Geometric Structure of the Spatial Grid: The Simple Cubic Lattice and the Exact Topology of the Vacuum (k-Foam Theory)

In previous iterations, k-Foam Theory intuitively described the fundamental structure of the vacuum as a "k = 6 regular-octahedron grid. " However, from a strictly crystallographic perspective, regular octahedra alone cannot seamlessly fill three-dimensional space without leaving gaps. This Working Paper refines and revisits the geometric description of the spatial grid. The skeleton of space is modeled as a Simple Cubic Lattice, wherein the 6 direct connection targets of each node (±x, ±y, ±z) form an octahedral arrangement. This redefinition helps address the 3D space-filling problem and provides a more robust geometric foundation for the theory's previous numerical observations: Key Geometric Clarifications: - Establishment of the Moore Neighborhood (Z = 26): The 26 topological elements (6 faces, 12 edges, 8 vertices) surrounding a node naturally correspond to the Moore neighborhood (3³ − 1 = 26) of a cubic lattice, reinforcing a possible structural basis for Planck-scale considerations. - A Geometric Interpretation of the Fine-Structure Constant (1/137): The probabilistic barrier cost (5³ = 125) for photon propagation is reinterpreted not as "3 independent axes, " but as the simultaneous coordination of 3 pairs of orthogonal connections at the cubic lattice intersection, selecting from 5 valid connection candidates. - Proton Configuration & Nuclear Force: The 3 quarks of a proton (RGB) can be modeled as orthogonal (90°) connections across adjacent nodes of the cubic lattice. This may help explain why the proton diameter (1. 68 fm) spans multiple cells, exceeding the single-cell direct connection distance (1. 414 fm). All existing numerical outputs (such as dₘax = 2. 0 fm, α⁻¹ = 137) remain consistent under this reinterpretation. "The skeleton of space is the cubic lattice. The arrangement of each node's connection targets forms an octahedron. " Both perspectives can be seen as complementary descriptions of the same underlying structure.