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The Modified Lyons–Sidorova Conjecture for Bounded-Variation Paths

For a continuous bounded-variation path with signature gg, logarithmic signature ll and increment vv, the modified Lyons–Sidorova conjecture predicts the structure of gg when R(l)=∞R(l)=\infty. The paper proves it: g=1g=1 when v=0v=0, and otherwise a prefix α\alpha of the centred path gives S(γ)=S(α)evS(α)−1S(\gamma) = S(\alpha)e^{v}S(\alpha)^{-1}.

Result
Proved(see note)
Status
Resolved
AI contribution
AI-assisted
Method
Argument
Field
Rough Paths, Path Signatures
Posed by
Horatio Boedihardjo; formulated systematically by Boedihardjo, Geng and Wang, after the original conjecture of Lyons and Sidorova
Year posed
2020
Years open
6y
Solved
2026-07-29
Model
ChatGPT
Vendor
OpenAI
Collaborators
Elena Boguslavskaya
Verification
Unreviewed
Publication
Preprint
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.

What the AI did

The author reports using ChatGPT for mathematical exploration, critical examination of arguments, testing of intermediate proof strategies, and drafting and editing, and states she reviewed and revised all AI-assisted material and takes full responsibility. The model is not credited with producing the proof.

Verification

No independent check, and the disclosure places the model in a supporting role rather than crediting it with the argument. Preprint, not refereed.

Source

arXiv

Discussion