Coordinate Closure Recovers Nonlinear Coupling in Coarse PDE Prediction

Coarse PDE prediction can lose a nonlinear contribution even when current samples identify the structure that produces it. If a coherent coordinate has a continuation beyond the grid cutoff, projecting it before multiplication deletes cross interactions that return to represented modes. We introduce coordinate closure (C0): recover the coordinate from the current field, retain the visible residual, evaluate the finite returned-band interaction from analytic coordinate coefficients, and allocate this forcing between coordinate and residual dynamics. A Fourier identity localizes the required continuation and gives an exponential tail envelope. On Michelson–Sivashinsky fields, the complete transition beats exact-product controls on all 48 compatible outer fields. A cloned-state mask isolates a smaller 2.7–3.3% tail effect in the two clean low-band cells and its resolution and rank-zero exits. An independently implemented Benjamin–Ono transition reproduces tail sensitivity and the resolution exit under a different coordinate law. In a parameter-matched FNO, the interaction feature lowers pooled current-field forecast error by 14.9%; the coordinate decomposition and history arms locate the gain in the explicit interaction token. Resolved and coordinate-free boundary fields place the learned benefit in the active, compatible regime. The results identify coordinate closure as a conditional same-observation representation for coarse nonlinear prediction.