A Rational φ-Ladder for Standard-Model Fermion Masses and Mixing Angles

This paper proposes a fundamental geometric origin for the twelve mass and mixing parameters of the Standard Model fermion sector, which are conventionally treated as free inputs. We demonstrate that these parameters are not independent constants but are locked into a single, coherent algebraic structure: a rational ladder of steps based on the golden ratio phi = (1 + sqrt (5) ) /2. This "phi-ladder" is anchored at the Higgs vacuum expectation value v = 246. 22 GeV and is characterized by a discrete granularity of n/15 in the exponent of phi. The denominator 15 is not a free fit parameter; it is uniquely determined by the stabilizer orders of the alternating group A5, corresponding to the icosahedral symmetry group. We derive thirteen precise identities that match experimental values for all nine charged fermions, neutrino mass-squared splittings, and key mixing angles. Notably, the top-quark mass is reproduced with a relative error of 5 x 10^-6, identifying it as a specific phi-rational step above the Higgs boson mass. Our model reinterprets the nature of matter and particle physics: fermions and bosons are not isolated building blocks but standing-wave resonances within a layered, fractal geometry. In this framework, particle accelerators do not split matter into fragments but provide the energy required to probe deeper, high-frequency layers of the underlying icosahedral architecture. The paper concludes with eight specific, falsifiable predictions for the next decade of experiments (including HL-LHC, KATRIN, and DUNE), offering a definitive test for this geometric unification of the Standard Model spectrum.