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AnuQuantumInterference
Phase mask operations with constructive/destructive interference
-/
import Mathlib
noncomputable section
open Real
-- ============================================================
-- Core Definitions
-- ============================================================
/-- Apply a phase mask: true = constructive (keep), false = destructive (negate) -/
def apply_phase_mask (w : ℝ) (constructive : Bool) : ℝ :=
if constructive then w else -w
/-- Phase shift values for a layer -/
def phase_shift (layer_idx : Nat) (n_layers : Nat) : ℝ :=
2 * Real.pi * (layer_idx : ℝ) / (n_layers : ℝ)
/-- Perturbation: small noise added to weights -/
def perturb (w : ℝ) (epsilon : ℝ) : ℝ :=
w + epsilon
/-- Pipeline: sequence of phase-masked operations -/
def pipeline (weights : List ℝ) (masks : List Bool) : List ℝ :=
List.zipWith apply_phase_mask weights masks
-- ============================================================
-- Theorems (6)
-- ============================================================
/-- Destructive interference negates the weight -/
theorem destructive_cancellation (w : ℝ) :
apply_phase_mask w false = -w := by
simp [apply_phase_mask]
/-- Constructive interference preserves the weight -/
theorem constructive_preservation (w : ℝ) :
apply_phase_mask w true = w := by
simp [apply_phase_mask]
/-- Phase shift values are in [0, 2*pi) for valid layer indices -/
theorem phase_shift_values (layer_idx n_layers : Nat)
(h_valid : layer_idx < n_layers)
(h_pos : n_layers > 0) :
0 ≤ phase_shift layer_idx n_layers ∧
phase_shift layer_idx n_layers < 2 * Real.pi := by
have h₁ : (layer_idx : ℝ) ≥ 0 := by exact_mod_cast Nat.zero_le layer_idx
have h₂ : (n_layers : ℝ) > 0 := by exact_mod_cast h_pos
have h₃ : (layer_idx : ℝ) < (n_layers : ℝ) := by exact_mod_cast h_valid
have h₅ : (layer_idx : ℝ) / (n_layers : ℝ) < 1 := by
rw [div_lt_one h₂]; exact_mod_cast h_valid
constructor
· simp [phase_shift]
positivity
· simp [phase_shift]
have : (layer_idx : ℝ) / (n_layers : ℝ) < 1 := h₅
have hpi : 0 < Real.pi := Real.pi_pos
nlinarith
/-- Small perturbation preserves sign structure -/
theorem perturbation_preserves_structure (w epsilon : ℝ)
(h_small : |epsilon| < |w|)
(h_w_pos : w > 0) :
perturb w epsilon > 0 := by
have h₁ : |w| = w := abs_of_pos h_w_pos
have h₂ : |epsilon| < w := by rwa [h₁] at h_small
have h₃ : -w < epsilon := (abs_lt.mp h₂).1
simp only [perturb]
linarith
/-- Pipeline produces output for every input with a matching mask -/
theorem pipeline_soundness (weights : List ℝ) (masks : List Bool)
(h_len : weights.length = masks.length) :
(pipeline weights masks).length = weights.length := by
simp only [pipeline, List.length_zipWith, h_len, min_self]
/-- Pipeline output fully determined by inputs (no hidden state) -/
theorem pipeline_completeness (weights1 weights2 : List ℝ) (masks1 masks2 : List Bool)
(h_w : weights1 = weights2)
(h_m : masks1 = masks2) :
pipeline weights1 masks1 = pipeline weights2 masks2 := by
rw [h_w, h_m]
end
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