Generic Identifiability and Directed Containment for Strongly Tree-Child Level-2 Networks under the Kimura Two-Parameter Model: The Principal Positive Domain and Strict Continuous Time

We classify regular full-dimensional stochastic containment among binary standard semi-directed strongly tree-child level-2 phylogenetic networks under the Kimura two-parameter (K2P) model. On the principal positive Fourier domain D₊ = {(s,g): 02s−1}, a directed containment germ exists if and only if the two labelled networks are isomorphic after independently redirecting ordinary three-cycle factors. The same condition is equivalent to a common full-dimensional regular germ; in particular, no proper one-sided containment occurs. It follows that the semi-directed topology is generically identifiable modulo ordinary triangle redirection, and that its structural triangle class is exactly reconstructible away from a proper algebraic exceptional set. The proof combines displayed-quartet inequalities and exact whole-map identities, an exact two-sector bridge-fibre theorem, physical marginal submersions, localization, and a bounded graph-to-algebra classification of cycle and theta factors. The bounded classification is computer-assisted: every directed primitive relation, rank exclusion, restoration parent, transport, and one-/two-port probe is represented by an exact certificate with independent replay and mutation evidence. The classification transfers to the strict continuous-time domain 0