Derivation of Planck's Constant from Continuum Hydrodynamics: Resolving the Ultraviolet Catastrophe via Elastic Substrate Cutoffs and Vacuum Pre-Stress

In standard quantum mechanics, Planck’s constant h is universally treated as an irre- ducible, axiomatic postulate introduced historically by Max Planck to resolve the ultra- violet catastrophe of blackbody radiation. Building upon our foundational ontological framework of the physical vacuum as a pre-stressed, fully occupied elastic continuum, we demonstrate that Planck’s constant is not an independent metaphysical constant, but an emergent scaling bridge coupling the continuum’s microscopic unit-cell geom- etry (ℓ0), characteristic phase impedance (Z0), intrinsic propagation velocity (c), and omnidirectional hydrostatic pre-stress (P0). Drawing a direct epistemological parallel to Peter Debye’s resolution of lattice specific heats in solid-state physics, we show that the Rayleigh-Jeans divergence was an artifact of assuming an infinitely divisible, contin- uous void. We derive the constitutive relation h= P0ℓ4 0/c from the action integral of an elementary displacement cell. Evaluating this relation reveals a two-tier physical hier- archy: (1) an absolute microscopic structural cutoff at the Planck length (ℓ0 ∼ 10−35 m) governed by gravitational vacuum energy density, and (2) an electrodynamic bound- ary at the Compton/Schwinger threshold (ℓcr ∼ 10−13 m) governing matter stability and pair creation. Furthermore, we establish the dynamical origin of the exponential Boltz- mann factor as a nonlinear impedance saturation of the substrate and show that high- energy dispersion corrections satisfy contemporary astrophysical bounds on Lorentz invariance violation.