Static updates: one-edge energy is a quadratic form in a unit-modulus phase

This theorem asserts the conjunction of the following two statements. First, for every pair of natural numbers $N,D$ with $N\ge1$ and $D\ge1$, every family of complex $D\times D$ matrices $A_{q,b}$ indexed by sites $0\le q<N$ and Boolean values $b$, every real array $w$ satisfying $w_{ab}=w_{ba}$, $w_{aa}=0$, and $0\le w_{ab}<2\pi$, every family of unitary complex $2\times2$ matrices $V_q$, every pair of distinct sites $j,k$, and every Hermitian matrix $H$ indexed by Boolean strings of length $N$, assume that the vector $t$ defined by $t_b=\operatorname{tr}(A_{0,b_0}\cdots A_{N-1,b_{N-1}})$ is nonzero. Put $Z=\|t\|^2$, let $(D_w)_{b,b}=\exp(i\sum_{a<c,\ b_a=b_c=\mathrm{true}}w_{ac})$, let $W_{a,b}=\prod_q(V_q)_{a_q,b_q}$, and define the vector $e$ by $e_b=\exp(-iw_{jk}) (D_wt)_b$ when $b_j=b_k=\mathrm{true}$ and $e_b=(D_wt)_b$ otherwise. Let $u$ equal $e$ on strings for which the conjunction of those two bits is false and zero elsewhere, and let $v$ equal $e$ on strings for which both bits are true and zero elsewhere. Define $H_V=W^\dagger HW$, $c_0=\operatorname{Re}\langle u,H_Vu\rangle+\operatorname{Re}\langle v,H_Vv\rangle$, $c_1=\langle u,H_Vv\rangle$, $\xi_z=W(u+zv)$, and $E_H(x)=\operatorname{Re}\langle x,Hx\rangle/\|x\|^2$. Then, for every complex $z$ with $|z|=1$, $\|\xi_z\|^2=Z$ and $E_H(\xi_z)=(c_0+2\operatorname{Re}(c_1z))/Z$; the number $z_* =1$ if $c_1=0$ and $z_*=-\overline{c_1}/|c_1|$ otherwise has modulus one; $E_H(\xi_{z_*})\le E_H(\xi_z)$ for every $|z|=1$; and if $c_1=0$, every such $z$ satisfies $E_H(\xi_z)=c_0/Z$. Second, for every natural number $s$, there exist a positive natural number $C$ and a family of edge programs $p_{N,D,m,(S_r),j,k}$ defined for all natural numbers $N,D,m$, all finite supports $S_r\subseteq\{0,\ldots,N-1\}$ for $0\le r<m$, and all sites $j,k$, with the following guarantees whenever $N\ge1$, $D\ge1$, $j\ne k$, and $|S_r|\le s$ for every $r$. Each such program has zero calls to its supplied product-evaluation function and work at parameter zero at most $C(m+1)ND^5$, where work charges $4N+1+B$ for each product-evaluation call at parameter $B$ and $1$ for every other instruction. For every $A,w,V$ as above and every family of Hermitian complex matrices $T_r$ indexed by Boolean assignments on $S_r$, provided $t\ne0$, define $H=\sum_r\widetilde T_r$, where $(\widetilde T_r)_{a,b}=(T_r)_{a|_{S_r},b|_{S_r}}$ if $a,b$ agree outside $S_r$ and is zero otherwise, and form $c_0,c_1$ by the formulas above for this $H$. Execute the program using tensor inputs $(A_{q,b})_{i,l}$, phase inputs $w_{ab}$ viewed in $\mathbb C$, rotation inputs $(V_q)_{a,b}$, term inputs $(T_r)_{a,b}$, and the identically zero product-evaluation function. Execution succeeds with a memory function $\mu:\mathbb N\to\mathbb C$; all four designated addresses are smaller than the instruction-list length; and their values are respectively $c_0$ viewed in $\mathbb C$, $c_1$, $Z$ viewed in $\mathbb C$, and $E_H(WD_wt)$ viewed in $\mathbb C$. Here scalar execution starts with all memory entries zero, writes successive instruction results at addresses $0,1,\ldots$, and permits complex constants, input reads, addition, multiplication, division, conjugation, $z\mapsto\exp(i\operatorname{Re}z)$, and product-evaluation calls; unwritten addresses read as zero and division by zero fails. The quantification allows $s=0$, empty supports, and $m=0$, in which case $H=0$; the distinct-site hypotheses cannot be satisfied when $N=1$, and the numerical hypotheses exclude $N=0$ and $D=0$ from these guarantees.