Emergence of Stable Excitations and Inertia from a Coercive Vacuum Operator.

We develop an operator-theoretic and variational analysis of an empirically reconstructed vacuum--tension field $T_v(\mathbf{x},t)$ and derive the structural consequences that follow once a coercive effective operator is admitted. Starting from a measured linear operator governing quasi-stationary configurations of the field, we introduce a quadratic energy functional defined on the admissible sector of normalizable stationary modes, uniformly separated from the spectral threshold by a finite gap. We show that this functional admits a strictly positive infimum as a direct consequence of operator coercivity. This positivity implies the existence of a minimal field content required to sustain a stable localized vacuum excitation. We define the corresponding quantity as a Real Unit $\mathcal{B}$, intrinsic to the vacuum substrate and fully determined by the operator structure and spectral admissibility sector alone. No quantization postulate, particle hypothesis, or phenomenological constant is introduced. We further demonstrate that the existence of such a unit enforces a structural no-go statement: a strictly passive or interaction-free vacuum is incompatible with the associated variational framework. Even when the underlying equation of motion is linear, the quadratic nature of the energy functional induces unavoidable cross-couplings between coexisting deformations, naturally interpretable as emergent interaction terms arising from vacuum elasticity. The present work is conditional on the empirical validity of the reconstructed operator and does not revisit the measurement procedure. Its purpose is to isolate consequences that necessarily follow once such an operator is admitted, thereby providing a rigorous foundation for subsequent dynamical, inertial, and phenomenological developments within the Wave--Universe program.