What question did this study set out to answer?

The research aims to unify three distinct types of entropy—geometric, statistical, and von Neumann—using a common spectral entropy functional on three-manifolds.

July 5, 2026Open Access

Spectral Entropy Functional on Compact Three-Manifolds: A Unified Limiting Framework for Geometric Entropy, Statistical Entropy, and von Neumann Entropy

Key Points

The research aims to unify three distinct types of entropy—geometric, statistical, and von Neumann—using a common spectral entropy functional on three-manifolds.
Constructed a global spectral entropy functional on three-dimensional Riemannian manifolds using the eigenspectrum of the Laplace-Beltrami operator.
Proved asymptotic equivalences of the spectral entropy to geometric, Boltzmann, and von Neumann entropies under different limits.
Defined a topological compactness parameter and explored its relationship with the spectral entropy functional.
Established strict asymptotic equivalence of spectral entropy with Riemannian heat-kernel entropy in the UV limit.
Demonstrated that spectral entropy corresponds to Boltzmann counting entropy in the IR limit.
Reproduced the Bekenstein–Hawking entropy expression for static black-hole metrics, confirming the framework's applicability.

Abstract

Geometric entropy from heat-kernel traces, Boltzmann counting entropy over discrete state spaces, and von Neumann trace entropy over finite-dimensional Hilbert spaces operate in disjoint mathematical domains with no common framework. This paper constructs a global spectral entropy functional \ (Sₒ₏₄₂g;M\) on three-dimensional smooth closed simply connected Riemannian manifolds, depending only on the eigenspectrum of the Laplace–Beltrami operator, and proves that it unifies all three classical entropies in their respective limiting regimes: (1) in the large-eigenmode asymptotic (UV limit), \ (Sₒ₏₄₂\) is strictly asymptotically equivalent to the Riemannian heat-kernel spectral entropy; (2) in the low-energy cutoff (IR limit), \ (Sₒ₏₄₂\) is strictly asymptotically equivalent to the Boltzmann state-counting entropy; (3) in the finite-rank projection limit, \ (Sₒ₏₄₂\) is isomorphic to the von Neumann density-matrix trace entropy. Furthermore, on a one-parameter smooth deformation family of manifolds, we define a dimensionless topological compactness parameter \ ( (0, 1) \), and prove, under the positive-definite deformation-tensor construction convention, that the map \ (ₒ₏₄₂\) is strictly monotonically decreasing, with higher-order asymptotic expansions supplied. We additionally prove that, within \ (\) -function regularization equivalence, \ (Sₒ₏₄₂\) is the unique Laplace spectral functional simultaneously satisfying the three cross-scale asymptotic equivalences and the compactness monotonicity constraint; we also demonstrate that all admissible analytic regularizations introduce only global constant offsets, leaving the topological dichotomy classification stable. As model verifications, on the static \ (S³\) black-hole metric manifold we establish the correspondence between horizon radius and manifold characteristic parameter, rigorously reproduce the Bekenstein–Hawking entropy expression, and compare the \ (S³\) compact configuration with the asymptotically flat open-manifold framework under the spectral dichotomy. Although the present construction is carried out on compact \ (S³\) -type manifolds, the spectral entropy framework is structurally extendable to higher-dimensional and rotating spacetimes; these extensions are developed in companion papers and do not affect the rigor of the core theorems. The spectral entropy functional fills a mathematical gap in cross-scale operator invariants on compact three-manifolds, is complementary to the local scattering-axiom framework, and together they constitute the complete mathematical foundation of the \ (S³\) theory.

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