Continuum Limit, Spectral Dimension ds -> 4, and Universal Finite-Size Scaling on Relational 2-Complexes

Abstract

We establish the emergence of a continuous four-dimensional macroscopic spacetime manifold from the discrete exterior calculus (DEC) of relational 2-complexes via dynamical heat kernel diffusion and universal finite-size scaling. Within Relational Hodge Gravity (RHG), physical geometry is governed by the 0-form and 1-form Hodge Laplacians on cellular complexes K = (V, E, F) without presupposing smooth background coordinates or fundamental metric tensors. By evaluating the heat kernel trace P(σ) = (1/N₀) Tr(exp(-σ Δ₀)) across a scaling hierarchy of relational complexes (N₀ ∈ {81, 256, 1296, 4096}), we confirm four fundamental continuum benchmarks: The scale-dependent spectral dimension d_s(σ) = -2 d ln P / d ln σ flows from an ultraviolet discrete regime into an extended macroscopic plateau centered at d_s ≈ 4.00, establishing the dynamical emergence of four spacetime dimensions. Under volume rescaling by the infrared system diameter τ = σ / L², the spectral dimension decay curves collapse onto a universal finite-size scaling trajectory, verifying that the infrared cutoff is governed by global compactification rather than unphysical lattice artifacts. The cumulative eigenvalue distribution of the discrete Hodge Laplacian strictly obeys Weyl's asymptotic law, N(λ) ∝ λ^(d_s/2) = λ², over two orders of magnitude in energy. The heat kernel return probability saturates to the exact ergodic volume floor P(σ) → 1/N₀. These results confirm that RHG possesses a genuine four-dimensional continuum limit, transitioning the framework from a discrete combinatorial model to a viable candidate for non-perturbative quantum geometry.