Prove linear-in-M nonzero support for FEM coupling vectors w_k

Prove that for the vectors w_k arising in the FEM analysis of u_k^T V u_{k+1} (where V is the finite element potential matrix and u_k are Laplacian eigenvectors), the minimal number of nonzero entries satisfies min_k ||w_k||_0 ≥ c M for some universal constant c > 0. Establishing this linear lower bound would strengthen the anti-concentration estimates and improve the resulting diversity bounds.

Background

The finite element diversity bound leverages an anti-concentration inequality for Bernoulli sums where the strength of the bound depends on the number of nonzero coefficients in w_k.

The current analysis only shows ||w_k||_0 ≥ 1; a linear-in-M lower bound would substantially tighten the probability estimates and the diversity guarantees.

References

Note that a stronger lower bound on $|\mathbf{w}_k|_0$ can improve the estimate; in particular, we conjecture that $\min_k |\mathbf{w}_k|_0$ scales linearly in $M$, which would substantially improve the result, but we leave it to future work to verify this.

— 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 Section 6 (Additional proofs), Proof of Theorem FEM