Lemma: Exact quadratic reduction

At finite $R$,

with no constant term, and

For bounded $T,T^{\prime }$ and pair tensors $K,K^{\prime }$,

Proof. The $(-+)$ and $(+-)$ terms are adjoint cross-species number-preserving terms. They give $\mathrm{d}\Gamma (T_R)$; moving the species past each other produces no scalar. For an elementary $P_R=g\otimes v$, $\frac{1}{2}{a}^{†}(K_R)={a_g}^{†}(g){a_{\varphi }}^{†}(v)$. Linearity and Hilbert tensor approximation prove the identity. The two species slots are orthogonal, giving the pair norm.

On the $n$-particle sector the number term is bounded by $n\Vert T\Vert$; pair creation and annihilation have bounds $\sqrt{(n+1)(n+2)}\Vert K\Vert$ and $\sqrt{n(n-1)}\Vert K\Vert$. The triangle inequality gives (eq:graph2). After compression, each of creation and annihilation has norm at most $\sqrt{M(M-1)}\Vert K\Vert$; the half-sum has the same upper bound. This gives (eq:compress2).