The GL Theory of Instantiation: Variational Selection and L-infinity Regularity of Adelic Ginzburg-Landau Vacua

We construct and analyze a globally coupled nonlinear Ginzburg-Landau theory over the idele class group, with local Archimedean and non-Archimedean kinetic terms and an arithmetic glue interaction coupling the places. We establish: (1) existence and uniqueness of a global weak minimizer modulo U(1); (2) L-infinity regularity of the vacuum via adelic Stampacchia truncation, replacing the failed tensorized Sobolev approach; (3) the glue Hessian reproduces the prime-sum structure of negative terms in the Weil explicit formula. Spectral identification with the Riemann zeros does not materialize — the arithmetic-spectral gap remains open and is precisely characterized. The paper presents complete variational and regularity results, and honest partial results toward the Riemann Hypothesis.