Direct Experimental Signatures of the Structural Constant L = 0.25 Across Physical Scales

A single structural constant L = 0.25, emerging from the spectral structure of a closed quartic variational functional, is tested across multiple independent physical domains. Starting from the Hessian spectrum Spec(H) = {1, L, L}, a hierarchy of observables is constructed without introducing external parameters or model-dependent assumptions. The analysis includes elastic proton–proton scattering (slope, radius, diffractive structure) measured by TOTEM, the presence of a C-odd component consistent with Odderon observations (TOTEM and DØ), and the extension of the baryonic spectrum to heavy doubly charmed states observed by LHCb. The same structural constant organizes intermediate and large-scale phenomena, including the leptonic hierarchy, the fine-structure constant, the gravitational hierarchy MP ~ MN L^{-32}, and the cosmological energy density ρΛ ~ L^{96}. In addition, gravitational-wave echo signals at ~43 Hz lead to an interferometric scale of ~3.46 km, consistent with existing detectors. The consistency of these results across independent experimental domains provides direct empirical support for a unified hierarchical organization governed by a single structural constant.