A geometry of fermions

This manuscript is the preprint "A geometry of fermions" by François Ritter (ritter.francois@gmail.com).

This preprint presents a geometrical and phenomenological framework addressing the Standard Model’s fermionic sector, including the fermion mass hierarchy, neutrino masses, CKM and PMNS flavor mixing, and quantum numbers. I base the framework solely on experimental inputs—measured fermion masses and quantum assignments—and introduce four discrete numbers (the seeds) from which quantum numbers and mass relations emerge.

The fermion mass law is established from simple linear patterns in a logarithmic space of mass ratios, forming a linear system that reproduces the full fermion mass hierarchy using only six constants. Quantum numbers are expressed as second-order polynomials in the seed parameters. Flavor mixing naturally arises from transitions in the seed space, providing a unified explanation for quark and lepton mixing structures. The model extends to Dirac-type neutrinos, generating extremely small masses through a geometrical singularity mechanism.

Key predictions include:

- A new charged lepton with electron-like properties and predicted mass 8.5938 MeV, consistent with a 2024 report of an “anomalon” at 8.5 ± 2.5 MeV.
- A third fermionic branch, the leonids, suggesting hidden or dark sectors.
- A fourth branch, the apex, restoring gauge anomaly cancellation and completing the geometrical fermion spectrum.

The framework also highlights the natural emergence of the golden ratio in left-right symmetric states, suggesting a deep algebraic symmetry underlying fermion organization, quantum numbers, mass hierarchy, and flavor structure.

This study was conducted independently and outside the primary research community. I am not submitting this work to peer review or arXiv at this time, but I request that anyone using or building on these results acknowledge this Zenodo preprint.

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v2.1 (2025/10/11), minor update : the factor (1/2) in the right-term of the mass law was unnecessary and it has been absorbed into the constants (lambda,alpha_prime,beta_prime). This change does not affect any numerical results; it only simplifies the expression of the law. The new constants are :
lambda_new = sqrt(lambda_old)
alpha_prime_new = 2*alpha_prime_old
beta_prime_new = 2*beta_prime_old