Time from Swirl: A Hydrodynamic Origin of Proper Time

Swirl--String Theory (SST) models matter as knotted swirl strings embedded in an incompressible, inviscid swirl condensate of effective density \(,=)\) and a local swirl energy density \[ _{} = {2}|^{2} +||^{2}) = )|^{2}}{_{\!}}}, )=dt_), \] leading to a ``redshifted'' Schr\"odinger evolution \( i)\,H[]\,_{} = {2}|^{2} -{2}||^{2}, \) we derive a vector Helmholtz equation \( +=0 \) whose bound solutions define discrete swirl shells and an effective core radius \( = -}{_{\!}} + +()dx^{j}, \] with \({2}||^{2}\), to summarize the impact of swirl on internal clocks and rulers while preserving a flat background condensate. Using the canonical SST constants and an axisymmetric reference profile, we obtain finite, strongly peaked swirl energy densities and a core swirl clock \(