[petalas]

Summary

A compatible partial Pauli contract has an exact low-length completion test. For h:U -> V, let D=(h-I)U and b((h-I)u,(h-I)v)=omega(u,(h-I)v).

rank D = 0                         -> 0 turns
rank D = 1, b nonzero / zero       -> 1 / 2 turns
rank D = 2, b nonsingular          -> 2 turns, or 3 if b is alternating
rank D = 2, b singular             -> at least 3 turns
rank D > 2                        -> at least rank D turns

Claim type: scoped algebraic proof, with a finite exact certificate. The first three lines classify every contract admitting at most two symplectic transvections and supply attaining words. Free Pauli sign corrections preserve the optimum. Turn count is a formal tariff, separate from native Gaussian paid rank. No new kappa or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. JoyboyBrian's #5051 supplies the partial signed compatibility contract and general completion construction; David Ganz / Dugongue's #4841 supplies native compiler context. Cao Yifan / intqwq, Cinna Davis, zxv, Jacob Alan Sussman and the retained native/owner/compiler lineage retain their credit. Classical symplectic transvection theory is credited; no literature-priority claim is made. No predecessor verifier or native compiler is imported, run or audited.

Evidence

• Before: Complete partial signed-Pauli contracts with literal Clifford lifts #5051 constructs a compatible completion using at most twice the source-span dimension in turns, without this exact low-length decision.
• After: PROOF.md derives the induced form, the complete two-turn cutoff and the sharp alternating three-turn boundary. Standalone code returns either an exact minimum and word or an explicitly labelled lower bound.
• Independent literal symplectic matrices and complete Cayley distances cover all 10,113 partial label contracts in dimensions 0, 2 and 4. All supplied words and 23,231 independent signed basis contracts pass. Sixteen fixed examples cover n=2 through n=9; eleven malformed contracts, an inconsistent system and three boundary fixtures exercise rejection.
• On CPython 3.12.15, macOS arm64, both registered warning-strict normal and optimized commands pass locally, timed at 0.568 and 0.567 seconds. Registered package replay, syntax/function-bound inspection, seven-file committed-byte/source-history preflight, official export integrity and a Git-free optimized exported replay pass at d32b46c. The verifier emits a generated receipt bound to all seven source hashes.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev research/partial-pauli-two-transvection/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error -X dev -O research/partial-pauli-two-transvection/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/partial-pauli-two-transvection
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/partial-pauli-two-transvection --check-source-history

A different CPython 3.12 path may replace the local path. Expected scientific marker: PASS_PARTIAL_PAULI_TWO_TRANSVECTION no_new_kappa. The publication check requires full Git history. There are no third-party requirements or live external inputs. Pending predecessor Git objects are citation-only and are not needed for replay.

The scientific base is 4c0a359, priority revision 7. Refreshed canonical main d809e98 leaves governing guidance and the selected record unchanged; #5051 has since merged. Final overlap covered 302 open PR bodies and 4,198 Discussion records with complete pagination, supplemented by main README and merged-PR searches. Sparse notes, arbitrary code diffs and unpublished work remain outside exhaustive coverage.

Limits and unrun obligations

The written theorem is dimension-uniform; exhaustive corroboration is only through n=2, and the executable API is bounded to n<=9. Dependent signed equations require the stated compatibility premise; finite sign checks cover independent generators. Singular rank-two forms and larger displacements receive only the stated lower bounds. Absolute target phase needs its own anchor. Native paid rank, an actual profitable caller, owner endpoints, routing, dirty restoration, precision, storage, stock, compiler/outer admission and a complete multiplication theorem remain unresolved. Lean, Docker and the selected scientific replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Determine the exact longer optimum for singular rank-two residual forms. Start with fixed source spaces for zero and rank-one forms; the missing step is an optimal enlargement of the displacement space.
• Price one literal native caller satisfying a partial signed contract, retaining its phase anchor and dirty-owner obligations. A short transvection word alone establishes no recursive-call saving.

Related open work

• research: price full P104 end-cut correction #3894 compiles signed-Pauli endpoint transports in a full P104 correction. This contribution decides a formal partial-map turn tariff and supplies no replacement endpoint recipe or correction invoice; no shared numerical value is computed.
• Draft research: retimed P111 query with no extra local recursive calls; full scatter pending #1525 studies a retimed P111 query with signed local native transitions and an unresolved scatter consumer. This contribution supplies no query/scatter implementation or native-call comparison.

The immediate partial-contract predecessor #5051 is merged. Recent #4859/#4864 concern words or outputs between Lagrangian charts and #4919 concerns unrestricted GL subspace triples; those differ from the prescribed pointwise partial Pauli map here.

Completion refresh read this account's other open PRs created since the CLAIM: #5080 concerns cyclic-polynomial inverse constants, #5083 concerns scalar-rank-capped GL2 swap words, and #5084 concerns norm-six dyadic decoder compression. None studies the partial-Pauli transvection cutoff or computes a competing value; no additional scientific computations were run.

CI notes

Both hosted checks passed at exact head d32b46c02387b3bc9dc0bea344b2c6271b08c5f9 in run 39924595948: planning in 4m7s and registered package replay in 1m17s. Run metadata confirms the matching head and successful completion of both jobs, including the unchanged-input gate. The registered replay executes both warning-strict normal and optimized commands. Checks were read once at completion. Local package, committed-byte/source-history and exported Git-free checks also passed. These bounded execution checks do not imply independent human or proof-kernel acceptance of the theorem. No CI failure was observed.

Merge danger

Door: two-way. Blast radius: package. Seven files are added in one new research directory. Shared code, frozen sources and the selected frontier are unchanged.