Fermion mass relations in one-parameter modular models

A bstract Modular flavour symmetries provide a possible organizing principle for the Standard Model Yukawa sector, by replacing generic couplings with a potentially small number of modular forms controlled by a single complex modulus. We study the extreme limit of this idea: one-parameter modular models (OPMs), in which each charged-fermion mass matrix is fixed by a single modular invariant contraction. We develop a systematic method to construct such models, showing that the one-parameter modular model (OPM) requirement is already highly constraining at the level of possible fermion hierarchies. In a concrete realization, the charged-lepton and down-quark sectors are controlled by the common modulus, leading to exact mass relations at the flavour scale, mₛ⁵=2√2md³mb²,0.5em mμ³=√2mₑmτ²,0.5em mₛ²mτ=√2mₑmb². m s 5 = 2 2 m d 3 m b 2 , m μ 3 = 2 m e m τ 2 , m s 2 m τ = 2 m e m b 2 . We show that, once renormalization-group evolution and selective supersymmetric threshold effects are included, these high-scale relations can be made compatible with low-energy charged-fermion data. Our results provide a working proof of principle for one-parameter modular models (OPMs) and point towards a possible route to the flavour puzzle through highly constrained constructions.