The Extended Zeta Function and Effective Planck Variations: Cosmological, Hydrodynamic, and Molecular Helical Manifestations

We present a unified framework rooted in the φ-Hurwitz extension of the Riemann zeta function. A small shift s → s + ε together with a global phase e iφ defines a two-parameter family that, when embedded into a helical geometry, remains Lorentz-covariant and induces an effective Planck variation ℏ eff = ℏ(1 + α h ε). This mechanism produces coherent, testable corrections across scales: an early-universe ∆N eff sufficient to account for the Hubble tension; dispersive quantum contributions in hydrodynamic balance; calibration via superfluid helium (in parallel with toroidal helical confinement); and measurable, isotopically sensitive shifts in molecular systems, including water and DNA. We emphasize the separation between a robust geometric core (helical embedding and Lorentz covariance) and speculative but falsifiable bridges (cosmology, fluid microstructure, biomolecular helices).