# Free boundary critical-dimension research benchmark

Family 367: The critical dimension for the one-phase Bernoulli problem. First source-informed review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Free boundary researchers can organize expected singularity behavior by dimension and examine whether a proposed solver test problem models a true one-phase energy minimizer.

## Applicability and commercial boundary

Critical dimension seven and the dimension n−7 singular-set bound concern minimizers. They do not certify a finite mesh output or apply to every free boundary equilibrium, higher-order model or local stationary point.

## Initial business decision

Research infrastructure. Commercial demand and profitability remain hypotheses. Research usefulness is not evidence of a buyer budget.

## Next verification action

Extract a concrete cone and identify a separately justified discretization or minimization procedure before a benchmark implementation.

## Evidence scope

The catalog statement and main-paper abstract or introductory theorem passages were individually reviewed. Available family-level scope documents were inspected, with attention to version differences and exclusions. This is a first application triage, not a full proof audit or independent Lean verification. Family 376 has nine paper abstracts inspected; selected scopes are not assumed to cover all nine papers in full. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Pinned sources

- [The critical dimension for one-phase Bernoulli minimizers](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026.pdf)
- [Selected formal scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/367.md)