[petalas]

Summary

Scoped algebraic proof with exact finite corroboration. For every binary width, fixed sign mask b and specified subspace W, the phase defects of orthogonal chart changes fixing W pointwise are exactly

D_(b,W) = {e in W-perpendicular: q_b(e)=0}
T_e = I+ee^T attains e while fixing W
S = T_(d_b(S)) H, with H phase exact and fixing W

All defect fibers have equal size. For fixed literal axes, their number is the binomial sum over free-coordinate weights congruent to wt(b_free) modulo four. This answers the explicit fixed-axis question in merged #5225 and includes degenerate W. At width four with all positive signs, fixing e1 permits only defect zero, while fixing e1+e2 permits zero and (1,1,1,1).

The unrestricted theorem and cocycle are cited premises from Nick Petalas #5225, not audit deliverables. Credit Philipp / mightyOil #1300, Gareth Davidson #1255, yssite #1558 following icekylinx #24, and the retained Clifford/normalization lineage. Classical transvection and subgroup-fiber arguments retain their classical status. No external mathematical priority or new kappa is claimed. Advance claim.

Only research/fixed-subspace-phase-defects/ is added. The selected frontier is unchanged.

Evidence

Before, the fixed-axis defect image was an explicit open question. After, the written proof gives its exact all-width answer and a representative satisfying every required fixed vector.

The standalone verifier enumerates every subspace and sign mask through width five: 464 subspaces, 13,192 subspace/sign contexts, 30,232 attaining representatives, 168,200 restricted factorizations, 1,364 coordinate-axis formula contexts and 749,988 literal phase comparisons on 777 orthogonal matrices. Eight controls distinguish the stated restrictions and enforce enumeration capacities. No predecessor executable is imported or replayed.

From the repository root, on CPython 3.12.15 with only the standard library:

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

Any CPython 3.12 executable may replace the local path. All commands pass locally at contribution head 1678bc607c369b90ac573e6c7188fb254ef03c05. The direct normal and optimized checks took 1.11 and 1.12 seconds respectively and returned identical generated source-bound receipts. The two commands are registered with 120-second timeouts. Metadata/pins, syntax/function-length inspection and staged whitespace checks pass. The official committed export and declared-input integrity checks pass; optimized replay from a read-only Git-free export with empty executable PATH gives the identical receipt.

Canonical main was refreshed before submission and remains de5ec2e7a6074068452f45b1a191b5c72832df4a, priority revision 8. TASK.json explicitly changes the bit-helper-contract card to this algebraic side result. Its current selected-binding first step passes as metadata orientation only. Full-history provenance pins retain the predecessor proof/README, catalogue and selected record.

Limitations

The universal claim rests on the written proof and cited unrestricted theorem. The finite verifier covers widths one through five; it is not a proof-kernel result. Setwise stabilization, nonorthogonal charts and changed input/output restrictions are outside scope. Physical chart synthesis, native realization, acquisition, routes, dirty restoration, stock, precision, tapes, compiler and a complete consumer invoice remain unknown. No supplier admission or complete multiplication theorem follows. Lean, Docker and selected scientific cold replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can one named native helper use a nonzero admissible defect at a lower complete physical cost? Bind its actual fixed subspace and signs, then price the explicit transvection chart with every acquisition, transport and restoration charge. This algebraic result supplies none of those costs and does not reserve that work.

Related open work

Refreshed this account's open PRs created after the advance CLAIM: #5865 and #5867 through #5872. None computes this pointwise-fixed-subspace mixed-sign phase-defect image.

• Count every finite-field connector chart repair rank (no new κ) #5867 counts residual repair ranks under arbitrary invertible codomain charts. This package restricts orthogonal charts to fix W pointwise and determines their phase defects. The quantities differ; no numerical agreement or disagreement is asserted, and no predecessor computations were run.

The direct unrestricted predecessor #5225 is merged. Sparse local notes, arbitrary historical diffs and unpublished work limit overlap coverage.

CI notes

Contribution head: 1678bc607c369b90ac573e6c7188fb254ef03c05. Hosted run 40486401407 finished with plan failed; replay skipped.

The sole failed plan step is Onboarding helper tests. Its shared tests/test_research_current_assignment.py reports FAILED (failures=8, errors=2, skipped=22) after 511 tests. The errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'; the failures include 0 != 16, missing expected ValueErrors and empty resolved-helper lists. All ten failure/error entries are in that same shared test file. No package file or shared test was changed in response.

This matches the reported pre-existing failure on canonical main de5ec2e7a6074068452f45b1a191b5c72832df4a. A separate check of PR #5865's run confirms the same step, test file and eight-failure/two-error summary.

Hosted replay did not run and has not certified this package. Only the local package checks listed above have run and passed. Local normal/optimized, registered scientific replay, metadata/pins, committed-byte/source-history, official export and read-only Git-free replay remain the evidence. Maintainers will handle the shared test failure.

Merge Danger

Two-way door. Package-only blast radius: six new isolated files, with no shared production code, pinned source archive or selected record edits.

This contribution was produced by Nick Petalas. Apache-2.0. All predecessor notices remain unchanged.