The dyadic denominator law for the phase constants of the Jacobi zeros

The asymptotic phase for the zeros of a Jacobi polynomial contains additive constants ${\kappa }_r$ that are not determined by the phase equation. We study their denominators as polynomials in $A={\alpha }^2$ and $B={\beta }^2$. We prove that the odd part of $\text{den}{\kappa }_r$ divides $\mathrm{lcm}(1,3,\dots ,2r-1)$ and that $2^{E_r}{\kappa }_r$ is $2$-adically integral, where $E_r=3r-1+{\nu }_2((r-1)!)$. The extremal coefficient is governed by the valuation law $${\nu }_2\text{\hspace{0.17em}⁣}\left(\sum _{j=0}^m\left(\genfrac{}{}{0px}{}{m}{j}\right)\frac{1}{2j+1}\right)=m+{\nu }_2(m+1),$$ which follows from the identity ${\sum }_{k\ge 0}k!\mathrm{/}(2k+1)!!=0$ in ${\mathbb{Q}}_2$. We also transform the conjectural sharp denominator law into a single coefficientwise statement. If $\mathrm{\Phi }$ is the Borel transform of the Legendre tangent and $W(t)=\sinh (2t)\mathrm{Im}\mathrm{\Phi }(t)\mathrm{/}{t}^2={\sum }_{m\ge 0}{w}_m{t}^{2m}$, then the sharp law is equivalent, with equality preserved at each index, to $((2m)!{)}^2{w}_m\in {\mathbb{Z}}_2^{\times }$ for every $m$. This final integrality statement (Conjecture~W below) has since been proved in the companion paper of this series, so the sharp denominator law holds in all orders; the present paper establishes the normal form and the valuation-exact transfer, and records the exact evidence and the structural obstructions that delimited that proof.