A source space, not a target algebra

Let ${\mathcal{P}}_{\mathrm{fin}}$ be the off-shell polynomial algebra of finite products of compact smooth linear smearings. Its degree-$n$ kernels are finite sums of smooth tensor products.

Let ${\mathcal{P}}_{\mathrm{mc}}$ be the classical space declared in XI-C: finite polynomials with compact distributional kernels $u_n$ obeying

where the causal cones include zero covectors and the total zero covector is excluded from a wavefront set. Include graded field slots when discussing free variations.

Then

are classical input statements. Neither ${\mathcal{P}}_{\mathrm{mc}}[[\hslash ]]$ nor its notation defines a quantum product.

For a possible on-shell formulation define

Here equality on free configurations is understood in the graded sense, including evaluations in finite Grassmann algebras. We do not identify ${\mathcal{N}}_{\mathrm{sol}}$ with a generated equation ideal without a separate proof.