A major conceptual step has recently been achieved within the No-Time Universe (NTU) framework. Until now, the NTU tensor was primarily interpreted as the effective source of gravitation emerging from the closure of pre-links. The new formulation provides a deeper geometrical interpretation.

The NTU tensor is now understood as the transformation matrix of the causal front as it propagates through the fractal surfaces generated by the NTU substrate.

The NTU substrate contains two complementary fractal surfaces:

the open quantum surface,$$\Sigma_Q = \chi \ell_P^2 N^{D_{\mathrm{eff}}}$$

and the closed gravitational surface,$$\Sigma_G = (1-\chi) \ell_P^2 N^{D_{\mathrm{eff}}}$$

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

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

The causal front continuously propagates through these surfaces. Within the open sector, unresolved pre-links generate a superposition of configurations. Within the closed sector, the shortest-action pre-links are selected and closed according to the NTU variational principle:$$\delta S_{\mathrm{NTU}} = 0$$

Each closed pre-link contributes to the rigidified gravitational surface.

The NTU tensor therefore becomes the mathematical representation of this selection and transformation process.

Its compact form is:$$T_{\mu\nu}^{\mathrm{NTU}} = -\frac{2}{\sqrt{-g}} \left[ \kappa_{\mathcal H} \ell_P^4 \mathcal B_{\mu\nu} – \kappa_{\ell} \mathcal C_{\mu\nu} \right]$$

where:$$\mathcal B_{\mu\nu}$$

describes the causal sweeping of the NTU hypervolume,

and:$$\mathcal C_{\mu\nu}$$

describes the closure matrix selecting the shortest-action pre-links. In this interpretation, gravitation is no longer viewed as a fundamental interaction. It emerges from the filtering action of rigidified fractal surfaces on the propagation of the causal front.

The NTU tensor simultaneously generates two complementary projections:$$T_{\mu\nu}^{\mathrm{NTU}} \longrightarrow G_{\mu\nu}$$

on the gravitational axis,

and:$$T_{\mu\nu}^{\mathrm{NTU}} \longrightarrow \psi$$

on the quantum axis.

The same geometrical object therefore underlies both General Relativity and Pilot-Wave Quantum Mechanics.

The NTU tensor can now be interpreted as the causal-front transformation matrix connecting the discrete NTU substrate to the continuous spacetime description observed in four dimensions.