Variational Origin of a Fundamental Physical Unit from a Measured Vacuum Field.

This preprint develops an operator-theoretic and variational route connecting a measured vacuum field to an intrinsic physical scale. Starting from a vacuum-tension field $T_v(\mathbf{x},t)$ reconstructed from ultraviolet-regulated turbulence, and from an empirically identified linear operator describing its quasi-stationary dynamics (finite propagation speed, inertia, damping, and an intrinsic spectral gap), we introduce a quadratic variational formulation for stationary configurations of the form $T_v(\mathbf{x},t)=\phi(\mathbf{x}) e^{i\omega t}$. Restricting to the admissible stationary sector selected by normalizability and by a uniform spectral separation from the threshold frequency, we show that the associated energy functional admits a strictly positive infimum. This non-vanishing lower bound implies that stable and normalizable vacuum excitations cannot be continuously scaled down to zero within the regime of validity of the measured operator. We then define the Real Unit $\mathcal{B}$ as the minimal $\mathrm{L}^2$ field content compatible with spectral admissibility and a fixed reference stability level $E_*$, thereby providing an intrinsic amplitude scale without particle identification, quantization postulates, or external phenomenological constants. The analysis is conditional on the empirical validity of the measured operator and isolates a structural consequence that necessarily follows once such an operator is admitted. This work is part of a strict and transversal research program, inspired by the Navier–Stokes resolution paradigm: starting from a measured field, identifying a well-posed effective operator, and extracting necessary mathematical, physical, and ontological consequences. The aim is not to introduce an interpretative model, but to construct a closed chain linking data, operator structure, and the emergence of intrinsic physical quantities.