max_ops_per_sec

formal source

 150
 151/-- For a system with energy E, the maximum number of operations per second is
 152    bounded by B × E (Bremermann's limit). -/
 153noncomputable def max_ops_per_sec (E : ℝ) : ℝ := bremermann_limit * E
 154
 155/-- **THEOREM IC-002.12**: Maximum computation rate scales with energy. -/
 156theorem max_ops_scales_with_energy (E : ℝ) (hE : E > 0) :
 157    max_ops_per_sec E > 0 :=
 158  mul_pos bremermann_limit_pos hE
 159
 160/-- **THEOREM IC-002.13**: A finite-energy system has a finite computation bound. -/
 161theorem finite_energy_implies_finite_computation (E M : ℝ) (hE : E > 0) :
 162    ∃ bound : ℝ, bound > 0 ∧ max_ops_per_sec E ≤ bound := by
 163  exact ⟨max_ops_per_sec E, mul_pos bremermann_limit_pos hE, le_refl _⟩
 164
 165/-! ## V. The RS Computation Bound from φ -/
 166
 167/-- **THEOREM IC-002.14**: φ > 1 (φ is greater than 1). -/
 168theorem phi_gt_one : phi > 1 := one_lt_phi
 169
 170/-- **THEOREM IC-002.15**: φ-based costs grow without bound as exponents increase.
 171    This means RS dynamics at high rung numbers require exponentially growing resources. -/
 172theorem phi_powers_unbounded (M : ℝ) : ∃ n : ℕ, phi ^ n > M := by
 173  obtain ⟨n, hn⟩ := pow_unbounded_of_one_lt M one_lt_phi
 174  exact ⟨n, hn⟩
 175
 176/-! ## Summary: Computation Limits from RS -/
 177
 178/-- Summary of computation limits derived in RS. -/
 179def computation_limits_summary : List String := [
 180  "IC-002.1: Fundamental tick τ₀ > 0 (minimum time quantum)",
 181  "IC-002.2: Maximum computation rate = 1/τ₀ > 0",
 182  "IC-002.4: φ is irrational (exact simulation requires transcendental precision)",
 183  "IC-002.7: No rational algorithm exactly computes φ",