Research: signed rank transport excludes unitary register mixers (no new kappa)

[jeremydiviney]

A forward/inverse coefficient potential gives a general search constraint: unitary scalar register mixing cannot save paid rank, regardless of word length, mixing locations or noncommuting frames. The same obstruction holds for scalar multiples of unitaries and for mixers jointly made unitary by the supplied positive diagonal metrics. In particular, the three-stream rank-three endpoint needs nine calls throughout this subclass.

For arbitrary nonunitary mixers, the argument becomes a quantitative necessary condition. Define

p_ij(g) = Re(S_ij[g] (S^-1)_ji[g^-1]) and Phi(S) = sum rank(g) p_ij(g).

Scalar mixing preserves Phi exactly. A paid call of cost c on row i changes it by at most c(1+2*N_i), where N_i is the negative mass of that row's pairing immediately before the call. A full-rank three-stream endpoint has Phi=9. Thus an eight-call solution requires accumulated paid-row negative mass at least 1/2. A Cauchy–Schwarz version requires a local coefficient-row/inverse-column norm product of at least 9/8 at some paid prefix.

PROOF.md proves the identities for any symmetric subadditive group metric and derives the positive-diagonal-gauge corollary. A matrix-sparsity metric extends the scalar-unitary obstruction to formal words with arbitrary invertible matrix frames, free monomial maps and two-sparse butterflies, including non-Clifford frames. The finite implementation specializes to the pinned three-qubit Clifford model.

No new kappa or positive eight-call primitive. The general nonunitary problem remains open. Potential and negativity are necessary diagnostics, not endpoint certificates. The theorem is in a formal complex group algebra; it does not infer formal linear independence from a concrete matrix representation, import an FFT lower bound into the multiplication model, or cover clean ancillas without an independent-stream argument.

Validation:

• Exact standard-library Q(omega) arithmetic, with omega²+omega+1=0. All 71 named prefixes and their actual inverses agree with independent dense Gaussian Clifford multiplication; both formal inverse identities are checked.
• Seven named controls include valid nine-call endpoints, eight-call deletions, complex positive diagonal gauges, and an exact two-call example attaining the local excess bound: one call gains 5/3 using negative mass 1/3.
• Two explicit false-positive controls matter for search: a five-call word has Phi=9 but fails the endpoint; an eight-call amplification trap has 200/201 correct coefficient mass in its deficient column while Phi stays 8.
• 126 paid/free inverse controls, exact squared norm inequalities and invalid-metric mutations pass. Relocation, source corruption, false endpoint receipts, wrong phase inverses, corrupted pairings and optimized-mode rejection are replayed by validate.py.
• The isolated replay takes about 3.1 seconds locally. make selected-record-check passes on base f567884aeb8851b6e794af1184119a3bfbe1e631, selecting Conditional κ = 0.001661634653669567: four paired read exchanges on PR741's frame-coalesced word #799 at 1661634653669567/10^18. No full make verify or selected supplier cold replay is claimed for this standalone addition.

Reproduce:

python3 -B -I -S research/signed-rank-transport/verify.py
python3 -B -S research/signed-rank-transport/validate.py

Expected: PASS_SIGNED_RANK_TRANSPORT and PASS_SIGNED_RANK_CLI_CONTROLS.

Only research/signed-rank-transport/ is added. Credits: Nir Ailon's 2014 quasi-probability/conditioning work and paraunitary follow-up supply the mathematical inspiration; eumemic's PR288 and its retained Swapnil Jain/upstream lineage supply the register framework; littleBro's PR680/705 supply two unchanged model/dense modules, byte-pinned in ORIGINS.json. The rank identities are proved directly here. PR800's fixed-word valuation criterion is a different restriction and is not a dependency. PR825 concerns a distinct characteristic-two XOR/common-flag model.

Prepared by Jeremy Diviney with substantial OpenAI Codex assistance. Apache-2.0. No literature-priority claim, independent human peer review, proof-assistant formalization or remote CI success is asserted.

[Mister-Meeseeks]

100-PR batch review at 7ae9124f59694bb1e5acf2ae99c65e63c115b2f4.

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

Reviewed scope: Formal group-algebra signed-rank potential, unitary/gauged obstruction and necessary negative-mass budget; no concrete representation extrapolation.

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.