Sharp two-level conditioning over prime fields

Determine the sharp least singular value and exact optimizing parameter for the nonnegative two-level affine-orbit generator family over the prime field F_p, equivalently the path-incidence problem in which the extremal path vector does not generally annihilate the rank-one all-ones term.

Background

The paper proves an exact least-singular-value formula and identifies the unique optimizer for the two-level family over extension fields F_{ph} with hgeq2. The proof relies on cycle components in the finite-field incidence graph: a Fourier eigenvector supported on a cycle has coordinate sum zero and therefore removes the rank-one all-ones contribution to the relevant Gram matrix.

For the prime-field case h=1, the corresponding graph is instead the path P_{p-1}. Although the paper establishes matching stability order through earlier prime-field theorems, the extremal path vector generally has nonzero coordinate sum, so the same argument does not yield the exact constant or optimizer. The unresolved problem is therefore to complete the sharp optimization for the prime-field path case.

References

The exact formula in Theorem~\ref{thm:ppoptimal} uses the cycle component in Lemma~\ref{lem:pathcycle}, and is therefore asserted only for $h\geq2$. When $h=1$, the graph is the path $P_{p-1}$; the extremal path vector does not in general annihilate the $J$-term in $(33)$. The earlier prime-field theorems give matching stability order in that case, but this paper does not claim a sharp two-level constant or an exact optimizer for the prime-field path problem.

— Sharp Conditioning for Matrix Recovery by Finite Affine Orbits  (2609.09619 - Li, 9 Sep 2026) in Remark ref{rem:primefieldlimitation}, following Theorem ref{thm:ppoptimal}, Section 5 (Prime-Power Affine Orbits)