The discrete geometric origin of elementary particles

The Standard Model of particle physics, established in the last century, is a fundamental theory that has successfully addressed numerous problems and profoundly influenced modern science. Despite its remarkable achievements, the Standard Model also presents significant unresolved issues, such as the origins of quark and lepton masses, the neutrino mass-generation mechanism, quark confinement, mixing angles, and CP violation. These limitations indicate the necessity for new theoretical frameworks, as many physical quantities within the Standard Model cannot be determined from first principles alone and currently rely on experimental input. While its overall framework is robust, obtaining constraints on these quantities from other perspectives is essential to enhance the theory's predictive power. Motivated by the study of the Riemann Hypothesis—a problem of profound difficulty—we note a potential connection with quantum gravity. The primes studied in the Riemann Hypothesis exhibit a loss of directional properties at a certain level, analogous to the fundamental absence of directionality at the smallest scales of spacetime. Guided by the duality between physical reality and mathematical theory, we propose a novel multipoint duality, distinct from conventional two-point duality. Within a new framework combining octahedral geometry and this duality, we generalize the Higgs field to a quaternion representation. This approach allows for a more unified description of Majorana and Dirac neutrinos. Notably, the theory introduces no free parameters and geometrically predicts the Weinberg angle as well as a ratio of neutrino squared-mass differences approximately equal to 33.