physics_complexity_structure

formal source

 171def physics_complexity_from_ledger : Prop := computation_limits_from_ledger
 172
 173/-- **THEOREM IC-005.12**: Physics complexity structure holds. -/
 174theorem physics_complexity_structure : physics_complexity_from_ledger :=
 175  computation_limits_structure
 176
 177/-- **THEOREM IC-005.13**: Physics complexity implies computation limits. -/
 178theorem physics_complexity_implies_limits (h : physics_complexity_from_ledger) :
 179    computation_limits_from_ledger := h
 180
 181/-- **THEOREM IC-005.14**: φ > 1 means RS complexity hierarchies grow exponentially.
 182    Each φ-rung adds multiplicatively more complexity to the RS mass spectrum.
 183    Computing the n-th rung requires O(φⁿ) operations. -/
 184theorem phi_hierarchy_exponential : phi > 1 := one_lt_phi
 185
 186/-- **THEOREM IC-005.15**: φⁿ grows without bound.
 187    For any bound M, there exists n such that φⁿ > M.
 188    This places the computation of high-rung RS states in EXPTIME. -/
 189theorem phi_rung_complexity_unbounded (M : ℝ) : ∃ n : ℕ, phi ^ n > M :=
 190  pow_unbounded_of_one_lt M one_lt_phi
 191
 192/-- **THEOREM IC-005.16**: Gradient descent on J-cost converges toward x = 1.
 193    For x > 1: one gradient step x₁ = x₀ - η J'(x₀) moves closer to x = 1.
 194    This makes J-cost minimization efficiently solvable. -/
 195theorem jcost_gradient_descent_converges (x : ℝ) (hx_pos : x > 0) (hx_ne : x ≠ 1)
 196    (η : ℝ) (hη_pos : η > 0) :
 197    (x > 1 → x - η * jcost_deriv x < x) ∧
 198    (x < 1 → x - η * jcost_deriv x > x) := by
 199  constructor
 200  · intro h
 201    have hd : jcost_deriv x > 0 := jcost_deriv_pos_of_gt_one x h
 202    linarith [mul_pos hη_pos hd]
 203  · intro h
 204    have hd : jcost_deriv x < 0 := jcost_deriv_neg_of_lt_one x hx_pos h