Masses Mass Genesis T10 Pattern Cyclic Shift Has Charged Species Label Simultane

The invariance of charge labels

A light pattern, a discrete record of eight values arranged in a cycle, can be shifted in time by applying the same basic tick to every position at once. The theorem hasChargedSpeciesLabel_simultaneousCyclicShift states that if such a pattern carries a label identifying it as a charged particle species, then after any simultaneous cyclic shift it still carries exactly that same label. The proof is immediate from the definition of the label: the label depends only on the pattern's topology, and the shift leaves the topology untouched.

This is one small piece of a larger construction in the framework's machine-checked library of formal theorems, which builds quotient infrastructure for classifying patterns by their orbits under repeated ticks. The shift itself preserves several other quantities as well: the support, the topology, the predicted mass, and the rest mass for stable patterns all remain unchanged. The framework uses this invariance to test the clock origin as a coordinate choice, meaning the starting point of the eight-tick cycle should not affect which particle species a pattern represents.

What the theorem does not claim is more limited than it might appear. It does not say that all patterns with the same label are related by a shift, nor that the label itself has been derived from more basic principles. The theorem only asserts invariance under the shift, not the existence or uniqueness of charged species labels. It also does not establish that any particular physical pattern is charged; it merely preserves the label if one is already present.

Within the framework, this invariance is a necessary consistency condition for the mass-generation story. If shifting the clock origin could change a particle's species label, the framework's account of masses would depend on an arbitrary choice of where the cycle begins. The theorem rules out that dependence for the labels it covers, a step toward treating the clock origin as physically irrelevant.

/-- Charged-row labels are unchanged because they read only topology. -/
theorem hasChargedSpeciesLabel_simultaneousCyclicShift
    {ψ : LightPattern (Fin 8)}
    {row : ChargedMassRow}
    (h : HasChargedSpeciesLabel ψ row)
    (k : Fin 8) :
    HasChargedSpeciesLabel (simultaneousCyclicShift k ψ) row := by
  simpa [HasChargedSpeciesLabel] using h

@[simp] theorem simultaneousCyclicShift_support
    (k : Fin 8) (ψ : LightPattern (Fin 8)) :
    (simultaneousCyclicShift k ψ).support = ψ.support :=
  rfl

@[simp] theorem simultaneousCyclicShift_topology
    (k : Fin 8) (ψ : LightPattern (Fin 8)) :
    (simultaneousCyclicShift k ψ).topology = ψ.topology :=
  rfl

/-- The topology-predicted mass is independent of clock origin. -/
theorem predictedMass_simultaneousCyclicShift
    (ψ : LightPattern (Fin 8))
    (k : Fin 8) :
    predictedMass (simultaneousCyclicShift k ψ) = predictedMass ψ :=
  rfl

/-- Stable rest mass is independent of clock origin. -/
theorem restMass_simultaneousCyclicShift_of_stable
    (ψ : LightPattern (Fin 8))
    (k : Fin 8)
    (h : StableClosedLightPattern ψ) :
    restMass (simultaneousCyclicShift k ψ) = restMass ψ :=
  restMass_evolvePattern_of_stable ψ k.val h

/-- Apply the same finite Rhat tick to every site window. -/
def simultaneousCyclicShift
    (k : Fin 8) (ψ : LightPattern (Fin 8)) :
    LightPattern (Fin 8) :=
  evolvePattern k.val ψ

What this page does not claim

The theorem does not derive the charged species label from more basic principles. The theorem does not establish that any particular light pattern is charged. The theorem does not show that all patterns with the same label are related by a shift.

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/T10PatternCyclicShift.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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