CartanPathPacking.lean`

1 declarations, 0 contain sorry, 0 declared as axioms

Ambient packing for coarse paths with controlled heights

Unit-step subdivision increases the height error by only the logarithmic coarse-path error. Its polynomial number of steps is absorbed by the previously proved Cartan word-packing bound.

theorem OAI.PolycyclicRecognition.CartanWordModel.exists_coarse_path_polynomial_packing line 45

Source documentation

Coarse paths with a common speed and error have the expected separated endpoint bound in a fixed enlargement of their height template.

theorem exists_coarse_path_polynomial_packing [ProperSpace G]
    (b₀ : Basis (Fin (finrank ℝ (cartanRootIdeal H))) ℝ (cartanRootIdeal H))
    (γ₀ : M.Bases) {ρ : ℝ} (hρ : 0 < ρ)
    (hφ : ∀ d, F.action d = M.adjointGroupAutomorphism T F.N.equiv F.ExpN F.isExpN
      (by rw [← show chartN = F.ExpN from funext hchartN]; exact chartN.bijective) dynkinN
      (cartanLog H F.D.equiv F.ExpD
        (by rw [← show chartD = F.ExpD from funext hchartD]; exact chartD.bijective) d))
    (hpath : ∀ g : G, ∃ γ : ℝ → G, γ 0 = 1 ∧ γ 1 = g ∧
      ∀ t ∈ Icc (0 : ℝ) 1, ∀ u ∈ Icc (0 : ℝ) 1,
        dist (γ t) (γ u) ≤ dist 1 g * |t - u|)
    {K : ℝ} (hK : 0 ≤ K)
    (hLip : ∀ x y, ‖(π x).toAdd - (π y).toAdd‖ ≤ K * dist x y) :
…

Mentions: OAI.PolycyclicRecognition.cartanRootIdeal, OAI.PolycyclicRecognition.cartanLog