bremermann_limit

formal source

 141
 142/-- The Bremermann limit: maximum operations per second per joule.
 143    B = 2/ℏ ≈ 1.9 × 10³⁴ operations per second per joule. -/
 144noncomputable def bremermann_limit : ℝ := 2 / hbar
 145
 146/-- **THEOREM IC-002.11**: The Bremermann limit is positive and finite. -/
 147theorem bremermann_limit_pos : bremermann_limit > 0 := by
 148  unfold bremermann_limit hbar
 149  norm_num
 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⟩