Fixed-Polynomial Alignment and Support-Adapted Reduction for Crouzeix's Conjecture

We study support-adapted dimensional reduction for Crouzeix's conjecture. For a fixed polynomial f, we prove that at smooth stationary points of the ratio \|f(A)\|/z∈ W(A)|f(z)|, the first variation of $\|f(A)\|$ aligns with the rank-one differential of the active support function. This yields the exact identity Su,v=c\,xθ^*xθ^*^* and a compression theorem reducing stationary degree-d problems to dimension at most $d+1$; in particular, quadratic and cubic stationary problems reduce to 3×3 and 4×4 compressions. We also show that the basic support-eigenvector space span_θ,Ax_θ,A^*x_θ\ always collapses to span_θ,Ax_θ\, so genuinely higher-dimensional support mechanisms begin at second order. In the full-rank case we identify a common orbit space for the singular data and show that the projected-rank condition is generic along analytic branches. Separately, we prove that every polynomial pair $(A,p)$ admits an invariant cyclic compression generated by a norm-attaining singular vector, and therefore that Crouzeix's conjecture is equivalent to its restriction to cyclic matrices. The support-adapted results then provide sharper low-degree reductions within that global cyclic framework.