TT discarded weights bound the post-sweep DMRG energy increase

There are finitely many local updates in the sweep. Each exact local solve preserves normalization and weakly decreases energy by [F2]; each intervening exact SVD split preserves the represented tensor and therefore its norm and energy by [F1]. Induction over this finite sequence shows that every intermediate state stays normalized and its energy does not exceed the preceding one. Hence ${\mathcal{E}}_H({\psi }_{\ast })\le {\mathcal{E}}_H({\psi }_0)$.

The tails ${\delta }_k$ are those recorded by [F3]. By [F4], $e:=\mathrm{\parallel }{\psi }_{\ast }-\tilde{\psi }{\mathrm{\parallel }}_F\le \tau$. The triangle inequality [F6] gives $1=\mathrm{\parallel }{\psi }_{\ast }{\mathrm{\parallel }}_F\le e+\mathrm{\parallel }\tilde{\psi }{\mathrm{\parallel }}_F$, so $\mathrm{\parallel }\tilde{\psi }{\mathrm{\parallel }}_F\ge 1-e\ge 1-\tau >0$ and $\varphi$ is defined.

By [F7], $\mathrm{\mid }1-\mathrm{\parallel }\tilde{\psi }{\mathrm{\parallel }}_F\mathrm{\mid }\le e$. Also, homogeneity [F6] and $\varphi =\tilde{\psi }\mathrm{/}\mathrm{\parallel }\tilde{\psi }{\mathrm{\parallel }}_F$ give $\mathrm{\parallel }\varphi -\tilde{\psi }{\mathrm{\parallel }}_F=\mathrm{\mid }1-\mathrm{\parallel }\tilde{\psi }{\mathrm{\parallel }}_F\mathrm{\mid }$. Therefore $\mathrm{\parallel }\varphi -{\psi }_{\ast }{\mathrm{\parallel }}_F\le \mathrm{\parallel }\varphi -\tilde{\psi }{\mathrm{\parallel }}_F+\mathrm{\parallel }\tilde{\psi }-{\psi }_{\ast }{\mathrm{\parallel }}_F\le 2e\le 2\tau \mathrm{.}$

Apply [F5] to the unit states $\varphi$ and ${\psi }_{\ast }$ to get ${\mathcal{E}}_H(\varphi )\le {\mathcal{E}}_H({\psi }_{\ast })+4\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\tau$. Combine this with step 1.1 to obtain the claimed bound relative to ${\psi }_0$. The same inequality gives the asserted upper bound on any post-rounding increase.