arXiv:2610.15220v2 Announce Type: replace Abstract: We analyse how errors in independently approximated adjoint waves propagate into the Hilbert Uniqueness Method (HUM) reconstruction of the minimum-$L^2$-norm interior control of a variable-coefficient wave equation. The central result is an identity: the error of the control assembled exactly from any approximate adjoints splits orthogonally into the distance of the minimum-norm control to their observation span and a moment-inconsistency term generated by their interior wave residuals. From it we derive a posteriori bounds for the returned control, including assembly and solve errors, and a continuous accept-or-refine test for the network directions returned by training; under continuous observability, a fully discrete estimate separates truncation, adjoint and assembly errors. The adjoints are computed here by adaptive Galerkin neural networks, but the bounds apply to any computed adjoints. In one-dimensional benchmarks with known minimum-norm controls, all inputs are enclosed by ball arithmetic. Without truncation the residual bounds exceed the true error by factors of 9 to 115; with sixteen adjoints outside the trial class, a truncated terminal space and all inputs computed from the target data, by about 1.08 to 6.50. In two and three dimensions, including a time-dependent coefficient, the neural controls closely reproduce reference controls of the same truncation, and at equal neuron counts trained hidden features reduce the error more than tenfold relative to random features. Classical solvers are much cheaper in all tested cases, and certification is confined to the one-dimensional benchmarks.