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

Abstract

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.

The idea in plain English

A phylogenetic tree describes an evolutionary history of branching lineages. A network can also include reticulation events, where lineages join. The paper constructs a tree and a network with two such events that predict exactly the same probability for every observed DNA pattern under the specified Kimura substitution models.

The equality is exact and uses strictly positive transition probabilities and nonzero inheritance weights. For these constructed histories, even an unlimited amount of sequence data from the model cannot distinguish the tree from the network.

These are families of exceptional coincidences, rather than a claim that most networks look like trees. Nearby tree distributions also have network realizations, but a generic choice of parameters on the network is not tree-equivalent.

The three-leaf construction can be inserted at one chosen internal vertex of any binary tree topology, giving examples with any number of leaves. The stronger continuous-time version allows each edge to have its own rates and duration; it does not impose a common rate matrix or molecular clock.

The precise result

An explicit three-leaf K2P tree and binary semi-directed strict level-two theta network have identical values for all 64 site-pattern probabilities. All edge transition entries are positive, both inheritance probabilities are 1/2, and the exact minimum pattern probability is 1188799/79626240.

This disproves the arbitrary-level K2P disjointness statement in Lemma 5.6 and the K2P part of Corollary 5.8 of Brits and collaborators’ Version 2, which were removed in Version 3. Since K2P is a submodel of K3P, the same example gives a negative answer to both high-level Kimura trinet-disjointness questions stated in Version 3.

A separate quartic K3P network parameter lies outside every globally character-relabelled K2P specialization, although its shared tree distribution itself is relabelled K2P. Local full-rank arguments additionally produce nearby shared distributions outside all those observable K2P symmetry strata.

The fixed theta maps have ambient ranks 9 under K2P and 15 under K3P. Near the witnesses, the tree-collision parameter loci have dimensions 17 and 23, and the network realizations of a fixed nearby tree distribution have dimensions 11 and 14, respectively.

Strictly positive-rate continuous-time versions exist edge by edge. A common-subtree argument gives parameter choices with the same full leaf-pattern distribution after replacing one chosen internal vertex of any labelled unrooted binary tree topology by one theta blob.

Why the result holds

The network has four retained-parent choices. The proof derives each corresponding Fourier monomial from the actual graph, including branches with no observed descendants, and averages them with the inheritance weights.

Special algebraic edge parameters make that averaged expression factor exactly into the three edge contributions of a tree. Exact ordinary-state Markov pruning provides a separate way to compare every pattern probability.

The quartic derivation uses the equal-coordinate comparison factor displayed in equation (17).

Nonzero Jacobian minors show that the unrestricted theta map is locally onto its effective Fourier space. The preimage and constant-rank theorems then give the collision dimensions and nearby tree coverage; an implicit-function argument moves into strict continuous-time chambers.

Attaching the same three conditional Markov subtrees applies the same linear map to both equal interface distributions. This preserves equality on any number of leaves after one theta insertion.

Scope and limitations

The theorem concerns a displayed-tree mixture substitution model with uniform root distribution and independent reticulation choices. It is not a theorem about every model of evolutionary history or the network multispecies coalescent.

The failure is pointwise distinguishability. Tree-equivalent theta parameters form a proper lower-dimensional subset, so the paper does not prove generic tree equivalence or generic nonidentifiability of all network topologies.

The theta admits no tree-child rooting. These examples do not contradict results restricted to tree-child or strongly tree-child networks, the retained level-one Kimura results, or the Jukes–Cantor tree–network theorem.

A K3P network parameter with three distinct edge eigenvalues need not produce an observably genuine K3P distribution. The closed-form quartic example has a relabelled K2P output; observable symmetry breaking is supplied by a separate local argument.

Edgewise continuous time allows different generators, rate ratios and durations on different edges. No common generator, molecular clock, global temporal compatibility or standard common-rate-matrix GTR conclusion is established.

The all-leaf theorem gives suitable parameter choices on every binary tree topology and one chosen vertex. It does not cover every preassigned tree distribution, independently composable replacements at several vertices, or genuine four-attachment blobs.

Zariski density excludes additional polynomial equalities only in the effective Fourier spaces. It does not fill the probability simplex, remove positivity inequalities or eliminate tree-model invariants.

Finite exact calculations verify the explicit witnesses and algebraic inputs. Local family and all-leaf conclusions also require the stated analytic and common-kernel proofs; stored verifier reports and AI checks are not formal proof certification or journal peer review.

Status and AI use

AI-assisted analysis was used for exploratory symbolic and numerical search, code assistance, adversarial checking and editorial review. The manuscript supplies exact constructions, source code and analytic arguments. Internal AI checks and verifier transcripts do not constitute conventional journal peer review.

Zenodo’s version field reads 1.2.6, while the deposited PDF and record description identify the manuscript as 1.2.7. The record was published on 28 August 2026; the title page prints August 2026. These original labels are preserved.

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Collaborate on this result

I welcome substantive collaboration on independent verification of the exact constructions and local geometry, comparison with other identifiability results, and preparation for journal review. Expertise in algebraic statistics, phylogenetic substitution models or exact symbolic computation would be especially useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Exact Tree–Theta-Trinet Collisions under the Kimura 2- and 3-Parameter Models (Zenodo version 1.2.6; deposited manuscript 1.2.7) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22136869

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Background and prior work

• Brits, Holtgrefe, van Iersel and Martin, On Tree–Network Distinguishability and Full Identifiability of Phylogenetic Networks, Version 2: The earlier Lemma 5.6 and K2P part of Corollary 5.8 claimed the general disjointness contradicted by the compact K2P example.
• Brits, Holtgrefe, van Iersel and Martin, Version 3: Removes those formal K2P claims, explains the leaf-order obstruction and asks about high-level K2P/K3P trinet distinguishability. Its level-one and Jukes–Cantor results remain distinct.
• Ardiyansyah, Distinguishing Level-2 Phylogenetic Networks Using Phylogenetic Invariants: Studies restricted nice level-two networks through variety noncontainment. Lemma 5.1 excludes all simple nice strict level-two trinets, so that setting does not cover this theta.
• Englander and collaborators, Identifiability of Phylogenetic Level-2 Networks under the Jukes–Cantor Model: Theorem 3.2 in the July 2026 version proves generic identifiability for binary triangle-free strongly tree-child level-two networks under JC, with different topology and model hypotheses.