Exact Derivation of Topological Casimir Tension in a Discrete Hydrodynamic Superfluid via Enstrophy-Bounded Euler-Maclaurin Regularization

First-Principles Foundation: The framework replaces traditional quantum field theory regularizations—such as Riemann zeta analytic continuation and ad-hoc subtraction of infinite bulk terms—with an exact, finite derivation grounded in classical continuum mechanics and discrete lattice dynamics. Superfluid Vacuum and Discrete Vortons: The physical vacuum is modeled as an inviscid, compressible superfluid supporting closed, quantized vortex filaments with a strictly finite spectral capacity of discrete modes. Dynamical Enstrophy Cutoff: The ultraviolet catastrophe is eliminated physically rather than formally. In three-dimensional inviscid flows, vortex stretching drives local enstrophy toward finite-time divergence, triggering topological reconnection and enforcing an upper bound at the vortex core scale. Acoustic Group-Velocity Flattening: Discrete nearest-neighbor coupling along the vortex chain produces acoustic dispersion where the group velocity matches the macroscopic sound speed in the long-wavelength limit and vanishes identically at the reconnection boundary due to Bragg backscattering. Euler-Maclaurin Topological Residue: Evaluating the bounded mode spectrum via the Euler-Maclaurin summation formula isolates the second Bernoulli number. The difference between the boundary group velocities generates the negative one-twelfth geometric residue directly, cleanly separating bulk volume energy and boundary chemical potential from the scale-invariant term. Mechanical Stability and Mass Floor: The scale-invariant residue produces an inward macroscopic Casimir tension that counterbalances outward hydrodynamic centrifugal expansion, stabilizing closed vortex solitons and providing a non-zero lower floor for physical mass excitations.