`phi_eq_one_add_inv_phi`

proved

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module: IndisputableMonolith.Masses.GapFunctionForcing
domain: Masses
line: 46 · github
papers citing: none yet

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IndisputableMonolith.Masses.GapFunctionForcing on GitHub at line 46.

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formal source

  43  gapAffineLogR a b c (Z : ℝ)
  44
  45/-- `φ = 1 + 1/φ`, used to normalize the unit-step condition. -/
  46lemma phi_eq_one_add_inv_phi : phi = 1 + (1 : ℝ) / phi := by
  47  have hphi_ne_zero : phi ≠ 0 := phi_ne_zero
  48  calc
  49    phi = phi ^ 2 / phi := by
  50      field_simp [hphi_ne_zero]
  51    _ = (phi + 1) / phi := by simp [phi_sq_eq]
  52    _ = 1 + (1 : ℝ) / phi := by
  53      field_simp [hphi_ne_zero]
  54
  55/-- Equivalent rewrite of `1 + 1/φ = φ`. -/
  56lemma one_add_inv_phi_eq_phi : 1 + (1 : ℝ) / phi = phi := by
  57  simpa using phi_eq_one_add_inv_phi.symm
  58
  59/-- Log rewrite at the canonical shift argument. -/
  60lemma log_one_add_inv_phi_eq_log_phi : Real.log (1 + phi⁻¹) = Real.log phi := by
  61  have hshift : (1 + phi⁻¹ : ℝ) = phi := by
  62    simpa [one_div] using one_add_inv_phi_eq_phi
  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) :

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