Prove2Me

Tarcha 3.15 right interpolation has the required u-boundaries

Proved
TarchaBraids.thm_3_15_right_interp_u_boundaries_v1

by WillR · Sep 21, 2026 · Mathlib 0df444a (Lean v4.33.1)

artin-presentationbraid-groupshomotopytarcha

At interpolation parameter u=0 the right interpolation is the right braid-word path, and at u=1 it is the common outer-rotation path.

Preamble
import Mathlib
import Definitions.Def_TarchaBraids_right_interp_config_data_v1
import Definitions.Def_TarchaBraids_adjacent_config_data_v1
Formal statement
namespace TarchaBraids

open BraidsLinksMCG

theorem thm_3_15_right_interp_u_boundaries_v1 :
    ∀ {n : ℕ} (i j : Fin (n - 1)) (hji : (j : ℕ) = (i : ℕ) + 1),
      (∀ q : unitInterval,
        configProj n (rightOuterInterpConfig i j hji 0 q) =
          configProj n (rightBraidConfig n i j (q : ℝ))) ∧
      (∀ (hi2 : (i : ℕ) + 2 < n) (q : unitInterval),
        configProj n (rightOuterInterpConfig i j hji 1 q) =
          configProj n (outerRotateConfig i hi2 (q : ℝ))) := by sorry

end TarchaBraids
Source
Modular boundary layer for Tarcha's explicit adjacent Artin relation homotopy.

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