Use pairwise operations

Use pairwise=True when every item on the left should combine with every item on the right. Inputs have shapes (..., N, left_lanes) and (..., M, right_lanes); the result has shape (..., N, M, output_lanes). Earlier dimensions broadcast. The item axes are not reduced.

import torch
from clifra import make_algebra

algebra = make_algebra(3, dtype=torch.float64)
vectors = algebra.layout((1,))
scalar = algebra.layout((0,))
pairwise = algebra.plan_product(
    left=vectors, right=vectors, output=scalar, pairwise=True,
)
x = torch.arange(18, dtype=torch.float64).reshape(2, 3, 3)
y = torch.arange(15, dtype=torch.float64).reshape(5, 3)
scores = pairwise(x, y)
assert scores.shape == (2, 3, 5, 1)
reference = algebra.geometric_product(
    x.unsqueeze(-2), y.unsqueeze(-3),
    left=vectors, right=vectors, output=scalar,
)
torch.testing.assert_close(scores, reference)
torch.testing.assert_close(scores.squeeze(-1), x @ y.T)

The last equality uses Euclidean vectors. In a mixed signature the scalar geometric product is the signed bilinear form, so it is not an ordinary coefficient dot product. Select output grades explicitly when only scalar scores or bivector relations are needed.

The pairwise axis convention is fixed when planning. For a one-item input, insert an item axis with unsqueeze(-2). For large item sets, split an item axis into chunks before execution, as the pairwise output still grows as \(NM\) regardless of coefficient compactness. Reduce results with ordinary PyTorch operations after deciding which item axis the reduction represents.