alpha_bounds

formal source

  91  nlinarith
  92
  93/-- **BOUNDS**: 0 < α < 0.1 -/
  94theorem alpha_bounds : alpha > 0 ∧ alpha < (0.1 : ℝ) := by
  95  constructor
  96  · exact alpha_positive
  97  · exact alpha_lt_0_1
  98
  99/-! ## Section C-005: Speed of Light c -/
 100
 101/-- **CALCULATED**: c = 1 (RS-native units). -/
 102theorem c_eq_one : c = 1 := by rfl
 103
 104/-- **CALCULATED**: c > 0. -/
 105theorem c_positive : c > 0 := by
 106  rw [c_eq_one]
 107  norm_num
 108
 109/-! ## Section C-006: Boltzmann Analog k_R -/
 110
 111/-- **CALCULATED**: RS-native Boltzmann constant k_R = ln(φ). -/
 112theorem boltzmann_analog_formula : ∃ (k_R : ℝ), k_R = Real.log phi := by
 113  use Real.log phi
 114
 115/-- **CALCULATED**: k_R > 0 since φ > 1. -/
 116theorem boltzmann_analog_positive : Real.log phi > 0 := by
 117  apply Real.log_pos
 118  exact one_lt_phi
 119
 120/-- **CALCULATED**: k_R < 0.5 since φ < 1.62 < e^0.5 ≈ 1.6487. -/
 121theorem boltzmann_analog_lt_half : Real.log phi < (0.5 : ℝ) := by
 122  -- We'll use a simpler approach: show ln(φ) < 0.5 from φ < 1.62
 123  -- This is true since 1.62 < e^0.5 ≈ 1.6487
 124  have h1 : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo