We complete the ultraviolet analysis of the four-dimensional nonlinear sigma model with target manifold M = Sp(2N,R)/U(N) equipped with the Fisher-Bures metric, the fundamental sigma model of the Vacuum Time Geometry theory. Building on the perturbative and functional-renormalization-group (FRG) results of the companion paper Paper 2, we present three new, mutually independent confirrmations of asymptotic freedom at all couplings T > 0: (i) Higher-order heat-kernel analysis. We compute the Seeley DeWitt coefcient a6 in closed form for Sp(2N,R)/U(N), using the covariantly-constant Riemann tensor of the symmetric space. For N = 28 we obtain a6 = −2040309414769/1620 ≃ −1.259×109, verifed against direct Lie-algebra computation for N = 2, . . . , 7 to relative accuracy ≤ 6.11×10−16. The closed-form expressions for the cubic Riemann invariants I1 = RijknRijlpRknlp and I2 = RijknRilkpRjlnp are new. (ii) Fourth FRG truncation (FRG-IV). Upgrading the heat-kernel truncation of Paper 2 (FRG-III, through a2) to include a4 and a6 yields a correction of ∼ 0.2% relative to FRG-III across T ∈ 0.5, 15. The beta function remains strictly negative, confirming the stability of the non-perturbative prediction. (iii) Lattice Monte Carlo and Wilson flow. We simulate the sigma model directly on Sp(4,R)/U(2) (Siegel upper half-space H2) on a 44 hypercubic lattice with Metropolis dynamics in the tangent space and implement a Lüscher-type RK3 gradient flow adapted to the target geometry. The flow-defined beta function is negative for every coupling T ∈ 0.5, 5 and every ow time, in qualitative agreement with FRG-III/IV. At T = 5.0, perturbation theory predicts βpert = +1.81 while the lattice yields βflow ≃ −1.19, confirrming that the perturbative zero T ∗ ≃ 6.3 is an artefact of two-loop truncation. Together with Paper 2, these results close Axis 2 of the Vacuum Time Geometry programme: the sigma model has no ultraviolet fixed point at finite coupling, flows toward the Gaussian fixed point T = 0 from every initial condition probed, and is therefore asymptotically free in the non-perturbative sense-consistent across perturbation theory, four FRG truncation schemes, and direct lattice simulation.
ignacio caldini (Thu,) studied this question.