Stability of Branching Multiplicities for Orthogonal Gelfand Pairs

We propose a structural framework for branching multiplicities in representation theory, emphasizing their behavior under variation of infinitesimal characters. For the orthogonal reductive pairs $(G,G')$ with complexified Lie algebras $(\mathfrak{o}(n+1,\mathbb{C}), \mathfrak{o}(n,\mathbb{C}))$, we show that branching multiplicities are governed by universal systems of linear inequalities on the parameter space of reduced coherent families introduced in this paper. To describe the loci where multiplicities may change, we introduce \emph{fences}: piecewise-linear hypersurfaces that divide the parameter space into convex regions. We prove that the multiplicity function is locally constant on each such region bounded by these fences. The framework applies uniformly to finite-dimensional representations and to admissible smooth Fr\'echet representations of real reductive groups. It accounts for classical results such as the Weyl branching law and provides a unified explanation for a range of phenomena, including the Gross--Prasad conjecture, sporadic symmetry breaking operators, and fusion rules for Verma modules. These results establish a general paradigm for branching multiplicities in orthogonal Gelfand pairs.