BQP to PP and PSPACE proof obligations in the exact QC14 basis

The established containments $\mathrm{B}\mathrm{Q}\mathrm{P}\subseteq \mathrm{P}\mathrm{P}$ and $\mathrm{B}\mathrm{Q}\mathrm{P}\subseteq \mathrm{P}\mathrm{S}\mathrm{P}\mathrm{A}\mathrm{C}\mathrm{E}$ remain proof obligations for the exact QC14 gate alphabet $H,T,T^{†},\mathrm{C}\mathrm{N}\mathrm{O}\mathrm{T}$, whose entries lie in $\mathbb{Q}({\zeta }_8)=\mathbb{Q}(i,\sqrt{2})$ (BQP for promise problems in the fixed QC14 model, Clifford+T gate library and exact matrices). The PP route must encode amplitudes and acceptance weights over this number field, prove the required GapP closure, and compare against the strict $1\mathrm{/}2$ threshold. The BQP thresholds $2\mathrm{/}3$ and $1\mathrm{/}3$ differ by $1\mathrm{/}3$; each is at distance $1\mathrm{/}6$ from $1\mathrm{/}2$ (equivalently, $2p-1$ has margin at least $1\mathrm{/}3$). A rational path count cannot simply replace the exact $\mathbb{Q}({\zeta }_8)$ computation, since the probabilities can contain $\sqrt{2}$. The PSPACE route needs exact-field or certified-precision recursive evaluation with explicit bit-length and space bounds for each arithmetic and threshold step. The source containment arguments are context, not local proof suppliers; no containment item is claimed here.