Compute norms, energies, and signed forms correctly

Use coefficient energy for a nonnegative penalty on stored coordinates. Use a signature-sensitive form when the algebraic quadratic form is the quantity of interest. They coincide on Euclidean vectors but not in general.

Operation	Quantity
lane_energy(A)	\(\sum_i a_i^2\)
lane_norm(A)	\(\sqrt{\sum_i a_i^2}\)
lane_dot_product(A, B)	\(\sum_i a_i b_i\)
lane_distance(A, B)	\(\sqrt{\sum_i(a_i-b_i)^2}\)
signature_norm_squared(A)	\(\langle A\widetilde A\rangle_0\)
scalar_product(A, B)	\(\langle AB\rangle_0\)
conjugate_scalar_form(A, B)	\(\langle\overline A B\rangle_0\)

import torch
from clifra import make_algebra

algebra = make_algebra(1, 1, dtype=torch.float64)
vectors = algebra.layout((1,))
x = torch.tensor([[1., 1.], [0., 2.]], dtype=torch.float64)
energy = algebra.lane_energy(x, input=vectors)
signed = algebra.signature_norm_squared(x, input=vectors)

torch.testing.assert_close(energy, torch.tensor([[2.], [4.]], dtype=x.dtype))
torch.testing.assert_close(signed, torch.tensor([[0.], [-4.]], dtype=x.dtype))
conjugate = algebra.conjugate_scalar_form(x, x, left=vectors, right=vectors)
torch.testing.assert_close(conjugate, -signed)

by_grade = algebra.lane_grade_energy(x, input=vectors)
assert by_grade.shape == (2, 3)
torch.testing.assert_close(by_grade[:, 1:2], energy)

Scalar quantities retain a final width-one axis and preserve batch dimensions. Grade energies and norms instead have a final axis of length \(n+1\), including zeros for absent grades. lane_energy(..., grades=(...)) restricts the coefficient penalty to selected grades. lane_grade_distribution normalizes grade energies with an additive denominator epsilon; its sum need not be exactly one.

The coefficient dot, distance, and conjugate form align differing declared layouts by canonical blade identity, treating absent lanes as zero. Their leading axes still broadcast normally.

A zero signed square need not mean zero coefficients; the first row above is a nonzero null vector. Use the Euclidean coefficient norm when you need a positive norm, or coefficient energy when you need a nonnegative loss.

If division by this form is required, choose the domain and singular-input policy explicitly. Squared coefficient energy also avoids the square-root derivative singularity of a norm at zero.