# Spectral geometry width research planner

Family 336: Spectral scalar curvature, Urysohn width, and macroscopic dimension. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry researchers can study dimension-reducing maps under the precise spectral scalar-curvature inequality.

## Applicability and commercial boundary

A continuous map with bounded entire fibers need not be efficiently constructible or preserve reconstruction accuracy of finite data. The quadratic-form hypothesis requires evidence.

## Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

## Next verification action

Inspect effective constants and a constructible example before treating the map as a compression tool.

## Evidence scope

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

[Spectral scalar curvature and uniform Urysohn width](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf).