Thermodynamic Emergence: Deriving the Cuboctahedral Vacuum from Geometric Saturation and Topological Ground States

We explore the thermodynamic and topological origins of the foundational space-

time geometry in the Selection-Stitch Model (SSM) [1, 2]. Rather than assuming a

background manifold, we posit that the continuous vacuum emerges from a discrete

quantum tensor network evolving to minimize its free energy while resolving the

geometric frustration of the early universe. In this framework, the minimal-energy

ground state of the Hamiltonian (T →0) is identified as the two-dimensional K = 6

hexagonal sheet. As the tetrahedral foam (K = 4) of Cosmic Inflation expands, it

thermodynamically seeks to saturate its open deficit angles. We derive a topologi-

cal free energy functional directly from a microscopic partition function, utilizing a

controlled cluster expansion bounded by the network’s hard-sphere exclusion limit.

This yields a physically motivated kinetic rate equation via non-conserved order

parameter dynamics. Drawing on the Kepler Conjecture [6], the absolute maxi-

mum density for 3D Euclidean packing caps the coordination at K = 12. While

this density is shared by Face-Centered Cubic (FCC) and Hexagonal Close-Packed

(HCP) lattices, the FCC lattice is selected during the Alder transition. Under an

adiabatic cosmological quench (Γ ≫H), its isotropic point-group symmetry maxi-

mizes the phononic density of states, making its vibrational entropy advantage [7, 8]

deterministic in the thermodynamic limit. Finally, we demonstrate that the FCC

Cuboctahedron is the unique low-energy variational minimum preserving the 2D

K = 6 zero-stress Euler state across its 3D bulk.