EXACT DERIVATION OF TOPOLOGICAL CASIMIR TENSION IN A DISCRETE HYDRODYNAMIC VACUUM VIA EULER-MACLAURIN REGULARIZATION

The zero-point Casimir energy, characterized by the zeta-regularized value ζ(−1) = −1/12, is conventionally derived within continuous Quantum Field Theory by discarding infinite bulk terms. We present
an exact, finite derivation grounded strictly in classical continuum mechanics. By modeling the vacuum
as a discrete, closed hydrodynamic vorton lattice with a finite spectral capacity N, the ultraviolet cutoff
is physically enforced by the divergence of local enstrophy. Applying the Euler-Maclaurin summation
formula to the bounded acoustic spectrum isolates the second Bernoulli number (B2 = 1/6), natively
generating the −1/12 topological residue without requiring analytical continuation or the subtraction
of infinities. This topological tension acts as an inward macroscopic pressure, stabilizing the discrete
manifold against pairwise hydrodynamic expansion.