Masses Mass Genesis Phi Rung Quantization Phi Rung Quantized Evolve Pattern Of S

Rung quantization under evolution

A recognition event, a discrete record of a pattern matching a rule, can carry a whole-number label called a rung. In the Recognition Science framework, a stable closed light pattern, one that persists under the framework's evolution, always has such a rung, and its characteristic scale is the golden ratio raised to that rung. The theorem phiRungQuantized_evolvePattern_of_stable proves that this rung quantization survives evolution: if a pattern starts stable, then after any number of evolution steps it is still quantized, with the same rung scale.

The proof is short because the pieces are already in place. A stable pattern has a rung by construction, and the framework proves that stability is preserved by evolution. The scale itself, phiRungScale, is defined as phi raised to the rung, so evolving the pattern does not change the rung and the scale stays fixed. A one-step change in the topology rung multiplies the scale by phi, which is the golden ratio, about 1.618, the number satisfying r² = r + 1.

This is a structural result, not a mass prediction. The framework separates the rung layer from the final mass law. The theorem does not prove that a pattern's rest mass equals the mass law's predicted value; that remains a later, separate step. The rung scale is a topology-derived quantity, independent of any empirical mass data, and it is invariant under the concrete evolution rule, not under all possible dynamics.

What this buys is a stable scaffold. If particle masses sit on a phi-power ladder, this theorem says the ladder's rungs are not washed out by evolution. A stable pattern keeps its place on the ladder, and stepping the topology rung by one moves the scale by exactly phi. The result is a machine-checked guarantee that the framework's quantization is a persistent feature, not an artifact of a single snapshot.

theorem phiRungQuantized_of_stable
    (ψ : LightPattern Λ) (_hψ : StableClosedLightPattern ψ) :
    PhiRungQuantized ψ :=
  ⟨rungOf ψ, rfl, rfl⟩

theorem phiRungQuantized_evolvePattern_of_stable
    (ψ : LightPattern Λ) (k : ℕ)
    (hψ : StableClosedLightPattern ψ) :
    PhiRungQuantized (evolvePattern k ψ) :=
  phiRungQuantized_of_stable
    (evolvePattern k ψ)
    (stable_evolvePattern_of_stable ψ k hψ)

/-- 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

/-- The existing mass-law prediction expands to the topology labels. This is
only the old formula evaluated on the pattern labels; it is not the M7 theorem
that invariant load equals that prediction. -/
theorem predictedMass_unfold_topological_labels
    (ψ : LightPattern Λ) :
    predictedMass ψ =
      Anchor.yardstick (sectorOf ψ) *
        (Constants.phi ^
          ((rungOf ψ : ℝ) - 8 + MassLaw.gap_correction (ZOf ψ))) := by
  rfl

What this page does not claim

This theorem does not assign a numerical rest mass to any specific particle. It does not claim that all light patterns are quantized, only stable closed ones. It does not identify the rung of a pattern with any particular particle species.

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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