affine_log_collapses_to_gap

formal source

 177         zero_normalization_forces_offset h0phi⟩
 178
 179/-- Under the normalizations, the affine-log family equals the canonical gap. -/
 180theorem affine_log_collapses_to_gap
 181    {a c : ℝ}
 182    (h0 : gapAffineLogR a phi c 0 = 0)
 183    (h1 : gapAffineLogR a phi c 1 = 1) :
 184    ∀ Z : ℤ, gapAffineLog a phi c Z = RSBridge.gap Z := by
 185  have hc : c = 0 := zero_normalization_forces_offset h0
 186  have ha : a = 1 / Real.log phi := unit_step_forces_log_scale h0 h1
 187  intro Z
 188  unfold gapAffineLog gapAffineLogR RSBridge.gap
 189  calc
 190    a * Real.log (1 + (Z : ℝ) / phi) + c
 191        = (1 / Real.log phi) * Real.log (1 + (Z : ℝ) / phi) := by
 192            simp [ha, hc]
 193    _ = Real.log (1 + (Z : ℝ) / phi) / Real.log phi := by
 194          simp [div_eq_mul_inv, mul_comm]
 195
 196/-- Three-point calibration gives direct collapse to the canonical gap. -/
 197theorem three_point_forces_canonical_gap
 198    {a b c : ℝ}
 199    (hb : 1 < b)
 200    (h0 : gapAffineLogR a b c 0 = 0)
 201    (h1 : gapAffineLogR a b c 1 = 1)
 202    (hneg1 : gapAffineLogR a b c (-1) = -2) :
 203    ∀ Z : ℤ, gapAffineLog a b c Z = RSBridge.gap Z := by
 204  have hbphi : b = phi := minus_one_step_forces_phi_shift hb h0 h1 hneg1
 205  have h0phi : gapAffineLogR a phi c 0 = 0 := by simpa [hbphi] using h0
 206  have h1phi : gapAffineLogR a phi c 1 = 1 := by simpa [hbphi] using h1
 207  intro Z
 208  simpa [hbphi] using affine_log_collapses_to_gap h0phi h1phi Z
 209
 210/-- Certificate structure for the three-point closure. -/