Masses Mass Genesis Stable Light Pattern Integrated Meaning Load Evolve Pattern

The invariant load

In the Recognition Science framework, mass does not enter as a fundamental constant but as a derived property of light patterns, which are discrete records of events distributed across a finite space. The framework's machine-checked library of formal theorems proves a conservation statement about these patterns: when a stable pattern evolves, its integrated meaning load, the total weighted content of the pattern, remains exactly the same at every step. The declaration integratedMeaningLoad_evolvePattern is the formal theorem that states this invariance for any number of evolution steps.

The proof rests on two earlier results. First, a stable light pattern is defined as one that is localized, carries a nontrivial neutral load, and closes into an eight-tick orbit under the cyclic-shift evolution Rhat. Second, the evolution operation evolvePattern preserves the pattern's support, topology, sector, rung, and load label at every step. Combining these, the theorem shows that the integrated meaning load is a true invariant of the evolution: integratedMeaningLoad (evolvePattern k ψ) = integratedMeaningLoad ψ for any natural number k and any light pattern ψ.

This invariant matters because the framework identifies the rest mass of a stable pattern with its integrated meaning load. A corollary theorem, restMass_evolvePattern_of_stable, then shows that rest mass is also conserved under evolution. The framework models a persistent particle as a light pattern bundled with its stability proof, and this bundle carries the same load and mass invariance. The eight-tick closure is the structural reason: after eight shifts the pattern returns to itself, so any quantity defined additively over the pattern must repeat.

What the theorem does not claim is equally precise. It does not establish the phi-rung quantization of masses or the mass-law identity; those remain separate targets in the framework. It does not assert that all light patterns are stable, only that localized nontrivial neutral-load patterns are. And it does not connect the invariant to any specific measured particle mass, which would require an additional empirical comparison against experimental data.

theorem integratedMeaningLoad_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    integratedMeaningLoad (evolvePattern k ψ) =
      integratedMeaningLoad ψ :=
  integratedMeaningLoad_evolve_invariant ψ k

/-- For the concrete eight-tick Rhat, stability reduces to locality plus
nontrivial neutral load. Later Rhat refinements can replace the closure
lemma without changing the downstream API. -/
theorem stable_iff_localized_nontrivial (ψ : LightPattern Λ) :
    StableClosedLightPattern ψ ↔
      LocalizedSupport ψ ∧ NontrivialNeutralLoad ψ := by
  constructor
  · intro hψ
    exact ⟨hψ.1, hψ.2.1⟩
  · intro hψ
    exact stable_of_localized_nontrivial ψ hψ.1 hψ.2

/-- Every light pattern closes after the concrete eight-tick cyclic-shift
Rhat evolution. This is the structural orbit-closure input for M2. -/
theorem closedRHatOrbit_period_eight (ψ : LightPattern Λ) :
    ClosedRHatOrbit ψ := by
  refine ⟨8, by norm_num, dvd_refl 8, ?_⟩
  intro x _hx
  exact cyclicShift_period_8 (ψ.window x)

@[simp] theorem evolvePattern_support
    (ψ : LightPattern Λ) (k : ℕ) :
    (evolvePattern k ψ).support = ψ.support := rfl

@[simp] theorem evolvePattern_topology
    (ψ : LightPattern Λ) (k : ℕ) :
    (evolvePattern k ψ).topology = ψ.topology := rfl

@[simp] theorem sectorOf_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    sectorOf (evolvePattern k ψ) = sectorOf ψ := rfl

@[simp] theorem rungOf_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    rungOf (evolvePattern k ψ) = rungOf ψ := rfl

@[simp] theorem ZOf_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    ZOf (evolvePattern k ψ) = ZOf ψ := rfl

theorem restMass_evolvePattern_of_stable
    (ψ : LightPattern Λ) (k : ℕ)
    (hψ : StableClosedLightPattern ψ) :
    restMass (evolvePattern k ψ) = restMass ψ := by
  have h_ev := stable_evolvePattern_of_stable ψ k hψ
  rw [restMass_eq_integratedMeaningLoad_of_stable (evolvePattern k ψ) h_ev,
      restMass_eq_integratedMeaningLoad_of_stable ψ hψ,
      integratedMeaningLoad_evolvePattern ψ k]

What this page does not claim

The theorem does not prove that every light pattern is stable, only that localized nontrivial neutral-load patterns are. The theorem does not assign a numerical value to any particle mass. The theorem does not establish the phi-rung quantization or the mass-law identity.

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/StableLightPattern.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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