Directional Information Theory from Geometric Coherence: The RTLI Framework

This work develops the information-geometric foundations of the Relational Thermodynamic Layered Index (RTLI), demonstrating that coherence can be understood not merely as a dynamical phenomenon, but as a geometric property of constrained parameter space. By endowing the RTLI coherence corridor [\lambda_{\mathrm{IR}}, \lambda_{\mathrm{UV}}] with a Fisher information metric, the parameter manifold acquires a natural Riemannian volume measure that modifies the effective counting of accessible states. As a consequence, the entropy of a system operating within the corridor is augmented by a geometric contribution, yielding the modified functional S_\lambda = H + \ln A(\lambda), \qquad A(\lambda) = (\lambda - \lambda_{\mathrm{IR}})(\lambda_{\mathrm{UV}} - \lambda), where H is the standard Shannon entropy. The logarithmic correction \ln A(\lambda) is not introduced heuristically; rather, it emerges uniquely from the Riemannian measure associated with the Fisher metric and represents the only entropy modification consistent with Boltzmann’s principle and additivity under shared geometric constraints. When the coherence parameter \lambda evolves according to slow-roll dynamics on the parameter manifold, the modified entropy exhibits a directional drift, \frac{dS_\lambda}{dt} = -2b(\lambda - \lambda_c), with \lambda_c denoting the midpoint (saddle point) of the coherence corridor. This drift persists even when the microstate distribution is stationary, establishing a geometric arrow of information that is distinct from thermodynamic irreversibility and arises purely from the structure of the parameter space. The framework yields concrete, falsifiable signatures across multiple domains. Geometric suppression near the corridor boundaries provides a potential mechanism relevant to the strong CP problem, offers a diagnostic lens for large-scale cosmological anomalies, and predicts coherence-dependent corrections to quantum information measures. Taken together, this work presents the first explicit information-geometric formulation of RTLI and identifies the coherence corridor as a unifying geometric structure with testable physical implications.