EMU Theory: A Discrete Topo-Computational Foundation via Operator Dynamics and Embedding Homotopy

We present the Excitation-based Model of the Universe (EMU), a rigorous mathematical framework that maps fundamental physical laws onto the spectral and topological properties of a discrete scalar substrate. By defining the vacuum state evolution via a linear Update Operatorˆ U on the Hilbert space ℓ2(V ), we achieve the following structural derivations: (1) Mass as Spectral Gap: Inertial mass is identified not as a primitive, but as the spectral gap µ induced by local topological perturbations in the evolution operator. (2) Fermions from Em- bedding Homotopy: We define particles as extended defects rather than points. By analyzing the configuration space of embeddings Emb(S2 , R3), we show that the fundamental group is the Loop Braid Group rather than the Symmetric Group, mathematically justifying Fermi-Dirac statistics in three dimensions. (3) Emer- gent Lorentz Covariance: We derive the dispersion relation from the discrete update rule, predicting specific Lorentz-violating corrections of order O((ka)2) ver- ifiable by high-energy astrophysical observations. (4) Spectral Dimension Flow: We formulate the spectral dimension DS (σ) of the substrate and predict a distinctive intermediate plateau (DS ≈ 2) driven by mediator topology. This work transforms "Digital Physics" from a qualitative hypothesis into a well-posed dynamical system amenable to spectral analysis and topological classification.