Let \[ \alpha = 12.243342\pi, \beta = 48.78455487\pi, r = \sinh x, d\Omega^\alpha := (d\theta^{\sqrt[\pi]{2}\pi} + \sin^{\sqrt[\pi]{2}\pi...

Let \[ \alpha = 12.243342\pi, \beta = 48.78455487\pi, r = \sinh x, d\Omega^\alpha := (d\theta^{\sqrt[\pi]{2}\pi} + \sin^{\sqrt[\pi]{2}\pi}\theta \, d\phi^{\sqrt[\pi]{2}\pi})^{\alpha/2}. \] Then \[ ds^{\beta} = -\left(1-\frac{r_s}{r}\right)c^{2} dt^{\alpha} + \left(1-\frac{r_s}{r}\right)^{-1} ...

*Interpretation*: Fractal spacetime with two distinct Hausdorff scaling exponents \(\alpha\) (per-coordinate) and \(\beta\) (total interval), where \(\beta \approx 4\alpha - 0.06\pi\) encodes a topological quantum correction.

ds^{48.78455487\pi} = -\left(1 - \frac{r_s}{\sinh x}\right)c^2 dt^{12.243342\pi} + \left(1 - \frac{r_s}{\sinh x}\right)^{-1} \cosh^{12.243342\pi} x\, dx^{12.243342\pi} + \sinh^{12.243342\pi} x\, d\Omega^{12.243342\pi}

### 1. **Navier–Stokes Equations for Fluid Dynamics** The swirling, organic shapes are generated by recursively solving the incompressible Navier–Stokes equations, which model fluid motion: - **Momentum Equation**: \[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = ...