Legendre compressions and an integrality conjecture for the Hörmander--Bernhardsson extremal function

We prove Conjecture~2 of Bondarenko, Ortega-Cerdà, Radchenko, and Seip for the three-term recurrence attached to the Hörmander–Bernhardsson extremal function $\phi$. More precisely, define $${\tilde{u}}_{-1}=0,{\tilde{u}}_0=1,$$ and $${\tilde{u}}_{n+1}=\frac{4n+2}{n+1}(n(n+1)-\lambda ){\tilde{u}}_n+\frac{4n}{n+1}x{\tilde{u}}_{n-1}\mathrm{.}$$ Then $${\tilde{u}}_n(x,\lambda )\in \mathbb{Z}[x,\lambda ](n\ge 0)\mathrm{.}$$ The proof is a determinant comparison in the scaled Legendre basis. After sign reversal and central-binomial normalization, the recurrence becomes exactly the continuant recurrence of a finite tridiagonal compression. In particular, if $T_n(a,\lambda )$ denotes the $n$th BOCRS tridiagonal truncation, then $${\tilde{u}}_{n+1}(a^2,\lambda )=\left(\genfrac{}{}{0px}{}{2n+2}{n+1}\right)\det T_n(a,\lambda )\mathrm{.}$$ As consequences, we derive that $${\left(\frac{\pi }{4C}\right)}^2\text{and}-\frac{L_{\tau }(1)}{2C}$$ are not simultaneously rational, where $C$ is the sharp point-evaluation constant for $P{W}^1$, $\pm {\tau }_n$ are the nonzero zeros of $\phi$, and $L_{\tau }(1)={\sum }_{n\ge 1}\frac{(-1{)}^n}{{\tau }_n}\mathrm{.}$ Finally, if we write $\phi (z)={\sum }_{n\ge 0}{c}_n{z}^{2n},$ then $$c_n\in C^n\mathbb{Z}[{\pi }^2,C,L_{\tau }(1)](n\ge 0)\mathrm{.}$$