Explanations

These chapters develop the design of clifra from tensor representation through geometric parameterization, numerical execution, and PyTorch integration.

1. Layout, Storage, and TensorContract explains how clifra represents a selected set of blades as an ordinary dense tensor.
2. Planning Policy as Dependency Injection explains resource limits and executor selection.
3. Geometric Parameterization describes geometric objects as learnable coordinate systems and positions the physics-informed bivector-field showcase within that method.
4. Why Bivector Coordinate Fields Work derives the generalized input, sampling, action, and inversion contracts behind the continuum-solver research package.
5. Bivector Exponential Methods separates exact low-dimensional formulas, matrix exponentiation, and spectral-local approximation.
6. Optimization for Geometric Parameters explains the built-in parameter dispatch and how to connect quasi-Newton, higher-order, or tangent-space methods.
7. Signatures and Algebraic Behavior distinguishes signed Clifford forms from positive coefficient-lane energy and relates both to differentiation.
8. Using clifra with PyTorch explains how clifra's algebraic machinery and the surrounding PyTorch system divide the work.

The chapters are independent of the tutorials. Each introduces the definitions needed for its own argument; the API reference provides the corresponding public interfaces.