Assumption: Constructed free partial product

The IX-Free-2 record, or the supported IX-C-ST predecessor when accepted with its stated inputs, supplies IXFreeTwo,IXCST:

1. The displayed free coefficients exist in ${\mathcal{P}}_{\Sigma }$, including required directional descendants.

2. With

   $$\mathcal{B}=\{B_f\},\mathcal{W}=\{W_g\},\mathcal{L}=\mathrm{span}\{B_f{W}_g,Q_{f,g}\},\mathcal{R}=\mathrm{span}\{W_g{B}_f,{Q^R}_{g,f}\},$$

   the actual algebraic tensor domains are

   $$D_{\mathrm{first}}=(\mathcal{B}\otimes \mathcal{W})+(\mathcal{W}\otimes \mathcal{B}),$$
   $$D_{\mathrm{second}}=(\mathcal{L}\otimes \mathcal{B})+(\mathcal{B}\otimes \mathcal{R}),D_{\mathrm{new}}=D_{\mathrm{first}}+D_{\mathrm{second}},$$
   $$D_{\mathrm{old}}={\mathcal{P}}_{\mathrm{mc}}\otimes {\mathcal{P}}_{\mathrm{mc}},D_{\mathrm{decl}}=D_{\mathrm{new}}+D_{\mathrm{old}},D_{\mathrm{new}}\cap D_{\mathrm{old}}=0.$$

   The old free coefficients are unchanged.

3. There is one linear coefficient assignment

   $$S={\star }_c:D_{\mathrm{decl}}⟶{\mathcal{P}}_{\Sigma }{\otimes }_{\mathbb{C}}\mathbb{C}[\hslash ]/({\hslash }^4).$$

   It is extended linearly to formal tensors. No arbitrary intermediate closure is supplied.

4. Both degree-two presentation maps and both degree-four presentation maps are injective. In particular, ${Q^R}_{g,f}\ne 0$ for nonzero $g\otimes f$.

5. For one common real parent $c$,

   $$\begin{array}{rl}{A}_c(f,g,f^{\prime })=& (B_f{\star }_c{W}_g){\star }_c{B}_{f^{\prime }}=B_f{\star }_c(W_g{\star }_c{B}_{f^{\prime }})\\ =& B_f{W}_g{B}_{f^{\prime }}+\hslash (G_{12}+G_{13}+G_{23})\\ & +{\hslash }^2(G_{12,23}+G_{12,13}+G_{13,23})+{\hslash }^3{\mathcal{N}}_c(f,g,f^{\prime })1(\mathrm{mod}{\hslash }^4).\end{array}$$

6. If $\sigma (F\otimes G)=G\otimes F$ and $\gamma (F\otimes G)=F^{\ast }\otimes G^{\ast }$, then

   $$d^{†}=\sigma \gamma d,{D_{\mathrm{decl}}}^{†}=D_{\mathrm{decl}},S(d{)}^{\ast }=S(d^{†}).$$

   Conjugation fixes formal $\hslash$.

7. Fixed-support continuity holds on the named generating families, denoted F9a. Actual-stage continuity F9b and quantum hypocontinuity are not supplied.