OpenAI · 009-01 · 2026-10-05 · Number theory
Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory
preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026
Family 009. Bogomolov–Pop and Milnor -theoretic reconstruction.
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
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from , , and their product. These data recover the perfect closure and constants when differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro- Galois data away from the characteristic.
We prove a mod-ℓ Bogomolov–Pop reconstruction theorem for function fields of transcendence degree at least two over arbitrary algebraically closed fields of characteristic different from ℓ. The groups and , together with their full bilinear product, determine the perfect closure and its constant field. Every compatible isomorphism of these data is induced by a field isomorphism up to a single scalar in , with only Frobenius ambiguity in the field isomorphism in positive characteristic.
Preprint source and Lean scope
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A Lean scope note is stored for family 009, but it does not name this preprint. It names 009-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.