we-solve (beta)

Math-2787-A1 · revision r1 · on Math-2787

OpenAI claim summary for Math-2787

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The source requires invertible $\mathbb F_p$-linear maps and equality $(\Theta\otimes\Theta)(R_K)=R_L$, rather than arbitrary maps preserving only some relations. With those conventions, its theorem gives the entire reconstruction bijection in this question for relative transcendence degree at least two. The field maps carry the named constant field onto the target constant field; the common nonzero scalar quotient appears on the Milnor side.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Milnor K reconstruction at the characteristic prime
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Milnor K reconstruction at the characteristic prime

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 009; 2026-10-05), the asserted result is that Milnor $K$-theory modulo the characteristic recovers such function fields from degree-two Steinberg relations, up to one nonzero scalar.

Mathematical statement of the claimed result

Manuscript 1.
Fix characteristic $p>0$. For any field $F$ of that characteristic put \[ V_F=F^{\times}/(F^{\times})^p,\qquad R_F=\operatorname{span}_{\mathbb F_p}\{[f]\otimes[1-f]:f\ne0,1\}. \] A linear isomorphism $\Theta:V_K\to V_L$ is compatible if $(\Theta\otimes\Theta)R_K=R_L$, equivalently preserving degree-two Milnor products. For finitely generated $K/k,L/l$ with algebraically closed constants of characteristic $p$ and both relative transcendence degrees at least two, the asserted reconstruction bijection is \[ \operatorname{Isom}(K,k;L,l)\xrightarrow{\sim} \operatorname{Isom}_{\rm M}(V_K,V_L)/\mathbb F_p^{\times}. \] On the left, field isomorphisms carry $k$ onto $l$; on the right compatible maps are identified under nonzero scalar multiplication. Only the first two Milnor groups modulo the characteristic are used.

Outline of the argument

For manuscript 1 (family 009; 2026-10-05), vanishing symbols are interpreted through derivations over the subfield of $p$th powers. An intersection calculation recovers projective lines; one common power correction aligns these lines, and projective incidence gives the field operations.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 009; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources