NAILONG: Freeze the Shared Lie Algebra of Generators

Two operator families can share a continuous symmetry and still differ in the residual that breaks it. The method extracts the shared Lie algebra of infinitesimal generators and freezes it on the target. A jointly equivariant predictor is capped by the group average; a trunk plus a free residual is already outside that class, so the trunk is a comparison, not a corollary of the ceiling. With the shared generators known in the control chart, freezing them and fitting the residual drops matched-design target OLS risk from $(s+r)n^2{\sigma }^2/N$ to $r{n}^2{\sigma }^2/N$. The algebra can be recovered from a finite source by identifying the span, aligning it to the target chart, and freezing it. On noisy generator OLS that procedure approaches the oracle residual rate at $\hat{\kappa }=3.72\pm 0.31$, against oracle $3.03$ and scratch $6.01$. On a separate labeled spherical field probe, generator regression followed by a freeze of $\Phi$ reaches residual-of-$\Phi$ $R^2=0.87$ on the lab axis; a jointly equivariant residual stops at the group-average fraction $0.034$.