# Almost-complex algebra

[All sections](../README.md)

## Modules

| Module | Verified result |
|---|---|
| [HopfObstruction](HopfObstruction.en.md) | Positivity and unboundedness of the scalar function 1/ε − 1 as positive ε approaches zero. |
| [HopfAlmostComplex](HopfAlmostComplex.en.md) | For J²=−I, exclusion of real eigenvalues on nonzero vectors and algebraic Nijenhuis identities. |
| [HopfIntegrabilityBarrier](HopfIntegrabilityBarrier.en.md) | A nonzero value rules out identical vanishing of an algebraic expression; bounds for a prescribed scaled scalar energy. |
| [HopfOctonions](HopfOctonions.en.md) | A seven-dimensional cross product specified by Fano triples: bilinearity, orthogonality, double-product identities, and tangent J²=−I. |

## Scope

The algebraic and scalar statements do not resolve the Hopf problem or establish a global geometric integrability obstruction.

[Contributing](../CONTRIBUTING.en.md) · [Validation](../VERIFICATION.en.md)