Minor-consistency regularizer for homogeneous-space coordinates

Integrate this into a symmetry-aware coordinate encoder or graph neural network. Assume a known set of n infinitesimal generator equations at each sample x, represented by A(x) with shape n-by-m; the network q_phi(x) supplies coordinates q, and autodiff supplies the Jacobian entries r=J_q(x). The target vector b(x,q) is computed from the generator action; for a prototype, use translations and rotations on R^2, or construct A from fixed graph or Lie-algebra generator matrices. During training compute A_j, b_j, and the damped projector P_perp=I-A_j(A_j^T A_j+epsilon I)^(-1)A_j^T. Add lambda*||P_perp*b_j||^2 to the task loss. If the matrices are tiny, explicitly enumerate every (m+1)-minor of L and compare normalized squared minors against the projector implementation; this validates the adaptation of the paper's rank criterion. Do not differentiate through an unstable exact pseudoinverse: use epsilon between 1e-4 and 1e-2 or solve the normal equations with Cholesky. The paper's contribution is the rank-equivalence condition; the projector residual is the differentiable numerical estimator. First experiment with an MLP encoding points on a rotated synthetic manifold and predicting a symmetry-invariant target, comparing a standard encoder, a conventional equivariance penalty, and this compatibility loss. Measure held-out transformation error, task accuracy, Jacobian singular-value conditioning, and optimization variance. Success is lower held-out transformation error at equal task loss and fewer exploding or ill-conditioned Jacobian batches.