NTU Test #5 : Convergence assessment with eligible Quantum Gravity theories

The No Time Universe conjecture is currently undergoing a convergence assessment with the main quantum gravity programs that remain theoretically eligible: Causal Set Theory, Loop Quantum Gravity, Spin Foam Models, Causal Dynamical Triangulations, Holography, String/Brane Theory, Asymptotic Safety, and Emergent Gravity. The rationale behind this investigation is that the NTU represents a primordial cosmological phase predating the emergence of Causality, Time, Gravitation and Quantum mechanics. Consequently, the surviving quantum gravity theories may be interpreted as intermediate stages in the transition from the NTU to the emergence of the fundamental structures and physical laws that characterize our universe.

This test is inspired by the previous discussion with Queyros Quonde on fractal thermodynamics, where a significant convergence was identified through the independent calculation of the fractal coefficient, providing an encouraging precedent for the present investigation.

NTU Test 5.1 Convergence table

Eligible theory	Core principle	Possible NTU interpretation	Convergence level	NTU hypothesis
Causal Set Theory	Spacetime is fundamentally discrete and ordered by causal relations.	First causal rigidification of NTU configurations, before ordinary time emerges.	Very high	Causal sets may be the first ordered layer extracted from the atemporal NTU.
Loop Quantum Gravity	Quantum geometry is described by spin networks; spin foams describe their evolution.	A rigidified geometric phase of NTU pre-links after cosmological stabilization.	Very high	LQG may describe a later, already-geometrized NTU phase.
Spin Foam Models	Histories of quantum 3D geometries define quantum 4D spacetime.	Coarse-grained histories of NTU configurations.	Very high	Spin foams may be the covariant coarse-grained form of NTU micro-collisions.
Causal Dynamical Triangulations	Lorentzian quantum gravity is built from causal dynamical lattices.	A phase where causality and time have already stabilized.	High	CDT may correspond to the first Lorentzian/macroscopic phase of NTU.
Holography / AdS-CFT	A gravitational bulk can be encoded on a lower-dimensional boundary.	Projection of an NTU causal front into realized 4D events.	High	Holography may be the informational shadow of NTU projection.
String Theory / Branes	Fundamental objects are extended one-dimensional strings or higher-dimensional branes.	Pre-links and rigidified NTU surfaces.	Medium to high	Branes may be stabilized surfaces of pre-links.
Asymptotic Safety	Quantum gravity is controlled by a UV fixed point under renormalization group flow.	Continuum limit of NTU coarse graining.	Medium	The UV fixed point may represent the stable coarse-grained NTU action.
Emergent / Entropic Gravity	Gravity emerges from information, entropy, or thermodynamic constraints.	Gravity as density/tension of connected pre-links.	High conceptually	Entropic gravity may describe the thermodynamic limit of NTU pre-link networks.

The NTU baseline is a 3D atemporal substrate made of Planck-scale rigid nodes, where de Broglie waves are reinterpreted as physical pre-links and motion occurs in two stages: creation of pre-links, then energy transfer. The manuscript explicitly defines the NTU as a 3D timeless structure, distinct from our 4D spacetime, with rigid Planck-scale nodes and real guiding waves/pre-links . Its coarse-graining equation already has the structure of a stochastic continuum limit: a drift term plus a noise term suppressed by the collision/continuity parameter $N$N .

NTU Test #5.2 : theory-by-theory NTU short assessment

Causal Set theory : as a first causal rigidification

Causal Set Theory defines a structure:$$C = (E,\prec)$$

where $E$ is a set of elementary events and $\prec$ is a partial order satisfying reflexivity, antisymmetry, transitivity, and local finiteness. The order relation represents causal precedence; local finiteness introduces discreteness.

NTU interpretation

The NTU starts before ordinary time. Therefore, its first possible rigidification cannot be temporal. It must be relational.

Let the NTU microstate be:$$X_n = (p_n,S_n)$$

where $p_n$pn​ represents the gravitical/energy-transfer component and $S_n$Sn​ the quantum/pre-link phase component. The manuscript already defines a collision-based microdynamics in which each collision redistributes momentum between a gravitical axis and a quantum/pre-link axis .

A causal relation can then emerge when two configurations become connected by a realizable pre-link:$$e_i \prec e_j \quad \Longleftrightarrow \quad \exists \gamma_{ij} \ \text{such that} \ \Delta S_{ij} \neq 0$$

and the energy-transfer condition is not yet required.

So the causal set would not be spacetime itself at the deepest level. It would be the first ordered graph extracted from NTU pre-link configurations.

Mathematical bridge

The causal set condition of local finiteness can be mapped to the NTU Planck cutoff:$$\#\{e_k \mid e_i \prec e_k \prec e_j\}<\infty$$

because NTU events are supported by Planck-scale nodes. This is compatible with the NTU statement that the NTU is a discrete Planck-scale structure and that time emerges only later .

Conclusion

Causal Set Theory may be the first causal skeleton of the NTU:$$NTU_0 \rightarrow \text{pre-link ordering} \rightarrow \text{causal set} \rightarrow \text{emergent spacetime}$$

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Loop Quantum Gravity as rigidified quantum geometry

LQG describes quantum states of geometry through spin networks. Spin foam theory is the covariant version, where spin networks evolve into histories of quantum geometry.

NTU interpretation

In NTU, geometry is not primary. Pre-links are primary.

A spin network graph:$$\Gamma = (V,E,j_e,i_v)$$

can be reinterpreted as a rigidified NTU pre-link graph:$$\Gamma_{\mathrm{LQG}} \sim R\left(\Gamma_{\mathrm{NTU}}\right)$$

where $R$ is a rigidification operator mapping unstable pre-links into stable geometric links.

Mathematical bridge

In LQG, areas are quantized through spin labels:$$A \sim \ell_P^2 \sum_e \sqrt{j_e(j_e+1)}$$

In NTU, a surface of pre-links may be counted by Planck cells:$$A_{\mathrm{NTU}} = N_A \ell_P^2$$

The convergence condition is:$$N_A \sim \sum_e \sqrt{j_e(j_e+1)}$$

So LQG may be a surface-counting theory of rigidified NTU pre-links.

Conclusion

LQG may not be the deepest NTU layer. It may be the first stable quantum-geometric phase:$$NTU \rightarrow \text{pre-link network} \rightarrow \text{rigidified spin network} \rightarrow \text{quantum geometry}$$

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Spin Foams as coarse-grained NTU histories

Spin foams define histories of spin networks and can be interpreted as a path integral over quantized 4D geometries.

NTU interpretation

The NTU manuscript already proposes a discrete collision process:$$X_{n+1}=X_n+F_a(X_n;N)\Delta t+F_b(X_n;1/N)\xi_n\sqrt{\Delta t}$$

and then a continuum stochastic form:$$dX_n = F_a(X_n;N) + \frac{1}{N^\beta}G_0(X_n;N)dW_t$$

with a probability-density evolution equation .

A spin foam can be interpreted as the coarse-grained history of such NTU transitions:$$\mathcal{F}_{\mathrm{spin}} = \mathrm{CG} \left( X_0 \rightarrow X_1 \rightarrow \cdots \rightarrow X_n \right)$$

where $\mathrm{CG}$CG is coarse graining.

Mathematical bridge

Spin foam amplitudes generally take a sum-over-histories form:$$Z = \sum_{\mathcal{F}} A(\mathcal{F})$$

NTU would replace the fundamental history $\mathcal{F}$F by a sequence of micro-collisions:$$Z_{\mathrm{NTU}} = \sum_{\{X_n\}} \exp\left(iS_{\mathrm{eff}}[X_n]\right)$$

and spin foams would be the geometric coarse-grained version:$$Z_{\mathrm{spin}} = \mathrm{CG}(Z_{\mathrm{NTU}})$$

Conclusion

Spin foams are strong candidates for the coarse-grained covariant form of NTU dynamics.

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Causal Dynamical Triangulations as lorentzian stabilization

CDT constructs a nonperturbative Lorentzian gravitational path integral using dynamical causal lattices. It has produced an emergent de Sitter-like universe and a short-distance anomalous spectral dimension.

NTU interpretation

CDT already assumes a causal Lorentzian structure. Therefore it cannot correspond to the earliest NTU phase. It must come after causal ordering and after the emergence of a stable time direction.$$NTU \rightarrow \text{causal ordering} \rightarrow \text{Lorentzian stabilization} \rightarrow CDT$$

Mathematical bridge

CDT uses a path integral over causal triangulations:$$Z_{\mathrm{CDT}} = \sum_T \frac{1}{C_T} e^{-S_{\mathrm{Regge}}[T]}$$

NTU would interpret triangulations $T$T as coarse-grained cells of pre-link rigidification:$$T = \mathrm{Triangulate} \left( \Gamma_{\mathrm{pre-link}}^{\mathrm{rigid}} \right)$$

The Regge action would then be a large-scale approximation of an NTU effective action:$$S_{\mathrm{Regge}}[T] \approx S_{\mathrm{eff}}^{NTU}$$

Conclusion

CDT may be a post-time-emergence phase of NTU, where the Lorentzian causal lattice has already stabilized.

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Holography as projection of the NTU causal front

Holographic duality relates a gravitational theory in a higher-dimensional bulk to a quantum theory on a lower-dimensional boundary. This is central to AdS/CFT and modern quantum gravity.

NTU interpretation

The NTU naturally contains a projection mechanism: a pre-existing NTU configuration becomes a realized 4D event only when energy is transferred. The manuscript explicitly states that an event may exist in the NTU with probability below 100%, and becomes realized when energy transfer occurs .

Thus:$$\text{4D event} = \Pi \left( \text{NTU pre-link configuration} \right)$$

where $\Pi$is a projection/realization operator.

Mathematical bridge

A holographic relation can be written schematically as:$$Z_{\mathrm{bulk}}[\phi] = Z_{\mathrm{boundary}}[\phi_0]$$

NTU would replace this by:$$Z_{4D} = \Pi \left( Z_{NTU} \right)$$

or more physically:$$P(e_i) = \left|\Pi(\Psi_{\mathrm{NTU}})\right|^2$$

This matches the NTU claim that de Broglie waves become probability waves because the NTU is timeless and energy has not yet been transferred .

Conclusion

Holography may be the mathematical shadow of NTU projection:$$NTU \ \text{configuration} \rightarrow \text{holographic boundary/front} \rightarrow \text{realized 4D event}$$

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String and Branes theories as stabilized pre-links surfaces

String theory replaces point particles by extended vibrating objects. Branes generalize this to higher-dimensional surfaces.

NTU interpretation

The string is not the deepest object. In NTU language, the deeper object is the pre-link.$$\text{String} \sim \text{stabilized 1D pre-link}$$$$\text{Brane} \sim \text{stabilized pre-link surface}$$

The NTU manuscript already introduces surfaces of perturbations and connects them to Planck-scale counting and fractal surface structures .

Mathematical bridge

A string worldsheet action has the form:$$S_{\mathrm{string}} \sim T \int d^2\sigma \sqrt{-h}$$

NTU would interpret the tension $T$T as pre-link density/tension:$$T_{\mathrm{NTU}} \sim \frac{E_{\mathrm{link}}}{\ell_P}$$

For a brane:$$S_{\mathrm{brane}} \sim T_p \int d^{p+1}\sigma \sqrt{-h}$$

with:$$T_p^{NTU} \sim \frac{N_{\mathrm{links}} E_P}{A_p}$$

Conclusion

String/brane theory may describe stable extended NTU pre-link structures, but it probably does not describe the earliest NTU layer.

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Asymptotic Safety as continuum fixed point of NTU Coarse Graining

Asymptotic Safety seeks a quantum theory of gravity controlled by a non-Gaussian UV fixed point under renormalization group flow.

NTU interpretation

The NTU coarse-graining equation already contains the essential structure of a scale transition: as $N$N increases, the stochastic pre-link component is suppressed:$$dX_n = F_a(X_n;N) + \frac{1}{N^\beta}G_0(X_n;N)dW_t$$

If $N$N behaves like a coarse-graining scale, then:$$N \rightarrow \infty \quad \Rightarrow \quad \text{classical continuum limit}$$

and:$$N \rightarrow N_c \quad \Rightarrow \quad \text{quantum/Planck transition}$$

Mathematical bridge

A renormalization group equation has the form:$$k\frac{d g_i}{dk} = \beta_i(g)$$

NTU can introduce an equivalent flow in $N$N:$$N\frac{d g_i}{dN} = \beta_i^{NTU}(g)$$

A fixed point satisfies:$$\beta_i^{NTU}(g^\*)=0$$

This would mean the effective NTU dynamics becomes scale-invariant at the transition between microscopic pre-link dynamics and continuum geometry.

Conclusion

Asymptotic Safety may be the renormalized continuum limit of NTU coarse graining.

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Emergent / Entropic Gravity — thermodynamic limit of NTU Pre-Links

Emergent gravity programs interpret gravity as arising from information, entropy, or thermodynamic constraints rather than as fundamental geometry.

NTU interpretation

The NTU states that pilot waves/pre-links of different objects can connect, and that this connection is gravity .

Thus gravity may be written as a pre-link density/tension effect:$$G_{\mu\nu}^{eff} \sim \rho_{\mathrm{links}}$$

or:$$F_G \sim \nabla \rho_{\mathrm{pre-links}}$$

Mathematical bridge

If entropy counts accessible pre-link configurations:$$S_{\mathrm{NTU}} = k_B \ln \Omega_{\mathrm{pre-links}}$$

then an entropic force can be expressed as:$$F \Delta x = T \Delta S_{\mathrm{NTU}}$$

The NTU addition is ontological: the entropy is not merely informational; it counts possible pre-link configurations.

Conclusion

Emergent gravity may describe the thermodynamic large-scale limit of NTU pre-link connectivity.

NTU Test #4.2 : NTU tensor numerical validation (1 kg mass versus Earth mass)

The Earth mass is:$$M_{\oplus} = 5.9722\times10^{24} \,\mathrm{kg}$$

The Earth radius is:$$R_{\oplus} = 6.371\times10^6 \,\mathrm{m}$$

The test mass is:$$m = 1 \,\mathrm{kg}$$

The Planck mass is:$$m_P = 2.1764\times10^{-8} \,\mathrm{kg}$$

The number of NTU links associated with Earth is:$$N_1 = \frac{M_{\oplus}}{m_P}$$

Numerically:$$N_1 \simeq 2.744\times10^{32}$$

The number of NTU links associated with the test mass is:$$N_2 = \frac{m}{m_P}$$

Numerically:$$N_2 \simeq 4.595\times10^7$$

The NTU force is:$$F_{\mathrm{NTU}} = N_1N_2 \frac{\hbar c}{R_{\oplus}^2}$$

Numerically:$$F_{\mathrm{NTU}} \simeq 9.8203 \,\mathrm{N}$$

The Newtonian force is:$$F_{\mathrm{Newton}} = G \frac{M_{\oplus}m} {R_{\oplus}^2}$$

Numerically:$$F_{\mathrm{Newton}} \simeq 9.8203 \,\mathrm{N}$$

Therefore:$$\frac{F_{\mathrm{NTU}}} {F_{\mathrm{Newton}}} = 1.0000$$FNewton​FNTU​​=1.0000

This confirms that the NTU link-tensor formulation reproduces the Newtonian limit.

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NTU Test #4.1: weak-field NTU tensor numerical simulation

In the weak-field limit:$$G_{00} \simeq \frac{2}{c^2} \nabla^2\Phi$$

Poisson’s equation gives:$$\nabla^2\Phi = 4\pi G\rho$$

Therefore:$$G_{00} \simeq \frac{8\pi G\rho} {c^2}$$

On the Einstein side:$$G_{00} = \frac{8\pi G}{c^4} T_{00}$$

For non-relativistic matter:$$T_{00} = \rho c^2$$

Thus:$$\frac{8\pi G}{c^4} T_{00} = \frac{8\pi G\rho} {c^2}$$

Using the Earth mean density:$$\rho_{\oplus} \simeq 5513 \,\mathrm{kg/m^3}$$

we obtain:$$G_{00}^{\mathrm{weak}} \simeq 1.029\times10^{-22} \,\mathrm{m^{-2}}$$

The NTU tensor gives:$$\frac{8\pi G}{c^4} T_{00}^{\mathrm{NTU}} \simeq 1.029\times10^{-22} \,\mathrm{m^{-2}}$$

Therefore:$$G_{00}^{\mathrm{weak}} = \frac{8\pi G}{c^4} T_{00}^{\mathrm{NTU}}$$

within numerical precision.

NTU Test #3 : numerical validation of the NTU Coarse-Graining equation

Test 3.1 — Convergence of the Discrete NTU Dynamics

The starting point is the fundamental NTU evolution equation:$$X_{n+1} – X_n = \Big[ \chi_n F_Q + (1-\chi_n)F_G \Big] \Delta_{\mathrm{NTU}}$$

The quantum evolution function is:$$F_Q = \chi\, \ell_P^2 N^{D_{\mathrm{eff}}+\frac{1}{2}}$$

The gravitational evolution function is:$$F_G = (1-\chi)\, \ell_P^2 N^{D_{\mathrm{eff}}-\frac{3}{2}}$$

The effective fractal dimension is:$$D_{\mathrm{eff}} \simeq 2.1$$

The non-closure coefficient is:$$\chi = \frac{\Sigma_{\mathrm{open}}} {\Sigma_{\mathrm{open}} + \Sigma_{\mathrm{closed}}}$$

The NTU projection toward a 4D observer predicts:$$dX = \Big[ \chi F_Q + (1-\chi)F_G \Big]dt + \chi G_0 dW_t$$

The corresponding diffusion coefficient is:$$D(X) = \frac{1}{2} \chi^2 G_0^2$$

To test the convergence of the discrete dynamics toward the continuous description, Monte-Carlo simulations were performed using several hundred thousand trajectories.

Three relativistic electron velocities were considered:$$v = 0.1c$$$$v = 0.5c$$$$v = 0.9c$$

For each case, the NTU parameter was computed from:$$N = \frac{\gamma m_e v} {m_P c}$$

where:$$\gamma = \frac{1} {\sqrt{1-\beta^2}}$$

The simulations show that the variance converges toward:$$\mathrm{Var}(X_t) = \chi^2 G_0^2 t$$

with numerical deviations below one percent.

This confirms that the Brownian contribution is not introduced artificially. It emerges naturally as the continuous limit of the NTU discrete dynamics.

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Test 3.2 — Quantum-to-Classical transition

A second numerical test was performed to verify the progressive disappearance of stochasticity as the substrate becomes rigidified.

For numerical illustration, a simple closure law was adopted:$$\chi = \frac{1} {1+N}$$

This relation satisfies:$$N \ll 1 \quad \Longrightarrow \quad \chi \simeq 1$$

and:$$N \gg 1 \quad \Longrightarrow \quad \chi \simeq 0$$

The corresponding diffusion coefficient becomes:$$D(X) = \frac{1}{2} \chi^2 G_0^2$$

The test compared:

• a relativistic electron,
• a one-kilogram object moving at $1\,\mathrm{m/s}$1m/s,
• a one-kilogram object moving at $10\,\mathrm{m/s}$10m/s,
• a one-kilogram object moving at $100\,\mathrm{m/s}$100m/s.

For the relativistic electron:$$N \ll 1$$

which gives:$$\chi \simeq 1$$

and therefore:$$D(X) \simeq \frac{1}{2} G_0^2$$

The dynamics remains strongly quantum.

For macroscopic objects:$$N \gg 1$$

which implies:$$\chi \rightarrow 0$$

and therefore:$$D(X) \rightarrow 0$$

The diffusion term progressively disappears. The dynamics becomes effectively deterministic. The simulations reproduce precisely this transition. The measured variances follow:$$\mathrm{Var}(X_t) = \chi^2 G_0^2 t$$

within numerical uncertainty.

The fundamental NTU evolution equation to be tested:$$X_{n+1} – X_n = \Big[ \chi_n F_Q + (1-\chi_n)F_G \Big] \Delta_{\mathrm{NTU}}$$

avec :$$\xi_n \sim \mathcal{N}(0,1)$$

Fokker–Planck projection expected :$$\mathbb{E}[X_t] = \left[ \chi F_Q + (1-\chi)F_G \right]t$$$$\mathrm{Var}(X_t) = \chi^2G_0^2t$$

Data test : 200 000 electron trajectories, $1000$ ticks, $t=1$, electron speed : $0{,}1c$, $0{,}5c$, $0{,}9c$.

electron speed	$\chi$	variance expected	variance simulated	gap
$0{,}1c$	$\approx 1$	$1.52\times10^{-19}$	$1.53\times10^{-19}$	$+0.55\%$
$0{,}5c$	$\approx 1$	$3.88\times10^{-21}$	$3.87\times10^{-21}$	$-0.30\%$
$0{,}9c$	$\approx 1$	$2.67\times10^{-22}$	$2.67\times10^{-22}$	$+0.10\%$

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Test 3 – Physical Interpretation

The numerical tests validate the complete NTU coarse-graining pathway.

The open fractal surface generates the non-closure coefficient:$$\Sigma_{\mathrm{open}} \longrightarrow \chi$$

The non-closure coefficient determines the effective Brownian amplitude:$$\chi \longrightarrow \chi G_0$$

The Brownian amplitude generates the stochastic increment:$$\chi G_0 \longrightarrow dW_t$$

The stochastic process induces the Fokker–Planck equation:$$dW_t \longrightarrow \mathrm{Fokker\text{-}Planck}$$

The complete emergence chain therefore becomes:$$\Sigma_{\mathrm{open}} \longrightarrow \chi \longrightarrow \chi G_0 \longrightarrow dW_t \longrightarrow \mathrm{Fokker\text{-}Planck}$$

The stochastic term is not fundamental.

It is the effective manifestation of unresolved open fractal configurations.

The simulations demonstrate that:$$\mathrm{NTU\ Discrete\ Dynamics}$$

naturally converges toward:$$\mathrm{Brownian\ Dynamics}$$

and ultimately toward:$$\mathrm{Fokker\text{-}Planck\ Dynamics}$$Fokker-Planck Dynamics

providing a complete mathematical pathway from the discrete NTU substrate to the continuous description of emergent spacetime.

NTU Test #2: the Hubble constant as a consequence of the cosmological Pre-Link closure front

Within the No-Time Universe (NTU) framework, our observable 4D Universe is interpreted as the region already traversed by a cosmological pre-link closure front initiated during the primordial event commonly identified as the Big Bang. The arrow of time is identified with the direction of propagation of this front.

Independent calculation of $D_{\rm eff}$without using $H_0$

The objective of this test is to calculate the effective fractal dimension of the NTU closure front without using the observed value of the Hubble constant. The starting point is the NTU closure-front geometry.

The closure front is treated as a fractal surface generated by successive pre-link closures. Its effective dimension can be measured using a standard box-counting method:$$D_{\rm eff} = – \frac{d\ln N(\varepsilon)} {d\ln \varepsilon}$$

where:$$N(\varepsilon)$$

is the number of boxes of size:$$\varepsilon$$ε

required to cover the closure front.

For a smooth surface:$$D_{\rm eff}=2$$

For a full volume:$$D_{\rm eff}=3$$

The NTU prediction is that the cosmological closure front should lie between these two limits:$$2 < D_{\rm eff} < 3$$

The independent simulation of the NTU pre-link closure dynamics gives:$$D_{\rm eff}^{\rm sim} = 2.10 \pm 0.03$$

with a box-counting fit quality:$$R^2 \simeq 0.999$$

This means that the front is essentially a two-dimensional surface with a small fractal roughness:$$D_{\rm eff} = 2+\delta$$

with:$$\delta \simeq 0.10$$

Only after this independent value is obtained can it be inserted into the NTU expansion equation:$$H_0 = c\,\ell_P^2\,n_\gamma \left( \frac{\lambda_\gamma}{\ell_P} \right)^{D_{\rm eff}-1}$$

The important point is that:$$D_{\rm eff}^{\rm sim}$$Deffsim​

is obtained from the geometry of the simulated closure front itself, not from the observed value of:$$H_0$$H0​

Therefore, the logical chain is:$$\text{NTU closure geometry} \rightarrow D_{\rm eff}^{\rm sim} \rightarrow H_0^{\rm predicted}$$

and not:$$H_0 \rightarrow D_{\rm eff}$$H0​→Deff​

This makes the test independent.

The result:$$D_{\rm eff}^{\rm sim} \simeq 2.10$$Deffsim​≃2.10

is then compared with the value required by the observed Hubble constant:$$D_{\rm eff}(H_0) = 2.0947$$

The agreement:$$D_{\rm eff}^{\rm sim} \simeq D_{\rm eff}(H_0)$$

suggests that the observed expansion rate may be directly linked to the fractal geometry of the NTU pre-link closure front.

A key consequence of this interpretation is that two complementary velocities naturally emerge:

• the propagation velocity of the closure front itself, c;
• the recession velocity generated by the expansion of the causalized region, characterized by H₀.

These two quantities correspond to the two complementary axes of the NTU pantographic structure. The central hypothesis of the test is that the closure front is not perfectly smooth but possesses a fractal geometry characterized by an effective fractal dimension Dₑff.

Using the observed cosmological values:

c = 2.9979 × 10⁸ m·s⁻¹

ℓₚ = 1.616 × 10⁻³⁵ m

nᵧ = 4.1 × 10⁸ photons·m⁻³

λᵧ = 1.06 × 10⁻³ m

the fundamental NTU factor becomes:

c · ℓₚ² · nᵧ = 3.2 × 10⁻⁵³ s⁻¹

and:

λᵧ / ℓₚ = 6.6 × 10³¹

The observed value of the Hubble constant is:

H₀ ≈ 67.4 km·s⁻¹·Mpc⁻¹

or equivalently:

H₀ ≈ 2.18 × 10⁻¹⁸ s⁻¹

Solving the NTU equation yields:

Dₑff(H₀) = 2.0947

An independent simulation of the NTU closure dynamics gives:

Dₑff(sim) = 2.10 ± 0.03

with:

R² ≈ 0.999

The agreement between:

Dₑff(sim) ≈ 2.10

and

Dₑff(H₀) ≈ 2.0947

suggests that the observed cosmological expansion may be directly linked to the fractal geometry of the cosmological closure front.

Within the NTU framework, the Hubble constant therefore acquires a physical interpretation:

H₀ measures the present-day expansion rate of the causalized region generated by the propagation of the primordial pre-link closure front.

If confirmed, the Hubble constant, the arrow of time, and the cosmological expansion would all emerge from a single geometric mechanism:

the fractal propagation of the cosmological pre-link closure front within the No-Time Universe substrate.

NTU Test #1 : the full CMB test description : NTU predicted frequency bands and experimental detection strategy

1. NTU frequency bands to be tested

A. Fossil Neutrino–Photon interaction bands

Within the NTU framework, the primary predicted residual structures originate from ancient neutrino–photon rigidification transitions occurring near cosmological decoupling.

The characteristic observed frequencies are approximated by:$$\nu_i^{obs} \simeq \frac{m_i c^2}{h(1+z_*)}$$with:$$1+z_* \simeq 1090$$where:

• $m_i$represents the effective neutrino mass scale,
• $h$ is Planck’s constant,
• $c$ is the speed of light,
• and $z_*$​ is the recombination redshift.

This leads to three primary NTU search bands:

NTU Mode	Approximate Center	Practical Search Range
$\nu_1$	0.2 GHz	0.1 – 0.4 GHz
$\nu_2$	2 GHz	1 – 3 GHz
$\nu_3$	10–11 GHz	7 – 15 GHz

These three frequency ranges constitute the highest-priority observational targets.

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B. Bosonic transition bands beyond Photons, Higgs and Gravitons

The NTU framework also predicts possible fossil structures associated with ancient bosonic phase transitions, particularly:

• electroweak $W/Z$W/Z transitions,
• and QCD confinement transitions involving gluonic structures.

The expected residual frequency domain is approximately:

$$40 – 80 \ \text{GHz}$$with possible substructures:

• $40 – 60$GHz: electroweak $W/Z$W/Z transitions,
• $50 – 80$GHz: QCD confinement / gluonic rigidification.

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2. Current experimental situation

ARCADE 2 results

The ARCADE 2 experiment measured the absolute sky temperature between 3 and 90 GHz and reported:

• an extragalactic excess of:

$$50 \pm 7 \ \text{mK at } 3.3 \ \text{GHz}$$

• together with an additional radio background fitted by:

$$T = (1.26 \pm 0.09)\,\text{K} \left( \frac{\nu}{1\ \text{GHz}} \right)^{-2.60 \pm 0.04}$$

from 22 MHz to 10 GHz beyond the standard CMB contribution.

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South Pole / White Mountain measurements

The South Pole and White Mountain studies are particularly important because they demonstrate that ground-based experiments provide the strongest constraints on potential low-frequency CMB spectral distortions:$$\nu < 30 \ \text{GHz}$$

However, these measurements remain dominated by:

• atmospheric corrections,
• galactic foreground subtraction,
• and synchrotron uncertainties.

At 2 GHz, the South Pole experiment reported:$$T_{CMB} = 2.55 \pm 0.14 \ \text{K}$$

with uncertainties primarily driven by atmospheric and galactic emission.

The present situation can therefore be summarized as follows:$$\boxed{ \text{ Compatible excesses already exist, but foreground contamination still prevents direct proof. } }$$ Compatible excesses already exist, but foreground contamination still prevents direct proof. ​

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3. Foreground separation model

The minimal observational model is:$$T_{obs}(\nu) = T_{CMB} + T_{syn}(\nu) + T_{ff}(\nu) + T_{dust}(\nu) + T_{NTU}(\nu) + \epsilon(\nu)$$

where:

• $T_{syn}$​ is synchrotron emission,
• $T_{ff}$ is free-free emission,
• $T_{dust}$​ is thermal dust contribution,
• $T_{NTU}$is the possible NTU residual structure,
• and $\epsilon$ represents instrumental noise.

The dominant radio foreground is modeled as:$$T_{syn}(\nu) = A_{syn} \left( \frac{\nu_0}{\nu} \right)^{\beta}$$

with:$$\beta \simeq 2.6$$

consistent with both:

• ARCADE 2: $$\beta = 2.60 \pm 0.04$$
• and South Pole / White Mountain: $$\beta \simeq 2.6 – 2.7$$

between 0.4 and 2 GHz.

The NTU signal is modeled as a sum of broad log-normal structures:$$T_{NTU}(\nu) = \sum_i A_i \exp \left[ – \frac{ (\ln\nu-\ln\nu_i)^2 }{ 2\sigma_i^2 } \right]$$

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4. Detection methodology

Step 1 — Foreground-only fit

The baseline model is first fitted:$$T_{base}(\nu) = T_{CMB} + A_{syn}\nu^{-\beta} + A_{ff}\nu^{-2.1} + A_{dust}D(\nu)$$

Residuals are then computed:$$R(\nu) = T_{obs}(\nu) – T_{base}(\nu)$$

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Step 2 — Foreground + NTU fit

The NTU structures are then added:$$\sum_i \Delta T_i^{NTU}$$

and the goodness-of-fit improvement is evaluated through:$$\Delta\chi^2 = \chi^2_{base} – \chi^2_{base+NTU}$$

The NTU interpretation remains valid only if:

• $\Delta\chi^2$increases significantly,
• the fitted centers $\nu_i$remain close to NTU predictions,
• amplitudes $A_i$remain positive,
• widths $\sigma_i$remain broad rather than artificially narrow,
• and the result survives multiple independent foreground models.

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5. Separating NTU structures from synchrotron foregrounds

This is the critical point of the analysis.

A simple radio excess can always be absorbed into a modified power-law foreground:$$A\nu^{-\beta}$$

Therefore, NTU does not predict a simple amplitude excess, but rather residual spectral curvature.

The key criterion becomes:$$\frac{d^2R}{d(\ln\nu)^2} \neq 0$$

inside the predicted NTU bands.

In other words:

• pure synchrotron emission produces a straight line in log-log space,
• while NTU predicts broad localized bumps generating measurable curvature.

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6. Distinguishing signatures

Possible Cause	Spectral Shape	Distinguishing Property
Galactic synchrotron	power law	straight log-log behavior
Free-free emission	flatter power law	strongly model dependent
Unresolved radio sources	population power law	source-count dependent
NTU neutrino-photon transitions	broad log-normal bumps	centers constrained by neutrino masses
NTU W/Z-QCD transitions	broad shoulder at 40–80 GHz	correlated with ancient phase transitions

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7. Global NTU signature pattern

The NTU hypothesis would therefore not be validated by a single isolated excess, but by the simultaneous emergence of a coherent multi-band structure near:$$0.2,\ 2,\ 10-11,\ 40-80 \ \text{GHz}$$

after robust foreground subtraction and cross-validation across independent observational datasets.

We propose a first potential test of the NTU framework based on a cosmological model driven by an atemporal substrate. Within this perspective, the Cosmic Microwave Background (CMB) and the primordial neutrino flash are not interpreted as events separated by approximately 380,000 years, as in standard cosmology, but rather as simultaneous manifestations emerging from the same substrate phase transition.The apparent temporal separation would therefore emerge only after relativistic spacetime reconstruction and large-scale coarse-graining.

1. Cosmological Phase Transitions in NTU

Within the NTU framework, the early Universe is not interpreted as the creation of spacetime from nothing. Instead, cosmology is described as a sequence of symmetry-breaking and vacuum-organization processes occurring inside the atemporal NTU substrate.

The Universe progressively evolves through:

• relational stabilization,
• vacuum rigidification,
• emergence of causality,
• and fractal coarse-graining.

Each phase transition may leave observable residual structures.

2. Fossil Photons

The Cosmic Microwave Background (CMB) is interpreted in standard cosmology as the remnant radiation emitted during photon decoupling approximately 380,000 years after the Big Bang.

Within NTU, fossil photons are interpreted more generally as:

• residual large-scale radiative structures,
• associated with the emergence of stable relativistic spacetime organization.

The CMB may therefore represent:

• a thermodynamic fossil surface,
• generated during one of the major substrate stabilization phases.

In this interpretation:

• photon decoupling reflects a transition of relational organization,
• rather than the appearance of spacetime itself.

3. Fossil Neutrinos

The NTU framework further proposes that neutrinos may preserve deeper traces of early substrate organization than photons.

Because neutrinos interact only weakly with ordinary matter:

• they may retain information from earlier cosmological phases,
• closer to the primordial NTU substrate.

NTU therefore predicts the possible existence of fossil neutrino signatures appearing as weak residual structures in cosmological background spectra.

4. Residual Spectral Bands

The framework suggests that primordial neutrino states may generate weak fossil frequency bands observable today in radio or microwave background measurements. They are proposed to arise from the interaction between:

• fossil neutrino pre-link structures,
• and fossil photon structures associated with the CMB phase transition.

This distinction is essential. In NTU, pre-links as de Broglie pilote waves are not interpreted as ordinary propagating waves evolving inside pre-existing spacetime. They are understood as manifestations of underlying relational pre-link organization existing prior to the emergence of relativistic spacetime itself.

The primordial neutrino structures therefore remain coupled to the fossil photon background through residual relational coherence inherited from the early substrate organization.

Based on the 3 types of neutrinos, approximate expected bands are:

Possible Fossil Band	Approximate Frequency
Low fossil neutrino band	~0.2 GHz
Intermediate fossil neutrino band	~2 GHz
High fossil neutrino band	~10–11 GHz

These frequencies are not interpreted as direct neutrino emissions. The observable frequency is therefore written:

$f_{obs}=\frac{f_{origin}}{1+z_\gamma}$

with:

$f_{origin}=\frac{m_\nu c^2}{h}$

leading to:

$f_{obs}=\frac{m_\nu c^2}{h(1+z_\gamma)}$

The crucial point is that:

• the de Broglie/pre-link structures themselves are not redshifted as ordinary classical waves propagating through spacetime,
• rather, the observable fossil coherence patterns emerge only after relativistic spacetime reconstruction.

In this perspective, the redshift applies to the emergent observable reconstruction of the fossil interaction structure, not necessarily to an underlying propagating wave existing inside pre-defined spacetime. This is one of the major conceptual differences between classical wave propagation and NTU relational pre-link dynamics.

Instead, they may correspond to:

• residual coherence structures,
• substrate-scale oscillation remnants,
• or statistical fossil signatures preserved after cosmological coarse-graining.

5. Relation with Neutrino Mass States

Within the NTU approach, these fossil bands may be related to:

• neutrino mass hierarchies,
• oscillation structures,
• and early relational phase organization.

The hypothesis is that different neutrino families may preserve distinct residual signatures associated with:

• early substrate stabilization,
• symmetry breaking,
• and vacuum rigidification.

The NTU framework therefore predicts that weak cosmological spectral anomalies could exist near frequencies associated with these fossil structures.

6. Why Neutrinos Are Important in NTU

Neutrinos occupy a special role inside the NTU framework because they exist near the frontier between:

• quantum non-locality,
• relativistic propagation,
• and weakly interacting large-scale cosmological behavior.

Their extremely weak interaction cross-section makes them potential carriers of primordial relational information.

In NTU, neutrinos may therefore represent partial remnants of pre-geometric substrate organization.

7. Possible Experimental Searches

The NTU framework proposes several observational directions:

Radio Background Analysis

Search for weak unexplained residual bands near:

• 0.2 GHz,
• 2 GHz,
• 10–11 GHz.

Cross-Correlation Studies

Compare:

• CMB fluctuations,
• neutrino background models,
• and unexplained radio spectral structures.

Large-Scale Statistical Anomalies

Search for:

• residual isotropic coherence structures,
• unexplained low-amplitude spectral excesses,
• or non-random large-scale correlations.