A constructive characterization of exactly trackable heat fluxes

We characterize the normal heat fluxes that can be produced on an observation patch by square-integrable Dirichlet data on a disjoint control patch. The domain is any bounded connected smooth Euclidean domain in dimension at least two. The characterization consists of explicit conditions on the target: causal smoothness, weighted Taylor coefficients computed from a Gaussian surface potential, agreement of the resulting series, and volume, boundary-trace, support and observation compatibility tests. No control or caloric extension is an unknown in these tests. The reconstructed parallel boundary values define the exact causal control norm. In this norm the squared integral of the separated heat-flux kernel gives the exact norm of point evaluation at each positive time. Every fixed target is little-o of this scale at the initial time, and no prescribed further decay factor holds for all targets.