[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. A partial Butson matrix A with n rows, n-1 orthogonal columns and entries among the mth roots of unity has normalized missing column

w = (nI-AA*)e_1, w_1=1
[A w]*[A w] = nI

The completion retains the original alphabet for every even m and n>=2, and every odd prime-power m and n>=3. For arbitrary odd m it uses at most the 2mth roots. The two-row input (1,1)^T proves the odd-alphabet dimension restriction is necessary.

This extends David Ganz's prime-order atlas B097 in #2849 and the prime pilot interface in #2855. The new step uses cyclotomic integrality and classical Kronecker theory; it does not audit or replay either predecessor. Classical number theory and matrix families retain credit. This contribution was produced by Nick Petalas. Only research/cyclotomic-column-completion/ is added. The selected frontier is unchanged.

Advance CLAIM. Base is canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. The formal-interface-lemma card is explicitly changed to this algebraic side result.

Evidence

Before, B097's proof and the pilot implementation cover prime order. After, PROOF.md gives the composite-alphabet implication with arbitrary matrix order under the stated hypotheses.

• All 314 dephased inputs in ten exact small domains pass. The census enumerates 3,879 candidate columns and 2,582 candidate matrices.
• Another 189 gauged tensor-Fourier erasures agree with independently retained missing columns. There are 32 one-parameter family erasures and four odd-alphabet two-row boundary examples.
• Total: 539 completions and 3,325 recovered entries, including 488 cases covered by the same-alphabet theorem. Nine malformed inputs reject and three output mutations fail the Gram oracle.
• Both warning-strict normal and optimized direct/registered commands pass on CPython 3.12.15, standard library only. The registered argv checks were timed locally at 0.117 s and 0.122 s, each below its 60-second limit. These are check durations, not performance comparisons.
• Metadata, six-file committed-byte/source-history preflight, official export integrity, and normal/optimized read-only Git-free export replays pass at a3f2b8f53dd4d31912254d40e578df8f63ed7164. All seven exported files remain unchanged. Generated receipts bind the package files by SHA256.
• Syntax and function-size inspection pass; the longest function is 26 lines. Managed-runtime allocation and dispatch deviations are documented.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/cyclotomic-column-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/cyclotomic-column-completion/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cyclotomic-column-completion
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cyclotomic-column-completion --check-source-history

Expected markers are PASS_CYCLOTOMIC_COLUMN_COMPLETION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Any CPython 3.12 path may replace the local path.

Limitations

This is a scoped algebraic proof with finite corroboration. More than one erased column, noisy data, physical preparation, paid consumption, dirty restoration, stock/precision/tape costs and supplier admission are unproved. No new kappa or complete multiplication theorem is claimed. Independent human/formal proof-kernel review and global literature priority are unclaimed. Lean, Docker and the selected cold replay were not run. The cheap selected source-binding orientation passed and is metadata evidence only.

Open next questions

1. Does original-alphabet completion also hold for odd composite m and n>=3? Start with m=15 in a dimension permitting mixed 3-cycle and 5-cycle vanishing sums. A root sum alone is insufficient; the candidate must be a full valid partial matrix.
2. Can the exact column formula give a certified noisy-input repair? Start with a perturbation bound and a root-rounding separation radius. Exact orthogonality and integrality remain the blocker here.

Related open work

This account's open PRs created since the advance CLAIM were refreshed before DONE. The two other new contributions study different quantities:

• #5866 classifies binary orthogonal-chart phase defects under a fixed-subspace constraint. It does not study missing columns over complex cyclotomic alphabets.
• #5867 counts finite-field connector repair ranks and minimum-rank chart density. It does not study the one-column alphabet-completion quantity.

Their titles and bodies were read. Neither computes a numerical value shared by this package; no additional computation was run to check their results. No open PR studying this one-column cyclotomic completion/alphabet quantity was found in the overlap searches. Merged #2849 and #2855 are the direct predecessors. Sparse local-note coverage is incomplete, and the searches do not cover unpublished work.

CI notes

At contribution commit a3f2b8f53dd4d31912254d40e578df8f63ed7164, Registered research packages run 40486181662 failed in plan, specifically the Onboarding helper tests step. The log identifies tests/test_research_current_assignment.py and ends Ran 511 tests / FAILED (failures=8, errors=2, skipped=22). The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the eight failed assertions are in the same test file.

All ten failure entries were compared with #5854's run 40478863245 and match exactly. Both contributions use main base de5ec2e7a6074068452f45b1a191b5c72832df4a. This confirms the reported shared onboarding failure. Maintainers will repair it; no shared tests or file outside this package were changed.

The replay job was skipped. Hosted replay has not certified this package; only local scientific checks have run. Local normal/optimized direct and registered checks, committed-byte/source-history preflight, export integrity and both read-only Git-free replays passed as recorded above. The mathematical result and package files are unchanged after publication.

Merge Danger

Door: two-way. Blast radius: one isolated research package. No shared executable, pinned source or selected record changes.