 181/-- **CONSTRUCTIVE INTERFERENCE**: When paths reinforce.
 182
 183    Occurs when phase_diff = 2πn (paths in phase). -/
 184def constructive_interference (path1 path2 : List Config) : Prop :=
 185  ∃ n : ℤ, interference_amplitude path1 path2 (2 * Real.pi * n) > 0
 186
 187/-- **DESTRUCTIVE INTERFERENCE**: When paths cancel.
 188
 189    Occurs when phase_diff = π(2n+1) (paths out of phase). -/
 190def destructive_interference (path1 path2 : List Config) : Prop :=
 191  ∃ n : ℤ, interference_amplitude path1 path2 (Real.pi * (2 * n + 1)) < 0
 192
 193/-- Constructive interference at phase 0. -/
 194lemma constructive_at_zero (path1 path2 : List Config)
 195    (_ : path1 ≠ []) (_ : path2 ≠ []) :
 196    constructive_interference path1 path2 := by
 197  use 0
 198  simp only [interference_amplitude, Real.cos_zero, mul_one]
 199  apply mul_pos
 200  · apply Real.sqrt_pos_of_pos
 201    exact mul_pos (path_weight_pos path1) (path_weight_pos path2)
 202  · norm_num
 203
 204/-! ## The Modal Manifold -/
 205
 206/-- **MODAL MANIFOLD**: The space of all possibilities with its natural structure.
 207
 208    M_modal is the manifold whose points are configurations and whose
 209    tangent space at each point represents "directions of possible change."
 210
 211    Key properties:
 212    - Dimension = 1 (value) + 1 (time) = 2
 213    - Metric = modal_distance
 214    - Curvature = possibility_curvature

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.