Gradient Differentials - Non-Equilibrium Entropy - Paper GD-III: The Independence of Kinetic Output and Informational Exhaust on the Discrete Relational Lattice

Essay III of the Gradient Differentials suite derives the differential structure of information production in the Gradient Fractal Field from first principles. It establishes the foundational independence of the kinetic and informational sectors: the per-Chronon entropy production rate dS/dτ = log₂(3) is an invariant of the discrete d = 3 lattice snap arithmetic, operating in strict structural separation from the suppressible kinetic output G(ρ). Three principal theorems are derived: (1) T.GD.ERS (The Two-Register Separation Theorem), proving that kinetic output and entropy rate occupy orthogonal operational registers; (2) T.GD.EPD (The Entropy Production Differential), establishing that the differential structure of entropy resides not in the rate (which is invariant) but in the total production per depth cycle H_fractal(n) = N_sat^(n-1) × (67/7) × log₂(3), which yields an exponentially scaling depth-differential dH/dn; and (3) T.GD.EZH (The Zero Horizontal Entropy Rate Differential), proving that the per-Chronon entropy rate is identically uniform across all 25 nodes of the Level-1 loxodrome. The paradigm shift of GD Essay III: the framework has two entropy differential structures, not one. The RATE differential is zero (the rate is a structural invariant of the d=3 lattice). The PRODUCTION differential is exponentially growing (the production is N_sat-scaled at each depth). These two results are not contradictory; they describe different aspects of the same lattice. Naming both precisely is the contribution of GD Essay III. Keywords: T.GD.ERS · T.GD.EPD · T.GD.EZH · T.GD.NED.ENT · Two-Register Separation · log2(3) invariant · H_fractal differential · Zero horizontal rate differential · Class T exponential production