The Consistency Functional 𝒦 (Part III): A Variational Framework Linking Quantum Theory, Thermodynamics, and Gravitation

This paper completes the Consistency Trilogy by bringing together the conceptual and mathematical developments introduced in The Consistency Principle (Oct 12, 2025) and The Consistency Functional 𝒦: The Unified Origin of Physical Law (Oct 16, 2025). It examines how core structures of physics—quantum dynamics, thermodynamic behavior, and spacetime geometry—may emerge as Euler–Lagrange conditions of a single variational relation,

δ𝒦(ρ, g) = 0,

where 𝒦 denotes a functional intended to measure informational-geometric self-consistency.

Building on the six axioms developed in earlier work, this final paper introduces two additional selection axioms that appear to single out specific informational and entropic structures:

Axiom 2′ (Metric Selection). Among monotone Riemannian metrics on quantum state space, the symmetric logarithmic derivative (SLD) quantum Fisher metric appears to satisfy full probabilistic and variational consistency.

Axiom 3′ (Entropy Selection). The von Neumann entropy emerges as the quantity most consistent with continuity, unitary invariance, additivity, and monotonicity under completely positive trace-preserving maps.

Under these assumptions, the stationary condition δ𝒦 = 0 is shown to lead to:

• Unitary quantum evolution (from δρ𝒦 = 0),

• Thermodynamic irreversibility (from CPTP monotonicity), and

• Einstein-like gravitational dynamics (from δg𝒦 = 0) in the appropriate geometric limit.

Within this framework, curvature, energetic behavior, and probabilistic evolution may be viewed not as independent postulates but as interconnected aspects of an underlying informational-geometric consistency relation. The paper concludes by formulating the Continuum Consistency Conjecture as a possible route toward a background-independent completion and outlines the broader Consistency Program as a direction for future work.

Together, Papers I–III suggest that the Consistency Principle may provide a coherent variational foundation capable of linking multiple domains of physical law.

Full manuscript currently under formal review; access temporarily restricted. DOI remains active for citation. 

Brenton E. Saba ORCID: https://orcid.org/0009-0006-8025-9662