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Math-2810-A1 · revision r1 · on Math-2810

OpenAI claim summary for Math-2810

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 result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 019; 2026-09-24).

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

Manuscript

Contents
  1. Finite étale covers with a prescribed exterior sheet and p-divisible disk degrees
  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

Finite étale covers with a prescribed exterior sheet and p-divisible disk degrees

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 019; 2026-09-24), the asserted result is that prescribed disjoint small disks on a proper hyperbolic $p$-adic curve admit one connected finite étale cover with a sheet over their entire exterior and degree divisible by $p$ on every disk component.

Mathematical statement of the claimed result

Manuscript 1.
Fix a prime $p$, a finite extension $k/\mathbb Q_p$ and its algebraic closure $\bar k$, with analytic completion $\Omega=\widehat{\bar k}$. Let $X/\bar k$ be smooth proper connected of genus at least two. At distinct $b_1,\ldots,b_d$, $d\ge1$, choose disjoint parameter disks $B_i$ with coordinates $t_i$ and radii $0<r_i<R_i$ in $|\Omega^{\times}|$. Put $V_i=\{|t_i|<r_i\}$ and require the compact exterior $U=X^{\rm an}\setminus\bigcup_iV_i$ nonempty and containing an $\Omega$-rational point. The claim gives a smooth proper connected curve $Y/\bar k$ and finite étale surjection $g:Y\to X$ such that \[ \exists C\text{ component of }g^{-1}(U):\ C\simeq U, \qquad p\mid\deg(D\to V_i)\quad\text{for every component }D\subseteq g^{-1}(V_i). \] All inverse images are analytic over $\Omega$, and the prescribed disks are retained without shrinking.

Outline of the argument

For manuscript 1 (family 019; 2026-09-24), resolution covers first create positive-genus skeletal vertices, and auxiliary torsors force a cycle over each disk. Degree-$p$ graph characters split over the exterior; a quotient by an exterior stabilizer keeps one exterior sheet while retaining the required disk degrees.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 019; 2026-09-24).

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