phi_in_observed_range

formal source

 116
 117    This proves the range property for the φ value itself. Once `ml_derived_value`
 118    is proven (showing ml_derived = φ), this immediately gives `ml_in_observed_range`. -/
 119theorem phi_in_observed_range : 0.5 < φ ∧ φ < 5 := by
 120  constructor
 121  · -- 0.5 < φ: Since φ = (1 + √5)/2 and √5 > 0, we have φ > 0.5
 122    unfold φ Constants.phi
 123    have h_sqrt5_pos : 0 < Real.sqrt 5 := Real.sqrt_pos.mpr (by norm_num : (5 : ℝ) > 0)
 124    linarith
 125  · -- φ < 5: Since φ = (1 + √5)/2 and √5 < 3, we have φ < 2 < 5
 126    unfold φ Constants.phi
 127    have h_sqrt5_lt_3 : Real.sqrt 5 < 3 := by
 128      rw [show (3 : ℝ) = Real.sqrt 9 by norm_num]
 129      exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
 130    linarith
 131
 132/-- **THEOREM (RIGOROUS)**: φ is strictly between 1 and 2. -/
 133theorem phi_bounds : 1 < φ ∧ φ < 2 := by
 134  constructor
 135  · -- 1 < φ: Since √5 > 1, we have (1 + √5)/2 > 1
 136    unfold φ Constants.phi
 137    have h_sqrt5_gt_1 : 1 < Real.sqrt 5 := by
 138      rw [show (1 : ℝ) = Real.sqrt 1 by norm_num]
 139      exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
 140    linarith
 141  · -- φ < 2: Since √5 < 3, we have (1 + √5)/2 < 2
 142    unfold φ Constants.phi
 143    have h_sqrt5_lt_3 : Real.sqrt 5 < 3 := by
 144      rw [show (3 : ℝ) = Real.sqrt 9 by norm_num]
 145      exact Real.sqrt_lt_sqrt (by norm_num) (by norm_num)
 146    linarith
 147
 148/-- The derived M/L is in the observed range [0.5, 5] solar units.
 149    Proof depends on the axiom ml_derived_value : ml_derived = φ. -/