Prove P104 phase-uniform gauge and unavoidable completion tag

[ToikY]

Result and scope

For the 104 published binary P104 labels at #3954 head f1e2839819959cd0534d008be91ed5b79ec55a6a, the explicit orthogonal transvection Ox=x+(x·420)420 sends every label to weight 3 modulo 4. All 10,816 tensor product lines then have weight 1 modulo 4, removing the predecessor's 5,376 gauge-dependent normalization mismatches.

The written proof also gives a limit: every orthogonal nine-dimensional gauge applied identically to both tensor factors and making those product phases constant has the same nonzero input quadratic character 420. The alternative 91 is excluded by preservation of the characteristic vector. No orthogonal-group enumeration is claimed.

For the explicit tensor gauge, the full native F boundary has a forced nonzero translation a=420⊗511+511⊗420, weight 40. Entrance/exit tags alternate under already completed uniform F calls on arbitrary data and all 72 dirty helpers. General non-involutive gauges distinguish the input character from its output image. A forced translation needs exactly four calls in the stated restricted one-bank odd-line C± alphabet. This is an ideal word tariff, not native/compiler or cross-role optimality.

Even granting all new gauge/tag work free, #3954's unchanged named scaffold retains excess 44,096. A chosen isolated bankwise correction adds 146,432 primitive calls; excess 190,528 is conditional on separately admitting each at one rank unit in the same ledger. Stop expansion of that failed scaffold. No supplier or new kappa.

Evidence

Public head: 75785ae0d1c4a8b1be00b30a07ec31106bb28937; tree 33c825e908495916dc237d2f47d9ced50c6fae19 equals the locally tested tree. Two focused commits add exactly 12 files under research/p104-phase-uniform-gauge/.

• Three byte-exact predecessor source exports with original Git blobs and independently hardcoded SHA256 authentication; no predecessor executable or pending Git object is needed.
• Primary exact binary-character checks and separate RREF/bit implementation:104 labels,10,816 product lines,262,144 dot pairs,81 tensor coefficients and22,680 all-role/basis checks.
• Written polarization/character and universal four-call arguments; separate small literal dyadic matrices, non-involutive conjugation control and585 three-bit short words corroborate the proofs.
• 60 primary API/CLI negative runs and eight independent damaged-source runs across normal/optimized Python.
• Registered Python 3.12 replay, exact-public-commit byte/source-history preflight and complete read-only Git-free export pass. All three standalone commands pass both normally and with -O; all 13 exported files retain bytes/modes/inodes. Cold export caught copied read-only directory permissions in private destructive fixtures; the surgical fix changes only those owned fixtures.
• No human/kernel, full predecessor/native-channel or selected cold audit claimed. Final-head hosted CI passes: planning selects only this package, all three registered commands succeed, and committed inputs remain unchanged. Logs bind head 75785ae to test merge 51c07fd against main 20ecf28 on CPython 3.12.15. This is execution evidence, not scientific admission.

Reproduce with standard-library Python 3.12:

python3.12 -B scripts/research_package.py --package research/p104-phase-uniform-gauge
python3.12 -B scripts/check_research_commit.py --package research/p104-phase-uniform-gauge --check-source-history

Local replay takes roughly 4 seconds on CPython 3.12.14; host-dependent.

Selection, overlap and remaining obligations

Handoff-first changed scope under revision 6 card p104-side-correction, with advance claim. Main 20ecf28 was refreshed by canonical fetch and live API checks, unchanged. Entry 662 open PR bodies including drafts; pre-submission 665 over seven pages plus latest 100 delta, repository helper and bounded live Discussion reads. Broader Discussion/closed-diff coverage remains partial.

Concurrent #3961 covers physical cross-axis encoding, #3962 coverage/helper sharing, and #3965 private-copy lifetimes. Their bodies were read; their proofs/executables were not replayed here. This gauge theorem does not remove those separate obligations. In particular its address maps fix constant functions, so they cannot alone remove a role-encoding defect already visible on that sector.

This is a partial gauge/boundary lemma, not the missing adjacent physical producer/consumer pair. A successor needs one actual chronological row/column pair, fiber-dependent tag reconciliation on every shared owner/helper and fixed returns, with a changed favorable complete invoice before expansion. Physical gauge routing, frames/native orientation/global phase, finite-width/signed-floor precision and recovery, ownership/holding/control/tapes, opposite orientation, all-size and supplier admission remain unproved or inherited. README/TASK give feasible first steps and blockers.

Credit

Chris R/Clear1nsight #3954 owns the retained labels, conventions and scaffold gate, following #3882/#3940/#3943. Retain 1337isnot1337/#3894, Rohan Arun/#1972, Chafik Boukhalfa/#1991, Adam Gall/#1940, Josie Elliston/#1071/E8, Aurel Prosz/Paureel, Zhihao Chen, Dominik Scholz, icekylinx, Sagar Thakrar/#1012, Douglas Colkitt, Jose Manuel, Vasek/#2744 and the retained native/compiler/caller/coverage lineage. #3943 owns its 960-ADD adapter; this does not reassign that construction.

New gauge, character obstruction, boundary and restricted tariff by ToikY with substantial OpenAI Codex assistance, including separate AI proof/test review contexts. Apache-2.0. Maintainer edits disabled; no merge or selected-frontier change.