Emergent Spacetime, Gauge Fields, and Matter Fermions in a Kelvin Foam Tessellation: A Unified Geometrodynamic Model

In this work, we formalize a complete geometrodynamic model of emergent spacetime based on the discrete tessellation of the Planck continuum (ℓP ≈ 1.616 × 10⁻³⁵ m). We model the quanta of the gravitational field (gravitons) as non-linear topological solitons—analogous to Wheeler geons—characterized by localized spatial domains of quantum curvature. While short-range packing considerations favor a local N = 12 icosahedral/dodecahedral configuration, the strict topological requirement of tiling the 3D spatial manifold ℝ³ without energy voids forces a structural phase transition toward a Kelvin Foam (curved truncated octahedron). We extend this geometric framework by incorporating: (1) Spin-1 bosons as rotational twist deformations algebraically derived from the Oh point symmetry group of the Kelvin cell, and (2) Spin-1/2 fermions as topological volume defects (solitons/Skyrmions) whose 4π spinorial property and chirality naturally emerge by evading the Nielsen-Ninomiya theorem via a domain-wall structure on the mixed cell faces. We derive an explicit covariant coarse-graining operator with volume element √-g d⁴x and demonstrate that the statistical cancellation of opposite-sign principal curvatures on domain walls yields a macroscopic Minkowski spacetime (G^(eff)_μν = 0). Furthermore, by analyzing wave propagation through the discrete Kelvin lattice, we analytically calculate the geometric factor ξ ≈ 1.24 appearing in Lorentz Invariance Violation (LIV) modified dispersion relations. Acknowledgments: A thank you to my family, and especially to my little dog Milo, my love, my life.