phi

formal source

   6open Real
   7
   8/-- Golden ratio φ as a concrete real number. -/
   9noncomputable def phi : ℝ := (1 + Real.sqrt 5) / 2
  10
  11namespace GapSeries
  12
  13/-- Gap-series coefficient (1-indexed by design via `n.succ`).
  14The conventional closed-form uses the series of `log(1+x)` at `x = z/φ`.
  15This definition is dimensionless and self-contained. -/
  16noncomputable def coeff (n : ℕ) : ℝ :=
  17  let k := n.succ
  18  ((-1 : ℝ) ^ k) / (k : ℝ) / (phi ^ k)
  19
  20/-- Finite partial sum (0..n-1) of the gap coefficients (evaluated at z=1).
  21This stays purely algebraic here; convergence and identification with
  22`log(1 + 1/φ)` can be proved in a companion module that imports analysis. -/
  23noncomputable def partialSum (n : ℕ) : ℝ :=
  24  (Finset.range n).sum (fun i => coeff i)
  25
  26/-- Generating functional F(z) := log(1 + z/φ).  -/
  27noncomputable def F (z : ℝ) : ℝ := Real.log (1 + z / phi)
  28
  29/-- The master gap value as the generator at z=1. -/
  30noncomputable def f_gap : ℝ := F 1
  31@[simp] lemma f_gap_def : f_gap = Real.log (1 + 1 / phi) := rfl
  32
  33end GapSeries
  34
  35namespace Curvature
  36
  37/-- Curvature-closure constant δ_κ used in the α pipeline.
  38Defined here as the exact rational/π expression from the voxel seam count. -/
  39noncomputable def deltaKappa : ℝ := - (103 : ℝ) / (102 * Real.pi ^ 5)