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
Abstract
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<s<1, s²<g<1. For every n≥3, two weakly but not strongly tree-child level-2 networks have continuous-time K2P images sharing a regular germ of dimension 4n−3, proving sharpness of strong tree-childness. This record is the complete v1.0.5-r1 priority and reproducibility package: the 26-page article, 24-page reader supplement, compile-complete five-file source archive, deterministic 495-member referee/verifier archive, external archive-qualification report, checksum sidecars, and dual-license notice. The manuscript source is v1.0.5; revision r1 repairs only an auxiliary probe-current semantic binding and changes neither the theorem, manuscript, PDFs, nor frozen classification. The clean verifier replay passed 41/41 layers, and the focused semantic mutation suite rejected 20/20 attacks. Exact source bindings: package tag k2p-same-referee-package-v1.0.5-r1; annotated tag object 6c9c89d38f4f4cdc9c328d8bb1237458c617136d; commit e2f6e32e6fe885e90c8e83a8c5b00785e663a4ae; referee archive SHA-256 4564cd1f8cd95f670a2e0d9619babaf3c343762cfd8ceeb190cd17df72802889. Article, supplement, and certificate data are licensed under CC BY 4.0; verifier and build code are licensed under MIT. No specific funding supported this work. The author declares no competing interests. Generative-AI assistance and its verification workflow are disclosed in the article. No mixed-sign K2P classification is claimed.r
