Research: determinant obstruction for rank-homogeneous register paths (no kappa)

[littleBro]

Eight fixed four-block Clifford words at m=3, W=3, s=8 remained unresolved by bounded support and polynomial searches. A determinant argument now excludes every invertible complex scalar realization of those words: when paths with the same interior group product have the same accumulated unary rank, the total unary rank must be at least W*m.

The written theorem allows arbitrary width and any number of scalar mixing blocks, subject to that explicit path-homogeneity hypothesis. The finite checker specializes to three-qubit, three-register formal projective Clifford words. This does not settle the full register problem or claim a new primitive or kappa.

The standalone corpus contains 73 literal words:

• Eight newly closed four-block follow-ups.
• Twenty-three further exclusions among 32 evolved words; nine fail the hypothesis and remain unclassified by this criterion.
• All 33 PR Scoped negative result: certify 33 fixed Clifford register words (no kappa claim) #680 words as regressions, recovered without support/torus solving.

The search consequence is concrete: repeated products alone are insufficient. A successful word below the W*m cost needs coincident interior products with different accumulated ranks.

Validation: python3 -I -B research/graded-path-obstruction/verify.py prints PASS_GRADED_PATH_OBSTRUCTION. Signed tableaus and exact Gaussian matrices agree on 1,377 paths; 73 polynomial determinant checks and exact nine-call positive controls pass. Forged homogeneity and rank certificates, damaged payload/expected data, and optimized execution are rejected. An isolated relocated package and a recompressed input reproduce identical results in about 0.63 seconds locally. The package uses only the Python standard library and writes no files.

The general claim rests on PROOF.md, not extrapolation from tests. No independent human review, proof assistant, full supplier/repository replay, remote CI, or multiplication theorem is claimed. All additions are under research/graded-path-obstruction/; the selected result and upstream files are unchanged.

Credits: eumemic's PR #288 and its retained Swapnil Jain/upstream framework lineage; same-author PR #680 supplies two unchanged attributed model files and regression words. Exact pins are in ORIGINS.json. Prepared by @littleBro with substantial OpenAI Codex assistance, Apache-2.0. No literature-priority claim. This is independent of whether #680 is merged.

[Mister-Meeseeks]

100-PR batch review at eeb0cd605415a73b792915aaa9185000f293c84a.

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

Reviewed scope: Rank-homogeneous path determinant obstruction: 64 literal formal words excluded, nine unresolved. Formal group algebra, not all circuit representations.

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.