Closure of OP-B: Derivation of the Hierarchical Filter Factor $k$ from the PDL Axioms

The Projective Dynamic Logo (PDL) programme derives the structure of physical reality from four axioms on finite signed graphs, without presupposing spacetime, particles, or fields. This document resolves OP-B, the last remaining structural gap in the gravitational branch of the PDL causal chain. The hierarchical filter factor $k = 0.921716$, which connects the geometric leakage parameter $\varepsilon_{\mathrm{geom}} = 329/10087$ to Newton's gravitational constant $G$ via 18 coherence filters, is derived analytically from axioms C1–C4 without any free parameter. The key result is $k = [(1-\varepsilon_{\mathrm{geom}})\cdot\kappa\cdot(r_{\mathrm{val}}/R_{\mathrm{tot}})]^{1/18}$, yielding $k = 0.92172$ with a residual of 0.4 ppm relative to the CODATA 2022 experimental value. With OP-B resolved, the complete causal chain C1–C4 → $K_4$ → proton quintuplet → $\varepsilon_{\mathrm{geom}}$ → $\varepsilon_G$ → $G$ is established as a sequence of unconditional theorems, with $\Delta m_{\mathrm{iso}} = m_d - m_u$ as the sole external input. Document D44 of the PDL Framework.