Assumption: IX-Free-2 entering release

Relative to its explicitly stated inherited premises, IX-Free-2 constructs coefficient maps

and one truncated partial product

extended over $\mathbb{C}[\hslash ]/({\hslash }^4)$. Its restriction to $D_{\mathrm{old}}$ is the unchanged old product modulo ${\hslash }^4$.

The following results are inherited, not reproved:

1. $q_L,q_R,p_L,p_R$ are injective, and

   $$\ker p_L=\ker p_R=\{0\}.$$

   The degree-four proof uses restricted sharp separation on $k_{00}=0$ and one-scalar, three-metric slot isolation.

2. All four prescribed degree-four coefficients descend to actual functionals.

3. Every nonzero sharp generator has nonzero advanced-trace charge at some smooth configuration. The partial contractions and admitted microcausal comparison kernels have zero charge. In particular,

   $$\mathcal{B}\cap {\mathcal{L}}_2=\mathcal{B}\cap {\mathcal{R}}_2=\mathcal{B}\cap {\mathcal{P}}_{\mathrm{mc}}=\{0\}.$$

4. The new domain is a direct sum of its six prescribed blocks. Its whole-sum old overlap vanishes:

   $$D_{\mathrm{new}}\cap D_{\mathrm{old}}=\{0\}.$$

   All coefficients glue, including the locked first stages.

5. $${\mathcal{C}}_{\mathrm{desc}}={\mathcal{C}}_{\mathrm{glue}}=\mathbb{R}.$$

   For each fixed real $c$, the product satisfies C-I1–C-I9 on $D_{\mathrm{decl}}$.

6. The two associations of the selected word exist and agree through ${\hslash }^3$. The product has actual-functional presentation independence and order-reversing involution.

7. F9a holds on the named fixed-support test families. F9b, quantum hypocontinuity, associativity for unspecified triples, and closure under unspecified intermediate products are not asserted.