From a Single Compensatory Principle to Physical Structure: Distinctions, Minimal Closures, Gauge Symmetry, Spatial Dimensionality, and Structural Dynamics

We derive the structural content of modern physics from a single pre-physical primitive — the act of distinction, governed by one compensatory condition (Axiom 0). All Global Entropic Consistency (GEC) - specific derivations are self-contained in this paper; standard mathematical results (Schur’s lemma, Peter-Weyl theorem, Clifford algebra classification, matroid axioms) are stated with precise references and applied explicitly. We prove: (1) Z2 and Z3 are the unique minimal closed structures (exhaustive case analysis with explicit definition of contradiction); (2) ZDB = In/n by two independent arguments; (3) the Standard Model gauge symmetry group SU (3)×SU(2)×U(1)Y with Ymin = 1 6 from the combined Z2 ⊗Z3 substrate, using Schur’s lemma and the GNS metric (the Clifford algebra derivation uses the correct minimal dimension N = 4 for Cl(3, 1), correcting an error in the previous version); (4) three spatial dimensions from a rank-4 uniform matroid, with explicit verification of all three matroid axioms; (5) the tetrahedron as the unique minimal 3D volume and IR cutoff; (6) the combinatorial area-volume scaling relation S ∼ N 2/3 from spherical geometry; (7) a structural analogue of E = mc2 in face-bonding dynamics. The central methodological thesis: physics derives relations, not convention-dependent numbers. Fermion masses, coupling constants, and cosmological parameters are realization data, not derivable from invariant axiomatics (Non-Invertibility Theorem)