EdgeLengthField

formal source

  73    records the edge length computed from the vertex log-potentials. The
  74    canonical conformal ansatz is `L e = a · exp((ε i + ε j)/2)`, which
  75    reduces to `a` at `ε ≡ 0` (flat vacuum). -/
  76structure EdgeLengthField (n : ℕ) where
  77  base_spacing : ℝ
  78  base_spacing_pos : 0 < base_spacing
  79  length : Fin n → Fin n → ℝ
  80  length_symm : ∀ i j, length i j = length j i
  81  length_pos : ∀ i j, 0 < length i j
  82
  83/-- The canonical conformal edge-length map:
  84    `L_{ij}(ε) = a · exp((ε_i + ε_j)/2)`.
  85
  86    In the flat vacuum `ε ≡ 0`, this reduces to `L_{ij} = a`. At leading
  87    order in small `ε`, `L_{ij}/a - 1 = (ε_i + ε_j)/2 + O(ε²)`. This is
  88    the standard conformal rescaling convention used when the recognition
  89    potential is identified with a scalar metric perturbation. -/
  90def conformal_edge_length_field {n : ℕ} (a : ℝ) (ha : 0 < a)
  91    (ε : LogPotential n) : EdgeLengthField n :=
  92  { base_spacing := a
  93  , base_spacing_pos := ha
  94  , length := fun i j => a * Real.exp ((ε i + ε j) / 2)
  95  , length_symm := by
  96      intro i j
  97      have : ε i + ε j = ε j + ε i := by ring
  98      simp [this]
  99  , length_pos := by
 100      intro i j
 101      exact mul_pos ha (Real.exp_pos _)
 102  }
 103
 104/-- At the flat vacuum `ε ≡ 0`, the conformal edge length is the base
 105    spacing `a`. -/
 106theorem conformal_edge_length_flat {n : ℕ} (a : ℝ) (ha : 0 < a)