Gradient Mechanics: The Dynamics of the Inversion Principle - Corpus Paper VI - The Reciprocal Necessity of Registration: The Derivation of the Transmissive Operator (η) from Informational Density (F )

Following the derivations of Resistance (Θ) from Constraint (C) in Paper IV and Drive (∆) from Systematization (E) in Paper V, this paper completes the primitive transformation triad by deriving the Transmissive Operator (η) from Registration (F ). We prove that when the Multiplicative Trap (G = E × C × F ) undergoes topological inversion to achieve dimensional consistency (G = E×C / F ), the primitive F must undergo a functional reversal from multiplicative co-dependence to divisive regulation. Through exhaustive logical analysis, we demonstrate that F in Phase I represents informational density—the resistive grain of the medium that thickens configuration space—while in Phase II, the system requires not density but its mathematical reciprocal: the transmissive capacity of the medium to conduct gradient resolution. This reciprocal relationship is η = 1 / F , the gain multiplier that scales net force into kinetic output. We eliminate all alternative formulations through dimensional analysis, conservation requirements, and the Zero-Product Property. The derivation establishes η ≈ 1.667 as the scalar-invariant transmissive constant, fixing the final parameter required for kinetic mechanics. This paper proves that η is not a measure of anthropic utility but the structural reciprocal of informational grain—the medium’s intrinsic capacity to amplify or dampen flux transmission based on its registration architecture.