Batch Conservation Laws, Nonintegrability, and the Geometry of Minibatch Switching

We study conservation laws for neural network training that are specific to a fixed batch of data. Associated to a batch $X$, we consider the distribution $V_X$ spanned by the parameter gradients of the network outputs on $X$. The batch functional dimension gives a pointwise upper bound on the number of independent batch conserved quantities, while the Frobenius integrability of $V_X$ determines whether this bound can be attained locally. The non-integrability of $V_X$ also has a direct dynamical manifestation: successive gradient flows on two minibatches can generate a second-order component tangent to the batch fibre that is absent from the corresponding combined-batch flow. We compute this discrepancy explicitly in terms of Lie brackets of the sample-gradient fields. We also give computable rank criteria for detecting non-integrability and examples of batch-specific conserved quantities arising from activation patterns in ReLU networks.