Small-time-aware qubitized simulator

Input. An exact Hermitian block encoding of $H\mathrm{/}\alpha$, $\alpha >0$, a rational time $t$, rational $0<ϵ<1$, and a terminating certified comparison of $\tau =\alpha \mathrm{\mid }t\mathrm{\mid }$ with $ϵ$, charged under the access model. For the $\tau >ϵ$ branch, supply all exact-block-mode inputs of Qubitized Hamiltonian simulator: the certified norm promise $\mathrm{\parallel }H{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le \alpha$, clean embedding, declared forward, inverse and controlled signal access, projector-phase circuits, coefficient preparation, SELECT, OAA reflections, phase synthesis, data access and classical arithmetic with their costs, and terminating certified scalar comparisons and rational-interval evaluations for every arithmetic test in that procedure. These capabilities must be supplied, not inferred from the block matrix or the single comparison of $\tau$ with $ϵ$.

Instructions.

1. If $\tau \le ϵ$, return the identity circuit on the system, with every work register left at zero. The query, gate, and bit costs of this branch are zero.
2. If $\tau >ϵ$, invoke Qubitized Hamiltonian simulator with $t,ϵ$, and return its circuit and error certificate. Propagate any INVALID, CERTIFICATE_ERROR, or IMPLEMENTATION_ERROR result without labeling it a successful simulation.

The comparison cost is recorded separately. The second branch is exactly the OP4 procedure; this wrapper does not modify OP4 or infer inverse, controlled, state-preparation, or phase-synthesis access.