On the Exponential Relaxation Limit of Vacuum Pressure Invariants and the Hydrodynamic Origin of Euler’s Number

<jats:p>Within the framework of the Fermionic Universe Hypothesis (FUH) and Project OCEAN, the physical vacuum is treated as a viscous fermionic condensate (VFC). This research note demonstrates that the phenomenological calibration constants regulating the background vacuum pressure P_ψ and the localized soliton pressure P local are governed by a strict exponential decay invariant. By applying a scale-space inversion layout, we show that the non-linear coupling of these macroscopic and microscopic pressure mantissas converges precisely to the classical relaxation quantum (1 – e^-1) ≈ 0.63212. Consequently, Euler's number 2.71828 emerges not as an arbitrary mathematical abstraction, but as a rigid hydrodynamic invariant defining the ultimate relaxation threshold of the vacuum lattice.</jats:p>