Pattern Recognition in Discrete Twisted Lattices A Descriptive Study of Scaling Behavior and Finite-Size Effects

This manuscript documents an iterative investigation into numerical scaling behaviors in discrete photonic lattices with anti-periodic boundary conditions — and the successive correction of three apparently robust findings. In the initial study (March 2026), coarse-resolution simulations suggested "perfect" mathematical constants; high-resolution analysis revealed these as artifacts of grid quantization. A subsequently observed √2 ratio between topologies was identified as an artifact of comparing different distance metrics (Manhattan vs. Euclidean); when harmonized, both topologies converge to the same scaling product (P ≈ 1.11072). A step-wise dependence of critical curvature Kc on ⌊L/2⌋ initially appeared robust, but sensitivity analysis (April 18, 2026) falsified this as well: replacing the hard, discontinuous boundary condition with smooth alternatives caused immediate oscillatory instability at K=0, and the ⌊L/2⌋ pattern vanished entirely. Extended simulations (April 26, 2026) up to L=128 (N=16,384) confirm that the scaling product P ≈ 1.11072 remains stable across both topologies within this specific CME implementation, though whether this reflects a universal feature or a consistent artifact of the numerical scheme remains an open question. Beyond its numerical findings, this study serves as a documented case of how computational tools — the grid resolution, the choice of metric, the implementation of boundary conditions — can generate convincing but false patterns. Each "discovery" was an artifact of the method used, not a property of the system studied. This carries implications beyond computational topology: in any field where measurement and comparison are central, the distinction between a law of nature and a signature of the instrument constant vigilance is advised. Full data, code, and the detailed falsification analysis are available at GitHub(KatharinaJacoby).