Eventual validity of the quasi-linear formula for generic two-row matrices

Establish that the quasi-linear formula for the generic column-number function \(\g(\Delta,2)\) holds for every integer \(\Delta\geq 385\), rather than only for \(385\leq\Delta\leq 1550\) and \(\Delta\geq 10^8\).

Background

The paper proves the exact formula $\g(\Delta,2)=\tilde\g(\Delta)$ for 385≤Δ≤1550385\leq\Delta\leq1550 and for every Δ≥108\Delta\geq10^8, where $\tilde\g(\Delta)$ is the piecewise-linear function determined by Δ mod 6\Delta\bmod 6. The value at Δ=384\Delta=384 does not satisfy this formula, so the conjectured threshold $385$ would be best possible.

The unresolved issue is to close the finite gap between $1550$ and 10810^8, which the authors state would require substantially new ideas beyond the estimates used in the paper.

References

We iterateConj.~4.14 and conjecture that \cref{thm:upper_bound_on_g_Delta2} holds for all $\Delta \geq 385$ instead of~$108$.

— On generic $Δ$-modular integer matrices with two rows  (2502.15394 - Kriepke et al., 21 Feb 2025) in Introduction, immediately after Theorem 1 (the theorem labeled \texttt{thm:upper\_bound\_on\_g\_Delta2})