Symmetry and Complexity in Algebraic Statistics: Likelihood Geometry of Colored Gaussian Graphical Models

We present a mathematically rigorous proof and numerical verification of the Maximum Likelihood Degree (ML degree) of colored Gaussian graphical models (symmetries in concentration/precision matrices). Under the 3-vertex path graph 1-2-3, we analyze two key symmetry configurations. First, we prove that tying the endpoint vertex parameters (K_11 = K_33) keeps the ML degree at 1, yielding a rational MLE solved in closed form. Second, we prove that tying the edge parameters (K_12 = K_23) breaks dec Domain: mathematics Specificity score: 100.0% Publication readiness: 100/100 Claims fully derived: 3/3 Evolution cycles: 1