# Rational point/Galois section correspondence ledger

Family 019: The local <i>p</i>-adic section conjecture and global consequences. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Arithmetic researchers can relate rational points to section conjugacy classes in the local curve and specified modular-curve global settings.

## Applicability and commercial boundary

Bijection is not a practical point enumeration or section construction algorithm, and the global conclusion is not all curves over Q.

## Initial business decision

Research infrastructure or conditional engineering until a computable implementation, effective bounds and a recurring buyer need are demonstrated. No commercial novelty or profitability is established.

## Next verification action

Inspect effective representation and a tractable modular-curve example.

## Evidence scope

The catalog statement was individually reviewed. Manuscript proof and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Étale covers with a prescribed exterior sheet](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf).