Even-index Zeilberger–Zudilin integrals are integer linear forms in $1$ and $\pi$

With $R_n$, $J_n$, $\mathrm{coef}_n$, $\Phi_n$ and $M_n$ as in the definition file PiIrrationality_ZZEvenForms, for every $n\ge1$ there are integers $U_n$ and $V_n$ with

This is the arithmetic of the Zeilberger–Zudilin construction at even index. The integrand has the even partial-fraction decomposition $R_n=P_n+\sum_{j=0}^{6n}c_{j,n}\bigl((5+t)^{-j-1}+(5-t)^{-j-1}\bigr)$ with $P_n\in\mathbb{Z}[t^2]$. Only the $j=0$ term produces $\pi$, through $\int(\tfrac1{5+t}+\tfrac1{5-t})\,dt=\pi i/2$. The multiplier $M_n=2^{4-5n}\operatorname{lcm}(1,\dots,8n)/\Phi_n$ clears all denominators: the $2$-adic and $5$-adic valuations of the Laurent coefficients give the power of $2$, and the deleted primes in $\Phi_n$ divide every relevant coefficient. The $\pi$-coefficient is $-M_nc_{0,n}/2$. The residue satisfies $c_{0,n}=16^n\,\mathrm{coef}_n/2$, by the substitution $z=(t+5)/(t-5)$, which turns $R_n(t)\,dt$ into $\tfrac12 16^n z^{-6n-1}S(z)^n\,dz$.