Nullspace-coordinate constrained operator blocks

Integrate this at the output and recurrent-update interface of a neural operator or latent PDE world model. Discretize the field on a fixed ambient grid with $n$ values. For each geometry, construct the sampled trace matrix $C$ and boundary vector $b$; compute a least-squares particular solution $u_p=C^TM^{-1}(CM^{-1}C^T)^{-1}b$ and a nullspace basis $V$ using an SVD or QR factorization of $C$, followed by weighted orthonormalization so that $V^TMV=I$. Pretrain ambient mechanism networks $F_i^\theta:\mathbb{R}^n\to\mathbb{R}^n$ on source geometries, then freeze them. At inference or fine-tuning time, encode the initial field as $z_0=V^TM(u_0-u_p)$, assemble $F(u)=\sum_i\alpha_iF_i^\theta(u)$, and update only $z$: u = up + V @ z; du = sum(alpha[i] * Fi(u) for i); z = z + dt * V.T @ M @ du. Decode with u=up+V@z at every step. Learn scalar or vector mechanism weights $\alpha_i$ in reduced coordinates if desired, but do not learn a soft boundary penalty. Compute $u_p$ and $V$ from the algebraic construction; estimate numerical conditioning, rank, and truncation tolerances empirically. First test on a small Fourier/MLP operator or graph neural operator for a 2-D diffusion or advection equation, pretraining on rectangles and evaluating on circles, annuli, and irregular masks. Compare against the same frozen blocks with unconstrained outputs plus projection, and against soft Dirichlet penalties, using equal parameters, rollout steps, and FLOPs. Pre-register these mechanism tests: (1) the maximum boundary residual $\max_k\|Cu_k-b\|_2$ must remain below $10$ times the SVD tolerance, while the penalty baseline should vary with its penalty coefficient; (2) at equal rollout length, reduced-coordinate updates should have lower long-horizon energy drift than projection-after-step updates, measured by $|\|u_k\|_M-\|u_0\|_M|/\|u_0\|_M$ when the test PDE conserves mass; (3) zeroing the nullspace adapter while retaining the same mechanism blocks must increase boundary residuals by at least two orders of magnitude. Falsify the transfer if exact residuals are not achieved, or if reduced coordinates provide no stability or unseen-geometry benefit after accounting for the QR or SVD overhead.