Horizontal Symmetry Quotient Layer

Insert the layer after an encoder produces feature vectors h_M for source objects and h_N for target objects, before attention, message passing, or classification. Specify a compact Lie-group action or learn its infinitesimal generators from known transformations. For each sample x and Lie-algebra basis element ξ_r, compute v_r(x)=∂_t[g_r(t)·x] at t=0 and stack these columns as V(x) in R^(d×r). Form the differentiable damped projector P=I−V(V^TV+εI)^−1V^T and replace h by Ph. If an explicit level constraint exists, compute a soft level weight w=exp(−||μ(x)−φ||²/τ) and apply the projection mainly to samples with large w; alternatively use P globally and add an invariance loss. Pseudocode: compute h; compute V; solve (V^TV+εI)u=V^Th; set h_q=h−Vu; feed h_q downstream; add lambda_orbit||f(g·x)−f(x)||² and optionally lambda_form||V^T∇_h s||². The theorem supplies the horizontality target, while V, μ, and the level residual are computed from the action or autodiff. Test rotations and translations on synthetic point clouds, then a small image or graph model against augmentation and an equivariant baseline. Measure transformed-input accuracy, orbit sensitivity, FLOPs, and gradient variance. Success is equal or better accuracy with substantially lower orbit sensitivity and at least 20% fewer downstream feature dimensions at matched accuracy.