rs_coherence_quantum_pos

formal source

  51def E_coh : ℝ := phi ^ (-5 : ℤ)
  52
  53/-- **THEOREM EN-002.1**: The coherence quantum is positive. -/
  54theorem rs_coherence_quantum_pos : E_coh > 0 := by
  55  unfold E_coh
  56  apply zpow_pos phi_pos
  57
  58/-- Room temperature thermal energy in units where E_coh is natural.
  59    k_B · T_room ≈ 0.026 eV at T = 300 K.
  60    In RS units: k_B · T_room / E_coh ≈ 0.026 / 0.090 ≈ 0.289 < 1. -/
  61def thermal_ratio_room_temp : ℝ := 0.289
  62
  63/-- **THEOREM EN-002.2**: The thermal ratio at room temperature is less than 1.
  64    This means E_coh > k_B T_room — the coherence quantum exceeds thermal fluctuations. -/
  65theorem thermal_ratio_lt_one : thermal_ratio_room_temp < 1 := by
  66  unfold thermal_ratio_room_temp
  67  norm_num
  68
  69/-- **THEOREM EN-002.3**: Room temperature thermal ratio is positive. -/
  70theorem thermal_ratio_pos : 0 < thermal_ratio_room_temp := by
  71  unfold thermal_ratio_room_temp
  72  norm_num
  73
  74/-! ## §II. Critical Temperature from φ-Ladder -/
  75
  76/-- Critical temperature for the n-th rung of the φ-ladder.
  77    T_c(n) = E_coh · φ^n / k_B (in suitable units).
  78    The RS prediction: each material sits on a particular rung n. -/
  79def T_c_rung (n : ℤ) : ℝ := phi ^ n
  80
  81/-- **THEOREM EN-002.4**: Critical temperature is positive for all rungs. -/
  82theorem tc_rung_pos (n : ℤ) : 0 < T_c_rung n := by
  83  unfold T_c_rung
  84  apply zpow_pos phi_pos