The Informational Arrow of Time: Deriving Thermodynamic Irreversibility from Lindblad Phase Relaxation

The emergence of thermodynamic irreversibility from fundamentally time-symmetric microscopic physical laws remains a foundational paradox in statistical mechanics. In this work, we present a self-contained theoretical framework operating over a composite Hilbert space Hsys ∼= HA ⊗ HT to resolve the arrow of time from first principles. We establish that while global evolution over the unobserved potential space HA is strictly unitary and time-symmetric under a self-adjoint recognitionoperatorKˆ,manifestdynamicsinspaceHT aregovernedbycontinuousLindbladphase relaxation driven by local informational density gradients I(x) ≡ Imax − svN(x). We prove that the directional increase of von Neumann entropy density (dSvN/dt ≥ 0) in manifest configurations is an exact algebraic consequence of Spohn’s inequality applied to the trace-preserving projection K → S. Finally, we derive a quantitative correction to quantum Poincar ́e recurrence times, demonstrating that while negligible in ordinary macroscopic regimes (∼ 10−40), it yields a strictly falsifiable prediction in ultra-high informational density regimes.