Abstract

We specify a Euclidean-calibrated family of finite quantum circuits and prove one operational boundary-approximation theorem. At refinement $j$, the spatial mesh is $\ell /j^2$, the permitted interaction radius is $\ell /j$, and one circuit layer takes $\ell /(cj)$, where $c>0$ is supplied. Every prescribed future-null displacement is approximated by a payload-independent sequence of local SWAP operations and waits. The spatial error is $O(j^{-2})$, the nonnegative arrival delay is $O(j^{-1})$, and the ideal logical channel is exactly the identity on a fixed finite-dimensional payload, including every untouched reference. All bounds are uniform on bounded-distance target families. The construction uses $O(j)$ ticks and $O(j)$ initialized registers; a matching leading-order register lower bound holds in the declared static-site model. Complete implementation errors are charged by channel telescoping. In particular, per-gate half-diamond error $o(j^{-1})$ is sufficient when preparations, spectators, waits, and output identifications are exact. The result concerns operational approximation of a prescribed causal boundary. It does not derive the metric, the speed, Lorentz covariance, a continuum field theory, or gravitational dynamics.