Strict Likelihood Multimodality and Septic Equations for Two-Component Birkhoff Models

We give a positive integer dataset on \(S_4\) whose likelihood on the closed two-component Birkhoff model has at least three distinct strict local maxima in distribution space. The dataset has \(2013\) observations and all three probability vectors have full support. One component of each displayed mode is a boundary component. An exact nonnegative-decomposition certificate and a compactness argument rule out the possibility that the modes are artifacts of a parameter chart or component-label switching. Separately, for every \(n \geq 4\), unequal positive parity-constant counts give at least three distinct global maximizers in the closed two-component model. We also determine the first nonzero homogeneous equation degree of the complex \(S_4\) secant variety: it is seven. A \(96\)-term septic and its \(24\) symmetry images supply local equations at the displayed smooth point. We do not determine the global defining ideal or assert modes with both component matrices strictly positive.