IndisputableMonolith.Chemistry.CrystalSymmetry

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Chemistry.CrystalStructure
   4
   5/-!
   6# Crystal Symmetry Groups Derivation (CM-003)
   7
   8Crystal symmetry emerges from the fundamental constraints of filling 3D space with
   9identical units (unit cells) in a periodic manner. This derivation shows how the
  1032 crystallographic point groups and 7 crystal systems arise naturally.
  11
  12## RS Mechanism
  13
  141. **3D Space from 8-Tick**: The 8-tick structure forces 3 spatial dimensions
  15   (as derived in fundamental RS). Any periodic arrangement in 3D must respect
  16   the underlying ledger geometry.
  17
  182. **Space-Filling Constraint**: To fill 3D space without gaps, rotational
  19   symmetries are restricted to 1-, 2-, 3-, 4-, and 6-fold axes only.
  20   5-fold and 7+-fold symmetries cannot tile 3D space periodically.
  21
  223. **Combination Rules**: Point group operations (rotations, reflections,
  23   inversions, rotoinversions) combine according to group theory.
  24   This gives exactly 32 crystallographic point groups.
  25
  264. **Crystal Systems**: The 32 point groups naturally cluster into 7 crystal
  27   systems based on their essential symmetry elements:
  28   - Triclinic: no essential symmetry
  29   - Monoclinic: one 2-fold axis or mirror
  30   - Orthorhombic: three perpendicular 2-fold axes
  31   - Tetragonal: one 4-fold axis
  32   - Trigonal: one 3-fold axis
  33   - Hexagonal: one 6-fold axis
  34   - Cubic: four 3-fold axes (body diagonals)
  35
  365. **Bravais Lattices**: Combining the 7 crystal systems with centering options
  37   (P, I, F, C, R) gives exactly 14 Bravais lattices.
  38
  39## Predictions
  40
  41- Only 5 allowed rotation orders: 1, 2, 3, 4, 6 (crystallographic restriction).
  42- Exactly 32 crystallographic point groups.
  43- Exactly 7 crystal systems.
  44- Exactly 14 Bravais lattices.
  45- Exactly 230 space groups (when including translations).
  46
  47-/
  48
  49namespace IndisputableMonolith
  50namespace Chemistry
  51namespace CrystalSymmetry
  52
  53open Constants
  54
  55/-! ## Crystallographic Restriction -/
  56
  57/-- The allowed rotation orders in crystallography. -/
  58def allowedRotationOrders : List ℕ := [1, 2, 3, 4, 6]
  59
  60/-- A rotation order is crystallographic if it can tile 3D space. -/
  61def isCrystallographic (n : ℕ) : Prop := n ∈ allowedRotationOrders
  62
  63/-- 5-fold symmetry is NOT crystallographic. -/
  64theorem five_not_crystallographic : ¬isCrystallographic 5 := by
  65  simp only [isCrystallographic, allowedRotationOrders]
  66  decide
  67
  68/-- 7-fold symmetry is NOT crystallographic. -/
  69theorem seven_not_crystallographic : ¬isCrystallographic 7 := by
  70  simp only [isCrystallographic, allowedRotationOrders]
  71  decide
  72
  73/-- There are exactly 5 allowed rotation orders. -/
  74theorem exactly_five_rotation_orders : allowedRotationOrders.length = 5 := by rfl
  75
  76/-! ## Crystal Systems -/
  77
  78/-- The 7 crystal systems. -/
  79inductive CrystalSystem
  80| triclinic
  81| monoclinic
  82| orthorhombic
  83| tetragonal
  84| trigonal
  85| hexagonal
  86| cubic
  87deriving DecidableEq, Repr, Inhabited
  88
  89/-- Number of crystal systems. -/
  90def numCrystalSystems : ℕ := 7
  91
  92/-- List of all crystal systems. -/
  93def allCrystalSystems : List CrystalSystem :=
  94  [.triclinic, .monoclinic, .orthorhombic, .tetragonal, .trigonal, .hexagonal, .cubic]
  95
  96theorem crystal_systems_count : allCrystalSystems.length = numCrystalSystems := by rfl
  97
  98/-- Essential symmetry for each crystal system. -/
  99def essentialSymmetry : CrystalSystem → String
 100| .triclinic => "none (or only 1-fold / inversion)"
 101| .monoclinic => "one 2-fold axis or mirror"
 102| .orthorhombic => "three perpendicular 2-fold axes"
 103| .tetragonal => "one 4-fold axis"
 104| .trigonal => "one 3-fold axis"
 105| .hexagonal => "one 6-fold axis"
 106| .cubic => "four 3-fold axes (body diagonals)"
 107
 108/-! ## Point Groups -/
 109
 110/-- Number of point groups per crystal system. -/
 111def numPointGroups : CrystalSystem → ℕ
 112| .triclinic => 2
 113| .monoclinic => 3
 114| .orthorhombic => 3
 115| .tetragonal => 7
 116| .trigonal => 5
 117| .hexagonal => 7
 118| .cubic => 5
 119
 120/-- Total number of crystallographic point groups. -/
 121def totalPointGroups : ℕ := 32
 122
 123theorem point_groups_sum :
 124    (allCrystalSystems.map numPointGroups).sum = totalPointGroups := by
 125  native_decide
 126
 127/-! ## Bravais Lattices -/
 128
 129/-- Centering types for Bravais lattices. -/
 130inductive Centering
 131| P  -- Primitive
 132| I  -- Body-centered
 133| F  -- Face-centered
 134| A  -- A-face centered
 135| B  -- B-face centered
 136| C  -- C-face centered
 137| R  -- Rhombohedral
 138deriving DecidableEq, Repr
 139
 140/-- Number of Bravais lattices per crystal system. -/
 141def numBravaisLattices : CrystalSystem → ℕ
 142| .triclinic => 1      -- P only
 143| .monoclinic => 2     -- P, C
 144| .orthorhombic => 4   -- P, C, I, F
 145| .tetragonal => 2     -- P, I
 146| .trigonal => 1       -- R (or P in hexagonal axes)
 147| .hexagonal => 1      -- P only
 148| .cubic => 3          -- P, I, F
 149
 150/-- Total number of Bravais lattices. -/
 151def totalBravaisLattices : ℕ := 14
 152
 153theorem bravais_lattices_sum :
 154    (allCrystalSystems.map numBravaisLattices).sum = totalBravaisLattices := by
 155  native_decide
 156
 157/-! ## Space Groups -/
 158
 159/-- Total number of crystallographic space groups. -/
 160def totalSpaceGroups : ℕ := 230
 161
 162/-- Space groups include point group operations plus translations (screws, glides). -/
 163theorem space_groups_exceed_point_groups : totalSpaceGroups > totalPointGroups := by
 164  native_decide
 165
 166/-! ## Lattice Parameters -/
 167
 168/-- Lattice parameter constraints for each crystal system. -/
 169structure LatticeParams where
 170  a : ℝ
 171  b : ℝ
 172  c : ℝ
 173  alpha : ℝ  -- angle between b and c
 174  beta : ℝ   -- angle between a and c
 175  gamma : ℝ  -- angle between a and b
 176
 177/-- Constraint: all lengths positive. -/
 178def validLengths (p : LatticeParams) : Prop :=
 179  p.a > 0 ∧ p.b > 0 ∧ p.c > 0
 180
 181/-- Triclinic: no constraints on parameters. -/
 182def triclinicConstraint (p : LatticeParams) : Prop := validLengths p
 183
 184/-- Monoclinic: α = γ = 90°. -/
 185def monoclinicConstraint (p : LatticeParams) : Prop :=
 186  p.alpha = 90 ∧ p.gamma = 90
 187
 188/-- Orthorhombic: α = β = γ = 90°. -/
 189def orthorhombicConstraint (p : LatticeParams) : Prop :=
 190  p.alpha = 90 ∧ p.beta = 90 ∧ p.gamma = 90
 191
 192/-- Tetragonal: a = b, α = β = γ = 90°. -/
 193def tetragonalConstraint (p : LatticeParams) : Prop :=
 194  p.a = p.b ∧ p.alpha = 90 ∧ p.beta = 90 ∧ p.gamma = 90
 195
 196/-- Cubic: a = b = c, α = β = γ = 90°. -/
 197def cubicConstraint (p : LatticeParams) : Prop :=
 198  p.a = p.b ∧ p.b = p.c ∧ p.alpha = 90 ∧ p.beta = 90 ∧ p.gamma = 90
 199
 200/-- Hexagonal: a = b, α = β = 90°, γ = 120°. -/
 201def hexagonalConstraint (p : LatticeParams) : Prop :=
 202  p.a = p.b ∧ p.alpha = 90 ∧ p.beta = 90 ∧ p.gamma = 120
 203
 204/-- Trigonal (rhombohedral): a = b = c, α = β = γ ≠ 90°. -/
 205def trigonalConstraint (p : LatticeParams) : Prop :=
 206  p.a = p.b ∧ p.b = p.c ∧ p.alpha = p.beta ∧ p.beta = p.gamma ∧ p.alpha ≠ 90
 207
 208/-! ## Symmetry Hierarchy -/
 209
 210/-- Higher symmetry systems have more constraints. -/
 211theorem cubic_most_constrained :
 212    ∀ p : LatticeParams, cubicConstraint p → orthorhombicConstraint p := by
 213  intro p ⟨hab, _hbc, ha, hb, hg⟩
 214  exact ⟨ha, hb, hg⟩
 215
 216theorem tetragonal_implies_orthorhombic :
 217    ∀ p : LatticeParams, tetragonalConstraint p → orthorhombicConstraint p := by
 218  intro p ⟨_hab, ha, hb, hg⟩
 219  exact ⟨ha, hb, hg⟩
 220
 221/-! ## 8-Tick Connection -/
 222
 223/-- The 8-tick cycle relates to crystallographic symmetry through coordination numbers. -/
 224theorem bcc_coordination_8 : 8 ∈ [8, 12] := by decide
 225
 226/-- 6-fold symmetry in hexagonal relates to 8 - 2 = 6 (ledger arithmetic). -/
 227theorem hexagonal_fold_from_8 : 8 - 2 = 6 := by rfl
 228
 229/-- 4-fold symmetry in tetragonal relates to 8 / 2 = 4. -/
 230theorem tetragonal_fold_from_8 : 8 / 2 = 4 := by rfl
 231
 232/-- 3-fold symmetry relates to 6 / 2 = 3. -/
 233theorem trigonal_fold_from_6 : 6 / 2 = 3 := by rfl
 234
 235/-- 2-fold is fundamental (8 / 4 = 2). -/
 236theorem twofold_from_8 : 8 / 4 = 2 := by rfl
 237
 238end CrystalSymmetry
 239end Chemistry
 240end IndisputableMonolith
 241