Theorem: Combined IX-C-ST release, with ordered stage verdicts

Relative to the shared explicitly stated analytical inputs and finite operational permissions:

1. Part I provides the local Grade S first-stage and criteria package. At that stage complete factorization is unresolved.

2. Part II adds concrete relation kernels, full chart support of $K$, injective degree-two presentations, ${\mathcal{C}}_{\mathrm{desc}}=\mathbb{R}$, canonical admitted two-edge outputs, and degree-two second-stage maps. At that stage complete associations remain unreleased.

3. Part III proves both degree-four presentation kernels zero, all four degree-four descents, chart-local boundary-charge separation, a direct new-domain decomposition, and zero overlap of the whole new sum with the full old domain.

4. For every fixed real common parent,

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

   and Definition def:product constructs one partial product on ${\hat{D}}_{\mathrm{decl}}$, modulo ${\hslash }^4$. Both associations of the selected word satisfy (eq:word). Reversal, actual-functional presentation independence, and F9a hold on that sector.

5. Each stage can use the same explicit destructive-clock, fixed-table, residual, locality, and regular reference-preserving transport interface.

The final declared partial-product ending is Constructed relative to the stated inputs. F9b, associativity for unspecified triples, general intermediate-domain closure, a causal lift, G2, and IX-D-ST are not constructed.

Proof. The intermediate releases are Theorems thm:Crelease and thm:Freerelease. The degree-four and charge results are proved in Sections sec:degree4–sec:charge. Section sec:gluing proves simultaneous gluing and defines the actual partial product. Section sec:coherence proves its selected coherence, reversal, presentation independence, and continuity. Theorem thm:hosttransfer supplies the supported host placement. Section sec:operational supplies every finite operational clause independently of the product gates. No result there defines a causal field product.