Geometric Renormalization of the Speed of Light and the Origin of the Planck Scale in a Saturation-Stitch Vacuum

Attempts to reconcile General Relativity with Quantum Mechanics often falter on the prob-

lem of discretizing spacetime without violating Lorentz invariance. In this work, we propose

the Selection-Stitch Model (SSM), a discrete vacuum framework based on a saturated

Face-Centered Cubic (FCC) lattice (K = 12). We demonstrate two fundamental geometric

renormalizations. First, we show that the physical speed of light (c) emerges as a renormal-

ization of the lattice hopping speed, specifically c= 4vlattice, due to the constructive interfer-

ence of the 12 nearest-neighbor paths on the Cuboctahedron unit cell. Second, utilizing this

relation, we derive the lattice spacing anot as a free parameter, but as the inevitable geomet-

ric limit where the vacuum’s elastic energy density encounters the Schwarzschild constraint.

Fixed by the packing efficiency of the FCC unit cell (V= a3/√2), we derive a fundamental

lattice spacing of a≈0.77lP . We explicitly derive the continuum limit, showing that Lorentz

invariance is restored for fermions up to O(a2) corrections due to the centrosymmetry of the

lattice. Furthermore, we demonstrate that this same lattice centrosymmetry enforces the

vanishing of the QCD topological charge density, naturally resolving the Strong CP problem

(θQCD = 0) without requiring an axion. Finally, we discuss the geometric obstruction to

point-particle quantization of gravity in this framework.