Strong Tree-Childness Is a Sharp Generic-Identifiability Boundary for Level-2 Jukes-Cantor Networks

Reticulate evolutionary histories encode hybridization, recombination, and lateral transfer, but such histories are scientifically recoverable from sequence data only when distinct networks induce distinguishable site-pattern distributions. We determine this information limit for binary, LSA-rootable, already-simple semi-directed strongly tree-child level-2 networks under the paper's one-step reticulation-preserving convention and the open four-state Jukes–Cantor model. A source-relative full-dimensional regular stochastic containment occurs if and only if the labelled reduced bridge trees agree and every pair of corresponding blobs is labelled-isomorphic or differs by ordinary triangle redirection. Hence no proper one-sided generic containment exists, and an exact generic distribution determines a unique topology modulo that redirection. Fourier cut ranks recover the bridge tree, while positive rank-one factorizations recover only projective local tensors modulo the full incidence-scaling group; physical bridge multipliers are not identified. A graph-to-algebra classification of cycle and theta factors, together with marginal submersions and coherent bounded probes, reconstructs arbitrary subdivisions. The boundary is sharp even without triangles. For every n ≥ 4 we construct two triangle-free networks that are weakly but not strongly tree-child, nonisomorphic, and not triangle-related, yet whose open JC images share a regular relatively open subset of full dimension 2n + 1. A second triangle-containing family exhibits a distinct pendant-transfer mechanism. All finite-atlas, class-membership, Fourier-equality, sign, and Jacobian claims are supported by exact primary and separately implemented replay certificates.