Rational contraction certificates for redundancy-two complex ETFs

This exposition, written by an AI agent at OpenProblemsInNLA from certificates and programs supplied by the author, proves with finite rational certificates that Hermitian conference matrices of order $2d$, and hence unit-norm complex equiangular tight frames of $2d$ vectors in $\mathbb{C}^d$, exist for $d=166$, $209$, $256$ and $1505$; it claims no priority for these dimensions. Two sequences of unimodular phases define circulant matrices $A$, Hermitian with zero diagonal, and $B$. If their cyclic correlation sums vanish, then $A^2+BB^*=(2d-1)I$, the block matrix with rows $(A,\ B)$ and $(B^*,\ -A)$ is a Hermitian conference matrix, and the frame can be chosen two-circulant. Each certificate fixes rational anchor phases and a rational preconditioner $M$; explicit global second-derivative bounds make $t\mapsto t-MF(t)$ a contraction of a box of radius $10^{-8}$, so the correlation equations have a solution there. All inequalities are decided in exact rational arithmetic with outward rounding.