Solver-coupled learning offers a structured alternative to direct state-to-state regression for global atmospheric forecasting, but its forecast quality depends on how accurately the embedded coefficient dynamics represent unresolved forcing and coarse-grid numerical effects. We represent these unresolved effects using a bounded transport-residual closure (BTRC) embedded in an explicit sub-stepping solver. Building on the reduced horizontal transport-coefficient dynamics of PASSAT, we separate a retained spherical advection–curvature operator from aggregate terms omitted, approximated, or numerically distorted by the reduced model, including pressure-gradient, Coriolis, frictional, subgrid, vertical-coupling, and discretization contributions. A graph network estimates a raw aggregate equation discrepancy, which is mapped through a scaled hyperbolic tangent to obtain the BTRC. The resulting closure is inserted into the coefficient-update pathway and modifies the transport coefficients used by subsequent atmospheric state updates. The bound limits the direct closure-induced coefficient increment per Euler sub-step to <em>δtτ</em>, providing a numerical safeguard against unconstrained learned forcing; it is not presented as a proof of global dynamical stability. The BTRC is shared across prognostic channels as a compact directional correction, while an auxiliary node-level gated wavelet branch refines multi-resolution graph features. Experiments on coarse-resolution ERA5 data under WeatherBench-style protocols show the clearest gains for temperature and 500-hPa geopotential (Z500). Near-surface wind fields exhibit more consistent improvements in ACC than in RMSE, and the latter gains are small or not uniformly positive. A residual-free control, <span style="position: relative;"><em>R<sub>t</sub></em> ≡ 0<sup class="up" style="position: absolute; top: 0px; left: 2px;">~</sup></span>, provides targeted evidence that the learned in-solver BTRC materially contributes to the long-lead improvements within the evaluated setting. These results support bounded transport-residual learning as a promising mechanism for improving autoregressive behavior in reduced solver-coupled atmospheric models.