Collision-free energy-kernel prototype flow

Integrate the flow into a prototype bank, VQ-style codebook, or latent-particle module rather than directly into network weights. Let an encoder f_theta produce a minibatch Y={y_b=f_theta(s_b)} in R^d, and maintain N trainable prototype vectors X={x_i}. At each training step, compute the vector field in the final formula. The first sum is over all j not equal to i, explicitly excluding self-interaction; the second sum treats the current feature minibatch as the empirical target mu. Update prototypes with Euler steps, either once per network step or with K=2 to 10 inner prototype steps. Use epsilon between 10^-3 and 10^-2 times the median pairwise feature distance to prevent exploding gradients for q<1, and optionally clip each prototype update to a fixed norm. Stop gradients through y_b for a codebook-only experiment; in a joint experiment, permit encoder gradients through the assignment or reconstruction loss but not through the prototype-flow update. Normalize features before applying the flow. The paper-derived components are the negative-distance kernel, the diagonal-free interaction, and the repulsive-attractive vector field. Estimate the target expectation with the minibatch and do not assume a theoretical global convergence rate. First test on a CIFAR-10 or MNIST encoder with N=64 prototypes. Compare vanilla EMA or k-means codebook updates, standard MMD-gradient updates, and this flow at matched update FLOPs. Measure dead-prototype fraction, minimum pairwise distance, feature reconstruction error, downstream linear-probe accuracy, and MMD to held-out features. Success means lower collapse and better feature coverage at equal prototype count without unstable norms or slower encoder loss descent.