Bi-Constructible pattern of weak and flavour mixing: implications for electroweak coupling constants

Abstract

We present semi-empirical evidence suggesting that weak and flavour mixing, at the most fundamental level, can be described in terms of the Euclidean geometry of regular polygons constructible with compass and straightedge, specifically, the pentagon and the heptadecagon, associated with Fermat primes -- a pattern referred to as Bi-Constructible. Our approach accurately reproduces quark and lepton mixing angles and offers indications that the Weinberg angle also fits naturally within this geometric framework. Concise Weak--Quark--Lepton Complementarity relations are derived. These findings suggest a semi-empirical unification pattern of weak and flavour mixing. The Standard Model gauge couplings g and g' admit elegant expressions involving the golden ratio, yielding a neat prediction for the fine-structure constant entirely in these terms.