Masses Mass Genesis T10 Green To Window Transport No Naturality Law Relation Eq

The corner transport limit

The declaration no_naturalityLaw_relation_eq_cornerModeWindowRelation is a formal proof of a negative result. It states that no naturality law, a rule that would connect particle sources to photon windows in a way that respects the framework's structure, can have its relation equal to the newly built corner-mode window relation. In plain terms, the map that was constructed from the twenty-seven sites of a torus to the eight ticks of a photon window is real, but it is not the missing physical law that the framework was looking for.

The construction itself is a model, a deliberate definitional choice. The framework's library, a machine-checked collection of formal theorems, starts with a finite torus that already carries Fourier modes. The eight corner characters are the modes whose exponent is nonzero on every axis with unit magnitude. A Gray-code clock, an eight-step cycle that changes one coordinate at a time, selects one corner character at each tick. Reading the posting dipole, the difference between two source events, through that character yields a window. The map is functional: each source gives exactly one window. It is not source-blind: different posting axes give different windows. This inhabits the exact class that a previous wall proved nonempty of obstructions but empty of examples.

The honest negative half is what the theorem in question proves. The corner readout of a single posting has full support on all eight ticks, meaning it is nonzero everywhere, and it is not equal to either gap window. So the corner transport cannot be the gap-one naturality law, and single-posting corner readout cannot supply raw phase-zero or phase-one support. The residual moves to a physical mechanism concentrating the settled octave's readout onto the anchor phases.

What this changes is the search space. The framework now knows that the answer is not this map, and it knows why: the map is too spread out. The next step is not to tweak the corner transport, but to find a mechanism that concentrates the readout. The theorem does not claim that the corner transport is the physical readout law, and it does not claim that no naturality law exists. It claims only that this particular relation is not it.

/-- The corner transport is not the relation of any gap-one naturality law:
its phase-zero window is not the gap-one window. -/
theorem no_naturalityLaw_relation_eq_cornerModeWindowRelation :
    ¬ ∃ law : SourceWindowNaturalityLaw3,
      law.relation = CornerModeWindowRelation3 := by
  rintro ⟨law, hrel⟩
  exact cornerModeWindowRelation_not_gapOne (hrel ▸ law.selects_gap_one)

/-- The corner transport is not the relation of any gap-one naturality law:
its phase-zero window is not the gap-one window. -/
theorem no_naturalityLaw_relation_eq_cornerModeWindowRelation :
    ¬ ∃ law : SourceWindowNaturalityLaw3,
      law.relation = CornerModeWindowRelation3 := by
  rintro ⟨law, hrel⟩
  exact cornerModeWindowRelation_not_gapOne (hrel ▸ law.selects_gap_one)

/-- The corner-mode source-window relation. -/
def CornerModeWindowRelation3 : SignedSourcePhotonWindowRelation3 :=
  fun source window => window = cornerModeWindowTransport source

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

theorem cornerModeWindowTransport_phaseSource_ne_zero
    (phase : Fin 8) (t : Fin 8) :
    cornerModeWindowTransport
      (phaseSignedPostingPhotonSource3 phase) t ≠ 0 := by
  rw [cornerModeWindowTransport_phaseSource_eq_axisSource]
  exact cornerModeWindowTransport_axisSource_ne_zero _ t

What this page does not claim

No claim is made that the corner transport is the physical readout law. No claim is made that no naturality law exists at all. No claim is made that the corner-mode window relation is equal to any gap window.

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)

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