Total Variation for the Lamplighter Walk on Z

Precise statement

For the switch-walk-switch walk on $\mathbb{Z}_2 \wr \mathbb{Z}$ started at $(0,0)$ and $(0,2)$, prove $\|P_t^x - P_t^y\|_{TV} \asymp t^{-1/2}$.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

QED (GPT-5.5)

Proved by the QED multi-agent system in decomposition mode; the expert who posed the problem provided no mathematical input beyond the statement.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

independent expert verified

Verified by the contributing domain expert; documented in the QED system paper.

Correctness: supported · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-05-15

   arXiv:2604.24021 - QED: an open-source multi-agent system for mathematical proofs

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.