Linear Approximations in Exponential Functions and the Deconstruction of Permutational Redundancies: A Dynamic Person-Capacity Model

This paper establishes the definitive mathematical formulation of the "Person-Capacity-Preference" model, originally conceptualized by Alper Pektas. While linear approximations fail to model the curved trajectories of exponential fields, this framework introduces a Dynamic Correction Coefficient (S(x,n)) that maps any linear estimation directly to its exact exponential reality. The model resolves the transition between "overflow redundancies" (at high fractional exponents) and "underflow voids" (at low fractional exponents), providing a unified algebraic correction for all real bases and exponents.