Theorem: The finite-width free algebra

At fixed preparation width, the polynomials of Theorem thm:rampadmission, the positive-width convolution polynomials, ordinary compact local polynomial fields and vertices, and their fixed smooth directional derivatives generate a unital associative free $\ast$-algebra in ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ,\kappa ]]$.

The adjoint is complex conjugation of symbols:

The quotient by $({\kappa }^3)$ is a consistent coordinate-order truncation. No field-degree truncation is imposed.

Proof. The admitted finite kernels obey the microcausal exclusion. For each contraction, a forbidden cancellation against the oriented Hadamard lines would force one input covector tuple wholly into one of the excluded causal cones. Thus the products and proper contractions are defined and their output kernels remain compact and microcausal.

On three factors, the constant even contraction operators ${\Gamma }_{12},{\Gamma }_{13},{\Gamma }_{23}$ commute. Both associations therefore equal

The same microlocal argument justifies the finite multicontractions. This is the polynomial Wick construction BF,Duetsch,HRV.

Hermiticity of $H$ proves the adjoint identity by reversing the contraction slots. Constants are the unit. Products can increase field degree, but each coefficient is still a finite polynomial. The formal parameter ideal $({\kappa }^3)$, unlike an arbitrary field-degree cutoff, is stable under multiplication.