[petalas]

Summary

A scoped mathematical proof and explicit circuit give the exact normalization tariff of a three-bank anticommuting-Pauli skew pairing: four cross-bank additions and one bank halving. No workspace is needed. Matching lower bounds hold even with any finite restored clean or dirty workspace, in the declared gate model.

K = [[0,I,X],[-I,0,Z],[-X,-Z,0]],  X²=Z²=I, XZ=-ZX

c += Z*a; c -= X*b; c = X*Z*c/2
 a -= X*c; b -= Z*c
swap(a,b); a = -a

The final banks are K^-1 times the input. The Pauli relabeling, bank swap and negation are explicit and outside the two resource counts. Pauli-weighted additions are granted by this model; their physical delivery is not declared free.

The odd-skew commutative/noncommutative rank separation is classical, as in Garg, Gurvits, Oliveira and Wigderson, Example 1.3, arXiv:1511.03730. This contribution claims the specified Pauli normalization circuit and its exact resource optimum, not that rank phenomenon. The result was produced by Nick Petalas. Ali Romman's #5615 supplies the scalar helper-pairing context and retains its P104 predecessors.

Only research/pauli-skew-pairing-normalizer/ is added. Canonical base is 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. The coverage-law card's changed scope is explicit. Advance CLAIM.

Evidence

• Before: the commuting scalar pattern has rank two. After: the Pauli pattern has the displayed dyadic inverse, normalized at the proved optimum. The rank distinction is credited as classical.
• Normal and optimized checks pass all 140 signed translation/character Pauli cases at widths 1..3 and all 3,000 input basis columns. A separate closed inverse and both dense inverse products agree with the literal circuit.
• All 81 scalar fixtures, the complete support-union domains through three additions with 3..6 banks, nine rejection controls and the extra unexpected-argument control pass. The universal lower bounds rest on the written support and integrality proofs, not finite extrapolation.
• Package metadata, final registered normal/optimized replay, committed bytes/source history/modes, syntax, indentation and function bounds pass locally. Local CPython 3.12.15 on macOS arm64 takes 2.59 and 2.98 seconds per scientific command, below the 120-second registered limits. These are check durations, not a speedup claim. No external Python linter is installed.
• Generated stdout receipts bind all five package files by SHA256. No receipt is handwritten. The verifier imports no predecessor program and writes no files.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/pauli-skew-pairing-normalizer/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/pauli-skew-pairing-normalizer/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/pauli-skew-pairing-normalizer
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/pauli-skew-pairing-normalizer --check-source-history --check-modes

Any equivalent CPython 3.12 path may be substituted. Expected markers are PASS_PAULI_SKEW_PAIRING_NORMALIZER, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This is an abstract operator-valued interface, with an exact tariff for the declared resources. It supplies no actual P104 prefix/suffix modules, causal cut, physical carrier, frame alignment, native delivery or complete paid/productive invoice. Unary relabeling costs and the physical implementation of operator additions remain payable. No actual helper split, new kappa, supplier admission or selected-frontier change is claimed.

Full selected replay, compiler/caller integration, rounded arithmetic, precision/tape costs, Lean, Docker, independent human review and formal proof-kernel acceptance were not run or claimed. The first scoped result ended the investigation.

Open next questions

1. Can an actual causal helper cut realize this pairing while preserving every arbitrary dirty input? Publish its exact prefix, suffix and legal charts before claiming a channel.
2. Can the four Pauli additions, halving and unary relabelings fit a favorable complete signed moment? Price all delivery, ownership and target debits before expanding the construction.

Related open work

• Exhaustion census: no PR5599-type helper split remains on the PR5907 word (no new kappa) #5994 excludes scalar zero-cost helper splits on the PR5907 word. This changes the coefficient alphabet in an abstract interface and neither contests that census nor supplies a new cut on that word.

• Exclude 1,981 more G20 helper-split contexts (no new κ) #5983 excludes additional common-frame scalar G20 split contexts. The present real operator-valued pairing has no claimed G20 realization or competing split count.

• Prove exact Pauli paid-history minimum for arbitrary targets: X/Y/Z squares need 2 returns, not 3 (no new κ) #5979 minimizes returned rows in a Pauli observation-history model. This minimizes elementary normalization gates for a specified three-bank matrix; it is a different model and quantity.

• Conditional κ 0.00309871 from a prepared P104 split helper #5606 prepares a concrete scalar split helper and prices its complete conditional invoice. This supplies no competing invoice or saving.

• Two successor hand-offs on the P104 word: conditional κ 0.0030428242 #5553 gives concrete scalar successor hand-offs on a P104 word. This supplies no physical successor schedule or competing target count.

• Prove exact joint Gaussian transition cost with dirty workspace (no new κ) #6001 proves joint costs for block-diagonal Gaussian multipliers in a coordinate gate model. This uses Pauli-weighted additions between three banks; no joint Gaussian transition tariff is recomputed or contradicted.

• Save twenty P104 scalar units and two events through dirty-output sharing (no new κ) #5999 shares an ordinary scalar form through a dirty P104 output. This normalizes a different operator-valued three-bank matrix and supplies no ordinary-word scalar allowance.

• Raise the partial-cache observer error floor to sqrt(7/92), exact over scalar witnesses (no new κ) #6006 optimizes scalar witnesses for a partial-cache observer. This supplies no observation-error floor or competing kernel norm.

• Prove one local rank-5 deletion per FULL lifts the selected complex invoice to the BIT saving (no new κ) #6010 prices optimistic paid/target edits to the selected invoice. This supplies no realized target split or competing invoice threshold.

• Bound dense cross-port thin-query capacity at 528 requests (no new κ) #5997 bounds cross-port observation capacity. This counts normalization gates in a specified operator interface, not scalar query-output dimensions.

The completion refresh inspected this account's fourteen other open PRs created since the CLAIM at 2026-10-27T21:06:39Z: #5997 through #6002 and #6004 through #6011. None studies this matrix or computes a competing normalization tariff. No additional mathematical comparison computations were run.

No additional computations were run to check these predecessors or parallel contributions.

CI notes

Both hosted checks passed at exact head 31688a5ebe4a57ba52086e982ee87a20b001dcf2: plan and replay (research/pauli-skew-pairing-normalizer), in run 40584901744. The completed check list was read once; the run conclusion is success and its head SHA matches the PR head. Hosted replay certifies the registered normal and optimized finite checks. The universal theorem remains the written proof, with the physical and admission limits above. All required local checks passed.

Merge Danger

Door: two-way.

Only a new isolated research package is added. The selected record and frozen predecessor files retain their bytes.

Blast Radius: package.