Level-1 semi-directed phylogenetic networks are fully identifiable under JC, K2P, and K3P at every non-boundary parameter point, and networks leave a detectable signature against trees.

Summary. The manuscript studies topological identifiability of phylogenetic networks from leaf-pattern distributions under the JC, K2P, and K3P Markov models of nucleotide substitution. The first main claim is that the semi-directed network parameter of a level-1 phylogenetic network is fully identifiable (at every point, not merely generically) on the open parameter space of probabilistic substitution rates and non-trivial mixing weights, modulo the equivalence of redirecting triangles. The second claim is that, under JC and K2P on the same parameter space, no phylogenetic network and phylogenetic tree induce the same leaf-pattern distribution unless the network is a tree possibly augmented by residual substructures called 2-blobs. The abstract further asserts broader consequences for identifiability under several coalescent-based models.

Significance. If the proofs hold as stated, the work advances algebraic phylogenetics by upgrading prior generic (measure-zero exceptional set) identifiability results for level-1 semi-directed networks under JC/K2P/K3P to full, pointwise identifiability on a biologically natural open parameter regime. The tree–network distinguishability theorem supplies a concrete signature of reticulation in leaf-pattern distributions, with a precisely delimited residual class (trees with 2-blobs). These are standard, well-posed questions in the field; the careful scoping (level-1, open rates and mixing weights, semi-directed equivalence modulo triangles) is appropriate and strengthens rather than weakens the claims. Broader consequences for coalescent-based models would further increase impact if made precise.