Fixed-feature invariant output layer

Integrate this at the output layer of a compact MLP used as a recurrent state updater, neural operator, or adapter, rather than replacing a full transformer. Inputs are sample coordinates, graph nodes, physical states, or latent positions $x_i$. Choose $N$ fixed hidden features $\phi_m(x)=\tanh(w_m^Tx+b_m)$ and construct $\Phi_{im}=\phi_m(x_i)$ once. Choose a known linear invariant represented by aggregation or quadrature weights $q\in\mathbb{R}^K$; for mass conservation, set $m=q^Ty_{\mathrm{initial}}$. Precompute $c=\Phi^Tq$, factor $\Phi^T\Phi+\lambda I$, and optionally cache its factorization. At each update, form the target $y_t$ from the baseline state update, solve $\hat\alpha=(\Phi^T\Phi+\lambda I)^{-1}\Phi^Ty_t$, then execute err=m-dot(c,hat_alpha); alpha=hat_alpha+c*err/(dot(c,c)+1e-12). Return $\Phi\alpha$ as the output. For multiple invariants, stack rows into $C\in\mathbb{R}^{r\times N}$ and use alpha=hat_alpha+C.T@solve([email protected],b-C@hat_alpha). Compute feature conditioning empirically from the singular values of $\Phi$; compute $c$ from known quadrature weights rather than learning it. First test a 2D Allen-Cahn or advection rollout and a graph diffusion task with a 2-layer, 256-feature baseline trained by Adam. Compare against full-network SGD and unconstrained random features at equal parameter count and rollout horizon. Measure wall-clock time, peak memory, task MSE, rollout stability, and invariant drift. Success means at least 3x faster output updates, invariant error near numerical precision, and no more than 5% accuracy loss. Also test transfer by reusing the frozen features on a second domain or initial condition without retraining hidden layers.