rs_true_classical_iff

formal source

 210    Equivalent to the old placeholder `def RSTrue (P : Prop) : Prop := P`. -/
 211def RSTrue_classical (P : Prop) : Prop := P
 212
 213theorem rs_true_classical_iff (P : Prop) : RSTrue_classical P ↔ P := Iff.rfl
 214
 215/-! ## RSReal: Existence in the Discrete Configuration Space -/
 216
 217/-- **RSReal**: A value x is "real" in the RS sense if:
 218    1. RSExists x (stable under J)
 219    2. x is in the discrete configuration space (quantized)
 220
 221    For now, we model discreteness as being algebraic in φ. -/
 222def RSReal (x : ℝ) : Prop :=
 223  RSExists x ∧ ∃ n m : ℤ, x = PhiForcing.φ ^ n * PhiForcing.φ ^ m
 224
 225/-- Unity is RSReal (trivially, as φ⁰ · φ⁰ = 1). -/
 226theorem rs_real_one : RSReal 1 := by
 227  constructor
 228  · exact rs_exists_one
 229  · use 0, 0
 230    simp [PhiForcing.φ]
 231
 232/-! ## The Meta-Principle as a Physical Theorem -/
 233
 234/-- **MP_PHYSICAL**: The Meta-Principle "Nothing cannot recognize itself"
 235    as a theorem about cost.
 236
 237    In the CPM/cost foundation, this is DERIVED, not assumed:
 238    - "Nothing" (x → 0⁺) has unbounded defect
 239    - Therefore "nothing" cannot be selected by cost minimization
 240    - Therefore "something" must exist (the unique x=1 minimizer)
 241
 242    This replaces the tautological "Empty has no inhabitants" with
 243    a physical statement about selection. -/