# Harmonic growth dimension comparison limitation

Family 361: Failure of integer-degree harmonic dimension comparison. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry researchers can flag the proposed Euclidean dimension upper bound violated on specially constructed nonnegative-Ricci metrics.

## Applicability and commercial boundary

The metric may depend on growth degree k, and exact integer growth classes differ from finite sampled basis dimensions. No new general numerical basis solver follows.

## 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

Extract a finite exact example and preserve the metric-dependence quantifier.

## 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).

[A counterexample to integer-degree harmonic dimension comparison](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026/paper.pdf).