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lemma proved tactic proof

unit_step_forces_log_scale

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formal statement (Lean)

  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

proof body

Tactic-mode proof.

  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. -/

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