---
id: "2026-10-06-finite-index-over-treeable-jkl-problem"
url: "https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/"
as_of: "2026-10-09T19:24:00+02:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 1
editor: Adam Bicz
human_review: null
version: null
---

As of: 2026-10-09 19:24 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz.
Canonical page: https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/

# Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation'

On 6 Oct 2026 Ruiyuan Chen (Warsaw), Anush Tserunyan (McGill) and Robin Tucker-Drob (Florida) posted arXiv 2610.08986, dedicated to Alexander Kechris's 80th birthday. They prove that every countable Borel equivalence relation containing a finite-index Borel treeable subequivalence relation is Borel treeable, answering Jackson–Kechris–Louveau's Open Problem 6.4(B) (2002) in its strongest, purely Borel form. As a corollary, strong treeability of locally compact groups passes to finite-index overgroups (new even for countable discrete groups). Their "Declaration of AI use" says they consulted ChatGPT 6 Astra Ultra after conceiving the approach and that "GPT provided a crucial observation which ensures the coherence of the forest obtained at stage n + 1 with that obtained at stage n".

## Key facts
- Theorem 1.2: if F ⊆ E are countable Borel equivalence relations, F has finite index in E and F is Borel treeable, then E is Borel treeable. Previously open even for index 2 in the probability-measure-preserving setting (treeing only on a conull set)
- Method: a median-subalgebra filtration on products of trees, extending the median-graph treeing algorithm of Chen–Poulin–Tao–Tserunyan (2025) to well-behaved increasing unions; answers Question 1.1, how to 'canonically combine' finitely many trees on the same vertex set into one tree-like structure
- Corollary 1.3: a locally compact second countable group with a strongly treeable finite-index open subgroup is strongly treeable (also for measure and Borel strong treeability); for unimodular groups this implies fixed price
- AI disclosure: 'We conceived of the approach to proving Theorem 1.2 described above prior to consulting GPT.' GPT 'supplied the following observation that we missed: for a given Q_{n+1}-class D, the partial equivalence relation Q_n|D is invariantly separable by half-spaces in M_D', which in the final proof 'remains implicit but becomes essentially definitional' (Theorem 4.1(c))
- Writing: 'The article was entirely written by the three named authors'; ChatGPT gave editorial suggestions after the first draft
- No expert reactions found on X, HN or blogs as of 9 Oct; not in OpenAI's 6 Oct catalogue

## What happened

Treeable equivalence relations play the role that free groups play among groups. Jackson, Kechris and Louveau asked in 2002
whether treeability survives finite-index extensions. They had shown that free actions of virtually free groups give
treeable relations, and the general question became a driver of geometric methods in the field.

Chen, Tserunyan and Tucker-Drob build the final tree in stages from a filtration of median algebras on a product of trees.
Their AI declaration is unusually precise: they had the staged approach and its index-2 setting, and ChatGPT 6 Astra Ultra
pointed out that each stage's relation is invariantly separable by half-spaces in the next stage's median graph. This was
the coherence property they needed, and after reformulation it became a definition.

## Why it matters

It settles a named problem from a standard reference on countable Borel equivalence relations, with direct consequences in
measured group theory (fixed price, strong treeability). The disclosure is a clear example of the "AI supplies the missing
lemma" pattern in research-level mathematics.

## What is disputed or not yet verified

- Verification: Unrefereed preprint by leading experts in the field

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 3.8 Flash (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [Chen, Tserunyan & Tucker-Drob: Finite index extensions of treeable equivalence relations are treeable (arXiv 2610.08986)](https://arxiv.org/abs/2610.08986) (arxiv.org, paper)

## Changes

- 2026-10-09 (filed): Created from the arXiv PDF

## Related

- 2026-10-06: [Woodin's question answered](https://postcutoff.com/e/2026-10-06-gch-above-strongly-compact-woodin-question/index.md)
- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)