Detection Ontology, Part 1: Index Stability of Geometric-Arithmetic Spectral Triples and the Golden-Ratio Diophantine Constraint

In this foundational paper of the Detection Ontology series, we address a long-standing "fine-tuning" problem in the Connes-Chamseddine spectral action formulation of the Standard Model: the origin of the number of fermion generations (Ng=3Ng=3).Standard noncommutative geometric models successfully reproduce the gauge group and particle content of the Standard Model but require the number of generations to be inserted as an external parameter. We propose an extension of the traditional geometric-finite spectral triple (A,H,D)(A,H,D) to include an arithmetic sector defined over the prime-gap lattice ΛpΛp.