NDS Chernoff coefficients including support endpoints

Let $M,J\in\mathbb N$ with $M>0$, allowing $J=0$, and let $\psi\in H=\ell^2(\mathbb N^J\times\mathbb N^M;\mathbb C)$ have norm $1$ and satisfy $\sum_i\psi(i,n)\overline{\psi(i,q)}=0$ whenever $n\ne q$. For $b\in\{0,1\}$, let $R_b\in[0,1]$, $\theta_b\in\mathbb R$, $r_b=\sqrt{R_b}$, and $t_b=\sqrt{1-R_b}$, with $R_0\le R_1$ and $r_0e^{i\theta_0}\ne r_1e^{i\theta_1}$. Define $L_b(n,k)=\prod_m\sqrt{\binom{n_m}{k_m}}e^{in_m\theta_b}r_b^{n_m-k_m}t_b^{k_m}$ for $k\le n$ componentwise, and zero otherwise; put $g_{b,k}(i,p)=\psi(i,p+k)L_b(p+k,k)$, let $v_{b,k}\in H$ equal this function when square-summable and zero otherwise, and set $\rho_b(x,y)=\sum_k g_{b,k}(x)\overline{g_{b,k}(y)}$, $P_{b,k}=\|v_{b,k}\|^2$, and $C_k=\langle v_{0,k},v_{1,k}\rangle$. A resolution $(a_n,u_n)$ of a kernel $K$ means summable nonnegative weights of sum $1$, vectors unit at positive weights and orthogonal at distinct positive weights, and $K(x,y)=\sum_n a_nu_n(x)\overline{u_n(y)}$, with zero-weight vectors unrestricted. Put $h(p,t)=0$ when $p=0$, and $h(p,t)=p^t$, using real powers, otherwise. Define

where the resolutions are of $\rho_0,\rho_1$. Define $E_{\mathrm{out}}$ as the infimum, over these resolutions and bounded complex-linear effects $E$, of $\tfrac12\sum_n a_n(1-q_E(u_n))+\tfrac12\sum_n b_nq_E(w_n)$, where $q_E(v)=\operatorname{Re}\langle v,Ev\rangle$, $\langle u,Ev\rangle=\overline{\langle v,Eu\rangle}$, and $0\le q_E(v)\le\|v\|^2$ for every vector. The real suprema and infima are total, taking value zero on empty defining sets. For each $s$, put

The theorem asserts that for every $s\in[0,1]$, the sequence $k\mapsto D_{s,k}$ is summable, $Q_s=\sum_kD_{s,k}$, and $E_{\mathrm{out}}\le Q_s/2$; that some $s\in[0,1]$ satisfies $Q=Q_s$ and $Q_s\le Q_t$ for every $t\in[0,1]$; and that $E_{\mathrm{out}}\le Q/2$. Both endpoints $s=0,1$ are included, and a zero resolution weight contributes zero even when its exponent is zero.