Dimensional Phase Transitions: The Unified Tensor Geometry of Quantum Entanglement and Cosmic Expansion

Abstract

We present a fundamental resolution to the problem of background dependence in modern physics through the Seonggil Universal Mathematical Tensor Framework(SUMTF) and Rough Operator Algebra (ROA). By discarding the assumption of a static spacetime manifold, we model the universe as a dynamic, topological tensor organism that autonomously folds and unfurls its dimensions. We mathematically demonstrate that at the microscopic scale, spacetime undergoes a sub-4D collapse: dimensional reduction to D_eff→2 recovers conformal symmetry where gravity evaporates into the Gauss-Bonnet topological invariant, while the ultimate D_eff→0 limit annihilates geometric distance, replacing it with a non-commutative von Neumann algebra that rigorously proves the ER=EPR conjecture. Conversely, on the macroscopic scale,we redefine Dark Energy not as a static vacuum energy, but as the “Topological Relaxation Tension” generated by the autonomous unfurling of higher-dimensional topological foliations. By embedding this dimensional unfurling into the FLRW metric, we show that the accelerated expansion of the universe is a mathematical imperative driven by holographic entropy conservation.