Sumsets regularity versus Castelnuovo-Mumford regularity for monomial curves

Let $A=\{0=a_0<a_1<\cdots <a_{k-1}=d\}$ be integers with greatest common divisor $1$, let $C_A$ be the projective monomial curve they define, and let $\mathrm{r}\mathrm{e}\mathrm{g}(A)$ be the Castelnuovo-Mumford regularity of its coordinate ring. Gimenez and González-Sánchez proved $\mathrm{r}\mathrm{e}\mathrm{g}(A)\le \sigma (A)+1$ for the sumsets regularity $\sigma (A)$; the reverse inequality $\sigma (A)\le \mathrm{r}\mathrm{e}\mathrm{g}(A)$ was proposed as an open conjecture in a 2024 talk. We show that it fails and measure by how much. Let $c_1,c_2$ be the conductors of the two numerical semigroups attached to $A$ and $N=\lceil (c_1+c_2)\mathrm{/}d\rceil$. For curves in ${\mathbb{P}}^3$ that are not arithmetically Cohen-Macaulay, $\mathrm{r}\mathrm{e}\mathrm{g}(A)\ge (c_1+c_2)\mathrm{/}d+2$, with equality exactly for the arithmetically Buchsbaum curves, and $\sigma (A)=\mathrm{r}\mathrm{e}\mathrm{g}(A)$. For arithmetically Cohen-Macaulay curves in ${\mathbb{P}}^3$ we prove $9N\le 10\mathrm{r}\mathrm{e}\mathrm{g}(A)-2$, with $9\mathrm{/}10$ best possible, and give an explicit family with $\sigma (A)-\mathrm{r}\mathrm{e}\mathrm{g}(A)$ unbounded. In ${\mathbb{P}}^4$ a family with a band-shaped Apéry set has $\mathrm{r}\mathrm{e}\mathrm{g}(A)\mathrm{/}\sigma (A)\to (2+\sqrt{2})\mathrm{/}4$ along the solutions of a Pell equation, and gluing carries this bound to every ${\mathbb{P}}^r$ with $r\ge 5$.