A Dissipative Informational Geometry of the Collatz Map : Informational Curvature and Subcritical Dynamics

The Collatz conjecture is a long-standing open problem in number theory characterized by simple local rules and unexpectedly complex global behavior. Despite extensive numerical verification and diverse analytical approaches, no classical proof of convergence is currently known. In this work, we present a structural reinterpretation of the Collatz map within Viscous Time Theory (VTT), a heuristic framework for dissipative information flow in discrete dynamical systems. Rather than addressing the conjecture through additive or probabilistic reasoning, we model Collatz iterations as trajectories evolving within an informational field governed by admissibility, curvature, and dissipation constraints. We introduce quantitative measures of informational curvature (ΔC), admissibility (Φα), and dissipation, and apply them to large-scale validated Collatz trajectories. Independent computational validation confirms that Φα remains strictly subcritical and that the mean informational curvature is negative across all tested initial conditions. These results indicate that local expansion events do not accumulate into global divergence, but are structurally confined within a stable informational basin. We classify the Collatz dynamics as a dissipative system operating in a non-additive informational regime, denoted Metamorphosis 4, in which classical conservation-based reasoning ceases to be applicable while identity persistence remains intact. This approach does not constitute a proof in the axiomatic sense, but provides a validated structural explanation for the absence of divergent trajectories. The proposed framework offers a new perspective on Collatz-type problems and suggests broader applicability to discrete dynamical systems exhibiting apparent paradoxical behavior.