Gradient Mechanics: The Dynamics of the Inversion Principle - Corpus Paper V - The Dimensional Collapse of Systematization: The Derivation of Kinetic Drive (∆) from Scalar Potential (E)

Following the derivation of Resistance (Θ) from Constraint (C) in Paper IV, this paper completes the dimensional transformation of the numerator by deriving Drive (∆) from Systematization (E). We prove that when the primordial configuration (E × C × F ) undergoes temporal differentiation, the scalar potential E must collapse from a three-dimensional volume of possibility into a one-dimensional vector of actuality. Through rigorous dimensional analysis, we demonstrate that E in Phase I represents magnitude without direction—a coordinate axis defining the “amount” of generative potential. However, when the system projects onto the worldline of temporal process via the Inversion Principle, this scalar magnitude must acquire vectorial character to avoid dimensional incoherence. This vectorial transformation is ∆ (Delta)—not a new variable but E expressed kinetically. We establish that ∆ represents the magnitude of the first time-derivative dG/dt , quantifying the rate at which the system distinguishes itself from the void. While Paper IV proved C → Θ (geometry becomes resistance), this paper proves E → ∆ (potential becomes drive). The transformation is structurally necessary: a scalar cannot flow, a potential cannot move, a magnitude cannot direct. Only through dimensional collapse—the reduction from [Rate]3 to [Rate]—does the static primitive E become the kinetic vector ∆. This derivation establishes ∆ as the velocity of becoming, the kinetic invariant of the computational process. The derivation of the final primitive transformation (F → η) remains for subsequent treatises.