A new numerical validation of the NTU coarse-graining equation has been completed.

This test represents an important milestone because it combines two previously independent developments:

• the fractal thermodynamics framework developed by Queirós-Conde,
• and the No-Time Universe (NTU) coarse-graining framework.

Both approaches independently converge toward the existence of an effective fractal dimension:$$D_{\mathrm{eff}} \simeq 2.1$$

Within fractal thermodynamics, this dimension characterizes the geometry of the quantum fractal paths.

Within the NTU framework, the same dimension naturally emerges from the fractal pilot surface generated by the pantographic transformation law acting on the pre-link substrate.

The numerical simulations were therefore performed using the common fractal coefficient:$$D_{\mathrm{eff}} \simeq 2.1$$

and the NTU surface law:$$\Sigma(N) = \ell_P^2 N^{D_{\mathrm{eff}}}$$

where:$$N = \frac{\lambda_{dB}} {\ell_P}$$

The resulting quantum and gravitational evolution functions are:$$F_Q = \chi\, \ell_P^2 N^{D_{\mathrm{eff}}+\frac{1}{2}}$$

and$$F_G = (1-\chi)\, \ell_P^2 N^{D_{\mathrm{eff}}-\frac{3}{2}}$$

The simulations confirm that the discrete NTU evolution equation:$$X_{n+1} – X_n = \Big[ \chi_n F_Q + (1-\chi_n)F_G \Big] \Delta_{\mathrm{NTU}}$$

naturally converges toward the continuous stochastic description:$$dX = \Big[ \chi F_Q + (1-\chi)F_G \Big]dt + \chi G_0 dW_t$$

and subsequently toward the associated Fokker–Planck equation:$$\frac{\partial P}{\partial t} = – \frac{\partial} {\partial X} \left\{ \left[ \chi F_Q + (1-\chi)F_G \right] P \right\} + \frac{1}{2} \frac{\partial^2} {\partial X^2} \left[ \chi^2 G_0^2 P \right]$$

The numerical results reproduce the expected variance:$$\mathrm{Var}(X_t) = \chi^2 G_0^2 t$$

with excellent agreement between the discrete simulations and the continuous analytical prediction.

Perhaps more importantly, this validation provides a direct physical interpretation of the Brownian term.

In the NTU framework, stochasticity is no longer introduced as an external mathematical assumption. It emerges from the fraction of unresolved open pre-link configurations:$$\chi = \frac{\Sigma_{\mathrm{open}}} {\Sigma_{\mathrm{open}} + \Sigma_{\mathrm{closed}}}$$

The Brownian process therefore appears as the effective 4D projection of a deeper fractal geometry rather than as a fundamental property of nature.

This new result strengthens the convergence between fractal thermodynamics, stochastic descriptions of physical systems, and the NTU framework, while providing a complete mathematical pathway from discrete substrate dynamics to the continuous Fokker–Planck description observed in emergent spacetime.