Information values and attainment for proper residual odd ranks

Determine the worst-frequency information values and establish whether the corresponding optima are attained for all proper residual odd output ranks not covered by the exact even-rank and co-rank-one results.

Background

The paper solves the information-design problem for every even output rank and for the even-aperture co-rank-one case. It proves that regular globally identifying designs are dense for every rank at least three and that the globally identifying and unconstrained suprema coincide, but it explicitly does not establish exact attainment for unresolved proper odd ranks. The open issue therefore concerns both evaluating the arbitrary-row-space objective and determining whether a maximizing design exists in those remaining odd-rank cases.

References

Open problems include the exact rank-two GI-constrained optimum, arbitrary-row-space values and attainment for proper residual odd ranks, the complete singular local limit experiment, exact minimax constants after fixing a leading normal form, and the transition as the gain annulus collapses.

— Compressed Single-Tone Frequency Estimation With Unknown Complex Gain: Singular Rates and Global Identifiability  (2608.30623 - Rasooli, 31 Aug 2026) in Section 6, Theorem 7 and Discussion and Limits

Does the floor hold on the strata $(h,\rho_G)$ with $\rho_G=1$ and $h\ge4$, and on the other open strata of Table~\ref{tab:strata} (\S\ref{sec:hadintro}, Conjecture~\ref{conj:w1})?

— Exact $T$-counts of Toffoli layers from an isotropy bound  (2610.01024 - Mazumder, 1 Oct 2026) in Open Problem 1, Section 4.2; Table 1 in Section 10