Main Conjecture: Algebraicity of average von Neumann entropy per site in gauged lattice models

Establish that, for a gauged lattice model consisting of a d-dimensional lattice with canonically isomorphic local graded Hilbert spaces V_x, local nonnegative Hermitian Hamiltonians H_U, and local BRST differentials Q_U acting on finite subsets U and satisfying [Q_U, Q_{U′}] = 0 and [H_U, Q_{U′}] = 0 for all translations U′, the low-temperature and thermodynamic limits T → 0 and N → ∞ yield that the average von Neumann entropy per site, lim_{N→∞} (1/N) log dim(ker H_N^{phys}), equals the logarithm of an algebraic integer, where H^{phys} is the induced Hamiltonian on the physical state space V^{phys} = H^*(V, Q).

Background

After constructing a gauged lattice framework that includes both a global Hamiltonian and a global gauge (BRST) differential compatible via commutation, the authors relate categorical entropy to von Neumann entropy of the physical sector of the lattice model. They show that two special limits recover the quantum lattice case and a cohomological case aligned with categorical entropy computations.

They then propose the Main Conjecture asserting algebraicity of the average entropy per site in the gauged setting—intended to unify their Theorem A (quantum lattice algebraicity) and the categorical entropy algebraicity via Condition B. The authors explicitly note that the proof of this conjecture remains unknown, marking it as an open problem.