minus_one_step_forces_phi_shift

formal source

  88
  89/-- Three-point calibration (`x = -1,0,1`) forces the affine-log shift to `b = φ`.
  90    The extra `b > 1` assumption encodes the physically relevant positive-shift branch. -/
  91lemma minus_one_step_forces_phi_shift
  92    {a b c : ℝ}
  93    (hb : 1 < b)
  94    (h0 : gapAffineLogR a b c 0 = 0)
  95    (h1 : gapAffineLogR a b c 1 = 1)
  96    (hneg1 : gapAffineLogR a b c (-1) = -2) :
  97    b = phi := by
  98  have hb_pos : 0 < b := lt_trans zero_lt_one hb
  99  have hb_ne : b ≠ 0 := ne_of_gt hb_pos
 100  have hplus_pos : 0 < 1 + (1 : ℝ) / b := by
 101    have hinv_pos : 0 < (1 : ℝ) / b := one_div_pos.mpr hb_pos
 102    linarith
 103  have hinv_lt_one : (1 : ℝ) / b < 1 := by
 104    simpa using (one_div_lt_one_div_of_lt (by norm_num : (0 : ℝ) < 1) hb)
 105  have hminus_pos : 0 < 1 - (1 : ℝ) / b := by
 106    linarith
 107  have hminus_ne : (1 - (1 : ℝ) / b) ≠ 0 := ne_of_gt hminus_pos
 108  have hc : c = 0 := by
 109    simpa [gapAffineLogR] using h0
 110  have h1' : a * Real.log (1 + (1 : ℝ) / b) = 1 := by
 111    simpa [gapAffineLogR, hc] using h1
 112  have hneg1_raw : a * Real.log (1 + (-1 : ℝ) / b) = -2 := by
 113    simpa [gapAffineLogR, hc] using hneg1
 114  have hneg1' : a * Real.log (1 - (1 : ℝ) / b) = -2 := by
 115    simpa [sub_eq_add_neg, div_eq_mul_inv, mul_assoc] using hneg1_raw
 116  have ha_ne : a ≠ 0 := by
 117    intro ha
 118    have h1'' := h1'
 119    simp [ha] at h1''
 120  have hscaled : a * (-2 * Real.log (1 + (1 : ℝ) / b)) = -2 := by
 121    calc