c_positive

formal source

 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
 125  -- We know e^0.5 > 1.62 (Taylor series: 1 + 0.5 + 0.125 + ... > 1.62)
 126  -- Therefore ln(φ) < ln(1.62) < 0.5
 127  have h2 : Real.log phi < Real.log (1.62 : ℝ) := by
 128    apply Real.log_lt_log
 129    all_goals nlinarith [phi_pos, h1]
 130  -- Now we need to show ln(1.62) < 0.5
 131  -- This is equivalent to showing 1.62 < e^0.5
 132  -- We use exp(0.5)^2 = exp(1) and exp(1) > 2.6244 = 1.62^2
 133  -- exp(1) > 2.6244 follows from exp(1) ≥ 8/3 ≈ 2.666 > 2.6244 (Taylor series: 1 + 1 + 1/2 + 1/6)
 134  have h3 : Real.log (1.62 : ℝ) < (0.5 : ℝ) := by
 135    -- Show 1.62 < exp(0.5) by showing 1.62^2 < exp(1)