Geometric Representations and Parameters

A layout defines the coefficient space. The same tensor representation can be used as fixed data, a differentiable input, or a PyTorch parameter. Its application role—model weight, sample, or operand—is independent of its grade.

The operation gives the coefficients their role. A vector may enter a product, define a reflection normal, or be projected onto a blade. A mixed-grade value can be added, reversed, or multiplied without interpreting it as a geometric transformation.

Vectors, blades, and mixed grades

A grade-1 layout contains the \(n\) vector coefficients. Its signed square comes from the signature; its coefficient length comes from ordinary Euclidean coordinates. Choosing one as a normalization rule does not make it the other.

A simple grade-\(k\) blade \(A=a_1\wedge\cdots\wedge a_k\) represents an oriented subspace with scale. Constructing it from vector factors ensures decomposability, although dependent factors can produce zero. Storing arbitrary grade-\(k\) coefficients is more general: a homogeneous multivector need not be a simple blade. A layout declaration alone cannot establish the assumptions required for a blade inverse or subspace projection.

A mixed-grade layout is a direct sum of coefficient spaces. For instance, grades 0 and 1 store a scalar and vector in one tensor. Reversal, projection, and geometric multiplication each give this value a well-defined meaning without additional parameter constraints.

import torch
from clifra import make_algebra

algebra = make_algebra(3, 0)
vectors = algebra.layout((1,))
scalars = algebra.layout((0,))
mixed = algebra.layout((0, 1))
v = torch.tensor([1.0, 2.0, 0.0])
a = mixed.convert(torch.tensor([2.0]), scalars) + mixed.convert(v, vectors)
assert a.shape == (4,)
square = algebra.geometric_product(
    a, a, left=mixed, right=mixed, output=mixed,
)
torch.testing.assert_close(square, torch.tensor([9.0, 4.0, 8.0, 0.0]))

Here \((2+v)^2=4+4v+v^2=9+4v\). The output selection retains the scalar and vector terms. This calculation uses only the specified grades and proceeds without introducing a transformation parameter.

Bivectors and explicit rotors

A bivector is a grade-2 value. It can represent an oriented area, enter a commutator, or generate an action. In dimensions four and above, a general bivector can contain several independent plane components; it need not be a single simple blade.

Exponentiation is one specialized use. For a bivector \(B\),

\[ R=\exp(-B/2),\qquad \widetilde R=\exp(B/2),\qquad R\widetilde R=1. \]

The identity follows from \(\widetilde B=-B\) and the commuting exponential factors. Generator coordinates are useful when varying an action smoothly. Explicit even-grade rotor coefficients are useful when composing, comparing, or exporting group elements. An arbitrary even multivector is not necessarily a rotor.

These representations are not interchangeable coordinate systems. The action identifies \(R\) with \(-R\), generator coordinates can be periodic, and a product of rotors need not have a unique bivector logarithm. Adding generators implements composition only when they commute. See Bivector Exponentials and Actions for the operation and its numerical evaluation.

Parameters and constraints

Any declared coefficient tensor can be a PyTorch parameter. A shared product operand of shape [lanes] broadcasts across samples; separate operands of shape [samples, lanes] receive separate gradients. Parameter ownership adds no Clifford semantics beyond the declaration and the operations using it.

Constraints follow the represented object. Ordinary mixed-grade coefficients may need no adjustment. A blade built from vector factors preserves decomposability under changes to those factors, but can become singular. A reflection normal must stay non-null for the exact reflection. An additive update to a stored rotor need not preserve rotor membership, whereas a bivector always defines its mathematical exponential.

A coefficient penalty constrains a representation. A signed invariant, projection residual, or transformed-value loss constrains a calculation. Choose these explicitly rather than treating every coefficient tensor as a generator or every grade as a manifold.

Composition and invariants

A sum or product combines represented values; an output layout selects grades. An intermediate projection can remove terms needed later, so a narrow result cannot always replace the complete value in another expression.

An exact reflection or rotor action preserves the signature-sensitive vector form, while a subsequent coordinate gate or generic tensor map need not. Likewise, a coefficient normalization changes scale, but verifying blade simplicity or a signed invariant requires an explicit check. Mathematical claims should refer to the complete calculation being used.