Gamma-Convergence of Free-Information Functionals on Quantum-Register Meshes

We establish a deterministic discrete-to-continuum theorem for free-information fields whose values are finite collections of quantum-register density operators. The local state space is the compact convex configuration space introduced in the companion finite theory, embedded in a finite-dimensional Hilbert space with the Hilbert–Schmidt metric. A mesh field carries two contributions: a finite-range quadratic register-mismatch energy and a local free-information density consisting of an energetic term, the relational-entropy hierarchy, and an optional log-determinant faithfulness barrier. Under explicit assumptions on cell quadrature, finite interaction range, directional second moments, discrete Poincaré compactness, and divergence consistency, the discrete Dirichlet forms Mosco-converge to an anisotropic continuum form. We then prove equicoercivity, the full Gamma-liminf inequality, a recovery theorem that respects the density-matrix constraints and the barrier, convergence of minima and almost minimizers, and mesh universality: any two admissible mesh sequences with the same limiting second-moment tensor have the same continuum functional. Natural Neumann and imposed Dirichlet boundary conditions are separated at the form level. In the convex regime the convergence strengthens to Mosco convergence of the complete functionals and therefore to convergence of resolvents and variational semigroups. We also give sharp failure statements: without a convergent directional tensor there need not be a unique full-sequence limit; the local continuum theorem does not solve the overlapping quantum marginal problem; and static Gamma-convergence alone does not derive a physical open-system dynamics. No random-substrate, Lorentzian, gravitational, gauge, particle-physics, cosmological, or empirical claim is made.