# Conformal derivative and boundary spectrum audit

Family 072: Brennan's conjecture and the integral-means spectrum. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Mapping researchers can review integrability and universal spectrum claims for conformal transformations.

## Applicability and commercial boundary

Strict exponent ranges and different bounded/unbounded univalent classes matter. Integrability alone does not give finite mesh-error rates or a production warp algorithm.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract an explicit reference map and its derivative integrals.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Brennan's conjecture and sharp inverse-square integral means](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf).