Proposition: Missing second multiplication

The admitted output ${\mathcal{Q}}_{f,g}\in {\mathcal{P}}_{\Sigma }$, the equality of regularized associations, and the parent finite-part assignment do not determine either sharp composition

Here $\mathsf{B}(f)$ denotes the actual admitted $B_f$, while ${\star }_{\Sigma }$ is only a placeholder for a future product. Such a construction still requires:

1. binary quantum maps on specified stages of the IX-A carrier, including multiplication of partial outputs by sharp insertions;

2. verification of the IX-A admission test for all resulting polynomial coefficients and directional descendants;

3. graph-specific subtraction and normalization conditions compatible with the fixed old products;

4. proof that those maps factor the calibration and satisfy the required coherence identities.

Proof. IX-A supplies the carrier and selected partial outputs, not their second quantum multiplication. VII-D assigns a numerical distribution to a particular fully contracted three-vertex graph. Its arguments are joint observation tests, not arbitrary intermediate elements of ${\mathcal{P}}_{\Sigma }$. Neither statement defines a binary map acting on ${\mathcal{Q}}_{f,g}$ and $B_{f^{\prime }}$.

A common subtracted limit of two existing regularized compositions specifies a ternary numerical coefficient. It does not factor that coefficient through the missing binary maps. Moreover, the constant fully contracted component is only one coefficient of a full product; it does not verify (eq:admission) for its nonconstant components. Therefore neither displayed sharp composition is determined.