[petalas]

Summary

Scoped proof with an exact finite certificate for the selected PR6030 native MOVE recipes. A single scalar phase choice per frame cannot remove every explicit phase. Even arbitrary width-dependent unit-complex frame gauges succeed at column multiplicity f exactly when 4 divides f.

3 -- phase i, rank 1 --> 2 -- phase i, rank 2 --> 299
3 ---------------- phase i, rank 3 -----------> 299
cycle residue = 1 mod 4
phase-deleted loop = -i I

One residual tag is necessary and sufficient on this triangle. The result closes the shared-frame scalar-gauge remedy for tensor-phase bookkeeping. It does not assert a defect in the correct phase-preserving source implementation. No new kappa or selected-frontier change is claimed.

The witness and gauge application were produced by Nick Petalas. Credit unpaidmoon / PR6293 for the selected tensor-phase question, DLamdata / PR5894 for existing absolute-phase coverage, Mark / Matarener's PR5974 and PR5442 for earlier lifts, DreamingOfClouds / PR6030 and the retained PR5907/5827/5599 word lineage, Chafik Boukhalfa's native channel, icekylinx's retained compiler and Ali Romman's independent Gaussian interpreter from PR1862/1931. The cycle-potential criterion and tensor scalar identity are classical. Full retained credit is in PROOF.md. Apache-2.0.

Advance CLAIM. Base is canonical main 2485805a6d47f36ea83ddde6327d453ac82d44dc, priority revision 8, card uniform-transfer-audit, refined to this changed-chart implication of native_all_columns. The selected source-binding first example passes at conditional kappa .003125804. Live claims, open PRs, account history, merged work and main object/formula searches were checked before reservation and refreshed before committing. Sparse local coverage and unpublished work remain outside that check.

Evidence

Before, the scalar-frame gauge remedy had no source-bound answer here. After, three actual selected MOVE pairs have phases 1/1/1 and ranks 1/2/3, so their edge equations have cycle residue one.

• Exact signed-Pauli and absolute-phase checks bind the three recipes to their chart operators. Both the full and phase-deleted loops are checked on every one of the 512 basis vectors, giving I and -i I respectively.
• All 64 fourth-root vertex assignments are enumerated at each width 1 through 4. The solution counts are 0/0/0/4. Compatible-triangle, tree, restored-loop, wrong-phase, reversed-sign and changed-source-byte controls pass.
• The written proof covers every positive f and arbitrary unit-complex vertex phases. Finite enumeration is corroboration, not the universal argument. The predecessor's full phase census is not replayed as this contribution.
• Warning-strict normal and optimized direct checks pass. The complete registered pair passes in 0.55 seconds locally; individual direct checks take about 0.14 and 0.12 seconds, below their 120-second limits. These are check durations, not performance claims.
• Committed bytes, modes and declared source history pass for all nine package/input files. Official export integrity and all three declared inputs pass. Both modes pass from a read-only export without Git and with empty PATH; all ten exported files, including its receipt, retain their bytes and modes. Scientific receipts are generated and bind package/input hashes.

From the repository root, using CPython 3.12 and the standard library:

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

Expected markers: PASS_MOVE_FRAME_PHASE_OBSTRUCTION, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES. Another CPython 3.12 path may replace the local interpreter path.

Limitations and unrun obligations

The theorem fixes the native recipes with literal phase gates deleted. Refactoring other gates, role-specific charts and history-dependent charts change its hypotheses. The witness edges occur on different roles, so this operator loop is not a chronological subroutine. No global minimum tag count or physical scalar tariff is proved.

No whole-word native replay, selected cold replay, physical caller, all-size implementation, Lean or Docker was run. All five selected framework assumptions remain. Independent human and formal proof-kernel review are unclaimed. No local registered command is left for CI to perform for the first time. No external static analyzer was available; warning-strict execution supplies the available diagnostics.

Open next questions

Can role-specific phase charts eliminate tags while preserving every actual common-chart ADD/COPY relation and endpoint normalizer? Start with these occurrences and add the real scalar-gate compatibility constraints. A complete compatibility graph and the physical scalar/restoration bill remain missing. This follow-on is unreserved.

Related open work

• PR6293 reports tensor-phase scaling and the full phase census for this same selected word. This contribution adds the scalar-frame gauge obstruction and exact width criterion, using only three witness edges; it does not recheck that census.

• PR6251 removes a scalar ADD from the same selected word while retaining every MOVE. This result concerns its native phase graph and changes no scalar event or invoice.

• PR6289 prices independent binary transvection charts on a different retained center-contact graph. Its paid-moment quantity differs from the scalar chart phases here; neither result is assumed to cover the other.

• PR6301 caches native-map setup for the historical PR5447 direct-history caller. It preserves operator phases and computes a different runtime quantity; it does not change or settle this selected-word gauge obstruction.

• PR6304 optimizes integer rounding for the rational phase (3+4i)/5. That arithmetic error differs from the exact scalar Clifford phases here; no numerical comparison is asserted.

• PR6306 answers PR6289's independent three-contact chart question on the retained center graph. Its paid-moment floor differs from this three-edge scalar phase criterion.

The completion refresh read all eight other open account PRs created since the CLAIM, PR6299 through PR6306. None computes this selected MOVE cycle residue or the scalar-frame width criterion. No additional computation was run to compare another contribution's reported number.

CI notes

Both hosted checks passed at exact head e2bec73e3f50e3b4e57a7a466eba2a3264d313ed in run 41105812315: plan in 4m43s and replay in 1m30s. The completed check list was read once; separate run metadata confirms completed status, success and the matching head.

All declared local checks and publication/export checks above also passed. Hosted replay certifies execution of the registered automated checks and source bindings within their stated scope. Independent human and formal proof-kernel review remain unclaimed. No CI failure remains to report.

Merge Danger

Door: two-way. Blast radius: isolated research package. Six additive files; no shared implementation, frozen source, predecessor credit or selected-frontier record changes.