Discovering Common Conservation Laws Across Gradient-Flow Environments

Common conservation laws remain invariant across a family of neural-network training problems. Tangency on finitely sampled fields is ambiguous: nearly parallel fields create false near-invariants, while functionally independent candidate rows can exceed the normal-space dimension available to common laws. We formulate this finite identification problem as *discover–certify–preserve*. Multi-environment neural deflation (MEND) jointly learns a vector-valued candidate from training fields. A held-out certificate then tests field tangency, pointwise rank, first-bracket compatibility, and codimension capacity on frozen candidates. Conditional matrix bounds connect residuals to normal-space error when candidate and direction singular margins, derivative accuracy, and support coverage are resolved. On shallow-network controls, the full certificate resolves all 40 predeclared boundary/null decisions on fresh tasks; matched stages show that brackets reject task-local candidates and capacity rejects the remaining overcounts. Preservation is a separate deployment question. The learned-law Conservation-Constrained Adam evaluation fails its formal utility gate because of pathwise conditioning events, whereas supplied exact attention laws are preserved at Adam-comparable loss. The evidence supports a finite, boundary-conditioned certification framework. Blind law-count recovery, nonquadratic discovery, and robust learned-law optimizer utility remain open evaluation problems.