IndisputableMonolith.Measurement.BornRuleLight

   1import Mathlib
   2
   3/-!
   4# Born Rule (Lightweight)
   5
   6Minimal algebraic lemma used in paper exports; avoids heavy dependencies.
   7-/
   8
   9namespace IndisputableMonolith
  10namespace Measurement
  11
  12open Real Complex
  13
  14/-- Born rule normalized: from recognition weights to normalized probabilities. -/
  15theorem born_rule_normalized (C₁ C₂ : ℝ) (α₁ α₂ : ℂ)
  16  (h₁ : Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₁‖ ^ 2)
  17  (h₂ : Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) = ‖α₂‖ ^ 2) :
  18  ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2 = 1 := by
  19  have hdenom : Real.exp (-C₁) + Real.exp (-C₂) ≠ 0 :=
  20    (add_pos (Real.exp_pos _) (Real.exp_pos _)).ne'
  21  calc ‖α₁‖ ^ 2 + ‖α₂‖ ^ 2
  22      = Real.exp (-C₁) / (Real.exp (-C₁) + Real.exp (-C₂)) +
  23        Real.exp (-C₂) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
  24          rw [← h₁, ← h₂]
  25      _ = (Real.exp (-C₁) + Real.exp (-C₂)) / (Real.exp (-C₁) + Real.exp (-C₂)) := by
  26          rw [div_add_div_same]
  27      _ = 1 := div_self hdenom
  28
  29end Measurement
  30end IndisputableMonolith
  31