Masses Mass Genesis Phi Rung Quantization Phi Rung Scale Evolve Pattern

The invariant rung scale

The declaration phiRungScale_evolvePattern states a simple invariance: for any light pattern and any number of evolution steps, the pattern's rung scale is unchanged by the evolution. In plainer terms, if a pattern has a certain scale assigned to it, then evolving that pattern, stepping it forward any number of times, leaves that assigned scale exactly the same. The proof is immediate from the definition: the scale is a function of the pattern's topology label, and the evolution operation does not alter that label.

This result lives within a larger framework called Recognition Science, which models physical structure from a ledger of recognition events. Here, a rung scale (a numeric scale attached to a stable pattern) is defined as a power of the golden ratio φ, where the exponent is an integer derived from the pattern's topology. The invariance theorem confirms that this scale is a stable attribute under the pattern's own evolution, not something that drifts or changes as the pattern develops.

The theorem does not claim that the pattern's mass is invariant, nor that the rung scale equals any measured mass. It only establishes that the scale itself is preserved under evolution. A separate theorem in the same module shows that stepping the topology rung by one multiplies the scale by φ, which is a different statement about how scales relate across patterns, not about evolution within a single pattern.

What this invariance enables is a clean separation: the rung scale can serve as a stable label for a pattern across its evolution, independent of any mass-law calculation. The framework's library proves this as a formal theorem, meaning it follows from the definitions with machine-checked certainty. The practical consequence is that when the framework later compares patterns or assigns masses, the rung scale provides a fixed reference point that evolution does not blur.

theorem phiRungScale_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    phiRungScale (evolvePattern k ψ) = phiRungScale ψ := by
  rfl

theorem phiRungScale_evolvePattern
    (ψ : LightPattern Λ) (k : ℕ) :
    phiRungScale (evolvePattern k ψ) = phiRungScale ψ := by
  rfl

/-- A one-rung topology step multiplies the rung scale by `phi`. -/
theorem phiRungScale_step
    (ψ χ : LightPattern Λ)
    (hstep : rungOf χ = rungOf ψ + 1) :
    phiRungScale χ = Constants.phi * phiRungScale ψ := by
  unfold phiRungScale
  rw [hstep]
  calc
    Constants.phi ^ (rungOf ψ + 1)
        = Constants.phi ^ (rungOf ψ) * Constants.phi ^ (1 : ℤ) := by
            rw [zpow_add₀ Constants.phi_ne_zero]
    _ = Constants.phi ^ (rungOf ψ) * Constants.phi := by
            simp
    _ = Constants.phi * Constants.phi ^ (rungOf ψ) := by
            ring

What this page does not claim

The theorem does not claim that the pattern's mass is invariant under evolution. The theorem does not claim that the rung scale equals any measured mass value. The theorem does not claim that evolution preserves the pattern's topology label in a physically meaningful sense beyond the formal definition.

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

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