Disproving Continuous Path Summation: Evidence for Discrete Hexagonal Lattice Routing and Mod 9 Phase Quantization in Optical Propagation

Abstract Recent experimental tests of Feynman's path integral formulation (e.g., Sci. Adv., DOI: 10.1126/sciadv.aeh1011) utilize discretized space-time frameworks to measure single-photon propagators, interpreting the results as evidence of particles traversing every possible path simultaneously. However, physical underdetermination dictates that such interpretations lack exclusivity if an alternative discrete mechanism yields observationally equivalent outcomes. In this work, we demonstrate that observed multi-path interference patterns and propagator amplitudes emerge naturally from deterministic routing across a discrete hexagonal lattice governed by a Mod 9 topological invariant and strict crystallographic symmetry. To resolve this underdetermination, we propose a definitive experimental protocol combining ultra-high-resolution sub-nanometer interferometry with full 360^\circ angular rotation scans. This setup tests for two distinct lattice signatures: 6-fold directional anisotropy in fringe spacing and quantized phase plateaus governed by modular algebraic boundaries. Detecting these signatures would falsify the continuous path-summation hypothesis and establish discrete lattice architecture as the underlying physical reality.