Research audit: exact path penalties for general idempotent labels (no new kappa)

[Curtis-D]

The historical cancellation audit's two-unit path bound is correct for its diagonal/common-form rank-one endpoints. Carrying that rule over to arbitrary idempotent or block labels needs a more precise penalty: rank(PQ) + rank(QP). This research audit proves the sharp endpoint identity, gives exact boundary examples, and verifies the retained compiler's complementary-block rank-additivity argument. No new kappa and no counterexample to the selected construction.

The useful distinction is concrete: a one-sided rank-one pair has excess 1; common-form rank-two labels can have excess 2, not automatically 4. An arbitrary common additive offset can erase the original excess, while the compiler's actual prior / I-prior split preserves it. Original side-edge checkers correctly reject both one-sided orientations.

Only research/idempotent-path-rank-audit/ is added. Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Seed 71, adversarial mathematical audit; three candidate scopes and a local claim are recorded. Pinned-issues queries found no coordination issue. The selected record was followed through its actual #626 construction and review; its exact conditional kappa remains 51110230636731/31250000000000000.

Reproduce with standard-library Python:

python3 -I -B research/idempotent-path-rank-audit/verify.py

Expected:

PASS_EXACT_IDEMPOTENT_PATH_RANK_AUDIT: 10 named witnesses; 1936 exhaustive pairs; 30 generated pairs; 400 original triple pairs; 24 compiled endpoint pairs

Validation:

• Self-contained general proof; exact kernel-basis witnesses; independent permutation-minor checks. No priority claim for the elementary rank identity.
• All 44 integer 2x2 idempotents in entries [-3,3] and all 1936 ordered pairs checked independently using explicit determinants. Finite coverage is not the unrestricted proof.
• 12 focused tests pass normally and under -O; all 10 original cancellation tests and 5 block-core tests pass; selected-record integrity passes.
• Normal, optimized, relocated and clean committed-source replays agree. Changed source pins, altered certificate penalties and existing output files are rejected.
• 23 exact source/input pins; source bytes match the base. upstream/, original checkers, selected frontier and unrelated artifacts remain unchanged.

The pre-submission refresh inspected 607 open PRs through #788. Related #717 concerns non-projection sink interfaces; #778 concerns Gram replay performance; #785 concerns nonlinear message pricing. None supplies this endpoint audit or is imported. My earlier #752 is independent verification tooling and is not a dependency.

The README scopes edge-disjointness, arbitrary internal matrices, common-form invertibility and tensor spectators explicitly. No full network rank sum, recursive recurrence, improved supplier, full make verify, supplier cold replay, CI pass or formal multiplication-theorem verification is claimed. Inherited asymptotic obligations remain.

Credit Douglas R. Colkitt's historical cancellation and projector/block interfaces. Prepared by Curtis Dransfield with substantial OpenAI Codex assistance, including a separate worker's determinant implementation and review; no independent human review is claimed. Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at f80382efd4cd3e37c67b2a966398b55bb56af53c.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: General idempotent endpoint penalty rank(PQ)+rank(QP), with complementary-block justification for retained compiler; endpoint bounds are not simultaneous circuit attainability.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.