Remove the large-M technical assumption from the finite-difference diversity bound in higher dimensions

Establish that the finite-difference discretization diversity result for random Schrödinger operators on [0,1]^D with separable Bernoulli potentials holds for all grid sizes M (i.e., without requiring M ≥ 9/(2 p(1−p))) when D > 1. Concretely, prove that the probability lower bound for the augmented sample set {A^(1), …, A^(N), −Δ_FD,D} having a trivial centralizer holds without this assumption for the matrix distribution induced by finite-difference discretization.

Background

Theorem FD provides an exponential-in-N lower bound for the probability that the augmented sample set of matrices arising from finite-difference discretization of Schrödinger operators has a trivial centralizer. The proof in dimensions D > 1 currently assumes M ≥ 9/(2 p(1−p)) for technical reasons.

The authors believe this assumption is not fundamental to the phenomenon and could be eliminated with a refined analysis, thereby strengthening the applicability of the diversity guarantee across grid sizes.

References

Finally, we remark that the condition that $M \geq \frac{9}{2p(1-p)}$ is required for technical reasons in the proof when $D > 1,$ but we conjecture that this condition can be removed with a more careful analysis.

— A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schrödinger Equations  (2601.12587 - Cole et al., 18 Jan 2026) in Subsection 3.1 (Schrödinger operators under finite difference discretization)