Proposition: No hidden quantum-family claim

The constructed field-only special fiber is exactly the free off-shell field algebra with differential $s_0$. The free causal quantization of its displayed linear equations is exactly ${\mathfrak{A}}_0$.

No flat family of interacting quantum algebras follows. In particular, the special fiber of the full antifield centralizer need not equal the centralizer of the specialized residual.

Proof. The field-only algebra lies entirely in ${\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}$, so specializing its formal coefficients gives every free field expression, with the kernel consisting of multiples of $\kappa$. Theorem thm:freeBRST gives its separate free causal presentation.

Kernels need not commute with specialization. For example, the kernel of multiplication by $\kappa$ on $\mathbb{C}[[\kappa ]]$ is zero, whereas the specialized zero map has kernel $\mathbb{C}$. Hence no specialization theorem for the entire centralizer can be inferred without additional hypotheses.