A Zero-Less Regularized Sum for the Riemann Zeta Function with n^4 Cutoff: Computational Framework and Mellin Correction

We propose a unifying, cross-disciplinary framework termed the Zero-Less Method, postulating that the coordinate 0 is not a valid physical or mathematical state, but rather a structural singularity. By systematically forbidding r=0, we regularize classical mechanics via a modified force law F = k/(r^2+L^2), capping the maximum force at F_max=k/L^2. In abstract algebra, this corresponds to operating exclusively in the punctured finite field F_5*={L,2L,3L,4L}, where the absorbing element 0 is excluded to preserve the validity of bijective mappings a -> b. We apply this principle to astrophysics using Gaia DR3 data (Taniguchi et al. 2026; Baba et al. 2024). The data reveals that the Sun and 6,594 solar twins migrated ~10,000 light-years from the hostile Galactic Center (R~5 kpc) to the quiet outer disk (R~8.5 kpc). We argue that the Galactic Center R=0 is a forbidden zone for habitability, and the corotation barrier acts as an astrophysical L_min. By setting L_min to the Planck length (l_P ~ 1.6 x 10^-35 m), the Zero-Less Method remains perfectly consistent with all current macroscopic observations (LIGO, LHC, Gaia N-body dynamics) while resolving fundamental singularities. We conclude that the exclusion of zero is a universal prerequisite for structural integrity, stability, and the survival of complex life.