[unpaidmoon]

Change and mathematical scope

One interface lemma. No new κ, no supplier admission, and no selected-frontier change. Selected κ stays 0.001672660197076918.

HyperplaneEnd/Labels.lean proves factor_iff_pre only at scale 1: if (N) holds and C is invertible, then L^{-1} T_U T_E^{-1} R'^{-1} = C ⊗ id if and only if R' = R. This package states the scalar extension

L^{-1} T_U T_E^{-1} R'^{-1} = λ (C ⊗ id)    iff    R' = λ^{-1} R

and checks it. A rank-1 residual can satisfy the scaled equation for an R' that is not λ^{-1} R, so invertibility of C is necessary. For monomial wrappers the only scales are fourth roots of unity, and the pre-wrapper phase field is the whole freedom.

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6. Task formal-interface-lemma, scope refined: the post-side phase-sharp lemma is already in research/hpe-post-wrapper-lemma, which explicitly does not claim this pre-side extension. research/hpe-prewrapper-binding (PR #2706) separates the pre-side implication and leaves the arithmetic unrun.

Item 24 of the chronological open-#1166 list after dropping tasks already delivered or performed by an open PR through #4123, then catalogue cards not already on that list. The full numbered list is in the #1166 finding text prepared with this PR.

Nearest overlap: Jeremy Kothe PR #593 (factor_iff_pre), Aaron Wang research/hpe-post-wrapper-lemma (post side only), Riquelme Soares PR #2706. This does not restate those results.

Validation

CPython 3.12, standard library only. Registered replay passed locally:

python3.12 -B scripts/research_package.py --package research/hpe-pre-wrapper-phase

PASS_REGISTERED_RESEARCH_PACKAGE research/hpe-pre-wrapper-phase outcome=completed

Evidence: ten equational replays over Q(i) at sizes 2 and 3, each with an invertible one-entry failure shift; one singular-C control; exact Z[i] circuits on certified pairs 1–3 (shipped and dead-bit pre-wrappers only at λ = 1; three phase shifts only at i^{-t}; four non-phase pre-wrappers miss all four roots; C8 rejected). --self-test rejects a phase-shifted shipped pre-wrapper as λ = 1 (it holds at i^3).

Not run: Lean/Mathlib 4.21.0, a selected cold replay, hosted CI. No cold (E = 0) pair is in Exact.lean.

Credit

Jeremy Kothe, PR #593, for factor_iff_pre and the certified pairs. Aaron Wang for the post-side phase lemma. Riquelme Soares, PR #2706, for the pre-side audit that left this arithmetic open. Gate semantics follow the five kinds in HyperplaneEnd/Clifford.lean.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review.

Open next questions

Kernel-check the scalar statement beside factor_iff_pre in a successor of Labels.lean. Blocked here by no Lean toolchain; the frozen file is not edited.

Bind one cold (E = 0) literal with the same gate semantics. Blocked because Exact.lean has no cold pair.

Workflow feedback (optional)

#1200 — no separate session report.