Strong Tree-Childness Is a Sharp Generic-Identifiability Boundary for Level-2 Jukes-Cantor Networks

Abstract

Reticulate evolutionary histories encode hybridization, recombination, and lateral transfer, but such histories are scientifically recoverable from sequence data only when distinct networks induce distinguishable site-pattern distributions. We determine this information limit for binary, LSA-rootable, already-simple semi-directed strongly tree-child level-2 networks under the fixed one-step reticulation-preserving semi-directed convention used here and the open four-state Jukes–Cantor model. A source-relative full-dimensional regular stochastic containment occurs if and only if the labelled reduced bridge trees agree and every pair of corresponding blobs is labelled-isomorphic or differs by ordinary triangle redirection. Hence no proper one-sided generic containment exists, and an exact generic distribution determines a unique topology modulo that redirection. Fourier cut ranks recover the bridge tree, while positive rank-one factorizations recover only projective local tensors modulo the full incidence-scaling group; physical bridge multipliers are not identified. A graph-to-algebra classification of cycle and theta factors, together with marginal submersions and coherent bounded probes, reconstructs arbitrary subdivisions.

The boundary is sharp even without triangles. For every n ≥ 4 we construct two triangle-free networks that are weakly but not strongly tree-child, nonisomorphic, and not triangle-related, yet whose open JC images share a regular relatively open subset of full dimension 2n + 1. A second triangle-containing family exhibits a distinct pendant-transfer mechanism. All finite-atlas, class-membership, Fourier-equality, sign, and Jacobian claims are supported by exact primary and separately implemented replay certificates.

The idea in plain English

An evolutionary network allows two ancestral lineages to merge, as they can during hybridization. Even with unlimited sequence data, two different networks might predict exactly the same distribution of DNA patterns. This paper studies when that ambiguity remains under the Jukes–Cantor substitution model.

The distinction is between a network that has a tree-child rooting and one whose every admissible rooting is tree-child. In a tree-child rooting, each internal vertex has at least one child that is not a merger. Under the paper’s precise graph convention, the stronger condition permits generic recovery of level-2 networks, apart from changing the orientation inside an ordinary triangle.

The surprising part is the boundary: allowing just the weaker rooting condition admits triangle-free examples that remain ambiguous on a full-dimensional open region of distributions. These are not isolated coincidences or cases where a parameter is forced to zero. The construction works for every number of leaves from four onward.

Earlier work already establishes generic recovery for triangle-free networks in the strong class and explains local triangle ambiguity. This paper extends the proposed classification to triangles at level two and supplies explicit weak-class counterexamples. It addresses an exact mathematical model; it is not a test of a particular biological history.

The precise result

The positive theorem concerns binary, LSA-rootable, already-simple semi-directed networks of level at most two. Reticulation arrowheads are preserved under the specified one-step root suppression, and every admissible rooting must be tree-child. The model uses a uniform root distribution and a displayed-tree mixture. Each edge has a Jukes–Cantor Fourier multiplier strictly between zero and one, and every inheritance probability is also strictly between zero and one.

Within that class, a regular neighborhood of a source model’s full dimension lies in another network’s stochastic image exactly when their labelled reduced bridge trees agree and corresponding blobs are labelled-isomorphic or differ by ordinary triangle redirection. Thus the directed full-dimensional containment relation has no proper one-sided cases.

Outside a proper algebraic exceptional set, the exact site-pattern distribution determines the semi-directed topology within this class, modulo ordinary triangle redirection. The bridge cuts themselves have a pointwise criterion throughout the strictly open parameter domain: their Fourier flattenings have rank at most four, while non-bridge splits have larger rank.

For every n ≥ 4, the paper constructs two labelled triangle-free level-2 networks that are weakly but not strongly tree-child, are nonisomorphic and not triangle-related, and share a regular relatively open region of JC distributions of dimension 2n + 1. A separate triangle-containing family shares such a region of dimension 2n.

The manuscript gives written reductions and a computer-assisted finite classification. The reader supplement explains the proof dependencies and explicit witnesses. The small source and entrypoint ZIPs accompany the paper; the full exact certificate and replay package is archived separately at DOI 10.5281/zenodo.22064121.

Why the result holds

First, Fourier rank conditions identify the bridges that connect the network’s reticulate components. Positive rank-one factorizations then separate the local component tensors, while keeping the full freedom to rescale their incidences. Those rescalings must be tracked because the physical bridge multipliers are not uniquely determined.

Each level-2 component reduces to a cycle or a theta shape with labelled attachment ports. The finite exact classification records the graph, all physical port arms, incoming roles and relevant marginal completions. Bounded probes recover the order of arbitrary subdivisions and test local containment modulo the allowed positive scaling.

The ordinary triangle ambiguity is realized on a regular physical open region. Compatible positive scalings then join local matches across the recovered bridge tree. Conversely, nonmatching local structures must be separated by the finite classification. This is the computational part on which the global theorem depends.

For the negative examples, explicit rational or algebraic parameter choices produce identical complete Fourier tensors on two different labelled networks. Rank witnesses and local parameter arguments extend those equalities to regular open regions. Attaching the same cherry at corresponding leaves extends the construction to larger numbers of taxa.

Scope and limitations

Generic topology recovery allows a proper algebraic exceptional set. The pointwise bridge-cut theorem does not turn the entire topology theorem into an everywhere-identifiable result.

The comparison class matters. The theorem does not classify all level-2 networks, arbitrary rooted refinements, or alternative semi-directed cleanup conventions. The weak-class counterexamples show that the strong hypothesis cannot simply be replaced by the whole weak class; they do not show that every weakly tree-child network is ambiguous.

Triangle redirection is a specific change of the reticulation placement and its arrowheads within a triangle, with the labelled underlying graph and outside attachments fixed. The theorem concerns full-dimensional regular overlap, not equality of entire stochastic model images. A particular distribution can distinguish triangle orientations even when the models overlap elsewhere.

Root location and all numerical parameters are not recovered. Positive factorization determines local tensors only up to the full incidence-scaling action, and physical bridge multipliers remain unidentifiable by that factorization.

The strictly open four-state JC model excludes zero or infinite edge lengths and inheritance probabilities zero or one. The result does not transfer automatically to K2P, K3P, coalescent models, rate mixtures, or sampled data with estimation error.

The constructive discussion assumes an exact full distribution with 4^n entries. Its bounded local probes do not provide a polynomial-time-in-n method for finite DNA samples. The bounded primary-literature comparison also does not establish worldwide priority.

Status and AI use

AI tools assisted the mathematics, code, drafting and internal review. The separately implemented replay is a second code implementation, not independent human peer review. The author is responsible for the work.

Zenodo lists version 1.1.7 with a publication date of August 23, 2026. The manuscript and reader supplement print only August 2026, without an exact day or version number.

Read the paper

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Supporting materials

• Full certificate and reproducibility record, version 1.1.7 · DOI 10.5281/zenodo.22064121
• Full certificate package (tar.gz, 94 MB) · DOI 10.5281/zenodo.22064121

Collaborate on this result

I welcome substantive collaboration on independent review of the graph conventions, the exact stochastic containment argument, or the finite certificate classification, as well as preparation for journal submission. Expertise in phylogenetic networks, algebraic statistics, and computer-assisted proof would be especially useful.

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

Cite this paper

Alec Kriebel. (2026). Strong Tree-Childness Is a Sharp Generic-Identifiability Boundary for Level-2 Jukes-Cantor Networks (Zenodo version 1.1.7) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22089373

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Background and prior work

• Englander et al., Identifiability of Phylogenetic Level-2 Networks under the Jukes–Cantor Model, v4 (2026): Theorem 3.2 establishes generic JC distinguishability for binary triangle-free strongly tree-child level-2 networks. Theorem 3.1 gives the level-1 triangle quotient; Corollary 2.12 already recovers blob-tree structure from exact JC data.
• Currie et al., Semialgebraic Conditions for Identifying Triangles in Phylogenetic Networks, v1 (2026): Theorem 3.2 proves that pairs of differently oriented triangle models have both full-dimensional overlap and full-dimensional differences. Local triangle ambiguity is prior work, and it does not mean whole-image equality.
• Brits et al., On Tree–Network Distinguishability and Full Identifiability of Phylogenetic Networks, v2 (2026): Theorem 4.9 supplies the pointwise level-1 parallel modulo triangle reticulation placement, including JC. This is the manuscript’s cited July 29 edition.
• Holtgrefe et al., Characterizing Semi-Directed Phylogenetic Networks and Their Multi-Rootable Variants (2026): Theorem 5 gives the no-omnian characterization of strong tree-childness. The present paper specializes this established structural idea to its binary, LSA-rootable fixed-graph convention.
• Huber et al., When Are Quarnets Sufficient to Reconstruct Semi-Directed Phylogenetic Networks? (2025): Lemma 4.2 and Figure 8 give the two underlying semi-directed level-2 generator shapes. The four directed theta cases here additionally record the incoming source position and ports.
• Ardiyansyah, Distinguishing Level-2 Phylogenetic Networks Using Phylogenetic Invariants, v1 (2021): Earlier invariant-based generic distinguishability for specified nice level-2 classes provides context; its class restrictions and algebraic statements differ from the open-image containment classification here.