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The Kannan-Tetali-Vempala conjecture for simple undirected graphs

Claims Theorem 1.1: for every $n\ge4$ and every graphical labeled degree vector $d$ , the lazy switch chain has total-variation mixing time at most $2n^8$ and, if $|\Omega_d|>1$ , spectral gap at least $[24n^2\binom n4]^{-1}$ . Corollary: an exactly uniform sampler for any graphical $d$ with expected polynomial bit running time, terminating almost surely. This settles the simple undirected formulation; the bipartite formulation was proved earlier by Fu, Qin and Wang (listed separately), and the directed formulation is not addressed. The exponent 8 is not claimed to be sharp.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Kannan-Tetali-Vempala conjecture for simple undirected graphs · KTV conjecture (undirected case)

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
model · ai model contributor · OpenAI

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Named evidence

VibeMathed record: The Kannan-Tetali-Vempala conjecture for simple undirected graphs

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction, Theorem 1.1 and Corollary 1.2, read against the conjecture as cited. The proof was not refereed; it adapts the Fu-Qin-Wang variance inequality from the bipartite case. Lean-checked on the release's own Comparator challenge SwitchChain together with its solution module OAI.Probability.SwitchChain.Main, both fetched at the pinned commit; the challenge is not listed in the release's formalization catalogue (lean/formalization.yaml), the statement was read here but not independently audited, and the development was not rebuilt here. The formal statements (OAI.Problem315.switch_chain_main, switch_chain_tv_bound, switch_connectivity) define the lazy kernel exactly as in the paper (holding probability 1/2, a uniform four-set and one of six ordered matching pairs) and state, for n >= 4 and every graphical d, mixing time at distance 1/4 at most 2 n^8, spectral gap at least 1/(24 n^2 C(n,4)) when there is more than one state, and switch connectivity. That is the headline claim. The exact uniform sampler (Corollary 1.2) is not formalized.

Source ↗

The Kannan-Tetali-Vempala conjecture for simple undirected graphs

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Kannan, Tetali and Vempala, Random Structures Algorithms 14 (1999)

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problem-record link supplied by the VibeMathed registry.

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Erdos et al., The mixing time of switch Markov chains: a unified approach (Conjecture 1.1)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Lean Comparator challenge SwitchChain (not in formalization.yaml)

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lean-proof link supplied by the VibeMathed registry.

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Lean solution module OAI/Probability/SwitchChain/Main.lean

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lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge SwitchChain (not in formalization.yaml)

lean artifact · pending

Artifact ↗

Lean solution module OAI/Probability/SwitchChain/Main.lean

lean artifact · pending

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Compute record

No linked compute attempts recorded.

Lineage and corrections

Lean Comparator challenge SwitchChain (not in formalization.yaml) evidence for this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: The Kannan-Tetali-Vempala conjecture for simple undirected graphs evidence for this event

Lean solution module OAI/Probability/SwitchChain/Main.lean evidence for this event

The Kannan-Tetali-Vempala conjecture for simple undirected graphs parent of this event

The Kannan-Tetali-Vempala conjecture for simple undirected graphs evidence for this event

Kannan, Tetali and Vempala, Random Structures Algorithms 14 (1999) evidence for this event

Erdos et al., The mixing time of switch Markov chains: a unified approach (Conjecture 1.1) evidence for this event

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