Exact Algebraic Prediction of the Spectral Dimension in Quantum Gravity from Tetrahedral Vacuum Structure.

Loop Quantum Gravity (LQG) and Causal Dynamical Triangulations (CDT) exhibit spectral dimension ds ≈ 2.0–2.2 at UV scales, computed numerically from spin-network coarse-graining and Monte Carlo path integrals. The geometric origin of this dimensional reduction has remained an open question for 15 years. We derive ds = ln 8 ≈ 2.0794 algebraically from a single axiom: the quantum vacuum maximizes holographic entropy density by selecting tetrahedral coordination (z = 4). Orientation duality (bidirectional information flow along network links) yields ds = ln(2z) exactly. This prediction, made independently of LQG/CDT data, falls within their empirical range with discrepancy < 4% and is obtained with zero adjustable parameters and zero numerical input from LQG or CDT simulations. The oriented heat-kernel formalism recently developed for directed Bethe lattices [Delporte 2024] provides rigorous support for the key derivation step. Originality: We present the first purely algebraic derivation of the spectral dimension from a geometric vacuum axiom. Previous results (LQG: Modesto 2009; CDT: Ambjørn et al. 2005) were numerical. The tetrahedral vacuum axiom and its connection to dimensional reduction are novel contributions not appearing in prior literature.