Abstract

During repeated attempts to derive the observed three-generation flavor hierarchy solely from an exact graph symmetry, every construction either preserved exact multiplet degeneracy or required external symmetry-breaking input. This suggested that the obstruction was intrinsic to symmetry-complete theories rather than to any particular graph construction, motivating the general no-go theorem proved here. A recurring claim across several research programs - discrete flavor-symmetry model building (Aₙ, Sₙ, Δ(27) family symmetries), Froggatt–Nielsen-type charge assignments, and combinatorial or graph-based approaches to emergent spacetime and particle content - is that the observed hierarchy of fermion masses, mixing angles, or analogous multiplet splitting’s can be derived from symmetry considerations alone, without an independently motivated symmetry-breaking sector. We show this is impossible in a precise, model-independent sense. If a theory is defined entirely by a symmetry group S, a covariant representation Γ of S, and dynamics Hₘᴇ covariant under S, then no numerical value can be derived within that theory for the splitting between components of a non-trivial irreducible multiplet of observables, beyond the trivial statement that the splitting is exactly zero when the theory's distinguished state is S-invariant. If the distinguished state instead spontaneously breaks S, the direction of breaking is undetermined by the theory, and even granting a direction, the typical magnitude of splitting predicted by a symmetry-neutral (Haar-random) prior is O(1) between multiplet components — so any large observed hierarchy is itself evidence of a further, non-symmetry input. The result is an exact operator-level sharpening of Curie's symmetry principle (1894) via the Wigner–Eckart theorem (1927), together with the standard vacuum-selection problem of spontaneous symmetry breaking (Goldstone 1961; Goldstone, Salam & Weinberg 1962). It applies to any “symmetry-complete” theory - continuum or discrete, field-theoretic or lattice-based and identifies exactly which additional, non-symmetry ingredient any such theory must supply.