Tarcha 3.15 right interpolation is the reflection of the left interpolation

Proved

TarchaBraids.thm_3_15_adjacent_right_interp_reflection_v1

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

artin-presentationbraid-groupsconfiguration-spacesymmetrytarcha

For adjacent strands, the three distinguished coordinates of the right interpolation are obtained by reflecting the corresponding left interpolation coordinates across the midpoint line at real coordinate i+2.

Preamble

import Mathlib
import Definitions.Def_TarchaBraids_adjacent_path_data_v1

Formal statement

namespace TarchaBraids

open BraidsLinksMCG

theorem thm_3_15_adjacent_right_interp_reflection_v1 :
    ∀ {n : ℕ} (i j : Fin (n - 1)) (hji : (j : ℕ) = (i : ℕ) + 1),
      (∀ (u q : ℝ),
        rightOuterInterpFun n i j u q (strandIdx i) =
          (((2 * (((i : ℕ) : ℝ) + 2) : ℝ) : ℂ) -
            leftOuterInterpFun n i j u q (strandIdxSucc j))) ∧
      (∀ (u q : ℝ),
        rightOuterInterpFun n i j u q (strandIdxSucc i) =
          (((2 * (((i : ℕ) : ℝ) + 2) : ℝ) : ℂ) -
            leftOuterInterpFun n i j u q (strandIdxSucc i))) ∧
      (∀ (u q : ℝ),
        rightOuterInterpFun n i j u q (strandIdxSucc j) =
          (((2 * (((i : ℕ) : ℝ) + 2) : ℝ) : ℂ) -
            leftOuterInterpFun n i j u q (strandIdx i))) := by sorry

end TarchaBraids

Source

Symmetry layer extracted from Tarcha's explicit adjacent Artin relation proof.

View graph