# Allen–Cahn classification and phase-field research reference

Family 375: De Giorgi's conjecture in dimension eight. First source-informed review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Phase-field researchers can compare entire monotone dimension-eight Allen–Cahn solutions and stable dimension-seven solutions with the planar heteroclinic classification.

## Applicability and commercial boundary

These are exact entire-space, monotonicity or stability statements. They do not classify arbitrary transient simulations, finite domains, noisy interfaces, nonstandard potentials or higher-dimensional models.

## Initial business decision

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

## Next verification action

Use the explicit planar heteroclinic as a reference and inspect whether a numerical test genuinely meets the entire-solution or stability assumptions.

## 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

- [A positive resolution of De Giorgi's conjecture in dimension eight](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/De-Giorgis-conjecture-in-dimension-eight-September-26-2026/article.pdf)