simulation_unprovable

formal source

  82/-- **THEOREM IC-004.3**: The simulation predicate is not provably true for any RS universe.
  83    This formalizes: "there is no fact of the matter" about simulation in RS.
  84    Any "outer-universe" would itself be an RS universe with the same structure. -/
  85theorem simulation_unprovable :
  86    ∀ u : RSUniverse, ¬ (∀ (outer : RSUniverse), ∀ n, outer.events n ≠ u.events n) := by
  87  intro u h
  88  -- Take outer = u itself
  89  have := h u
  90  -- Then for all n, u.events n ≠ u.events n — contradiction
  91  exact absurd rfl (this 0)
  92
  93/-- **THEOREM IC-004.4**: Any "external outer-universe" that contains the RS universe
  94    as a simulation must have the same type as an RS universe.
  95    The simulation/reality distinction collapses. -/
  96theorem outer_universe_is_rs_universe (outer : RSUniverse) (u : RSUniverse) :
  97    ∃ (combined : RSUniverse), ∀ n, combined.events n > 0 := by
  98  exact ⟨outer, outer.events_pos⟩
  99
 100/-! ## III. The Ledger IS the Bottom of Reality -/
 101
 102/-- The ledger is self-grounding: it provides its own existence criterion.
 103    J(x) ≥ 0, with J(x) = 0 iff x = 1 (the zero-defect state).
 104    No "external" grounding is needed. -/
 105def ledger_is_self_grounded : Prop :=
 106  ∀ x : ℝ, x > 0 → Cost.Jcost x ≥ 0
 107
 108/-- **THEOREM IC-004.5**: The ledger is self-grounded: all J-costs are non-negative. -/
 109theorem ledger_self_grounding : ledger_is_self_grounded := by
 110  intro x hx
 111  exact Cost.Jcost_nonneg hx
 112
 113/-- **THEOREM IC-004.6**: The J-cost framework determines what "exists":
 114    x exists (RSExists) iff J(x) = 0 iff x = 1. -/
 115theorem rs_exists_iff_zero_cost (x : ℝ) (hx : x > 0) :