unit_step_forces_log_scale

formal source

  70  simpa [gapAffineLogR] using h0
  71
  72/-- Unit-step calibration fixes the log scale coefficient. -/
  73lemma unit_step_forces_log_scale
  74    {a c : ℝ}
  75    (h0 : gapAffineLogR a phi c 0 = 0)
  76    (h1 : gapAffineLogR a phi c 1 = 1) :
  77    a = 1 / Real.log phi := by
  78  have hc : c = 0 := zero_normalization_forces_offset h0
  79  have hlog_ne : Real.log phi ≠ 0 := ne_of_gt (Real.log_pos one_lt_phi)
  80  have hmul_raw : a * Real.log (1 + phi⁻¹) = 1 := by
  81    simpa [gapAffineLogR, hc] using h1
  82  have hmul : a * Real.log phi = 1 := by
  83    calc
  84      a * Real.log phi = a * Real.log (1 + phi⁻¹) := by
  85        rw [log_one_add_inv_phi_eq_log_phi]
  86      _ = 1 := hmul_raw
  87  exact (eq_div_iff hlog_ne).2 hmul
  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