no common numerical result is asserted.
• Research: actual Turyn profiles have asymmetric block rank 9d (no new kappa) #1037 studies characteristic-two asymmetric block-fitting rank for actual Turyn profiles. This certificate concerns local-similarity classes on a different fixed mask, without a Turyn or physical-cost comparison.

Before DONE, this account's open PRs created since the CLAIM were refreshed: #5865 through #5874. Their titles and bodies were read. No other PR studies this total-seven C5 classification, and no common computed quantity disagrees. No extra computation was run to check another contribution. The live inventory and recent handoffs were also checked before committing; these searches exclude unpublished work and do not establish global literature priority.

CI notes

At head 125d03c17d24b6299c82b956439e9c2958206309, run 40487305685 has plan failed; replay skipped. The plan job's only failed step is Onboarding helper tests. All ten failure/error entries identify unchanged tests/test_research_current_assignment.py. The log ends Ran 511 tests and FAILED (failures=8, errors=2, skipped=22), exit code 1.

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight failed assertions concern frozen-card command resolution, missing current/historical references and sparse omission lists. This matches the reported shared main de5ec2e7a onboarding failure affecting #5854 through #5863 and #5865 through #5873. Maintainers will repair it; no shared tests, implementation or files outside this package were changed.

Hosted replay has not certified the package; only local package checks have run. The local registered commands, normal/optimized direct replays, controls and committed-byte/source-history preflight passed as recorded above. The package and contribution commit are unchanged.

Advance CLAIM. Canonical source base: de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8.

Merge Danger

Door: two-way. Blast radius: isolated research package. Nine new files and a bounded opt-in replay manifest; no shared code, predecessor package or selected record changes.