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

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 \(Sspec[Δ_g;M]\) 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), \(Sspec\) is strictly asymptotically equivalent to the Riemannian heat-kernel spectral entropy; (2) in the low-energy cutoff (IR limit), \(Sspec\) is strictly asymptotically equivalent to the Boltzmann state-counting entropy; (3) in the finite-rank projection limit, \(Sspec\) 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 \(τ∈(0,1)\), and prove, under the positive-definite deformation-tensor construction convention, that the map \(τspec\) is strictly monotonically decreasing, with higher-order asymptotic expansions supplied. We additionally prove that, within \(ζ\)-function regularization equivalence, \(Sspec\) 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 \(S^3\) 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 \(S^3\) compact configuration with the asymptotically flat open-manifold framework under the spectral dichotomy. Although the present construction is carried out on compact \(S^3\)-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 \(S^3\) theory.