Signatures and Algebraic Behavior

The signature of \(Cl(p, q, r)\) defines the square of each generating basis vector:

\[ e_i^2 = \begin{cases} +1 & 1 \le i \le p,\\ -1 & p < i \le p + q,\\ 0 & p + q < i \le p + q + r. \end{cases} \]

These signs propagate through every Clifford product. They are separate from the positive coefficient geometry that clifra also exposes for losses, diagnostics, and stable optimization.

Signed forms and lane energy answer different questions

For compact coefficients \(x_i\), lane energy is

\[ E_{\mathrm{lane}}(x) = \sum_i x_i^2. \]

It is positive definite and independent of metric signs. It measures the size of a coefficient tensor in its declared layout. lane_energy, lane_norm, lane_distance, and the per-grade lane functions use this geometry.

A Clifford form is built from algebraic products. signature_norm_squared computes

\[ Q(x) = \langle x\widetilde{x}\rangle_0, \]

with signs determined by the signature and blade grades. Depending on the form and convention, its value can be positive, negative, or zero for a nonzero multivector. Null directions in a degenerate signature make the last case fundamental rather than exceptional.

Quantity	Signature-sensitive	Positive definite	Typical use
Lane energy	No	Yes	Coefficient scale, regularization, stable distances
Per-grade lane energy	No	On the selected grade	Grade distribution and diagnostics
Signature or conjugate scalar form	Yes	No in general	Algebraic invariants and metric-aware constraints
Magnitude derived with abs	Yes, before abs	Nonnegative, but not a norm in every signature	Stable scale based on a signed form

Calling both quantities a norm without qualification hides an important design choice. A signed form represents the algebraic metric. Lane energy represents a Euclidean geometry on the coordinate lanes used by the optimizer.

Signature determines product behavior

A planned product stores fixed left-lane positions, right-lane positions, output positions, and coefficients. The coefficients include permutation signs, metric signs, and zero products from null basis directions. Planning removes interactions that are known to be zero and execution performs the remaining gather, multiply, and reduction operations.

The resulting tensor program is signature-specific even though its runtime operations look like ordinary PyTorch arithmetic. Changing \(Cl(3, 0, 0)\) to \(Cl(1, 2, 0)\) changes more than coefficient names: it changes basis products, signed forms, exponential behavior, and potentially executor eligibility.

Forward and backward use the same algebra

Autograd differentiates the actual tensor program used in the forward pass. For an exact planned product with fixed coefficients,

\[ y_k = \sum_{i,j} c_{ijk} a_i b_j, \]

the backward pass contains the same signature-dependent coefficients \(c_{ijk}\):

\[ \frac{\partial L}{\partial a_i} = \sum_{j,k} c_{ijk} b_j \frac{\partial L}{\partial y_k}. \]

Backpropagation retains the algebraic signs from the forward calculation; the gradient follows that signed tensor program, with no substituted positive metric.

Using lane energy as a loss or regularizer introduces a positive Euclidean objective on coefficient space. That choice changes the optimization geometry and therefore the path taken by learning, while leaving the model's Clifford product unchanged. An indefinite algebra can therefore use a positive coefficient-space objective and still retain its original signature.

Use lane quantities for coefficient scale and distances. Use a signed form for a signature-sensitive invariant or constraint. Taking abs of a signed form yields a nonnegative quantity but does not make it a positive-definite norm.

import torch
from clifra import make_algebra

algebra = make_algebra(1, 1)
vectors = algebra.layout((1,))
x = torch.tensor([1.0, 1.0])
assert algebra.lane_energy(x, input=vectors).item() == 2.0
assert algebra.signature_norm_squared(x, input=vectors).item() == 0.0

This nonzero vector is null for the signed form. Its coefficient energy remains positive.

Scalar forms and output axes

scalar_product(A, B) computes \(\langle AB\rangle_0\). conjugate_scalar_form(A, B) computes \(\langle\overline A B\rangle_0\), where the bar is Clifford conjugation, not complex conjugation. Reversal, conjugation, and no involution give different signs. Even in a Euclidean signature, the scalar part of a bivector square is negative while its coefficient energy is positive.

Lane dot products and distances align layouts by blade identity before comparing coefficients. Lane energies and scalar forms preserve all leading axes and return a final singleton axis. Per-grade energy returns \(n+1\) entries indexed by grade, including zeros for grades absent from the layout. That final axis is a list of grade measurements, not a multivector layout.

For zero input, every grade energy is zero. lane_grade_distribution divides by total energy plus eps, so its entries sum to slightly less than one for nonzero data and to zero for all-zero data. It is a stabilized coefficient summary, not a probability distribution inferred from the algebra.

Numerical behavior at zero and near null values

lane_energy is a polynomial in real coefficients. lane_norm takes its square root, which has a nonsmooth norm point at the zero vector. No epsilon is added by that helper. If a differentiable objective needs smooth behavior there, squared coefficient energy often expresses the intended penalty without a square root.

Signed cancellation is a separate issue. Large coefficients can produce a small signed form in an indefinite algebra, making relative error large. Inversion divides by such a form and can magnify that error. Strict geometry rejects an exactly zero denominator; it does not set a conditioning threshold. Stabilized geometry changes the denominator near zero, and autograd follows that changed numerical expression.

Compare quantities in the precision and range used by the application. Coefficient energy can reveal large values hidden by signed cancellation; the signed form still determines the algebraic invariant being checked.