Fixed and adaptive orthogonal pinching have the same length

Let $\varphi(d)$ be the least $q$ for which fixed orthogonal matrices $O_1,\dots,O_q$ satisfy $\Delta(X)=\frac1q\sum_iO_iXO_i^T$ for every real $d\times d$ matrix $X$. Let $a(d)$ be the corresponding length when the $O_i$ may depend on $X$. Both equal $h(d)$, the least number of rows of a $\pm1$ matrix $H$ with $H^TH=h(d)I_d$. The upper bound uses the diagonal sign matrices given by the rows of $H$. For the lower bound, a single test matrix $D+K$ is used, with $D$ diagonal with distinct entries and $K$ skew-symmetric with all off-diagonal entries nonzero. It forces every conjugating matrix in any representation to be a diagonal sign matrix, and their sign patterns form a matrix with orthogonal columns. For $d\ge3$, $h(d)$ is a multiple of four and at least $d$. A sign matrix of order twelve from quadratic residues modulo $11$ gives $\varphi(d)=12$ for $9\le d\le12$; in general $\varphi(4m)=4m$ exactly when a Hadamard matrix of order $4m$ exists.