# Localized homotopy finite-generation contract

Family 313: Finite generation for the $`K(n)`$-local sphere. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Record degreewise finite generation over p-adic integers for the specified localized sphere and finite-complex homotopy groups.

## Applicability and commercial boundary

Degreewise finiteness gives neither a uniform rank bound nor a terminating general computation procedure.

## Initial business decision

Research reference or conditional symbolic/verification engineering. No recurring buyer need, practical implementation or profitability has been demonstrated. Defer a standalone commercial product until a constructive example and a measurable workflow improvement exist. These records also identify unsupported inferences that an evidence platform could flag.

## Next verification action

Inspect effective bounds at one prime, height and degree.

## Evidence scope

The repository catalog statement was individually read. Main-paper construction, effectiveness and selected formal-scope comparison remain queued for this record. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Finite generation for the K(n)-local sphere](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf).