[petalas]

Summary

Scoped all-rank mathematical proof with finite corroboration. Every nonzero Gaussian-integer module invariant under every local P104 butterfly and single-bit i phase is a Gaussian scalar multiple of Clear1nsight’s #3382 Newton lattice:

O[A,B] = M_2(O)  ->  tensor matrix units give M_(2^r)(O)
E_r^-1 L = J^(2^r), J an ideal of Z[i]  ->  L = alpha B_r

For every nonzero scalar alpha and rank r>=1, alpha*(1+i)^max(r,4)*x_1*...*x_r belongs to L and has integral quarter outside L. Consequently a different uniform invariant lattice cannot repair quarter-ADD closure in this fixed alphabet. A source-dependent divisibility filtration remains possible. No new kappa or selected-frontier change.

The initial advance claim overlapped merged #3382’s exactness/maximality/phase theorem. The public scope correction credits that result. This PR claims only the subsequent integral-order rigidity theorem and its universal quarter-closure consequence; predecessor results are cited, not re-audited.

The contribution was produced by Nick Petalas. Credit Clear1nsight #3382 for the Newton lattice, maximal integral entrance and coefficient intertwiners; Ross Richardson #3068 for the closure boundary; Tekalign #3025/#3033 for lifting; Nick Petalas #3035/#3037, Arham Tahir #3046, ywz65 #2586 and the retained P104/native/geometry lineage. Matrix-unit and principal-ideal arguments are classical; their application to this fixed integral order is the new step.

Evidence

• Four exact local matrix-unit words and all sixteen multiplication laws pass. The displayed identity has Gaussian-unit coefficient, so no denominator is inverted.
• Every one of the 21,844 tensor matrix-unit actions through rank seven passes. The written tensor proof establishes the operator claim; the dense action fixtures alone would not.
• A complete census of all 67 additive subgroups of (Z[i]/2)^2, imposing i and both generators, leaves exactly zero, the coordinatewise pi ideal, and the whole module.
• 168 signed scalar-ideal generator checks and 49 integral-quarter counterexamples pass. The C-only skew-lattice control demonstrates why the phase hypothesis matters.
• CPython 3.12.15 / macOS arm64, standard library only: normal and optimized warning-strict checks pass, about 3.1 seconds each (120-second limit). Registered metadata/replay and committed-byte/source-history checks pass for all six package files at bfdea71502f026071e7dda46d2e15f540d3306cb. Each verifier run generates a receipt bound to all six file hashes.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B -W error research/clifford-lattice-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -B -W error research/clifford-lattice-rigidity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/clifford-lattice-rigidity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/clifford-lattice-rigidity --check-source-history

Use an equivalent CPython 3.12 elsewhere. Success markers: PASS_CLIFFORD_LATTICE_RIGIDITY, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The theorem requires Gaussian-module closure and every local butterfly/phase generator. It does not classify a smaller actual alphabet, nonadditive/correlated domains or states exact only after a completed word. Quarter closure is the uniform implication f in L and f/4 integral => f/4 in L; a stronger source entrance 4L -> L is not excluded.

PR3382’s all-rank maximality and coefficient intertwiners are cited mathematical premises. No predecessor scientific replay, selected cold replay, native/physical realization, full caller, codec/storage/tape tariff, recursive invoice, Lean/Docker run or complete multiplication theorem is supplied. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Bind a divisibility filtration through quarter ADDs to one actual caller, with its exact scalar coefficients, phases, codec, independent targets and full native/storage bill.
2. Fix a smaller caller-specific generator alphabet and test whether a useful nonhomothetic domain survives; removing a phase already permits a skew lattice. A favourable complete cost is still required.

Related open work

Completion refresh inspected all nine other open account PRs created since the advance CLAIM: #5962 (chronological width), #5963 (C3 call tariffs), #5964 (rounded P104 errors), #5965 (Eisenstein stage tail), #5966 (theta sharing), #5967 (Clifford tensor thresholds), #5968 (four-call observation adapters), #5969 (P104 scalar rewriting), and #5970 (borrowed-shear tariffs). None computes this invariant-module classification or uniform integral-quarter closure. No extra computations were run to check another contribution.

• Bound rounded error on the two #5833 P104 words: 20-bit guard, envelope below 97B (no new κ) #5964 bounds rounded errors on two later P104 gauge words; it does not classify invariant Newton lattices or supply this exact entrance contract. No shared number is computed here.
• Prove the full n-qubit Clifford tensor threshold is at least 2n+1 (no new κ) #5967 studies pointwise tensor independence of the full Clifford group, rather than integral invariant modules of its local generators. No Gram threshold is computed or compared here.
• Save six more scalar units after P104 sign normalization (no new κ) #5969 removes a same-chart scalar ADD from the Raise conditional kappa to 0.00310045 with independent helper splits #5599/Simplify the accepted P104 word by 974 scalar primitives per call #5750 words. It preserves paid profiles and does not classify invariant entrance lattices or quarter closure. This PR computes no scalar allowance total.

Merged #3382 is the direct mathematical predecessor; its exactness result is retained with full credit.

CI notes

At commit bfdea71502f026071e7dda46d2e15f540d3306cb, run 40541750695 was cancelled: the plan job reached its 10-minute job timeout during “Onboarding helper tests”, before package selection. Its annotation states, “The job has exceeded the maximum execution time of 10m0s.” Package selection and replay were skipped. This package was not exercised in CI; hosted certification is still pending.

Similar runner timeouts hit #5961 / run 40538450233 and #5968 / run 40540481534 this hour; both plans were cancelled during checkout and their replays skipped. Other PRs passed in the same period. The job-level runner timeout occurred before this package could be selected. All required local checks recorded above passed. Maintainers handle re-runs; this contribution has no new push to retrigger CI.

Merge Danger

Door: two-way.

Blast radius: isolated. Six new research-package files; no shared code, frozen evidence, selected frontier or other contribution changes.