Family 069-01 as a source record

The referenced catalogue identifies Family 069-01 as Global quantum geometric Langlands at irrational level, dated 4 October 2026 OpenAI069Catalogue. That concrete date and title govern the present attribution.

The publicly accessible introduction gives the shifted convention

The complex power describes a differential-operator twisting, not a positive fractional power of a Hilbert-space operator OpenAI069Introduction.

For a connected simple complex algebraic group $G$, a smooth projective connected complex curve $X$, and $c\in \mathbb{C}\setminus \mathbb{Q}$, the source claims an equivalence

of presentable complex-linear DG categories. Here $r$ is the lacing number, and the full root datum fixes the global form.

The stated characterization includes localization–Whittaker comparison, compatibility with families of marked points and collisions, and interchange of central-torsor actions with the corresponding transgression local systems OpenAI069Introduction.

The source introduction and supplied excerpts have been consulted for attribution. No independent proof verification is asserted. The catalogue explicitly separates this claim from Cheung's registered series and reports no Lean scope note for this family OpenAI069Catalogue.

The order $d$ of the finite cyclic group introduced below is not the lacing number $r$. No shifted level $c$ is needed for our finite construction.