Unrestricted Kronecker trace estimation: polynomial real simulation, parallel factor access, and a bounded-probability lower bound

We study relative trace estimation of arbitrary real positive-semidefinite matrices from adaptive, exact, real rank-one tensor matrix-vector queries. A degree-$q$ interpolation identity simulates one complex product query with $q+1$ real product queries. It also yields a counterexample, already for local dimension two, to Conjecture 23 of Meyer, Swartworth and Woodruff (arXiv:2502.08029v2): a fixed span of $q+1$ real product vectors has squared projection at least $2^{1-q}$ on every real unit product vector. For the promise $M=A_1\otimes\cdots\otimes A_q$ with PSD factors of order $n$, a normalization-and-shift construction gives simultaneous ordinary matrix-vector access to all factors, hence exact recovery in $n$ queries and a relative-error estimator using $O(\sqrt q/\varepsilon)$ queries. An adaptive Wishart posterior argument, combined with log-concave anti-concentration, gives a matching bounded-probability lower bound for this subclass. Blocking yields a general lower bound, while interpolation and ordinary-query simulation give general upper bounds.