North-East Lattice Paths with Few Collinear Vertices

Both bounds move, and the gap stays enormous: the lower bound rises from $\exp(\Omega(\log^2 k))$ to $\exp(\Omega(k^{1/3}))$ and the upper falls from $\exp(O(k^4))$ to $\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness, a sharper line-counting step, or a different model entirely.

Both bounds move, and the gap stays enormous: the lower bound rises from $\exp(\Omega(\log^2 k))$ to $\exp(\Omega(k^{1/3}))$ and the upper falls from $\exp(O(k^4))$ to $\exp(O(k^2))$, so $A(k)$ is still undetermined between an exponent of $k^{1/3}$ and one of $k^2$. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness, a sharper line-counting step, or a different model entirely.