Phi-Toric Tessellations as Spinfoam Boundary States: Loop Quantum Gravity in B.O.O.F. Cosmology

Abstract

The Big Boof Theory (B.O.O.F.) introduced the Phi-torus as a fundamental geometric scaffold for a unified cosmological framework, proposing a tuning force as the universal regulatory principle governing oscillatory stability from quantum to cosmic scales. The present paper provides the rigorous mathematical formalization that extends and clarifies those claims. Drawing upon Rubinstein's recent work on Kähler–Einstein metrics, toric Fano stability, and the convex-complex geometry interface (2024–2025), we demonstrate that the tuning force admits a precise mathematical identity: the barycenter condition Bc(P) = 0 on the polytope P associated with Phi-toric tessellations. We further construct a spinfoam partition function Z_Φ constrained to Phi-toric tessellation configurations, and demonstrate that its semiclassical limit recovers Regge calculus on a triangulated manifold with golden-ratio metric constraints. The Ponzano–Regge amplitude formulation is extended to Phi-toric vertex geometry, providing a computable criterion for geometric stability consistent with SU(2) representation theory. Together, these results elevate B.O.O.F. from a geometrically motivated cosmological proposal to a mathematically grounded framework with quantitative predictive structure.