phiTetrahedralAngle

formal source

  30Note: This angle is mathematically identical to the sp³ hybridization angle in
  31chemistry and appears in any tetrahedral geometry. It is derived here from
  32first principles via the φ-lattice structure. -/
  33noncomputable def phiTetrahedralAngle : ℝ := Real.arccos (-(1 / 3 : ℝ))
  34
  35/-- Rotor pitch family: integral log-spiral pitch `kappa : ℤ`. -/
  36abbrev RotorPitch := IndisputableMonolith.Spiral.SpiralField.Params
  37
  38/-- Rotor radial path: log-spiral scaling on the φ lattice.
  39
  40`r(θ) = r0 · φ^{(κ·θ)/(2π)}`. -/
  41noncomputable abbrev rotorPath := IndisputableMonolith.Spiral.SpiralField.logSpiral
  42
  43namespace SpiralLemmas
  44
  45open IndisputableMonolith.Spiral
  46
  47/-- `logSpiral` is nonzero whenever `r0` is nonzero (since φ^x > 0). -/
  48lemma logSpiral_ne_zero
  49    {r0 θ : ℝ} {P : SpiralField.Params} (hr0 : r0 ≠ 0) :
  50    SpiralField.logSpiral r0 θ P ≠ 0 := by
  51  classical
  52  unfold SpiralField.logSpiral
  53  -- `φ > 0` hence `φ^exp > 0`.
  54  have hφpos : 0 < IndisputableMonolith.Constants.phi :=
  55    IndisputableMonolith.Constants.phi_pos
  56  have hrpow_ne : Real.rpow IndisputableMonolith.Constants.phi
  57        ((P.kappa : ℝ) * θ / (2 * Real.pi)) ≠ 0 := by
  58    exact ne_of_gt (Real.rpow_pos_of_pos hφpos _)
  59  exact mul_ne_zero hr0 hrpow_ne
  60
  61/-- Closed-form step ratio for the log-spiral: it depends only on `Δθ` and `kappa`.
  62
  63This is the mathematical kernel behind the "discrete pitch families" idea.