Masses Mass Genesis T10 Green To Window Transport Corner Mode Window Transport P

The phase-axis identity

The declaration cornerModeWindowTransport_phaseSource_eq_axisSource is a theorem in the framework's machine-checked library of formal theorems. It states that for any of the eight phase labels, the transport map sends the phase-labeled posting source to exactly the same window as it sends the corresponding axis-labeled posting source. In plainer terms: if you start with a source described by its phase (one of eight tick positions) and run it through the transport, you get the same eight-complex-tick window as when you start with the source described by its axis (one of three spatial directions) and run it through the same transport. The theorem is proved by definitional equality, meaning the two expressions reduce to the same object by unfolding definitions; it is not a substantive physical law but a structural identity within the model.

The identity matters because it sits inside a larger construction. The transport map itself is a definitional choice, a MODEL, that takes a signed posting source (a torus carrier with twenty-seven sites) and produces a photon window (eight complex ticks). The framework's library proves that this map is functional, meaning each source yields exactly one window, and that it is not source-blind, meaning distinct posting axes give distinct windows. The phase-axis identity is one of the properties that make the map well-behaved: it shows that the phase labeling and the axis labeling are not independent degrees of freedom but are locked together by the transport. This locking is what the framework calls a naturality constraint, and the identity is a piece of that constraint.

What the identity does not claim is equally important. It does not say that the phase-labeled source and the axis-labeled source are the same object; they are different sources with different internal structure. It only says their images under the transport are equal. It also does not say that this transport is the physical readout law for particle masses. The framework's own documentation is explicit on this point: the corner transport is a canonical MODEL construction, and no claim is made that it is the physical readout law. The identity is a theorem about a chosen model, not a statement about which model nature uses.

The identity also does not imply that the corner transport can supply the raw phase-zero/one support that the framework's earlier results showed is missing. The library proves separately that the corner readout of a single posting has full support on all eight ticks and is not equal to either gap window. So the phase-axis identity does not close the gap; it leaves open the question of what physical mechanism concentrates the settled octave's readout onto the anchor phases. The identity is a clean structural fact, but it is a fact about the model's internal consistency, not about the physical world.

/-- The phase-bearing source is the axis source at the Gray phase axis. -/
theorem cornerModeWindowTransport_phaseSource_eq_axisSource
    (phase : Fin 8) :
    cornerModeWindowTransport (phaseSignedPostingPhotonSource3 phase) =
      cornerModeWindowTransport
        (axisSignedPostingPhotonSource3 (grayPhaseAxis3 phase)) :=
  rfl

/-- The phase-bearing source is the axis source at the Gray phase axis. -/
theorem cornerModeWindowTransport_phaseSource_eq_axisSource
    (phase : Fin 8) :
    cornerModeWindowTransport (phaseSignedPostingPhotonSource3 phase) =
      cornerModeWindowTransport
        (axisSignedPostingPhotonSource3 (grayPhaseAxis3 phase)) :=
  rfl

/-- Corner-mode readout of the signed posting source: at each tick, the
posting dipole is paired with the inverse corner character selected by the
Gray vertex of that tick. Canonical MODEL construction; the induced relation
properties below are THEOREM. -/
def cornerModeWindowTransport
    (source : SignedPostingPhotonSource3) : PhotonWindow :=
  fun t =>
    (torusFourierMode (grayCornerMode3 t)
        ((torusSiteEquivFin 3).symm source.event.1))⁻¹ -
      (torusFourierMode (grayCornerMode3 t)
        ((torusSiteEquivFin 3).symm source.event.2))⁻¹

theorem cornerModeWindowRelation_functional :
    IsFunctionalSignedSourcePhotonWindowRelation3
      CornerModeWindowRelation3 := by
  intro source left right hleft hright
  exact hleft.trans hright.symm

structure T10GreenToWindowTransportCert : Prop where
  transport_is_dipole_transform :
    ∀ (source : SignedPostingPhotonSource3) (t : Fin 8),
      cornerModeWindowTransport source t =
        torusSourceTransform
          (fun p =>
            ((dipole ((torusSiteEquivFin 3).symm source.event.1)
                ((torusSiteEquivFin 3).symm source.event.2) p : ℝ) : ℂ))
          (grayCornerMode3 t)
  relation_functional :
    IsFunctionalSignedSourcePhotonWindowRelation3 CornerModeWindowRelation3
  relation_not_source_blind :
    ¬ IsSourceBlindSignedSourcePhotonWindowRelation3
      CornerModeWindowRelation3
  missing_class_inhabited :
    ∃ relation : SignedSourcePhotonWindowRelation3,
      IsFunctionalSignedSourcePhotonWindowRelation3 relation ∧
        ¬ IsSourceBlindSignedSourcePhotonWindowRelation3 relation
  phase_window_every_tick_nonzero :
    ∀ (phase : Fin 8) (t : Fin 8),
      cornerModeWindowTransport
        (phaseSignedPostingPhotonSource3 phase) t ≠ 0
  not_gap_one :
    ¬ CornerModeWindowRelation3
      (phaseSignedPostingPhotonSource3 0) gapOneTwoPhaseMode
  not_gap_two :
    ¬ CornerModeWindowRelation3
      (phaseSignedPostingPhotonSource3 0) gapTwoTwoPhaseMode
  not_gap_one_naturality_law :
    ¬ ∃ law : SourceWindowNaturalityLaw3,
      law.relation = CornerModeWindowRelation3
  single_posting_no_anchor_support :
    ∀ phase : Fin 8,
      ¬ RawPhotonPhase01Support
        (cornerModeWindowTransport
          (phaseSignedPostingPhotonSource3 phase))

theorem cornerModeWindowRelation_not_gapOne :
    ¬ CornerModeWindowRelation3
      (phaseSignedPostingPhotonSource3 0) gapOneTwoPhaseMode := by
  intro h
  have h2 := congrFun h 2
  have hzero : gapOneTwoPhaseMode 2 = 0 := by
    simp [gapOneTwoPhaseMode]
  rw [hzero] at h2
  exact cornerModeWindowTransport_phaseSource_ne_zero 0 2 h2.symm

What this page does not claim

The phase-labeled and axis-labeled sources are the same object; they are distinct sources with equal images under the transport. The corner transport is the physical readout law for particle masses; it is a definitional model construction. The identity supplies the raw phase-zero/one support that the framework's earlier results showed is missing.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Masses/MassGenesis/T10GreenToWindowTransport.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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