The strong core of the variational–spectral program: from coercivity in mobile geometry to the universal second-order classification

This upload gathers the strongest core of the variational–spectral program developed around quadratic residual energies, spectral coercivity, and the structural boundary between fixed and mobile functional geometries. At its center lies a unified mathematical framework for variational functionals of the form F (u) = ||R (u) ||², where the local second-order theory is governed by the closed quadratic form q (h) = ||Lh||² and by the associated self-adjoint operator A = L*L, with L = DR (u*) the linearization at a zero of the residual. The collection develops three tightly connected directions. First, it establishes universal second-order classification results showing that coercivity, stability, and local quadratic structure are canonically encoded by the operator A = L*L. Second, it analyzes the sharp boundary of this universality, proving that the classification generically breaks at third order because higher-order jets depend on nonlinear data that are not determined by A. Third, it studies coercivity for energies measured in solution-dependent metrics, identifying the precise obstruction created by mobile geometry and decomposing bilateral coercivity into three independent mechanisms: linear spectral coercivity, uniform weight non-degeneracy, quantitative nonlinear subordination. These abstract results are instantiated in conformal curvature problems on compact hyperbolic surfaces, where the canonical operator is the Jacobi-type operator J = -Δg0 + 2κ0, and where perturbative and extended bilateral coercivity estimates are proved for curvature residual energies. The program clarifies how spectral gaps govern local rigidity, how canonical operators emerge from geometric residuals, and why global coercivity may fail when the energy geometry itself moves with the solution. Overall, this deposit represents the most robust mathematical nucleus of the project: a coherent framework connecting quadratic residual functionals, spectral classification, bilateral coercivity, and the mobile-geometry obstruction in geometric analysis and mathematical physics.