First-order term-list simulator

Input: A finite term-list access input with $H={\sum }_{j=1}^m{H}_j$, a finite time code $t$ in the expression language of Term-list Hamiltonian access, a rational tolerance $0<ϵ<1$, a certified rational $C_2\ge 0$ satisfying $C_2\ge {\sum }_{1\le j<\mathrm{\ell }\le m}\mathrm{\parallel }[H_j,H_{\mathrm{\ell }}]\mathrm{\parallel }$, and a supplied certified rational upper bound $R\ge 0$ satisfying $R\ge t^2{C}_2\mathrm{/}ϵ$. For $t\mathrm{\ne }0$, put $r=\max \{1,\lceil R\rceil \}$ and $h=t\mathrm{/}r$; the access declarations must supply each term call at this exact signed code $h$, with its implementation and costs. A call may be directly supported at $h$ or supplied by separately declared inverse or time-reversed access, with that route's own costs. When $t\mathrm{\ne }0$, every approximate call must have a certified error, and the sum of those errors over all $rm$ calls must be at most $ϵ\mathrm{/}2$. The time-code and call conventions are those of Term-list Hamiltonian access.

Output: The composed unitary circuit for the declared product order, together with its query, gate, ancilla, coefficient-bit, description-bit, and classical-generation cost record.

Instructions:

1. Compare the exact value of the supplied time code with $0$. If it is $0$, return the identity circuit.
2. Otherwise set $r=\max \{1,\lceil R\rceil \}$ using rational arithmetic and set $h=t\mathrm{/}r$ as a supported finite expression code.
3. For each of exactly $r$ repetitions, execute the declared term calls chronologically in order $m,m-1,\dots ,1$. By Denotational semantics of a finite layered circuit, later executed gates multiply on the left. The ideal exact-call operator for one repetition is therefore $e^{-ih{H}_1}\cdots e^{-ih{H}_m}$ of First-order Lie product formula; with supplied implementations the actual operator is $V_1(h)\cdots V_m(h)$, with the declared call errors accumulated separately. If the call at $h$ is supplied through inverse or time-reversed access, use that separately declared route and its cost and error record; do not infer it from forward access alone.
4. Keep circuit-generation, coefficient, and all oracle costs in their separate declared coordinates. When $t\mathrm{\ne }0$ and calls are approximate, retain their per-call error records; admissibility requires their sum over all $rm$ calls to be at most $ϵ\mathrm{/}2$.

The procedure makes no claim that the required term calls or their uniform generator exist for an arbitrary matrix.