Use unary and grade operations

Reversal changes the order of basis-vector factors. Grade involution changes the sign of odd grades, and Clifford conjugation combines the two. On a grade-\(k\) component their signs are respectively \((-1)^{k(k-1)/2}\), \((-1)^k\), and \((-1)^{k(k+1)/2}\). They preserve the coefficient space unless an output layout selects grades.

import torch
from clifra import make_algebra

algebra = make_algebra(3, dtype=torch.float64)
mixed = algebra.layout((0, 1, 2))
bivectors = algebra.layout((2,))

x = torch.arange(1, mixed.dim + 1, dtype=torch.float64)
grades = mixed.grade_indices_tensor()
reverse = algebra.plan_unary(op="reverse", input=mixed)
torch.testing.assert_close(reverse(reverse(x)), x)

conjugated = algebra.clifford_conjugation(x, input=mixed)
torch.testing.assert_close(
    conjugated, algebra.grade_involution(reverse(x), input=mixed),
)
selected = algebra.grade_projection(x, input=mixed, output=bivectors)
torch.testing.assert_close(selected, x[grades == 2])
assert selected.shape == (3,)

grade_projection requires output; it returns that layout's compact lanes unless a canonical output contract is supplied. This differs from zeroing lanes with x * mixed.grade_mask((2,)), which keeps the original layout and width. Use projection when downstream operations should declare a smaller coefficient space.

pseudoscalar_product(x, input=layout) computes right multiplication by \(I=e_1e_2\cdots e_n\) and defaults to complementary output grades. It is not multiplication by \(I^{-1}\). In a degenerate signature \(I\) is not invertible, so this operation should not be assumed to define an invertible duality. The planning counterpart is plan_pseudoscalar_product.