Triangle Hypersurfaces and a Sharp Identifiability Boundary for Level-2 K3P Networks

Abstract

We classify regular full-dimensional stochastic containment among binary standard semi-directed strongly tree-child level-2 phylogenetic networks under the Kimura three-parameter (K3P) model. On the principal positive Fourier domain, a directed containment germ exists if and only if the labelled reduced trees of blobs agree and corresponding complete factors are either labelled-isomorphic or ordinarily triangle-redirected, with coherent boundary transports. The same condition is equivalent to a common full-dimensional regular germ and remains exact in strict continuous time. Thus no proper one-sided containment occurs in the strong class, and the semi-directed topology is generically identifiable and exactly reconstructible outside a proper exceptional set, modulo ordinary triangle redirection. The three ordinary K3P triangle orientations have generic normalized rank 14, share the same irreducible eight-term quartic hypersurface H₁₄ in normalized three-leaf Fourier space, and meet in a common strict continuous-time smooth rank-14 germ. The bounded residue consists of fourteen four-port directed relation orbits—nine polynomially separated and five directed-rank separated—plus two separately separated sink swaps. Exact restoration and coherent one- and two-port probes extend the bounded classification to arbitrary labelled subdivision words. Strong tree-childness is sharp against weakening to weak tree-childness. For every n ≥ 3, two weakly but not strongly tree-child networks have strict continuous-time K3P images sharing a common full-dimensional regular germ of dimension 6n − 3. This is the first Zenodo/DOI-bearing archival release of version 1.0.0 of the complete K3P level-2 classification and its reproducibility evidence. It contains the article, reader supplement, compile-complete source archives, canonical full proof archive, compact verifier, independent-referee replay package, exact manifest, checksums, citation metadata, and a file-level license notice. The deposited bytes correspond to immutable Git commit 0b76dd8e38f262ebe9ba8c1d23281853e334fef2, annotated tag k3p-level2-identifiability-v1.0.0 resolving to that commit, and full-archive SHA-256 4f84417f40b4e5ddae80b36d87dc7bb6d00389a573a58d0b82a2592c9f7c6403. The archive includes the previously completed all-producer regeneration evidence bound to unchanged mathematical inputs, together with successful exact and independently implemented replays, rigorous interval arithmetic where required, and fail-closed mutation tests. The present release changes bibliography, nonmathematical administrative/public-release prose, and release engineering only; unchanged multi-hour mathematical producers were not rerun during this dependency-scoped reseal. No empirical data set is used. Preprint; not peer reviewed by a journal. Generative-AI assistance and its verification workflow are disclosed in the article. Article, supplement, figures, documentation, and mathematical certificate data are licensed under CC BY 4.0; original verifier and build code are licensed under MIT, as mapped in LICENSES.md. No specific funding supported this work. The author declares no competing interests.