On the Scope and Structure of This Manuscript

1.1. Background and Motivation

• Toroidal structure: The circular cross-section represents a slice through the 4D torus at an instance of space-time, showcasing the model’s fundamental geometry.
• Everted universe and black holes: Central to the image is the "everted universe," surrounded by a distinct event horizon. Beyond this horizon, HTUM proposes a region composed entirely of what the model considers to be a singular, unified black hole continuum. This concept illustrates HTUM’s unique perspective that all black holes are fundamentally interconnected, forming a continuous boundary of the observable universe. This approach challenges conventional views of black holes as separate entities, instead presenting them as integral parts of the universe’s toroidal structure.
• Matter distribution: Color gradients within the everted universe visualize the spatial distribution of matter across cosmic structures. The gradient progresses from the densest structures near the torus "walls" (composed entirely of event horizons) to the least dense regions at the center.
• Cosmic components: The legend identifies various components such as stars, dense gas, planetary nebulae, and interstellar medium, providing an intuitive understanding of density variations across the slice.
• Temporal representation: This static image represents a single moment in the model. The full simulation depicts Temporal evolution by animating slices moving through the 4D structure.

1.2. Roadmap of the Paper

1.3. Significance of HTUM in Cosmology

2.1. The Lambda-CDM Model and Cosmic Evolution Scenarios

2.2. Historical Context

2.3. Limitations of the Lambda-CDM Model

2.4. The Cosmological Constant Problem

2.5. Addressing Limitations with HTUM

Table 1. Comparison of Lambda-CDM model and HTUM

2.6. HTUM and Cosmological Challenges

2.7. Implications and Future Directions

2.8. Conclusion

3.1. Conceptual Framework

3.2. Toroidal Structure of the Universe

3.3. Mathematical Formulation of the Toroidal Structure

3.4. Advanced Topological Concepts for the Toroidal Structure

3.5. HTUM and the Flatness Problem

3.6. TQFT and the Hyper-Torus

3.7. Challenges in Visualizing and Conceptualizing a 4DTS

3.8. The Nature of Dark energy and Dark matter in HTUM

3.9. Detailed Mathematical Framework for Dark Matter and Dark Energy Interactions

3.10. Nonlinear Probabilistic Nature of Dark Matter and Dark Energy

3.11. Specific Predictions for Observables

3.12. Large-Scale Homogeneity and Isotropy in HTUM

3.13. Observational Tests and Experimental Proposals

3.14. Implications for HTUM and Cosmology

3.14.9. Extended Framework and Comparisons

Comparison with Current Dark Matter Models

Current dark matter models primarily focus on particle-based explanations, such as Weakly Interacting Massive Particles (WIMPs) or axions [80]. HTUM, however, proposes a fundamentally different approach:

• Wave function localization: In HTUM, dark matter is viewed as a nonlinear phenomenon that contributes to the localization of the universal wave function. This contrasts with particle models by suggesting that dark matter is an intrinsic property of the quantum universe rather than a distinct particle species.
• Dynamic distribution: Unlike static dark matter haloes in standard models, HTUM suggests a dynamic distribution that evolves with the universe’s wave function. This could potentially explain observed anomalies in galactic rotation curves that challenge conventional dark matter models [384].
• Quantum entanglement: HTUM proposes that dark matter’s effects are deeply connected to quantum entanglement on a cosmic scale. This could provide a new perspective on the "small scale crisis" in cosmology, where observations of dwarf galaxies seem to conflict with simulations based on cold dark matter models [110].

Relation to Current Dark Energy Models

HTUM’s approach to dark energy also differs significantly from current models, such as the cosmological constant or quintessence fields:

• Quantum superposition maintenance: In HTUM, dark energy is conceptualized as a nonlinear probabilistic phenomenon that helps maintain quantum superposition states. This contrasts with the static energy density of the cosmological constant model or the slowly varying scalar fields in quintessence models [117].
• Wave function collapse dynamics: HTUM suggests that dark energy plays a crucial role in the dynamics of wave function collapse on a cosmic scale. This could potentially address the coincidence problem in cosmology, explaining why dark matter and dark energy densities are of the same order of magnitude in the present epoch [551].
• Toroidal structure influence: The model proposes that dark energy’s behavior is intimately linked to the universe’s toroidal structure. This geometric connection could provide a new perspective on the flatness problem and the apparent accelerating expansion of the universe [364].

Table 2. Dark matter and dark energy properties in HTUM vs. Standard Models

Table 2. Dark matter and dark energy properties in HTUM vs. Standard Models

Property	HTUM	Standard Models
Nature	Nonlinear probabilistic phenomena	Particle-based or constant field
Dynamics	Wave function localization (DM)	Static dark matter halos
	Quantum superposition maintenance (DE)	Static energy density or slow-rolling field
Distribution	Dynamic, evolving with wave function	Static or slowly evolving
Quantum effects	Deeply connected to entanglement	Not typically considered
Cosmic scale effects	Addresses "small scale crisis"	Challenges at small scales
Coincidence problem	Potentially resolved	Unexplained
Geometric connection	Linked to toroidal structure	No specific geometric connection
Mathematical framework	Nonlinear Schrödinger equation	Particle physics or field theory
Observational predictions	Modified structure formation	Standard structure formation
	Unique CMB and gravitational signatures

Mathematical Framework for Nonlinear Probabilistic Phenomena

To mathematically describe these nonlinear probabilistic phenomena, we can extend the previously introduced framework:

$$i\hslash {\displaystyle \frac{\partial \psi }{\partial t}}=\widehat{H}\psi +f_{DM}\left(\psi \right)\psi +f_{DE}\left(\psi \right)\psi$$

(65)

where $\widehat{H}$ is the standard Hamiltonian, and $f_{DM}\left(\psi \right)$ and $f_{DE}\left(\psi \right)$ are nonlinear functionals representing the effects of dark matter and dark energy, respectively. These functionals could take forms such as:

$$f_{DM}\left(\psi \right)={\alpha |\psi |}^2+\beta {\nabla }^2{|\psi |}^2$$

(66)

$$f_{DE}\left(\psi \right)={\gamma |\psi |}^4-\delta {\displaystyle \frac{{\partial }^2}{\partial t^2}}{|\psi |}^2$$

(67)

where $\alpha$, $\beta$, $\gamma$, and $\delta$ are coupling constants determining the strength of dark matter and dark energy effects.

Observational Consequences

The HTUM framework for dark matter and dark energy leads to several potentially observable consequences:

• Galactic dynamics: The nonlinear nature of dark matter in HTUM could lead to unique signatures in galactic rotation curves and galaxy cluster dynamics that differ from predictions of standard dark matter models [209].
• Cosmic web structure: The interplay between dark matter’s wave function localization and dark energy’s superposition maintenance could result in distinctive patterns in the cosmic web structure, potentially observable through large-scale structure surveys [348].
• Cosmic microwave background (CMB): The quantum nature of dark energy in HTUM might lead to specific imprints on the CMB power spectrum, particularly at large angular scales [74].

Experimental Approaches

Testing HTUM’s predictions regarding dark matter and dark energy will require a multifaceted approach:

• Advanced gravitational lensing studies: Precise measurements of gravitational lensing effects could reveal the nonlinear and quantum nature of dark matter distribution predicted by HTUM [541].
• High-precision cosmological surveys: Next-generation surveys like LSST and Euclid could provide data on large-scale structure and cosmic expansion that could be compared with HTUM predictions [296,340].
• Quantum experiments: While challenging, experiments exploring quantum effects at larger scales could provide insights into the quantum nature of dark energy proposed by HTUM [93].

3.15. Addressing the Cosmological Constant Problem

3.15.8. Testable Predictions

This model makes several testable predictions:

• Anisotropies in the CMB corresponding to the toroidal structure [547].
• Specific relationships between vacuum energy and the universe’s large-scale structure [63].
• Potential observable effects in high-energy particle physics experiments [29].

This framework provides a rigorous mathematical foundation for HTUM’s approach to the cosmological constant problem. It connects the model’s fundamental concepts to observable phenomena and allows for empirical testing and refinement.

Figure 3 illustrates the three-dimensional representation of the TVEM function $\mathsf{\Gamma }(x,y)$ over a section of the torus. The bell-shaped surface with subtle undulations demonstrates how the TVEM function varies across the toroidal geometry. This visualization helps elucidate the spatial distribution of the vacuum energy modulation proposed in HTUM.

Figure 4 presents a heatmap of the vacuum energy density as modulated by the TVEM function. The bright center surrounded by concentric rings of decreasing intensity illustrates the non-uniform distribution of vacuum energy in the HTUM framework. This visualization supports our hypothesis that the TVEM mechanism can naturally suppress extreme values of vacuum energy [564].

These figures clearly represent how HTUM addresses the cosmological constant problem through the TVEM function. These visualizations show that the non-uniform distribution of vacuum energy suggests a mechanism for reconciling the vast discrepancy between observed and theoretically predicted vacuum energy values [128]. This approach aligns with recent efforts to understand the nature of vacuum energy and its implications for cosmology [379].

Table 3. Comparison of HTUM and $\mathsf{\Lambda }$CDM approaches to the cosmological constant problem

Table 3. Comparison of HTUM and $\mathsf{\Lambda }$CDM approaches to the cosmological constant problem

Feature	HTUM	$\mathit{\Lambda }$CDM
Vacuum energy modulation	TVEM function	Not addressed
Effective cosmological constant	Scale-dependent	Constant
Suppression mechanism	Topological constraints	Not present
	and averaging effect
Observable consequences	CMB anisotropies,	Standard CMB and
	modified structure growth	structure formation
Parameter space	$\{k_1,k_2,k_3,k_4,\alpha \}$	$\mathsf{\Lambda }$
Numerical approach	Monte Carlo integration	N/A
Data fitting method	Bayesian (MCMC)	Various
Testable predictions	Specific CMB patterns,	Standard cosmic
	vacuum energy-structure	evolution
	relationships

3.16. Visualization and Analysis of the TVEM Function

3.17. Quantum Field Theory in Toroidal Space

3.18. HTUM and Quantum Foundations

3.19. Emergence of Observed Dimensions

Table 4. HTUM Structure Formation: Simulation Parameters and Results

3.20. Dynamical Relaxation

3.21. Comparative Analysis

3.22. Observational Consequences

3.23. Theoretical Implications

3.24. Addressing the Nature of the Singularity and Time

3.25. Experimental Implications and Testable Predictions

3.26. Simulation Results and Discussion

3.26.6. Conclusion

These simulation results significantly strengthen the HTUM hypothesis by demonstrating complex quantum behavior in a 4D toroidal structure. The simulations reveal patterns consistent with HTUM’s key predictions, particularly regarding the universe’s topology and the nature of dark matter and dark energy.

Table 5. Key Features of the Hyper-Torus Universe Model (HTUM)

Table 5. Key Features of the Hyper-Torus Universe Model (HTUM)

Feature	Description	Implications
4D toroidal structure	Universe as a 4D torus	Finite but unbounded universe
TVEM function	Modulates vacuum energy	Addresses cosmological constant problem
Dark matter/energy	Nonlinear probabilistic phenomena	Unified quantum treatment
Wave function collapse	Amplified at event horizons	Emergence of classical reality
CMB predictions	Specific anisotropy patterns	Testable via observations
Quantum-classical transition	Scale-dependent process	Explains macroscopic behavior
Gravitational waves	Unique signatures predicted	Potential for empirical validation

Future work should focus on extending simulation times, varying initial conditions, and developing more refined measures of topological quantum effects. Additionally, as discussed in Section 3.25.1, comparing these results with observational data, particularly from cosmic microwave background (CMB) studies, will be crucial for further validating HTUM.

These findings open new avenues for understanding the universe’s fundamental structure and offer a promising framework for resolving long-standing cosmology and quantum gravity puzzles.

4.1. Foundational Principles

4.2. Quantum Mechanics Axiomatization

4.3. Gravity Axiomatization

4.4. Quantum Field Theory on $T^4$

• Axiom 16 (Field Operators): Quantum fields on $T^4$ are operator-valued distributions $\mathsf{\Phi }\left(x\right)$ that satisfy the periodicity condition $\mathsf{\Phi }(x+L_i)=\mathsf{\Phi }\left(x\right)$ for each dimension i.
• Axiom 17 (TVEM-Modified Propagators): The propagator for a scalar field on $T^4$ is given by

  $$G(x,y)=\langle 0|T\left\{\mathsf{\Phi }\left(x\right)\mathsf{\Phi }\left(y\right)\right\}|0\rangle =\sum _n{\displaystyle \frac{e^{i{k}_n·(x-y)}}{k_n^2-m^2+i\epsilon }}·\mathsf{\Gamma }\left(x\right)\mathsf{\Gamma }\left(y\right)$$

  (98)

  where $k_n$ are the discrete momenta allowed by the torus topology.

4.5. Dark Matter and Dark Energy

• Axiom 18 (Nonlinear Probabilistic Nature): Dark matter and dark energy are described by nonlinear functionals of the universal wave function: ${\rho }_{\mathrm{DM}}\left[\mathsf{\Psi }\right]$ and ${\rho }_{\mathrm{DE}}\left[\mathsf{\Psi }\right]$.
• Axiom 19 (TVEM-Dark Sector Coupling): The evolution of dark matter and dark energy densities is governed by:

  $${\displaystyle \frac{d{\rho }_{\mathrm{DM}}}{dt}}=-3H{\rho }_{\mathrm{DM}}+{\mathsf{\Gamma }}_{\mathrm{DM}}\left(\mathsf{\Gamma }\right){\rho }_{\mathrm{DM}}$$

  (99)

  $${\displaystyle \frac{d{\rho }_{\mathrm{DE}}}{dt}}=-3H(1+w){\rho }_{\mathrm{DE}}+{\mathsf{\Gamma }}_{\mathrm{DE}}\left(\mathsf{\Gamma }\right){\rho }_{\mathrm{DE}}$$

  (100)

  where ${\mathsf{\Gamma }}_{\mathrm{DM}}\left(\mathsf{\Gamma }\right)$ and ${\mathsf{\Gamma }}_{\mathrm{DE}}\left(\mathsf{\Gamma }\right)$ are TVEM-dependent interaction terms.

4.6. Emergence of Spacetime and Dimensions

• Axiom 20 (Dimensional Emergence): The effective dimensionality of spacetime $D_{\mathrm{eff}}$ is energy-dependent:

  $$D_{\mathrm{eff}}\left(E\right)=4-\alpha ·exp(-E/E_P)$$

  (101)

  where $E_P$ is the Planck energy and $\alpha$ is a model parameter.
• Axiom 21 (Emergent Time): Time emerges as a parameter describing the evolution of entanglement entropy:

  $${\displaystyle \frac{d{S}_{\mathrm{ent}}}{dt}}=k_B\sum _i{\gamma }_i\langle L_i^{†}{L}_i,\rho \rangle$$

  (102)

  where $L_i$ are Lindblad operators describing the system-environment interactions.

4.7. Quantum Information and Holography

• Axiom 22 (Toroidal Holographic Principle): The information content of any region A in $T^4$ is bounded by:

  $$S\left(A\right)\le {\displaystyle \frac{\mathrm{Area}(\partial A)}{4G_N}}+S_{\mathrm{topo}}$$

  (103)

  where $S_{\mathrm{topo}}$ is a topological entropy term specific to $T^4$.
• Axiom 23 (Entanglement-Geometry Duality): The entanglement entropy between regions A and B is related to the geometry of $T^4$:

  $$S(A:B)={\displaystyle \frac{\mathrm{Area}\left({\gamma }_{AB}\right)}{4G_N}}+f\left(\mathsf{\Gamma }\right)$$

  (104)

  where ${\gamma }_{AB}$ is the minimal surface separating A and B, and $f\left(\mathsf{\Gamma }\right)$ is a TVEM-dependent correction term.

4.8. Unified Mathematical Operations

• Axiom 24 (Generalized Operator): There exists a unified mathematical operator U on $T^4$ such that:

  $$U(a,b,\alpha ,\beta )=\alpha a+\beta b+f(\alpha ,\beta )ab+g(\alpha ,\beta )(a/b)$$

  (105)

  where f and g are smooth functions, encompassing addition, subtraction, multiplication, and division as special cases.

4.9. Cosmological Dynamics

• Axiom 25 (Modified Friedmann Equations): The evolution of the scale factor $a\left(t\right)$ on $T^4$ is governed by:

  $${\left({\displaystyle \frac{da}{dt}}\right)}^2/a^2=(8\pi G/3)({\rho }_m+{\rho }_r+{\rho }_{\mathrm{TVEM}})-k/a^2+{\mathsf{\Lambda }}_{\mathrm{eff}}/3$$

  (106)

  $${\displaystyle \frac{d^{2}a}{d{t}^2}}/a=-(4\pi G/3)({\rho }_m+2{\rho }_r-2{\rho }_{\mathrm{TVEM}})+{\mathsf{\Lambda }}_{\mathrm{eff}}/3$$

  (107)

  where ${\rho }_{\mathrm{TVEM}}$ is the TVEM-modulated vacuum energy density.