Chemistry Quasicrystal Penrose Frequency Ratio

The frequency ratio

Quasicrystals are materials whose atoms are arranged in a pattern that is ordered but never repeats periodically. The most famous mathematical model is the Penrose tiling, which covers a plane with two shapes of rhombus, one thick and one thin, without ever repeating exactly. A classical fact about this tiling is that the ratio of the area of a thick rhombus to a thin one is the golden ratio φ, about 1.618. A less obvious companion fact, also classical, is that in a large patch of the tiling the number of thick tiles divided by the number of thin tiles also approaches φ. This frequency ratio is what the declaration penrose_frequency_ratio formalizes.

The declaration itself is a theorem in the framework's machine-checked library of formal theorems. It states that the Penrose frequency ratio equals the golden ratio, and the proof is direct: the definition of the ratio is the golden ratio, so the statement follows by unfolding definitions. The same file also proves that the inverse ratio, 1/φ, equals φ − 1, and that this value lies between 0.6 and 0.65. These are exact statements about the constant, not approximations or measurements.

In Recognition Science, the framework models quasicrystal stability through an energy proxy. The proxy is E(r) = (r − 1/φ)², a convex function that is minimized exactly when r = 1/φ. The framework proves that this minimum is zero and that no other ratio can beat it. This ties the appearance of φ in tiling geometry to a stability condition: any deviation from the golden ratio inverse increases the energy proxy, which the framework reads as structural strain. The framework's claim is that this energy proxy is the right way to see why φ dominates quasicrystal structure, not that real quasicrystals have been measured to follow this exact law.

What the declaration does not claim is just as important. It does not prove that any real quasicrystal exists with exactly this frequency ratio; that is an empirical matter. It does not show that other irrational ratios, such as √2 or √3, cannot also appear in stable quasicrystals; the framework lists that as a falsification criterion. It does not even claim that the energy proxy is the true physical energy of a quasicrystal; that is a modeling choice. The theorem is a statement about a mathematical constant inside a defined framework, and its reach stops at the boundary of that framework.

/-- Penrose frequency ratio (thick/thin count in large tiling) = φ. -/
theorem penrose_frequency_ratio : penrose_ratio = Constants.phi := rfl

/-- Stability: energy is minimized at the golden ratio ratio. -/
theorem quasicrystal_stable (x : ℝ) : tiling_energy phi_ratio ≤ tiling_energy x := by
  dsimp [tiling_energy, phi_ratio]
  have : (0 : ℝ) ≤ (x - (1 / Constants.phi)) ^ 2 := sq_nonneg _
  simpa using this

/-- The minimum energy is exactly zero. -/
theorem min_energy_zero : tiling_energy phi_ratio = 0 := by
  simp only [tiling_energy, phi_ratio, sub_self, sq, mul_zero]

/-- 1/φ = φ - 1 (fundamental φ identity). -/
theorem phi_ratio_identity : phi_ratio = Constants.phi - 1 := by
  rw [phi_ratio]
  have hphi_sq := Constants.phi_sq_eq
  have hphi_pos := Constants.phi_pos
  -- φ² = φ + 1 implies φ(φ-1) = 1, so 1/φ = φ - 1
  have h : Constants.phi * (Constants.phi - 1) = 1 := by
    calc Constants.phi * (Constants.phi - 1)
        = Constants.phi^2 - Constants.phi := by ring
      _ = (Constants.phi + 1) - Constants.phi := by rw [hphi_sq]
      _ = 1 := by ring
  field_simp
  linarith [h]

/-- φ ≈ 1.618, so 1/φ ≈ 0.618. -/
theorem phi_ratio_bounds : 0.6 < phi_ratio ∧ phi_ratio < 0.65 := by
  rw [phi_ratio_identity]
  constructor
  · have h := Constants.phi_gt_onePointSixOne
    linarith
  · have h := Constants.phi_lt_onePointSixTwo
    linarith

What this page does not claim

This answer does not claim that real quasicrystals have been measured to follow the exact φ frequency ratio. This answer does not claim that the energy proxy is the true physical energy of a quasicrystal. This answer does not claim that other irrational ratios cannot appear in stable quasicrystals.

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/Chemistry/Quasicrystal.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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