Masses Mass Genesis T10 Physical Sigma Zero Selection Exists Settled Current Phy

A boundary that does not choose

A physical theory that derives particle masses must say which state of its fields is the one that exists. The Recognition Science framework models this as a ledger, a discrete record of recognition events, where each configuration of matter and light carries a cost, a number measuring how far it is from a zero point. Its mass genesis account works with a settled boundary: a specific arrangement of a phase posting, a photon, and a Q3 matter pattern that is consistent with the current physics of the framework.

The declaration exists_settledCurrentPhysicalBoundary_photonZero_patternNonzero is a proved theorem in the framework's machine-checked library of formal theorems. It establishes a precise fact: there exists a settled boundary model in which the photon's propagating mode has zero total cost, while the matter pattern's load-recognition cost is not zero. In plain language, the current boundary data are not enough to force the matter pattern to be the special zero-cost representative. The photon can be at its zero point while the pattern is not. A kernel-checked witness, an explicit example, separates the two costs.

The theorem is a wall, not a gap. It proves that the current physical boundary class does not select the sigma-zero pattern. A related theorem states the same wall in another form: current compatibility does not force the photon window's load to match the topology-selected matter amplitude. The framework's library also proves that a sigma-zero-enriched interface, one that adds the missing field, is consistent and inhabited. A positive rescaling of the unit settled gap-one worldline supplies a witness. That proves non-vacuity, not selection.

What the declaration does not claim is as important as what it proves. It does not say the sigma-zero pattern is impossible, only that the current boundary does not pick it out. It does not prove that the amplitude equation fails; it shows that a missing selection law is the exact open premise. If a model carries the pattern-specific sigma-zero field, the amplitude equation, the raw canonical mode, and same-topology pattern rigidity follow as theorems. The open question is a 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.

/-- **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

/-- **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)

/-- 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 theorem does not prove that the sigma-zero pattern is impossible. The theorem does not claim the amplitude equation is false or inconsistent. The theorem does not identify which physical law provides the missing selection.

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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