project

formal source

 100  res_pos : 0 < resolution
 101
 102/-- The projection map: a deterministic state maps to an observed state. -/
 103noncomputable def project (obs : Observer) (x : ℝ) : Fin obs.resolution :=
 104  ⟨(Int.toNat (Int.floor (x * obs.resolution))) % obs.resolution,
 105   Nat.mod_lt _ obs.res_pos⟩
 106
 107/-- **Theorem**: Multiple distinct states map to the same observation.
 108    This is the origin of "apparent randomness." -/
 109theorem projection_lossy (obs : Observer) :
 110    ∃ x y : ℝ, x ≠ y ∧ project obs x = project obs y := by
 111  use 0, 1
 112  constructor
 113  · norm_num
 114  · simp [project]
 115
 116/-! ## Determinism Resolution -/
 117
 118/-- **The Determinism Theorem (F-007 Resolution)**:
 119
 120    1. The universe is deterministic: unique J-cost minimizer at each step.
 121    2. Apparent randomness arises from finite-resolution observation.
 122    3. "Quantum randomness" is a feature of the OBSERVER, not reality.
 123
 124    This dissolves the determinism-vs-randomness debate:
 125    - Reality IS deterministic (unique cost minimizer)
 126    - Observations APPEAR random (projection through finite resolution)
 127    - Both sides of the debate are correct, about different things -/
 128theorem determinism_resolution :
 129    (∀ x : ℝ, 0 < x → x ≠ 1 → 0 < LawOfExistence.defect x) ∧
 130    (∃! x : ℝ, 0 < x ∧ LawOfExistence.defect x = 0) := by
 131  constructor
 132  · intro x hx hne
 133    exact LawOfExistence.defect_pos_of_ne_one hx hne