Chemistry Quasicrystal Phi Ratio Identity

The identity and its limits

The golden ratio φ, approximately 1.618, is the positive solution to the equation φ² = φ + 1. A classical consequence is that its reciprocal, 1/φ, equals φ − 1, which is approximately 0.618. This is not a deep physical claim; it is an algebraic identity that follows directly from the defining quadratic equation. The Recognition Science library contains a machine-checked proof of this identity, named phi_ratio_identity, which verifies that 1/φ = φ − 1 using only the definition of φ and the fact that φ² = φ + 1.

The identity matters in the framework because the reciprocal 1/φ appears as a natural ratio in quasicrystal geometry. Quasicrystals are aperiodic tilings with long-range order but no translational symmetry, and the golden ratio appears in their structure: in Penrose tilings, the ratio of thick to thin rhombus areas is φ; in icosahedral symmetry, five-fold axes involve φ; and in diffraction patterns, spots appear at φ-related positions. The framework's library defines a quantity called phi_ratio as 1/φ, and the identity proves this equals φ − 1. This is a definitional choice, not a discovery about nature.

The library also defines a simple energy proxy for tiling stability: E(r) = (r − 1/φ)². This convex function is minimized when r equals 1/φ, and the minimum value is exactly zero. A theorem in the library, quasicrystal_stable, proves that this energy at 1/φ is less than or equal to the energy at any other real number. This is a mathematical fact about the square function, not a physical law. The framework models quasicrystal stability as arising from this ratio minimizing the energy proxy, but that modeling choice is not itself a proved theorem about real materials.

In Recognition Science, the framework proposes that this algebraic identity and the energy minimum explain why stable quasicrystals have tile ratios involving φ, why φ appears in diffraction spacings, and why icosahedral quasicrystals dominate. These are predictions with named falsification criteria: if stable quasicrystals were found with tile ratios not involving φ, or if other irrational ratios like √2 or √3 produced equally stable structures, the prediction would fail. The identity itself, however, is purely algebraic. It does not prove that real quasicrystals are stable, nor that φ is the only possible ratio, nor that the energy proxy reflects actual physical forces.

/-- 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]

/-- 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]

/-- Convex energy proxy minimized at `phi_ratio`. -/
def tiling_energy (x : ℝ) : ℝ := (x - phi_ratio) ^ 2

/-- The golden ratio inverse 1/φ = φ - 1. -/
def phi_ratio : ℝ := 1 / Constants.phi

What this page does not claim

The identity does not prove that real quasicrystals are physically stable. The energy proxy is a modeling choice, not a derived physical law. The framework does not claim φ is the only possible stable ratio for 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)

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Derived articles

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