Sum-of-squares proof barrier for Dirichlet polynomial large values

Show that for the Dirichlet polynomial matrix M_Dir with T = N^α (1 < α < 2), there exists a critical σ_crit(α) > 1/2 such that for any degree D, ε > 0, and constant C, and for all sufficiently large N, there is no degree-D sum-of-squares proof of the bound |W| ≤ C N^{α + 1 − 2σ − ε} whenever σ ≤ σ_crit(α).

Background

Motivated by average-case complexity barriers for random matrices, this conjecture proposes an analogous SOS limitation for the explicit Dirichlet matrix M_Dir, suggesting that any proof surpassing the operator-norm bound in certain regimes must be complex in a precise SOS sense.

If true, it would imply that Conjectures of Montgomery cannot be proved by constant-degree SOS and would guide future attempts toward methods that exploit number-theoretic structure beyond algebraic certificate frameworks.