Following the recent clarification of the NTU stochastic coefficient as the ratio of non-closed pre-links, a second major improvement has now been achieved.

The NTU framework now possesses a complete coarse-graining structure, including:

• a discrete evolution law at the substrate level,
• a physical interpretation of the stochastic term,
• explicit evolution functions for the quantum and gravitational branches,
• a direct projection toward Brownian dynamics,
• and the associated Fokker–Planck equation.

The key conceptual advance is that the quantum and gravitational branches are no longer represented by abstract functions. Both are now derived from the same underlying fractal geometry generated by the pantographic transformation law of the NTU.

The fractal surface is generated from the dimensionless parameter:$$N = \frac{\lambda_{dB}} {\ell_P}$$

where:$$\lambda_{dB}$$

is the de Broglie wavelength and$$\ell_P$$

is the Planck length.

The associated fractal coefficient is:$$N_k = \sqrt{\frac{1}{N}}$$

The resulting fractal surface is:$$\Sigma(N) = \ell_P^2 N^{D_{\mathrm{eff}}}$$

where:$$D_{\mathrm{eff}} \simeq 2.1$$

is the effective fractal dimension obtained from fractal thermodynamics.

The open and closed sectors are then defined as:$$\Sigma_Q = \chi\, \Sigma(N)$$$$\Sigma_G = (1-\chi)\, \Sigma(N)$$

with:$$\chi = \frac{\Sigma_{\mathrm{open}}} {\Sigma_{\mathrm{open}} + \Sigma_{\mathrm{closed}}}$$

representing the fraction of non-closed pre-links.

This construction provides a unified geometrical origin for both quantum and gravitational evolution.

Most importantly, the Brownian term is no longer introduced as an external stochastic assumption. It emerges naturally from the inability of a 4D observer to resolve the open fractal configurations of the NTU substrate.

This establishes a continuous derivation:$$\text{NTU Discrete Dynamics} \longrightarrow \text{Brownian Projection} \longrightarrow \text{Fokker–Planck Dynamics}$$

without introducing any arbitrary stochastic coefficient.

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