phi_powers_unbounded

formal source

 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",
 184  "IC-002.8: Landauer energy k_B T ln(2) > 0 (cost to erase 1 bit)",
 185  "IC-002.9: Landauer bound scales linearly with temperature",
 186  "IC-002.11: Bremermann limit = 2/ℏ > 0 (operations/second/joule)",
 187  "IC-002.14: φ > 1 (RS hierarchies grow exponentially)"
 188]
 189
 190/-- IC-002 Status Certificate -/
 191def ic002_certificate : String :=
 192  "═══════════════════════════════════════════════════════\n" ++
 193  "  IC-002: COMPUTATION LIMITS — STATUS: DERIVED\n" ++
 194  "═══════════════════════════════════════════════════════\n" ++
 195  "✓ tick_pos:                 τ₀ > 0\n" ++
 196  "✓ max_rate_pos:             1/τ₀ > 0\n" ++
 197  "✓ computation_limits:       Irrational φ (core constraint)\n" ++
 198  "✓ no_exact_phi_computation: ∀ q : ℚ, q ≠ φ\n" ++
 199  "✓ landauer_energy_pos:      k_B T ln(2) > 0\n" ++
 200  "✓ landauer_scales_with_temp: monotone in T\n" ++
 201  "✓ bremermann_limit_pos:     B = 2/ℏ > 0\n" ++
 202  "✓ phi_powers_unbounded:     φⁿ → ∞\n"