The discrete skeleton of the Standard Model from a minimal subfactor: an economy result and the boundary where it stops

The discrete gauge architecture of the Standard Model — the gauge algebra, the fermion representations, the hypercharge spectrum, and the generation count — is derived from a single structural input: the index-two subfactor, selected by a minimality principle. The derivation costs exactly three independent inputs: one minimality selection (the minimal Jones index, with the generating Ising fusion category following as a consequence by the A–D–E classification in the amenable setting), one imported translation (the Connes–Chamseddine noncommutative-geometry dictionary), and one empirical fact (the chirality of the weak interaction). A no-go theorem establishes the boundary where the construction stops: the discrete skeleton is the tracial (timeless) sector; masses, couplings, and continuous mixing parameters enter only with a non-trivial modular flow, which the minimal structure does not carry. This is companion paper I of a quartet; companion paper II (the bipartite hinge) develops the boundary; companion paper III (the closure vector) derives the forced mixing amplitudes; companion paper IV (the exhaustion theorem) proves the forced content is complete