Ambient Polytope & Local Coordinates: All 125 visualizations (81 SNIEP and 44 RNIEP) examine matrix order n = 5 inside the normalized McDonald–Neumann trace polytope 1 ≥ λ₂ ≥ λ₃ ≥ λ₄ ≥ λ₅ ≥ -1 with nonnegative trace 1 + λ₂ + λ₃ + λ₄ + λ₅ ≥ 0. Each 2D Polygon or 3D Polyhedron projects the affine hull of the listed extreme vertices onto an orthonormal basis (Local X, Local Y, and Local Z).

• 7 Realizable Vertices (exact in SNIEP & RNIEP): a0=(1,1,1,1,1), a1=(1,1,1,1,-1), a2=(1,1,1,-1,-1), b1=(1,0,0,0,-1), b2=(1,1,0,-1,-1), c_10=(1,-1/4,-1/4,-1/4,-1/4), and c_21=(1,1,-1/2,-1/2,-1).
• 2 Unrealizable Vertices (excluded from NIEP(5)): d_20=(1,1,-2/3,-2/3,-2/3) (violates the Laffey–Meehan trace-zero condition s₂² ≤ 4s₄) and e_12=(1,1/2,1/2,-1,-1) (negative third power sum s₃ = -3/4 < 0).
• What “Realizable Region” Means: Shaded Realizable Region meshes and Boundary Points are empirical numerical optimization estimates computed by alternating-projection / nonlinear matrix searches over 5 × 5 symmetric nonnegative (SNIEP) or general nonnegative (RNIEP) matrices—not formal necessity/sufficiency proofs. Hovering over boundary points displays the candidate matrix and its L₂ spectral residual.
• Exact Reference Curves: Wireframe outlines mark the exact linear trace-polytope slice, and curves such as s₁³ - s₃ = 0 plot exact algebraic power-sum loci for comparison against the numerical boundary.