Prime Harmonic Spectral Geometry (PHSG): A Complete, Parameter-Free Framework for Fundamental Physics

The Standard Model (SM) of particle physics, despite its unparalleled empirical success, is fundamentally incomplete, relying on 19+ free parameters whose origins remain unexplained. This “parameter problem”—a challenge recognized since the early days of quantum field theory and highlighted by concepts like Dirac’s large number hypothesis—and the persistent lack of a quantum theory of gravity are central challenges in modern physics. This monograph presents the Prime Harmonic Spectral Geometry (PHSG), a complete and self-contained framework for a parameter-free unification of fundamental physics. PHSG synthesizes three foundational concepts: (1) an octonionic spectral triple from Noncommutative Geometry (NCG), providing a unified geometric origin for spacetime, gravity, and the Standard Model; (2) Kolmogorov-Arnold-Moser (KAM) stability theory, postulated as a physical principle to explain fermion mass hierarchies and mixing matrices as dynamically stable configurations; and (3) number theory, specifically the harmonic properties of prime and Lucas numbers, to anchor the absolute scales of physical parameters. The framework is built on a minimal set of axioms and a four-step calculational blueprint. It rigorously derives the values of all Standard Model parameters—including fermion masses, mixing angles, gauge couplings, and Higgs properties—from first principles, without any free parameters. The derivation is now fully self-consistent, with all previously empirical KAM correction factors and scale anchors derived analytically from the underlying geometry and number theory. A comprehensive two-loop Renormalization Group (RG) analysis is integrated, enabling the precise evolution of bare parameters from the unification scale to experimental energies. The framework also provides complete derivations for the strong coupling constant, the cosmological constant, dark matter properties, and gravitational constants, resolving previous unvalidated sectors and addressing long-standing cosmological puzzles. A key prediction is a geometric contribution to the tau lepton’s anomalous magnetic moment, a falsifiable signature of physics beyond the Standard Model. The PHSG implies that classical spacetime is emergent from a discrete, noncommutative Planck-scale geometry, offering a concrete and UV-finite framework for quantum gravity. This work establishes PHSG as a complete, coherent, and parameter-free theory of fundamental physics, directly addressing and refuting criticisms leveled against other “prime-based” models by providing explicit, transparent, and empirically validated derivations.