unit_step_forces_log_scale

formal source

  69
  70/-! ## Step 2: g(1) = 1 forces a = 1/log(φ) (given c = 0 and b = φ) -/
  71
  72lemma unit_step_forces_log_scale
  73    {a c : ℝ}
  74    (h0 : gapAffineLogR a phi c 0 = 0)
  75    (h1 : gapAffineLogR a phi c 1 = 1) :
  76    a = 1 / Real.log phi := by
  77  have hc : c = 0 := zero_normalization_forces_offset h0
  78  have hlog_ne : Real.log phi ≠ 0 := ne_of_gt (Real.log_pos one_lt_phi)
  79  have hmul_raw : a * Real.log (1 + phi⁻¹) = 1 := by
  80    simpa [gapAffineLogR, hc] using h1
  81  have hmul : a * Real.log phi = 1 := by
  82    calc
  83      a * Real.log phi = a * Real.log (1 + phi⁻¹) := by
  84        rw [log_one_add_inv_phi_eq_log_phi]
  85      _ = 1 := hmul_raw
  86  exact (eq_div_iff hlog_ne).2 hmul
  87
  88/-! ## Step 3: g(-1) = -2 forces b = φ (the key theorem)
  89
  90This is the paper's Theorem 4.2: setting u = 1/b, the condition
  91`(1 - u)(1 + u)^2 = 1` expands to `u^2 + u - 1 = 0`, giving u = 1/φ. -/
  92
  93theorem minus_one_step_forces_phi_shift
  94    {a b c : ℝ}
  95    (hb : 1 < b)
  96    (h0 : gapAffineLogR a b c 0 = 0)
  97    (h1 : gapAffineLogR a b c 1 = 1)
  98    (hneg1 : gapAffineLogR a b c (-1) = -2) :
  99    b = phi := by
 100  have hb_pos : 0 < b := lt_trans zero_lt_one hb
 101  have hb_ne : b ≠ 0 := ne_of_gt hb_pos
 102  have hplus_pos : 0 < 1 + (1 : ℝ) / b := by