For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2: CC^*-C^*C=2F\}$. We give the exact value of $\kappa_d$ on the whole inertia-$(m,2)$ stratum (positive eigenvalues $a_1\ge\dots\ge a_m>0$, two negative eigenvalues $-b_2\ge -b_1$, any number of zeros) for every $m\ge2$ and every $d$: $\kappa_d=\max(F_0,\dots,F_{m-1},G_1,G_3,\dots)$ for explicit linear forms. The lower bound sums Lidskii–Wielandt and Weyl inequalities with one Pieri-type Horn triple (Littlewood–Richardson coefficient $1$); the upper bound is an explicit $2\times2$-block weighted shift whose feasibility is a one-parameter interval recursion. Both bounds are dimension-free, so the value is invariant under zero padding, and the chamber structure is explicit. Every optimizer has rank $m$ or $m+1$, the minimal optimal rank equals $\max(n_+,n_-)=m$ in every dimension, and rank $m+1$ occurs exactly in one open chamber after padding, where the optimal spectrum is not unique. By $F\mapsto-F$ the same formula covers inertia $(2,n)$.
Research paperScreenedARR-2026-3WVWP20DY09HFASA · v1
Let $p$ be prime and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\mathbb C^p$. We prove that every nonzero linear combination $C$ of $\ell$ Weyl operators satisfies $\dim\ker C\le\ell-1$, i.e. $\operatorname{rank}C+s_W(C)\ge p+1$, where $s_W$ counts the nonzero Weyl coefficients. The collinear case is Tao's uncertainty principle on $\mathbb Z/p$ (Chebotarev's theorem on Fourier minors). The general case follows from the reduction-modulo-$(1-\omega)$ mechanism of Frenkel and Tao after two further observations: Weyl combinations stay integral under the non-unimodular Clifford conjugations $F'Q_s$, which realise all $p+1$ line directions of $\mathbb F_p^2$, and, in the nonvacuous sparsity range, a pigeonhole over those directions produces a nonzero fibre sum. The bound is sharp for every $\ell\le p$ and fails in every composite dimension. As a self-contained corollary, for one use of the $p^2$ Weyl channels the minimal perfect list length over probes of Schmidt rank $r$ is exactly $p-r+1$. An exact certificate refutes a natural "line conjecture".
Research paperScreenedARR-2026-02AGP452839C09SH · v1
In cyclic Weyl dimension d=3^k, k at least two, we classify every reduced pure-probe state of rank strictly below 7d/9 permitting exact channel discrimination with lists of at most three labels. The low-rank stratum consists of 150 relative interiors indexed by affine lines over the field of three elements, with explicit ranks, spectra, unique positive decompositions and minimum factor counts. Exactly twelve states permit lists of two. We obtain exact-rank list minima at ranks 4d/9 and 5d/9, and show that the structural cutoff 7d/9 is sharp. The result concerns a fixed cyclic Weyl basis and exact zeros; higher-rank and approximate classifications remain open. Exact finite-geometry and matrix certificates accompany the written proofs.
Research paperScreenedARR-2026-0WESHW4YMM9FG8EK · v1
We classify the ranks of nonzero operators with at most three terms in a fixed cyclic Weyl basis in every prime-power dimension. A central-sector formula valid in arbitrary cyclic dimension computes trinomial nullity from commuting powers and a common block multiplicity; nonzero three-term coefficients permit at most two singular sectors. The formula explains composite-dimension departures from the prime-power bound. For exact-rank pure probes in one-use Weyl channel list discrimination, we classify all reduced states below rank 3d/4 in dyadic dimensions: exactly thirty flat half-rank states. Positive sums of sparse squares produce ranks impossible for an individual three-term factor, including an exact three-label family of rank 13d/16. We also determine an exact four-label family of rank 5d/8 and the complete feasible three-label rank set in dimension sixteen. Classical Weyl representation and covariance tools are attributed; exact algebraic certificates and replay programs accompany the proofs.
