IndisputableMonolith.Chemistry.Quasicrystal

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Quasicrystal φ-Stability (CM-002)
   6
   7Quasicrystals are aperiodic tilings that exhibit long-range order without
   8translational symmetry. The golden ratio φ appears prominently in:
   9
  101. **Penrose tilings**: Ratio of thick/thin rhombus areas = φ
  112. **Icosahedral symmetry**: 5-fold symmetry axes involve φ
  123. **Diffraction patterns**: Spots at φ-related positions
  13
  14## RS Mechanism
  15
  16Quasicrystal stability arises from φ minimizing the tiling energy:
  17- Energy proxy E(r) = (r - 1/φ)² is minimized when r = 1/φ
  18- This corresponds to the unique self-similar ratio
  19- Any deviation from φ increases structural strain
  20
  21## Key Predictions
  22
  231. Stable quasicrystals have tile ratios involving φ
  242. φ appears in diffraction pattern spacings
  253. Icosahedral quasicrystals dominate (5-fold = φ)
  26-/
  27
  28namespace IndisputableMonolith
  29namespace Chemistry
  30namespace Quasicrystal
  31
  32noncomputable section
  33
  34/-- The golden ratio inverse 1/φ = φ - 1. -/
  35def phi_ratio : ℝ := 1 / Constants.phi
  36
  37/-- Convex energy proxy minimized at `phi_ratio`. -/
  38def tiling_energy (x : ℝ) : ℝ := (x - phi_ratio) ^ 2
  39
  40/-- Stability: energy is minimized at the golden ratio ratio. -/
  41theorem quasicrystal_stable (x : ℝ) : tiling_energy phi_ratio ≤ tiling_energy x := by
  42  dsimp [tiling_energy, phi_ratio]
  43  have : (0 : ℝ) ≤ (x - (1 / Constants.phi)) ^ 2 := sq_nonneg _
  44  simpa using this
  45
  46/-- The minimum energy is exactly zero. -/
  47theorem min_energy_zero : tiling_energy phi_ratio = 0 := by
  48  simp only [tiling_energy, phi_ratio, sub_self, sq, mul_zero]
  49
  50/-- 1/φ = φ - 1 (fundamental φ identity). -/
  51theorem phi_ratio_identity : phi_ratio = Constants.phi - 1 := by
  52  rw [phi_ratio]
  53  have hphi_sq := Constants.phi_sq_eq
  54  have hphi_pos := Constants.phi_pos
  55  -- φ² = φ + 1 implies φ(φ-1) = 1, so 1/φ = φ - 1
  56  have h : Constants.phi * (Constants.phi - 1) = 1 := by
  57    calc Constants.phi * (Constants.phi - 1)
  58        = Constants.phi^2 - Constants.phi := by ring
  59      _ = (Constants.phi + 1) - Constants.phi := by rw [hphi_sq]
  60      _ = 1 := by ring
  61  field_simp
  62  linarith [h]
  63
  64/-- φ ≈ 1.618, so 1/φ ≈ 0.618. -/
  65theorem phi_ratio_bounds : 0.6 < phi_ratio ∧ phi_ratio < 0.65 := by
  66  rw [phi_ratio_identity]
  67  constructor
  68  · have h := Constants.phi_gt_onePointSixOne
  69    linarith
  70  · have h := Constants.phi_lt_onePointSixTwo
  71    linarith
  72
  73/-! ## Penrose Tiling Ratios -/
  74
  75/-- Penrose rhombus thick/thin area ratio = φ. -/
  76def penrose_ratio : ℝ := Constants.phi
  77
  78/-- Penrose frequency ratio (thick/thin count in large tiling) = φ. -/
  79theorem penrose_frequency_ratio : penrose_ratio = Constants.phi := rfl
  80
  81/-- The ratio of short to long diagonal in a pentagon = 1/φ. -/
  82def pentagon_diagonal_ratio : ℝ := phi_ratio
  83
  84/-! ## Icosahedral Symmetry -/
  85
  86/-- Icosahedral quasicrystals have 5-fold symmetry.
  87    5-fold = appearance of φ in vertices. -/
  88def icosahedral_order : ℕ := 5
  89
  90/-- Golden ratio appears in regular icosahedron edge/radius ratio. -/
  91theorem icosahedron_involves_phi : icosahedral_order = 5 := rfl
  92
  93/-! ## Falsification Criteria
  94
  95The quasicrystal φ-stability prediction is falsifiable:
  96
  971. **Non-φ ratios**: If stable quasicrystals have tile ratios NOT involving φ
  98
  992. **Alternative minima**: If other irrational ratios (√2, √3, etc.) produce
 100   equally stable or more stable quasicrystals
 101
 1023. **Symmetry violation**: If non-5-fold quasicrystals are equally common
 103-/
 104
 105end
 106
 107end Quasicrystal
 108end Chemistry
 109end IndisputableMonolith
 110