Output wavefront and a precisely defined admission interface

For the independent even-sector interface, define

A compact polynomial functional has finitely many symmetric kernels $t_n\in {\mathcal{E}}^{\prime }(M^n)$. It is microcausal if

where the product causal cones allow zero components but exclude the all-zero cotangent vector. Let ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ denote this specified even polynomial space.

Fix bidistribution $H$ for which the finite contractions are defined, and set

For polynomials the sum is finite at each pair. Associativity follows from the commuting contraction operators, provided all indicated products are defined. This is the standard deformation-quantization interface FredenhagenRejzner.

Now supply a named free solution space and an explicitly specified two-sided ${\star }_H$-ideal ${\mathcal{I}}_{\mathrm{sol}}$. Define

The ideal property is a hypothesis to verify, not a consequence of merely calling the ideal “on shell.”

This defines a complete abstract admission interface. It does not identify ${{\mathcal{E}}_{\mathcal{H}}}^{\mathrm{if}},{Q_H}^{\mathrm{if}}$ with XII-A's full graded carrier ${\mathcal{E}}_H,Q_H$ without its exact field multiplet, free equations, ideal, normalization, and graded contraction rules.