Discrete Interaction Lattices and Split-Step Symmetry A Structural Correspondence Between Tukutuku Weaving and Quantum Walk Models

Darren Cudd

Published Mar 27, 2026• 0 citations

Abstract

This paper introduces a discrete interaction lattice derived from the directed geometry of tukutuku weaving and formulates it as a two-component unitary dynamical system. It is shown that the lattice recovers the 1+1 dimensional Dirac equation in the continuum limit, establishing a rigorous connection to relativistic quantum dynamics. At the discrete level, however, a structural obstruction emerges: a sign-flip interface fails to support a localized zero-mode due to broken chiral symmetry. This failure is identified as a fundamental limitation of the single-step model, leading to the necessity of a period-2 staggered lattice and a symmetrized split-step formulation. The work provides a mathematically grounded framework while clearly distinguishing proven results from heuristic structural correspondence.

DOI: 10.5281/zenodo.19247851

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