RSReal

formal source

 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. -/
 244theorem mp_physical :
 245    (∀ C : ℝ, ∃ ε > 0, ∀ x, 0 < x → x < ε → C < defect x) ∧  -- Nothing is infinitely expensive
 246    (∃! x : ℝ, RSExists x) ∧  -- There exists exactly one existent thing
 247    (∀ x, RSExists x → x = 1)  -- That thing is unity
 248  := ⟨nothing_cannot_exist, rs_exists_unique, fun x hx => (rs_exists_unique_one x).mp hx⟩
 249
 250/-- The Meta-Principle forces existence: since nothing is not selectable,
 251    something must be selected. -/
 252theorem mp_forces_existence :