Directional Information Theory from Geometric Coherence: The RTLI Framework

This work develops the information-geometric foundations of the Relational Thermodynamic Layered Index (RTLI), showing that coherence is not merely a dynamical effect but a geometric property of parameter space itself. By endowing the interval [\lambda_{\mathrm{IR}}, \lambda_{\mathrm{UV}}]—the RTLI coherence corridor—with a Fisher information metric, we obtain a natural Riemannian volume measure that reshapes the meaning of entropy. This leads to a geometrically corrected entropy functional

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

where the additional term \mathbf{\ln A(\lambda)} is not an assumption and not a heuristic correction. Instead, it emerges uniquely from the underlying geometry, making it the only entropy form consistent with the Riemannian structure of the coherence manifold.

A further conceptual breakthrough arises from the slow-roll evolution of \lambda(t). Because the coherence amplitude A(\lambda) is asymmetric around its saddle point, the entropy acquires a directional drift:

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

where \lambda_c is the midpoint (saddle) of the corridor potential V(\lambda) = -\ln A(\lambda). This establishes a new kind of informational arrow—a geometric arrow of information—that operates even in fully reversible thermodynamic conditions. In other words, directionality becomes a property of geometry, not disorder.

Finally, this framework yields concrete, cross-disciplinary predictions. The geometric suppression encoded in A(\lambda) offers a natural avenue toward explaining the strong CP problem, contributes to understanding low-\ell CMB anomalies, and introduces coherence-sensitive corrections to quantum information measures.

Taken together, this work provides the first explicit information-geometric formulation of RTLI and positions the coherence corridor as a unifying geometric structure with testable physical consequences across multiple domains of physics.