`zero_normalization_forces_offset`

proved

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module: IndisputableMonolith.Masses.GapFunctionForcing
domain: Masses
line: 66 · github
papers citing: none yet

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IndisputableMonolith.Masses.GapFunctionForcing on GitHub at line 66.

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formal source

  63  simp [hshift]
  64
  65/-- Neutral normalization fixes the additive offset. -/
  66lemma zero_normalization_forces_offset
  67    {a c : ℝ}
  68    (h0 : gapAffineLogR a phi c 0 = 0) :
  69    c = 0 := by
  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) :

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