[petalas]

Summary

Scoped algebraic proof: for every n=4k+2>=2, the shortest tuple in F2^n with zero nonconstant quadratic phase and full affine span has exactly 2n+2 columns. The new unbounded domain starts at n=6, with an explicit 14-column witness. This answers the zero-character part of the other-axis-residues question in Ali Romman's #5635.

n = 4k+2
f <= 2n+1: impossible by a two-coset Gaussian-integer sum
f = 2n+2: attained by k eight-column code blocks and a six-column block

Using the cited #5635 cuspidal cancellation lemma, its fixed zero-phase residual first becomes visible at L_r=2n+2+((r-(2n+2)) mod 4). At six axes these widths are (16,17,14,15) in residue order (0,1,2,3). These are first-visible widths of that residual, not claimed complete representation-kernel thresholds. No new kappa or selected-frontier change.

The contribution was produced by Nick Petalas. Credit Ali Romman #5635/#5369/#5529/#5622, Joseph Anglada #5542, Hiroki KUMAZAKI #5522, Mihir Srivastava #5521, Jose Manuel, ywz65, Chris R and the retained native/phase lineage. Doubly-even codes, character orthogonality, Gauss sums and Green's character theorem are classical. No global literature-priority claim is made. Predecessor results are cited and built on, not audited or replayed.

Evidence

Before: #5635 settles the n=1 mod 4 family and explicitly leaves the other axis-count residues open. After: the written proof settles the exact zero-phase length for n=2 mod 4, with a source-bound standalone verifier.

Local CPython 3.12.15 normal and optimized checks pass with identical receipts. Each registered command takes about 0.17 seconds, below its 120-second limit. The finite checks enumerate 132 padded witnesses through n=130, 20 independent literal-column witnesses, 1,092 codewords, 33 exact Gaussian-integer contradictions, two other-residue controls and 18 rejection controls. The complete ordered two-axis census covers 5,461 tuples and 174,752 literal phase checks, with exactly 90 full-span zero-phase tuples at length six and none shorter. No six-axis exhaustive census is claimed.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/zero-phase-even-residue
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/zero-phase-even-residue --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/zero-phase-even-residue/verify.py

On another machine substitute a CPython 3.12 executable. Standard library only. Expected markers are PASS_ZERO_PHASE_EVEN_RESIDUE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Metadata/pins, committed bytes and modes, full source-history preflight, official export integrity with declared-input coverage, and warning-strict normal/optimized replay from a read-only Git-free export with empty PATH all pass at dedb8bd4a53ecf7b36975941a872173ed8f2a1ba. All nine exported files, including the export manifest, retain their bytes and modes. An actual cited-source corruption is rejected. The verifier emits hash-bound receipts and writes no files.

Base and final upstream refresh are canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. Advance claim. Live PR/Discussion checks and separate main/merged searches found no matching published result; sparse local notes, arbitrary historical diffs and unpublished work remain overlap limits.

Limitations and unrun obligations

This is a scoped written proof with finite corroboration. The residual corollary inherits #5635's cuspidal cancellation lemma and its classical character-theory input. No large affine group or complete phase-character family is enumerated. The all-character upper bound, complete native cutoff, efficient residual expansion, physical inputs/returns, paid invoice, compiler, precision, tapes and stock remain open. All five framework assumptions retain their status. No selected cold replay, Lean, Docker, independent human acceptance or formal proof-kernel review is claimed.

Related open work

No open PR found in the refreshed published search computes this zero-phase tuple minimum or its residual's first-visible widths. Nearby work has distinct scopes:

• Prove odd-axis Clifford tensor Gram lower bound above one half #5883 proves a lower Gram bound for the full Clifford tensor family; it does not compute monomial zero-phase tuple lengths.

• Prove identity scratch does not unlock full monomial two-FULL routing. #5843 studies identity scratch in monomial two-FULL routing; its graded rank obstruction is a different invariant.

• Exclude scratch-reading two-FULL completions (no new κ) #5891 permits scratch-reading FULL shears in that routing problem; it does not determine a native tensor test cutoff.

• Prove the selected #5599 scalar instructions lift at every column mul… #5837 and Audit existing phase coverage for selected P104 MOVE maps #5894 concern all-column lifting and existing phase-check coverage for selected P104 maps; this package is an algebraic zero-character construction, with no selected operator replay.

• Prove exact constant-weight phase COPY error (no new κ) #5903 computes the exact COPY approximation error for a fixed constant-weight phase-history alphabet. It shares Gaussian monomial tensor operators, but its response projection error differs from this package's zero-phase tuple length and first-visible residual width.

Completion refresh inspected this account's nine other open PRs created since the advance CLAIM: #5897, #5898, #5899, #5900, #5901, #5903, #5904, #5905 and #5906. The phase-history relationship with #5903 is recorded above. The other contributions study contiguous helper storage, clean bank exchange, tail-gate block length, Newton residual precision, invoice frontiers, dyadic observation acquisition, one-pass child buffering and a fixed two-burst query schedule. None computes this zero-phase tuple minimum or this residual's first-visible widths. No common-instance numerical disagreement was identified; no extra mathematical computations were run to validate or compare their results.

Open next questions

• Does every phase character have a full-affine-span realization at the proposed six-axis widths? Start with affine character-orbit reduction and a checked constructive upper bound. The present result only handles the zero character.
• What are the complete native test cutoffs in the remaining higher-axis residue classes? A matching all-character upper certificate is required; an attaining zero-phase tuple alone is insufficient.

CI notes

At exact head dedb8bd4a53ecf7b36975941a872173ed8f2a1ba, run 40501260249 finished with plan failed/cancelled; replay skipped. The plan job lasted 10m03s and was cancelled at its ten-minute job time limit while actions/checkout was still running the repository git fetch. Its checkout log reaches Fetching the repository, starts git fetch, and ends with The operation was canceled. Setup Python, onboarding tests and package selection were all skipped. No repository check or package command ran.

Separately, canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a has the known shared onboarding-test failure reported across recent PRs. Direct inspection of PR #5904's plan log confirms ten failure/error entries in unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight assertions concern frozen-card/current-route metadata, missing references and sparse path expectations. This is a separate main-base issue, not the failure reached by this PR's checkout-cancelled run. No rerun, empty commit or shared-file change was made.

The replay job was skipped. Hosted replay has not certified this package; only the local scientific and publication checks listed above have run. Local normal, optimized, registered, committed-byte/mode/source-history and read-only export checks passed. The hosted status is not scientific acceptance.

Merge Danger

Door: two-way. The change is an isolated research package.

Blast radius: local. Shared code, frozen sources and selected-frontier files are unchanged.