Exact Tree–Theta-Trinet Collisions under the Kimura 2- and 3-Parameter Models

We construct exact distributions shared by a phylogenetic tree and a binary semi-directed strict level-two theta network under the Kimura two-parameter (K2P) and three-parameter (K3P) substitution models. The compact three-leaf K2P construction lies in the strict stochastic interior and is an exact counterexample to the formal K2P lemma and the K2P part of the corresponding global corollary removed between Versions 2 and 3 of Brits, Holtgrefe, van Iersel, and Martin. Because K2P is nested in K3P, this collision alone answers both high-level Kimura tree–trinet disjointness questions posed in Version 3 negatively. We additionally give an exact K3P theta realization whose network parameter lies outside every globally character-relabelled K2P specialization, while openly identifying its shared distribution as globally character-relabelled K2P. Full-rank Jacobians show that the fixed theta images are Zariski dense in the effective K2P and K3P Fourier spaces. Locally, the collision loci have dimensions 17 and 23 and project submersively onto the tree models, with 11- and 14-dimensional fixed-output network fibers. Consequently, a relatively open dense set of nearby K3P tree distributions outside every globally character-relabelled K2P submodel also has theta realizations. These conclusions persist in edgewise strictly continuous-time chambers, where generators and rate ratios may vary by edge and no clock or global timing is imposed. Finally, a common-subtree argument replaces any chosen internal vertex of any labelled unrooted binary tree by one theta blob while preserving the full leaf-pattern distribution; hence exact strict-interior collisions occur for every number of leaves. For the compact K2P and quartic K3P witnesses, exact verifiers separately reconstruct the retained-graph Fourier coordinates and perform ordinary-state Markov pruning in every pattern. Preprint; not peer reviewed by a journal. This record archives version 1.2.7 of the full manuscript together with its exact certificates, standard-library verifiers, replay transcripts, source files, and integrity manifests. The deposited files correspond to Git commit 19cf11f65ed448cf031842b666e4bfc7e02a9ab7 and canonical archive SHA-256 7c603e15f6afb7546f618b88bbefdc52ee264b656399f6656fc742aeabb786c7. Zenodo's record metadata documents when this version was uploaded and made publicly available through the repository. No empirical data set is used. AI-assisted exploratory analysis, code assistance, adversarial checking, and editorial review are disclosed in the manuscript's “Provenance and AI-assisted methods” section.