Masses Mass Genesis T10 Physical Sigma Zero Selection

The selection gap

The T10 module in the framework's machine-checked library of formal theorems examines a specific question about mass genesis: does the current physical boundary, the interface between a matter pattern and a photon, force the matter pattern to be the one with zero load-recognition cost, a discrete record of events where the cost of recognizing a pattern is zero? The module proves it does not.

The current boundary model includes a source-attached phase posting, a physical photon with a zero-cost propagating mode, and a settled anchor shape. The library proves a sharp separation: the photon's zero cost belongs to its propagating mode, not to the matter pattern's load cost. A kernel-checked witness exhibits a settled boundary model where the photon mode has zero cost while the matter pattern's load cost is nonzero. The current boundary class therefore cannot force the pattern to be the sigma-zero representative.

The module also proves the surviving physical interface is nonempty. A positive rescaling of the unit settled gap-one worldline reaches load sigma zero and supplies every current boundary field. Once a model carries that pattern-specific sigma-zero field, the amplitude equation, raw canonical mode, and same-topology pattern rigidity follow as theorems.

The exact open premise is a selection law that marks the actual settled boundary model as the sigma-zero member of its scale orbit. Existing physical compatibility proves consistency of that choice, not selection. The module names this as the remaining physical selection premise, a target for further derivation.

/-- **Physical boundary wall.** The current settled physical boundary class
does not force the matter pattern to be the sigma-zero representative. -/
theorem currentSettledPhysicalBoundary_does_not_force_patternSigmaZero :
    ¬ ∀ model : SettledCurrentPhysicalBoundaryModel3,
      loadRecognitionCost model.base.pattern = 0 := by
  intro hall
  obtain ⟨model, _hphoton, hpattern⟩ :=
    exists_settledCurrentPhysicalBoundary_photonZero_patternNonzero
  exact hpattern (hall model)

/-- **Zero-cost mismatch witness.** There is a settled current physical
boundary model whose photon propagating mode has zero cost while its matter
pattern has nonzero load-recognition cost. -/
theorem exists_settledCurrentPhysicalBoundary_photonZero_patternNonzero :
    ∃ model : SettledCurrentPhysicalBoundaryModel3,
      Physics.PhotonAsZeroCostMode.totalModeCost
          model.base.photon.zero_cost_carrier = 0 ∧
        loadRecognitionCost model.base.pattern ≠ 0 := by
  obtain ⟨ψ, a, ha, hcarrier, hwin, _hunique, hne⟩ :=
    uniqueGroundStateScale_does_not_force_currentAmplitude
  let model :=
    settledCurrentPhysicalBoundaryModelOf ha hcarrier hwin
  have hpattern : loadRecognitionCost ψ ≠ 0 := by
    obtain ⟨E⟩ := hcarrier
    intro hzero
    exact hne
      ((settledAnchor_loadRecognitionCost_zero_iff_amplitude
        E hwin (le_of_lt ha)).1 hzero)
  refine ⟨model, model.base.photon.zero_cost, ?_⟩
  exact hpattern

/-- The sigma-zero-enriched physical interface is consistent and inhabited.
A positive scale of the unit settled gap-one worldline supplies a witness.
This proves non-vacuity, not selection of that witness by current boundary
data. -/
theorem physicalSettledSigmaZeroModel_nonempty :
    Nonempty PhysicalSettledSigmaZeroModel3 := by
  obtain ⟨Eseed⟩ := q3MatterCarrier_worldlinePattern_gapOne
  obtain ⟨c, hc, hload⟩ :=
    exists_scale_loadNormalizedToTopology Eseed
  let ψ :=
    scalePattern c (worldlinePattern gapOneTwoPhaseMode)
  have Eψ : Q3ClosedPatternEvidence ψ :=
    scalePattern_q3ClosedEvidence c hc Eseed
  have hcarrier : Q3MatterCarrier ψ := ⟨Eψ⟩
  have hunit :
      SettledAnchorWindow 1 (worldlinePattern gapOneTwoPhaseMode) := by
    unfold SettledAnchorWindow
    funext t
    simp [worldlinePattern, commitSettlementWindow_zero_eq_gapOne]
  have hwin : SettledAnchorWindow c ψ := by
    simpa [ψ] using scalePattern_settledAnchor c hunit
  have hcost : loadRecognitionCost ψ = 0 := by
    exact (loadRecognitionCost_eq_zero_iff_loadNormalizedToTopology Eψ).2
      hload
  exact ⟨
    { boundary :=
        settledCurrentPhysicalBoundaryModelOf hc hcarrier hwin
      pattern_sigma_zero := hcost }⟩

What this page does not claim

The module does not prove that the current physical boundary is inconsistent or empty. The module does not derive the selection law itself; it only proves consistency of that choice. The module does not claim that the photon's zero-cost propagating mode is the same as the matter pattern's load cost.

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/T10PhysicalSigmaZeroSelection.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.

Derived articles

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