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 >= 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. Corpus identifier: AIM-COMPUTATION-0073. This unrefereed, AI-assisted preprint includes exact arithmetic certificates and a written self-audited proof. It is a completed scoped result, not a claim that every part of the original source problem is solved. No independent certification, formal verification or absolute priority claim is made. PDF, reproducible source code and a verification report are included.