eightTickKernelParams

formal source

  61}
  62
  63/-- Eight-tick aligned kernel parameters with c = 49/162. -/
  64noncomputable def eightTickKernelParams (tau0 : ℝ) (h : 0 < tau0) : KernelParams := {
  65  alpha := alphaLock,
  66  C := 49 / 162,
  67  tau0 := tau0,
  68  tau0_pos := h,
  69  alpha_nonneg := le_of_lt alphaLock_pos,
  70  C_nonneg := by norm_num
  71}
  72
  73/-! ## Kernel Definition -/
  74
  75/-- The ILG kernel function:
  76  w(k, a) = 1 + C · (a / (k τ₀))^α
  77
  78We use max with a small ε to avoid division issues when k τ₀ = 0. -/
  79noncomputable def kernel (P : KernelParams) (k a : ℝ) : ℝ :=
  80  1 + P.C * (max 0.01 (a / (k * P.tau0))) ^ P.alpha
  81
  82/-- Simplified kernel when k = 1 (reference wavenumber). -/
  83noncomputable def kernelAtRefK (P : KernelParams) (a : ℝ) : ℝ :=
  84  1 + P.C * (max 0.01 (a / P.tau0)) ^ P.alpha
  85
  86@[simp] lemma kernelAtRefK_eq (P : KernelParams) (a : ℝ) :
  87    kernelAtRefK P a = kernel P 1 a := by
  88  simp [kernelAtRefK, kernel, one_mul]
  89
  90/-! ## Basic Properties -/
  91
  92/-- Kernel is always positive for valid parameters. -/
  93theorem kernel_pos (P : KernelParams) (k a : ℝ) : 0 < kernel P k a := by
  94  unfold kernel