From Scalar Certificates to Spectral Bounds: Radial Pulsations, Discrete Hardy Forms, and SU(N) Lattice Gauge Theory

We examine the operator content of three scalar certificate families and supply explicit spectral comparisons where the stated hypotheses permit them. For radial stellar pulsations, we derive a Newtonian work identity and a self-adjoint general-relativistic form, distinguish homologous trial values from fundamental eigenvalues, and recover a post-Newtonian structural benchmark. For a discrete Dirichlet Hardy-type form with constant confinement, we obtain the complete finite spectrum and the sharp box-uniform spectral floor; direct action variations identify an obstruction to a proposed physical coupling identification. For the SU(N) Hamiltonian lattice gauge model at fixed lattice spacing, a full-space single-plaquette comparison gives an explicit gap for all N at the stated coupling, and an application of Yarotsky's weak-interaction theorem gives a sufficiently strong-coupling window uniform in N and open-lattice volume, including the canonical infinite-volume physical GNS representation. The paper separates analytic proofs, applications of established external theorems, Lean-certified scalar results, and finite numerical checks. It does not assert a stellar mass cutoff, physical selection of the coupling kappa=1/2, a numerical many-plaquette threshold, or a continuum Yang-Mills mass-gap solution. The supplementary archive includes proof notes, Lean sources, reproducible computation scripts, results, and provenance records. The manuscript is CC BY 4.0; repository software in the supplement retains its AGPL-3.0-or-later license.