On a heuristic discrete‑geometric ansatz for Yukawa coupling constants and the fine‑structure constant

the Standard Model, fermion masses are determined by Yukawa coupling constants, yet the pattern of these couplings still lacks a fundamental theoretical explanation. In this paper, we tentatively propose a discrete geometric phenomenological hypothesis: the Yukawa coupling satisfies yf = 2−k/24, where k is a positive integer. This discrete mapping is intended as a rigid “sampling” of experimental masses rather than a “fitting”; in essence, we use a logarithmic scale with a clearly defined step to measure the masses of Standard Model particles. The sampling formula Ek = E0 ×2−k/24 introduces no adjustable free parameters: the Higgs vacuum expectation value E0 = 246220 MeV is taken from experiment, the scale factor 2−1/24 originates from the self-similar scaling property of the 24-dimensional RFW orthotope, and the integer k is the reading of the particle’s experimental mass on this logarithmic scale. Within this framework, the masses of 15 Standard Model particles can be sampled as a set of integers K, with an overall absolute mean deviation of about 0.737%. We interpret the sampled mass as the geometric bare mass within an ultraviolet-complete quantum field theory framework, while the physical observed mass is obtained after further radiative corrections.The mathematical background of this scaling law can be traced to the 24-dimensional Leech lattice and its Turyn tripartite construction E8⊕E8⊕E8. The dynamics are tentatively understood as a generational collision–fusion–fission reaction: k+(k+2) → 2×(k+1)+γ. From this reaction channel, we derive a geometric mass-defect relation and, combined with a coherent amplification factor from the tripartite structure, obtain a geometric expression for the fine-structure constant. We perform a global scan over the discrete parameter space 1 ≤ D ≤ 26,1 ≤ N ≤ 13; amongall 338 integer combinations, only (D,N) = (24,9) reproduces the experimental fine-structure constant with a precision of 0.048%, while the next-best solution deviates by 1.123%.It should be emphasised that all the above results are phenomenological observations and conjectures, not rigorous derivations from first principles. The purpose of this work is to report a set of structured, experimentally testable phenomenological regularities and to offer a discrete geometric perspective on the origin of Yukawa couplings, different from previous approaches. The observations made here are still preliminary; all inferences are falsifiable, and their validity awaits future astronomical observations and laboratory experiments.