 185  env_modes : ℕ
 186
 187/-- Typical superconducting qubit. -/
 188def typicalSCQubit : QubitParams := {
 189  t1 := 50e-6,        -- 50 μs
 190  t2 := 70e-6,        -- 70 μs
 191  temperature := 0.015,-- 15 mK
 192  env_modes := 10     -- Estimated
 193}
 194
 195/-- Predicted decoherence time for the typical qubit. -/
 196noncomputable def predictedQubitDecoherence : ℝ :=
 197  decoherenceTime ⟨typicalSCQubit.env_modes, 0.5, by norm_num, by norm_num⟩
 198
 199/-! ## The Gap-45 Threshold in Practice -/
 200
 201/-- Number of modes to cross from quantum to classical (Gap-45 crossover).
 202    For τ_target = 1 s (human scale), coupling = 1:
 203    N ≈ 45 × ln(10) / ln(φ) ≈ 45 × 2.3 / 0.48 ≈ 215 -/
 204noncomputable def gap45CrossoverModes : ℝ :=
 205  criticalModes tau_bio 1.0
 206
 207/-- Approximation of Gap-45 crossover modes as a rational.
 208    N ≈ ln(τ_bio/τ₀) / ln(φ) ≈ ln(1/(5.4e-44)) / 0.48
 209    ≈ 99.3 / 0.48 ≈ 207
 210
 211    Since τ₀ ≈ 5.4×10⁻⁴⁴ s, τ_bio = 1 s:
 212    ln(1/5.4e-44) ≈ 44 × ln(10) ≈ 44 × 2.3 ≈ 101
 213    ln(φ) ≈ 0.48
 214    N ≈ 101/0.48 ≈ 210 -/
 215def gap45CrossoverApprox : ℚ := 210
 216
 217/-- **THEOREM**: The Gap-45 crossover occurs at approximately 100-300 modes. -/
 218theorem gap45_crossover_range :

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