# Galois homomorphism reconstruction reference

Family 031: Uchida’s conjecture for open homomorphisms of Galois groups. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Arithmetic researchers can record equivariant field embeddings corresponding to the specified open Galois homomorphisms.

## Applicability and commercial boundary

Possibly infinite extensions and abstract topological groups may have no efficient input encoding; a unique embedding need not be computably cheap.

## 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 a restricted finite-data example and exact representation requirements.

## 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).

[Open Homomorphisms of Global Solvably Closed Galois Groups](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf).