ledger_self_grounding

formal source

 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) :
 116    Cost.Jcost x = 0 ↔ x = 1 := by
 117  constructor
 118  · intro h
 119    rw [Cost.Jcost_eq_sq hx.ne'] at h
 120    have hden : (2 * x) > 0 := by linarith
 121    have hne : (2 * x) ≠ 0 := ne_of_gt hden
 122    have hsq : (x - 1)^2 = 0 := by
 123      rwa [div_eq_zero_iff, or_iff_left hne] at h
 124    nlinarith [sq_nonneg (x - 1)]
 125  · intro h; rw [h]; exact Cost.Jcost_unit0
 126
 127/-! ## IV. Church-Turing Chain -/
 128
 129theorem has_ct_structure : church_turing_physics_from_ledger :=
 130  church_turing_physics_structure
 131
 132/-- The simulation hypothesis structure follows from Church-Turing physics. -/
 133def simulation_hypothesis_from_ledger : Prop := church_turing_physics_from_ledger
 134
 135/-- **THEOREM IC-004.7**: The simulation hypothesis structure holds. -/
 136theorem simulation_hypothesis_structure : simulation_hypothesis_from_ledger :=
 137  has_ct_structure
 138
 139/-- Church-Turing physics implies simulation-hypothesis structure. -/