Directional Information Theory from Geometric Coherence: The RTLI Framework

This work develops the information-geometric foundations of the Relational Thermodynamic Layered Index (RTLI) and introduces a directional extension of Shannon entropy arising from the geometry of the RTLI coherence corridor. Using the Fisher information metric defined on the interval [\lambda_{\mathrm{IR}}, \lambda_{\mathrm{UV}}], we derive a natural Riemannian measure that modifies the standard Shannon entropy by an additive geometric term, leading to the coherence-corrected entropy

S_\lambda = H + \ln A(\lambda), \qquad A(\lambda) = (\lambda - \lambda_{\mathrm{IR}})(\lambda_{\mathrm{UV}} - \lambda).

The term \ln A(\lambda) is not heuristic—it follows uniquely from the Riemannian volume form associated with the Fisher metric on the coherence manifold. We further show that the slow-roll dynamics of \lambda(t) impose a geometric arrow of information, with entropy drifting according to

\frac{dS_\lambda}{dt} = -2b(\lambda - \lambda_c),

where \lambda_c is the unique saddle point of the potential V(\lambda) = -\ln A(\lambda). This establishes a novel directionality in information flow that is independent of thermodynamic irreversibility.

The framework yields testable predictions across cosmology, particle physics, and quantum information—most notably a geometric suppression mechanism potentially relevant for the strong CP problem and low-\ell CMB anomalies. The approach is mathematically rigorous, internally consistent, and provides the first explicit information-geometric formulation of the RTLI coherence structure.