Entanglement-Algebraic Spacetime and the Selection of Non-Commutative Geometry

This paper is archived as a speculative research work. Finite non-commutative geometry has proven remarkably effective as a description of internal physical structure, yet its finite component—the choice of algebra, reality structure, and spectral organization—remains largely axiomatic. In this work we provide a structural explanation for its appearance by placing it within a pre-geometric framework based on a static scalar admissibility field, as formulated in Entanglement–Algebraic Spacetime (EAS). The EAS scalar field encodes admissibility, stiffness, and persistence without reference to spacetime, dynamics, or quantization. Treating changes of descriptive resolution as admissible deformations, we derive a set of non-negotiable requirements on any representational interface: preservation of persistent structure, quadratic encoding of misalignment, closure of admissible perturbations, and canonical equivalence across non-unique coarse-grainings. These requirements force a spectral, rather than coordinate-based, interface description and sharply restrict admissible internal symmetries. We show that persistence forbids accidental degeneracy and requires irreducible protection of doublets, selecting a quaternionic structure. Independently, canonical equivalence across coarse-grainings forces complexification as the minimal continuous, norm-preserving gauge. When combined with a triadic closure condition, these constraints uniquely select the finite algebra C ⊕ H ⊕ M₃(C), coinciding with the finite internal geometry used in non-commutative geometric models. From this perspective, finite non-commutative geometry is identified not as a fundamental ontology, but as the minimal spectral interface compatible with scalar admissibility. The Higgs field acquires a natural interpretation as an interface-level stiffness deformation, and the restricted form of internal symmetries is explained by persistence rather than phenomenology. This work reverses the usual explanatory arrow: non-commutative geometry is shown to work because it captures the unique stable interface selected by a pre-geometric scalar kernel.