Higher Verlinde fiber functors in positive characteristic
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Definitions and mathematical statement
Statement recorded in the supplied source.
For every prime p, every algebraically closed field k of characteristic p and every finite symmetric tensor category C over k, do there exist $n\geq 1$ and an exact faithful k-linear symmetric tensor functor $C\to Ver_{p^n}(k)$?Short English statement
For every prime p, every algebraically closed field k of characteristic p and every finite symmetric tensor category C over k, do there exist $n\geq 1$ and an exact faithful k-linear symmetric tensor functor $C\to Ver_{p^n}(k)$?Sources
- S1 Original question source or mathematical context. http://aimpl.org/fusioncat/10/.
- S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2012/47.
- S3 Original question source or mathematical context. https://doi.org/10.4171/owr/2024/20.
- S4 Original question source or mathematical context. https://doi.org/10.4171/owr/2026/9.
- S5 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2514
The proposed functor is faithful and exact, respects the symmetric tensor structure, and is linear over the chosen coefficient field. Every finite category in the stated positive-characteristic class is assigned some finite Verlinde level, with no fixed uniform level required. This is presented as a claimed affirmative answer.
Higher Verlinde fiber functors in positive characteristic
OpenAI's claim
OpenAI presents a proof in the selected manuscript. Every finite symmetric tensor category in characteristic $p>0$ is claimed to admit an exact faithful symmetric tensor functor to a finite higher Verlinde category, including at $p=2$.Mathematical statement of the claimed result
Result 1.
For each prime $p$, algebraically closed characteristic-$p$ field $k$, and finite symmetric tensor category $\mathcal C$ over $k$, the claim supplies an integer $n\ge1$ and an exact faithful symmetric tensor functor \[ \mathcal C\longrightarrow\operatorname{Ver}_{p^n}(k). \] The target is the higher Verlinde category at level $p^n$; the level may depend on $\mathcal C$, rather than being a universal fixed choice.Outline of the argument
A representation-theoretic restricted-tilting assertion supplies the higher Frobenius construction. Classification and reconstruction then identify a suitable finite level of the Verlinde tower and produce the faithful fiber functor; the tower level can vary with the input category.Formal verification
Family 208 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, Fiber functors for finite symmetric tensor categories in positive characteristic, 2026-09-24; citation key
OAI:Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.