Entropy Based Central Limit Theorem

This paper interprets the Central Limit Theorem (CLT) as the out- come of successive lossy projections imposed by measurement and aggregation, rather than as a primitive universality principle. Measurement is modeled as a non-injective projection from a high-dimensional system onto a reduced representational space, inducing probability and entropy through the collapse of distinguishable states. Within this framework, we derive the generalized Gamma distribution as the stationary form produced by irreversible asymmetry accumulation under stochastic dynamics with injection, decay, and scale-dependent fluctuations. This Gamma-type geometry represents the most informative distribution preserved after the first projection induced by measurement. We then show that averaging constitutes a second projection that suppresses all higher-order cumulants under normalization, yielding Gaussian convergence as a limiting case. Gaussianity thus emerges as a represen- tational shadow of the underlying Gamma geometry, with the rate of convergence governed by the Gamma shape parameters. This situates the CLT within a theory of projection, asymmetry, and information loss.