Chemistry Quasicrystal Phi Ratio Bounds

The golden ratio bound

Quasicrystals are materials whose atoms arrange in a pattern that is ordered but never repeats periodically. Their diffraction patterns show sharp spots, yet the structure lacks the translational symmetry of ordinary crystals. The golden ratio φ, roughly 1.618, appears throughout their geometry: in Penrose tilings the ratio of thick to thin rhombus areas equals φ, and icosahedral quasicrystals display five-fold symmetry axes that involve φ.

A useful quantity is the inverse of the golden ratio, 1/φ, which equals φ − 1 and is approximately 0.618. The Recognition Science library defines this as phi_ratio and proves a simple numerical fact about it: the value lies strictly between 0.6 and 0.65. This is the content of the declaration phi_ratio_bounds. The proof is a machine-checked theorem, meaning a computer verified every step from the definition of φ and basic arithmetic. It does not rely on measurement or approximation; it is an exact statement about a real number.

In Recognition Science, quasicrystal stability is modeled through an energy proxy. The library defines tiling_energy(x) = (x − 1/φ)², a convex function that reaches its minimum exactly at x = 1/φ. A theorem confirms that this energy at the golden ratio inverse is less than or equal to the energy at any other real number, and another theorem shows the minimum value is exactly zero. These results are formal, but they rest on a chosen model: the energy proxy is a definition, not a derived law of physics.

What phi_ratio_bounds does not claim is broader than what it does. It does not assert that real quasicrystals must contain the golden ratio, nor that the energy proxy describes actual atomic forces. The library itself lists falsification criteria: if stable quasicrystals were found with tile ratios not involving φ, or if other irrational ratios such as √2 produced equally stable structures, the prediction would fail. The theorem only pins down a numerical interval for a defined constant; the physical claim that φ governs quasicrystal stability remains a hypothesis with named tests.

/-- 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

/-- 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

What this page does not claim

This does not claim that real quasicrystals must contain the golden ratio. This does not claim the energy proxy is a derived law of physics rather than a definition. This does not claim the golden ratio 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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