Matroid-selected circular latent coordinates

Integrate this at the representation bottleneck of a VAE, world model, or supervised encoder. Assume each training example has a dictionary of \(m\) candidate angle observables \(\theta_j(x)\in\mathbb R/\mathbb Z\), plus a neighborhood graph \(G=(V,E)\) built from the input samples. First compute wrapped edge differences \(a_j(i,k)=\operatorname{wrap}_{[-1/2,1/2]}(\theta_j(x_k)-\theta_j(x_i))\) on graph edges. Estimate the candidate Gram matrix with \(G_{jk}=\sum_{(i,k)\in E}w_{ik}a_j(i,k)a_k(i,k)\). Obtain an estimated persistent cohomology basis, or use a precomputed target matrix \(Q\in\mathbb R^{m\times r}\) whose columns span the detected \(H^1\) directions; set \(v_j=Q_{j,:}\) and use \(M=Q^TGQ\). Sort candidates by \(E_j=v_j^TMv_j\), greedily retain a candidate only if the smallest singular value after appending it exceeds a tolerance such as \(10^{-5}\). For every retained coordinate, inject \([\cos(2\pi\theta_j(x)),\sin(2\pi\theta_j(x))]\) into the latent vector. During training, add a prediction or reconstruction loss on these pairs, and optionally penalize unexplained persistent directions using \(\|R\|_F^2\), where \(R=I-V(V^TMV)^\dagger V^TM\). The mathematical quantities are the wrapped cochains, Gram matrix, energies, rank test, and projection residual; graph bandwidth, edge weights, rank tolerance, and loss coefficient are empirical hyperparameters. The first experiment should use a torus or synthetic pendulum dataset with known angles, a small VAE baseline with two unconstrained latent scalars, and the selected circular latents. Measure reconstruction error, angle decoding error, latent traversability, and the number of latent dimensions needed for the same error. Success is lower-dimensional representation with equal or better reconstruction and substantially higher correlation to the known physical angles, while the residual should correctly flag a dictionary that omits one torus angle.