Arithmetic-spectral derivation of Standard Model particle masses Prime numbers and the Dirac operator as the source of the mass spectrum of W, Z, H bosons, the t quark and nuclei

The Standard Model describes particle interactions with remarkable precision, yet it treats the values of particle and nuclear masses as free inputs. This work explores a different possibility: that these masses emerge from a hidden arithmetic and spectral structure based on prime numbers and the Dirac operator. We introduce the Dirac–F2 model, in which simple products and ratios of prime numbers generate a dense “hologram” of candidate mass scales, and a discrete Dirac operator with a mass profile built from these values acts as a spectral filter that selects a small set of physically relevant eigenvalues. Using a large scale scan over the first one hundred thousand primes, we construct an F2 hologram that systematically matches many known particle and nuclear masses, including W, Z, Higgs, top quark, proton, neutron, muon, light nuclei such as carbon 12 and iron 56, and a hadronic structure around 600 MeV. Typical relative deviations are between one part in a thousand and a few parts in ten thousand. When these F2 values are used as a mass profile in a three dimensional lattice Dirac operator, the resulting eigenvalues reproduce the same experimental masses with comparable precision. The Z boson appears as an arithmetic spectral resonance: the F2 hologram itself prefers an empty region near 87.5 GeV, and only a small stochastic perturbation of the mass profile, of order a few percent, shifts the corresponding Dirac eigenvalue to the physical Z mass near 91 GeV. We further show that the F2 hologram exhibits structured attractors and energy gaps that differ strongly from those obtained from random prime like fields. Random control tests demonstrate an excess of physically meaningful hits in the real prime data at the level of several standard deviations, and particularly strong Higgs related peaks. Extended “hyperon hunter” and hadronic hologram modules reveal a coherent baryon ladder that links heavy quarks, hyperons, nucleons, and light nuclei, in agreement with known quark content. The model also singles out a small number of especially sharp arithmetic attractors that may correspond to new or subtle states, such as candidates near 17, 34, 149, 600 and 1465 MeV, and a “shadow” neutron slightly above the neutron mass, which we qualitatively relate to existing anomalies including the X17 signal and the neutron lifetime puzzle. The Dirac–F2 framework does not replace the Standard Model; instead, it proposes that its input masses may be constrained by an underlying arithmetic spectral skeleton, and offers concrete numerical structures that can be confronted with further theoretical and experimental work.