Exact linear reduction for rational dynamical systems

The paper proves correctness of Algorithm 1 via the invariant-subspace characterization of lumpings and the fact that sampled Jacobians spanning the full space imply invariance under all Jacobians; minimality follows from Step 2’s construction. The candidate solution reproduces the same structure: (i) an equivalence between lumpings and J(x)-invariant row spaces, (ii) a lemma that evaluations span the same space as coefficient matrices, (iii) conditioning on the event that the samples span, and (iv) an explicit minimality argument among constrained lumpings. This matches Proposition 7 and Lemmas 12 and 14 in the paper and fills in the same logical chain.