CUP-Ω∗: A Covariant GKLS–Einstein–Langevin Universal Equation for Thermodynamically Consistent Quantum–Informational Dynamics in the CUCE/Spinoza/Hilbert Framework

Introduction

A recurring technical challenge in both quantum information science and artificial intelligence is to design dynamics that are simultaneously expressive and stable. In quantum platforms, stability means admissibility of the evolution (complete positivity and trace preservation) and compatibility with thermodynamic constraints, often expressed by detailed balance and entropy production inequalities for quantum dynamical semigroups [1-4]. In machine learning, stability is often enforced by requiring monotone descent of a loss or free-energy functional, echoing information–thermodynamic constraints on computation [5-7]. The CUCE/Spinoza/Hilbert framework proposes that these motifs are not merely analogous but operationally unified: a single dynamical principle should govern reversible (Hamiltonian) change, irreversible organization (thermodynamics), and informational constraints (observer/prior).

The refined universal equation CUP- provides a concrete proposal in this direction by embedding completely positive opensystem evolution into the Tomonaga–Schwinger (TS) hypersurface formalism, and by allowing a stochastic semiclassical gravitational coupling consistent with the Einstein–Langevin approach [8-11]. This manuscript has three goals. First, we state a self-contained definition of CUP and identify minimal axioms guaranteeing covariance and finite-step complete positivity. Second, we establish a thermodynamic stability structure in which quantum relative entropy to a unified target state is a Lyapunov functional, yielding secondlaw monotonicity and exponential convergence under primitivity. Third, we provide falsifiable and quantitative predictions suitable for experimental tests in engineered open quantum systems and for conceptual transfer to AI as a template for stable “learning-like” dynamical flows.

Results

State Functional on Hypersurfaces and the Local Generator

Consistency Axioms 

Main Theorems: Covariance, CPTP, and Lyapunov Stability

Coupling to Stochastic Semiclassical Gravity

Figure 3: Schematic Lyapunov Descent: Relative Entropy to the Target State Decreases Monotonically

Under Detailed Balance and Converges Exponentially Under Primitivity

Visual Summary

Figures 1–3 provide schematic visualizations of the local TS update, integrability, and Lyapunov descent.

Quantitative Predictions and Falsifiability

Implications for Artificial Intelligence

Methods

From Coarse-Grained TS Dynamics to Covariant GKLS

Detailed Balance, Modularity, and Entropy Production

AI-Assisted Drafting Statement

Conclusion

and simulation platforms. The framework also provides a mathematically well-defined

bridge to AI via Lyapunov-stable descent under admissibility constraints [27].

Declarations

Author contributions

V.M.G. conceived the CUP-formulation, developed the theoretical structure

and proofs, and wrote the manuscript.

Competing interests

The author declares no competing interests.

Data availability

No datasets were generated or analyzed during the current study.

Code Availability

No simulation code is associated with this manuscript. All LATEX source code,

including TikZ/PGFPlots figures, is provided with this submission package.

Acknowledgements

The author thanks the open scientific community for foundational results on

quantum dynamical semigroups, detailed balance, and stochastic semiclassical

gravity, and acknowledges the use of AI tools for language and LATEX drafting support.

References