adelic analytic torsion in infinite dimension: von neumann traces, finite-level envelopes, and l2 spectral density

We construct an adelic analytic torsion invariant in infinite dimension for automorphic representations of reductive groups over \mathbb{Q}. The invariant is defined through the von Neumann trace of a projected heat semigroup on the adelic Hilbert space \mathcal{H} = L^2(G(\mathbb{Q}) \backslash G(\mathbb{A})), regularized via a spectral zeta function canonically associated with the Fuglede-Kadison determinant. The central unconditional result is analytic. Under uniform bounded geometry, uniform coefficient control, and a positive spectral gap along an analytically admissible tower \{X_j\} of finite adelic levels, the Rodrigues-Ray-Singer and Quillen envelope estimates give |\log T_j(\pi)| \leq B_{\text{ad}} \text{Vol}(X_j) uniformly in j. Under the standard L^2-determinant approximation hypothesis, the normalized finite-level torsion densities converge and the adelic torsion satisfies |\log T_{\mathbb{A}}^{(2)}(\pi)| \leq B_{\text{ad}}. The dimension-two case, arising when finite adelic levels are compact Shimura curves, is handled by a Quillen-Rodrigues envelope for the Dolbeault complex proved in Appendix A. This appendix provides a complete Mellin analysis for real dimension two, including explicit control of the critical logarithmic coefficient. No arithmetic identification with special values of L-functions is assumed or proved. The paper establishes the analytic foundation on which such identifications can be built: construction, regularization, finite-level approximation, and uniform boundedness of the adelic torsion density.