Symmetry-Stratified Identifiability of Strongly Tree-Child Level-2 Phylogenetic Networks: A Unified Theory for JC, K2P, and K3P

The classical four-state group-based DNA models form three symmetry strata on G = Z₂ × Z₂: Jukes–Cantor (JC), Kimura 2-parameter (K2P), and Kimura 3-parameter (K3P) tie the three nonzero Fourier characters in one, two, and three blocks, respectively. Using exact version-bound results from three companion classifications, this work gives a unified theorem for leaf-labelled binary standard semi-directed strongly tree-child networks of level at most two. On the strict principal positive-eigenvalue domain, and separately in strict edgewise continuous time, a source-relative regular full-dimensional containment from N to N′ on the same finite labelled leaf set exists exactly when the networks differ only by coherent ordinary-triangle redirection. The same condition characterizes a common regular germ full-dimensional in both images. Consequently, under any one known fixed model, a generic exact complete leaf site-pattern distribution identifies the same triangle-redirection quotient among networks in this class on the same leaf set. The synthesis also proves uniform three-leaf triangle geometry. With r ∈ {1,2,3} character-tying blocks, the normalized ambient dimension is a_r = r(r+7)/2, the Jacobian rank is d_r = 5r−1, the regular local fibre dimension under the standard six-edge triangle parameterization is r+2, and the ambient codimension is binomial(r−1,2). It interprets the K3P H₁₄ conormal through the sign representation, proves that full-dimensional containment cannot point from a richer classical model to a poorer one, establishes monotone target-model refinement for triangle-equivalent networks, and classifies all 144 ordered cross-model relations among the labelled three-leaf tree and three ordinary-sunlet orientations. Strong tree-childness is shown to be a sharp class-level boundary through model-specific weak-class examples. The reproducibility package contains the article, technical supplement, source, four exact bounded replays, deterministic build files, verification manifests, dependency bindings, and provenance. The replays check the synthesis-specific finite and symbolic calculations; the imported global JC, K2P, K3P, and tree–theta results remain bound to their separate DOI records. The paper does not claim equality of complete stochastic images, boundary coverage, recovery under an unknown substitution model, bounded finite-sample sufficiency, complete numerical-parameter identification, all binary level-2 networks, or level-3 results. Preprint; not peer reviewed by a journal. Generative-AI assistance and its verification workflow are disclosed in the article. No specific funding supported this work. The author declares no competing interests.