Spectral gap of the Hamiltonian constraint in 2+1D quantum gravity with positive cosmological constant: Numerical Turaev-Viro spectra on the torus

We present a comprehensive numerical and analytical study of the Hamiltonian constraint spectrum for 2+1D quantum gravity with a positive cosmological constant \( Λ ∝ 1/k^2 \) on a spatial torus. Using the Turaev-Viro topological quantum field theory formalism at roots of unity \( q = eπ i / (k+2) \), we exactly diagonalize the Hamiltonian operator on the minimal triangulation (up to \( k=12 \)) and on refined 2×2 and 3×2 periodic lattices (at \( k=3 \) and partially at \( k=4,5 \)). The ground state exhibits the predicted topological degeneracy of \( k+1 \), and we confirm the existence of a non-zero physical excitation gap \( Δphys \) for any finite \( Λ \). Statistical scaling analysis demonstrates that the physical gap follows a smooth power-law decay, scaling as \( Δphys ∝ Λ0.895 ± 0.010 \). Importantly, the asymptotic offset is statistically consistent with zero, meaning the gap vanishes in the flat-space limit \( Λ → 0 \). We therefore characterize this as an IR-regulated gap or curvature-induced gap, not a fundamental topological mass. Analysis of the spatial wavefunctions reveals that the vacuum state acts as a topological condensate favoring an equilibrium curvature, while the first excited state represents a macroscopic quantum "breathing mode" of the spatial geometry.