log_one_add_inv_phi_eq_log_phi

formal source

  54lemma one_add_inv_phi_eq_phi : 1 + (1 : ℝ) / phi = phi :=
  55  phi_eq_one_add_inv_phi.symm
  56
  57lemma log_one_add_inv_phi_eq_log_phi : Real.log (1 + phi⁻¹) = Real.log phi := by
  58  have hshift : (1 + phi⁻¹ : ℝ) = phi := by
  59    simpa [one_div] using one_add_inv_phi_eq_phi
  60  simp [hshift]
  61
  62/-! ## Step 1: g(0) = 0 forces c = 0 -/
  63
  64lemma zero_normalization_forces_offset
  65    {a c : ℝ}
  66    (h0 : gapAffineLogR a phi c 0 = 0) :
  67    c = 0 := by
  68  simpa [gapAffineLogR] using h0
  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