Mapping the 12-Neighbor Truss: From Discrete Geometry to Continuum Fields in a 13-Point Lattice

This paper develops a mathematical mapping from a discrete 13-point lattice to continuum field equations that resemble those of the Standard Model and gravitation. By modeling a central node surrounded by 12 neighbors in an FCC-like coordination, the author derives how local node degrees of freedom (position, orientation, and velocity) map to density, gauge-like phases, and momentum flow. Key contributions include: Elasticity to Gauge Fields: Mapping the 12-neighbor truss energy to U(1), SU(2), and SU(3) symmetries. Strong Force Geometry: Identifying the 8 traceless shear modes of the lattice with SU(3) "color" interactions. Physical Constants: Interpreting the fine-structure constant α as a geometric ratio of torsional to radial stiffness. Gravity and Regularization: Proposing a refraction-based model of gravity where lattice discreteness acts as a natural regulator to prevent singularities.