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All-level NANUQ circularity and exact displayed-split support

Holtgrefe et al. proved that the original NANUQ distance of a binary, semi-directed LSA, outer-labeled planar, galled level-two bloblet is circular decomposable, with support exactly the splits of its displayed trees. Their Section 6 asks whether this extends to multiple blobs at level two, conjectures circularity for level-three bloblets, and asks about a parametric distance-family extension. Does the circularity and exact-support statement persist at every finite level in the same structural class?

Result
Proved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Computation
Field
Mathematical phylogenetics; circular split systems
Posed by
Holtgrefe et al., Distinguishing Phylogenetic Level-2 Networks with Quartets and Inter-Taxon Quartet Distances (2025), Theorem 4.7 and Section 6; DOI 10.1007/s11538-025-01549-4.
Year posed
2025
Years open
1y
Solved
2026-09-29
Model
OpenAI Codex (GPT-6 family)
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Preprint
Significance
12 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Original NANUQ is circular decomposable at every finite level and with any number of blobs in the binary semi-directed LSA, galled, outer-labeled planar class (at least four taxa). Its positive split support equals the union of displayed-tree splits. This answers the source's level-two multi-blob and level-three circularity questions. A six-label reduction, exact finite certificate and composition identity remove the level bound. The local four-score family has exact universal anchor-positivity domain s=o=1, 1/2<=a<=1, 0<=c<=a; five labels are the sharp testing cutoff. Contiguous multicopy trees admit an extreme-copy reduction. Raw-matrix error <1/2 is optimally correctable by rounding; exact support uses a supplied correct order. Planarity cannot be dropped for exact support. Full all-level Lean verification, network uniqueness, statistical inference and unrestricted ARGs are not claimed.

What the AI did

The human collaborator selected the biological open-problem direction, pressed for an all-level theorem and stronger generalizations, and requested honest source correspondence and publication. Parallel Codex research chats developed the proofs, finite certificates, Lean components, counterexamples and exposition. Separate agents audited the structural reduction, composition, exact arithmetic, support transfer and parameter domain. These are internal project reviews, not independent human expert verification. Supporting software was executed, with a second implementation and a separate actual-tree-selection support checker.

Verification

Checked here on 30 September 2026 as far as the artefacts allow. The complete all-level theorem has a written computer-assisted proof with separate structural, finite, composition and support audits; two implementations cover 84,076 plane-tree and duplication instances, 122 distinct quartet systems and 24,667 anchor coefficients; the distance-based verifier checks 11,848,859 physical-copy quartets and a second verifier 2,525,210 globally consistent occurrence selections. The submitter states plainly that the complete theorem is computer-assisted and that the Lean endpoints are narrower than it, which is why this is Unreviewed rather than Lean-checked: the headline result is not the formalised part. The site did not rerun the verifiers. The repository's nanuq-finite workflow and its Verify workflow are green on the correct repository. The source URL was corrected: the submitted address omitted the repository's trailing dash and returned 404.

Sources

Submitted by ZestyDingo473 on

Changelog2 changes

Discussion