We present a covariant, completely positive and thermodynamically consistent refinement of the universal equation within the CUCE/Spinoza/Hilbert framework, denoted CUP-?*. The formulation unifies Tomonaga–Schwinger evolution on Cauchy hypersurfaces with a modular Gorini–Kossakowski–Lindblad Sudarshan (GKLS) generator that obeys detailed balance in the GNS metric with respect to a unified thermodynamic target σ*. We prove (i) foliation independence under local commutation, (ii) complete positivity of the finite-step propagator, (iii) existence of a global Lyapunov functional  DS ensuring the second law, (iv) primitivity with a unique attractor σ*, and (v) local conservation via consistent coupling to the Einstein–Langevin equation with conserved stochastic sources. We further outline falsifiable predictions with quantitative protocols in superconducting circuits and optomechanics.

• Covariant Open Quantum Systems; Quantum Markov Semigroups; Detailed Balance; Tomonaga Schwinger Equation; Stochastic Gravity; Information Geometry; Thermodynamic Learning 1

Gallardo VM (2026) The Universal Refined Equation (CUP-?*): Covariant GKLS Dynamics, Tomonaga–Schwinger Integrability, and Ein stein–Langevin Coupling in the CUCE/Spinoza/Hilbert Framework. J Phys Appl and Mech 3(1): 1016.

Spinoza’s monism can be operationally reformulated in Hilbert-space language: one substance, many modes as observable algebras. Within this CUCE/Spinoza/Hilbert programme, the CUP-?∗ equation supplies a universal dynamical law that is (a) covariant at the level of foliation by Cauchy hypersurfaces, (b) completely positive at finite steps, and (c) thermodynamically consistent via a modular target capturing both KMS equilibrium and observer/prior information through an affine geometric mean. Our presentation emphasizes rigorous mathematical structure and empirical consequences.

Let ρ[Σ] be the state functional on a Cauchy hypersurface Σ. The local Tomonaga–Schwinger (TS) evolution at x ∈ Σ reads

The jump operators are modular with respect to the unified thermodynamic target 

Where denotes the affine geometric mean. Equation (1) couples to semiclassical gravity via a conserved stochastic source through 

Axioms

(Causal) Local commutation. For spacelike separated x, y, the local superoperators commute:  and  (CPTP) Bochner positivity.  and the rates  arise from positive-definite (Bochner) environment correlators so that  (GNS) Detailed balance. With modular jumps (2), the generator is symmetric in the GNS inner product  (Prim) Primitivity. The set  generates the full local ∗-algebra  in the spectral basis of  implying a unique faithful stationary state. 

(Cons) Conservation and gauge. Fα are BRST-invariant; (4) uses conserved noise with fluctuation–dissipation relations.

Global Lyapunov functional and the second law

Theorem 1 (Monotonicity and exponential convergence). Under axioms (GNS) and (CPTP),  If in addition (Prim) holds, then  exponentially with a rate bounded below by the GNS spectral gap  Proof. Modularity (2) implies GNS symmetry [1]:  Hence   is positive in this metric and coincides with the gradient of  yielding monotonicity. Primitivity makes the zero eigenspace one-dimensional and opens a spectral gap; the GNS Poincaré inequality then gives exponential decay of Φ.

Finite-step complete positivity

Theorem 2 (Finite-step CPTP).  come from positive-definite correlators (Bochner), the finite-step propagator  Proof. The GKLS form ensures complete positivity of infinitesimal maps with Kossakowski matrix positive semidefinite [2,3]. The kernel  makes the finite-step map a convex average of CP maps. Trace preservation follows from the Lindblad form. 

Primitivity and uniqueness of the attractor

Theorem 3 (Unique fixed point). If  generates the full local algebra, the semigroup is primitive: ker  and the stationary state is unique. Proof. Irreducibility implies that the commutant of  andard quantum semigroup theory then yields uniqueness of the fixed point and a positive spectral gap. 

TS integrability and causality

Theorem 4 (Foliation independence). If  and  Satisfies the hypersurface deformation algebra, then the TS evolution is independent of the chosen foliation Σ.

Proof. Adapt Schwinger’s argument: local commutators integrate to zero over spacelike-separated elements, ensuring path-independence of the ordered exponential along deformations of Σ. 

Conservation and Einstein–Langevin coupling

Theorem 5 (Local conservation). With conserved noise  and fluctuation–dissipation relations, and including the Lamb-shift  the renormalized stress tensor satisfies  and (4) is consistent with Bianchi identities.

Falsifiability and quantitative predictions

We outline protocols that access the modular structure and the conserved stochastic back-reaction:

1. Entropy production bound in a qubit. Engineering {Fa} to be the full set of matrix units on a superconducting qubit makes the dynamics primitive. The relative-entropy half-life obeys  we predict  .

2. For Equilibrium test of the unified target. For a 5 GHz qubit at T = 50 mK, the KMS factor is  and the excited-state fraction is  Any deviation explained by  in (3) can be estimated by tomography.

3. Choi test of finite-step CPTP. Reconstruct the Choi matrix of the propagator over a finite TS “slab”; positivity must hold within uncertainties fixed by the noise kernel.

4. Optomechanical probe of conserved noise. For a membrane of mass  the on-resonance displacement noise floor is  at room temperature; a conserved Einstein–Langevin contribution at the level of of thermal noise would be marginally resolvable with state-of-the-art interferometry.

Limiting regimes

The principal asymptotic limits of CUP-?∗ and the corresponding recovered theories are summa- rized in Table 1. 

Figures (TikZ/PGFPlots)

GNS detailed balance and modular jumps. Choosing  self-adjoint in the GNS inner product and pins  as the unique fixed point under (Prim). This identifies the flow with a gradient flow for  in the sense of quantum information geometry [1] (Figure 1).

Figure 1 Local TS block: unitary plus modular GKLS applied on a surface element.

Finite-step CPTP with Bochner kernels. Bochner positivity guarantees the Kossakowski matrix is positive semidefinite; the finite-step propagator is a convex mixture of infinitesimal CPTP maps, hence CPTP (Figure 2).

Figure 2 Local integrability under spacelike separation..

Einstein–Langevin consistency. Conserved noise ensures  Bianchi identities then ensure compatibility of (4) (Figure 3). 

Figure 3 Typical Lyapunov descent of relative entropy with rate bound 2λgap.

Data availability: No datasets were generated or analysed for this theoretical study.

Code Availability: All LaTeX/TikZ/PGFPlots code to reproduce the manuscript is included in the accompanying project.

Author Contributions: VMG conceived the CUP-?∗ framework, developed the mathematical proofs and wrote the manuscript.

Acknowledgements

We thank the broader communities working on GKLS dynamics, information geometry and stochastic gravity for foundational insights.

1. Carlen EA, Jan Maas. Gradient flow and entropy inequalities for quantum markov semigroups with detailed balance. J Functional Analysis. 2017; 273: 1810-1869.
2. Gorini V, Kossakowski A, Sudarshan ECG. Completely positive dynamical semigroups of n-level systems. J Mathematical Physics. 1976; 17: 821-825.
3. Lindblad G. On the generators of quantum dynamical semigroups. Communications in Mathematical Physics. 1976; 48: 119-130.