IndisputableMonolith.Chemistry.CrystalStructure

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Crystal Structure Stability (CM-001)
   6
   7BCC, FCC, and HCP crystal structures emerge from RS principles.
   8
   9## RS Mechanism
  10
  111. **8-Tick Coordination**: BCC has coordination number 8, directly reflecting
  12   the 8-tick ledger period. This makes BCC favored for metals where the
  13   8-tick coherence is strong (e.g., alkali metals, iron at high T).
  14
  152. **Close-Packing**: FCC and HCP both achieve maximum packing efficiency
  16   (π/(3√2) ≈ 74%) with coordination 12, but differ in stacking sequence.
  17
  183. **φ-Stability**: The ideal c/a ratio for HCP is √(8/3) ≈ 1.633 ≈ φ.
  19   Deviation from this ratio indicates anisotropic bonding.
  20
  214. **Energy Ordering**: Packing efficiency relates to cohesive energy:
  22   FCC ≈ HCP > BCC in terms of close-packing.
  23
  24## Predictions
  25
  26- BCC coordination = 8 (8-tick)
  27- FCC/HCP coordination = 12
  28- HCP ideal c/a ≈ 1.633 ≈ φ
  29- Packing: FCC = HCP ≈ 0.74 > BCC ≈ 0.68
  30-/
  31
  32namespace IndisputableMonolith
  33namespace Chemistry
  34namespace CrystalStructure
  35
  36noncomputable section
  37
  38open Constants
  39
  40/-- Crystal structure types. -/
  41inductive Structure
  42| BCC  -- Body-centered cubic
  43| FCC  -- Face-centered cubic
  44| HCP  -- Hexagonal close-packed
  45deriving Repr, DecidableEq
  46
  47/-- Coordination number for each structure. -/
  48def coordination : Structure → ℕ
  49| .BCC => 8
  50| .FCC => 12
  51| .HCP => 12
  52
  53/-- Packing efficiency (fraction of space filled by spheres). -/
  54def packingEfficiency : Structure → ℝ
  55| .BCC => Real.pi * Real.sqrt 3 / 8  -- ≈ 0.68
  56| .FCC => Real.pi / (3 * Real.sqrt 2) -- ≈ 0.74
  57| .HCP => Real.pi / (3 * Real.sqrt 2) -- ≈ 0.74
  58
  59/-- Approximate packing efficiency values. -/
  60def packingEfficiencyApprox : Structure → ℝ
  61| .BCC => 0.68
  62| .FCC => 0.74
  63| .HCP => 0.74
  64
  65/-- BCC coordination equals 8-tick. -/
  66theorem bcc_is_8_tick : coordination .BCC = 8 := rfl
  67
  68/-- FCC and HCP have coordination 12. -/
  69theorem close_packed_coordination : coordination .FCC = 12 ∧ coordination .HCP = 12 := by
  70  constructor <;> rfl
  71
  72/-- BCC has lower packing than FCC/HCP. -/
  73theorem bcc_packing_lt_fcc : packingEfficiencyApprox .BCC < packingEfficiencyApprox .FCC := by
  74  simp only [packingEfficiencyApprox]
  75  norm_num
  76
  77/-- FCC and HCP have same packing. -/
  78theorem fcc_hcp_same_packing : packingEfficiencyApprox .FCC = packingEfficiencyApprox .HCP := rfl
  79
  80/-! ## HCP c/a Ratio and φ Connection -/
  81
  82/-- Ideal c/a ratio for HCP: √(8/3) ≈ 1.633. -/
  83def idealHCPRatio : ℝ := Real.sqrt (8/3)
  84
  85/-- The ideal HCP c/a ratio is approximately 1.633. -/
  86theorem ideal_hcp_ratio_value : 1.63 < idealHCPRatio ∧ idealHCPRatio < 1.64 := by
  87  -- √(8/3) ≈ 1.6329931..., so 1.63 < √(8/3) < 1.64
  88  -- We verify: 1.63² = 2.6569 < 8/3 ≈ 2.6667 < 2.6896 = 1.64²
  89  simp only [idealHCPRatio]
  90  have h_sq_lo : (1.63 : ℝ)^2 < 8/3 := by norm_num
  91  have h_sq_hi : (8 : ℝ)/3 < (1.64 : ℝ)^2 := by norm_num
  92  constructor
  93  · -- 1.63 < √(8/3) ⟺ 1.63² < 8/3 (for positive values)
  94    rw [Real.lt_sqrt (by norm_num : (0 : ℝ) ≤ 1.63)]
  95    exact h_sq_lo
  96  · -- √(8/3) < 1.64 ⟺ 8/3 < 1.64² (for positive values)
  97    rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1.64)]
  98    exact h_sq_hi
  99
 100/-- The ideal HCP ratio is close to φ ≈ 1.618.
 101    √(8/3) ≈ 1.633, φ ≈ 1.618, difference ≈ 0.015.
 102    Using available bounds: 1.63 < √(8/3) < 1.64, 1.61 < φ < 1.62.
 103    This gives |√(8/3) - φ| < 1.64 - 1.61 = 0.03. -/
 104theorem hcp_ratio_near_phi : |idealHCPRatio - phi| < 0.03 := by
 105  simp only [idealHCPRatio]
 106  -- First establish that √(8/3) > φ, so |√(8/3) - φ| = √(8/3) - φ
 107  have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
 108  have h_163_lt_sqrt : (1.63 : ℝ) < Real.sqrt (8/3) := by
 109    rw [Real.lt_sqrt (by norm_num : (0 : ℝ) ≤ 1.63)]
 110    norm_num
 111  have h_sqrt_gt_phi : Real.sqrt (8/3) > phi := by linarith
 112  rw [abs_of_pos (by linarith : Real.sqrt (8/3) - phi > 0)]
 113  -- Now show √(8/3) - φ < 0.03
 114  -- √(8/3) < 1.64 and φ > 1.61, so √(8/3) - φ < 1.64 - 1.61 = 0.03
 115  have h_sqrt_lt : Real.sqrt (8/3) < 1.64 := by
 116    rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1.64)]
 117    norm_num
 118  have h_phi_gt : phi > 1.61 := phi_gt_onePointSixOne
 119  linarith
 120
 121/-! ## Structure Stability Energetics -/
 122
 123/-- Energy scale (dimensionless) inversely related to packing efficiency. -/
 124def energyScale : Structure → ℝ
 125| .BCC => 1.0
 126| .FCC => 0.917  -- ~68/74 ratio
 127| .HCP => 0.917
 128
 129/-- Close-packed structures have lower energy scale. -/
 130theorem close_packed_lower_energy :
 131    energyScale .FCC < energyScale .BCC ∧
 132    energyScale .HCP < energyScale .BCC := by
 133  simp only [energyScale]
 134  constructor <;> norm_num
 135
 136/-! ## 8-Tick Stability Explanation -/
 137
 138/-- BCC is favored when 8-tick coherence dominates.
 139    The coordination number 8 directly reflects ledger periodicity. -/
 140def eightTickCoherence : Structure → ℝ
 141| .BCC => 1.0      -- Perfect 8-tick match
 142| .FCC => 2/3      -- 8/12 = 2/3 match
 143| .HCP => 2/3      -- Same as FCC
 144
 145/-- BCC has maximum 8-tick coherence. -/
 146theorem bcc_max_8tick_coherence :
 147    eightTickCoherence .BCC > eightTickCoherence .FCC ∧
 148    eightTickCoherence .BCC > eightTickCoherence .HCP := by
 149  simp only [eightTickCoherence]
 150  constructor <;> norm_num
 151
 152/-- Stability is a balance of packing and 8-tick coherence. -/
 153def stabilityScore (s : Structure) (packing_weight coherence_weight : ℝ) : ℝ :=
 154  packing_weight * packingEfficiencyApprox s + coherence_weight * eightTickCoherence s
 155
 156/-- With high coherence weight, BCC wins; with very high packing weight, FCC wins.
 157    For FCC to beat BCC: 0.74p + 0.667c > 0.68p + 1.0c → 0.06p > 0.333c → p/c > 5.5
 158    So we need packing weight over 5× the coherence weight. -/
 159theorem stability_tradeoff :
 160    stabilityScore .BCC 0.3 0.7 > stabilityScore .FCC 0.3 0.7 ∧
 161    stabilityScore .FCC 0.9 0.1 > stabilityScore .BCC 0.9 0.1 := by
 162  simp only [stabilityScore, packingEfficiencyApprox, eightTickCoherence]
 163  -- BCC (0.3, 0.7): 0.3 * 0.68 + 0.7 * 1.0 = 0.204 + 0.7 = 0.904
 164  -- FCC (0.3, 0.7): 0.3 * 0.74 + 0.7 * (2/3) ≈ 0.222 + 0.467 = 0.689
 165  -- So BCC > FCC with high coherence weight ✓
 166  -- BCC (0.9, 0.1): 0.9 * 0.68 + 0.1 * 1.0 = 0.612 + 0.1 = 0.712
 167  -- FCC (0.9, 0.1): 0.9 * 0.74 + 0.1 * (2/3) = 0.666 + 0.067 ≈ 0.733
 168  -- So FCC > BCC with very high packing weight ✓
 169  constructor <;> norm_num
 170
 171/-! ## Metal Preferences -/
 172
 173/-- Metals that prefer BCC (8-tick dominant): alkali metals, Fe, Cr, W, Mo. -/
 174def prefersBCC : List ℕ := [3, 11, 19, 37, 55, 26, 24, 74, 42]  -- Li, Na, K, Rb, Cs, Fe, Cr, W, Mo
 175
 176/-- Metals that prefer FCC: Cu, Ag, Au, Al, Ni, Pb. -/
 177def prefersFCC : List ℕ := [29, 47, 79, 13, 28, 82]
 178
 179/-- Metals that prefer HCP: Mg, Zn, Ti, Zr. -/
 180def prefersHCP : List ℕ := [12, 30, 22, 40]
 181
 182/-- Alkali metals prefer BCC (strongest 8-tick coherence). -/
 183theorem alkali_prefer_bcc : ∀ Z, Z ∈ [3, 11, 19, 37, 55] → Z ∈ prefersBCC := by
 184  decide
 185
 186end
 187
 188end CrystalStructure
 189end Chemistry
 190end IndisputableMonolith
 191