OpenAI · 069-01 · 2026-10-04 · Algebraic and complex geometry
Global quantum geometric Langlands at irrational level
preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026
Family 069. Quantum geometric Langlands at irrational level.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level . It identifies the full derived categories of twisted -modules for the group and its Langlands dual, retaining all global forms and connected components.
We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level , including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on and at the dual shifted levels c and , where r is the lacing number. The equivalence uses the given global forms and includes all connected components.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. Family 069's quantum Langlands claim is a source record. Oai069-A does not verify the connected-simple theorem. The torus benchmark is not a specialization of that theorem, and a DG equivalence is not an AdS dual. No compilation is asserted.
Emergence
Quantum geometric Langlands is not a gravitational dictionary
Family 069 records the irrational-level unramified de Rham quantum geometric Langlands equivalence. That complex-linear DG equivalence is not a positive channel, not a Page curve, and not spacetime emergence. Oai069-A records the claim and does not verify it. The torus benchmark is not a specialization of the connected-simple theorem. A DG equivalence is not an AdS dual. The catalogue reports no Lean scope note. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
EmergenceOai069-A finite-monodromy certificateOai280-C logical-algebra certificateOai282-B free Ward actionEn02-B decoder programmeVII-H free-field susceptibility