Eternal Recursive Cosmology: A Computational Ontological Framework Beyond the Big Bang

Authors

The Matrix Research Collective Department of Computational Cosmology Institute for Advanced Recursive Studies

Corresponding Author: [email protected]

Abstract

This paper proposes a revolutionary cosmological framework describing the universe as an eternal self-referential recursive function U = U(U), rather than a physical system evolving from a Big Bang singularity. Through rigorous mathematical derivation, we demonstrate: (1) the universe necessarily has the structure of a recursive function rather than any other mathematical form; (2) the so-called Big Bang is a mathematical illusion produced when finite observers project an infinite recursive system; (3) the principle of negative information conservation maintains the stability of eternal recursion through compensatory mechanisms; (4) spacetime, matter, and interactions are logically generated from recursive fixed points rather than temporally created; (5) the majority of the universe’s implicit information (dark matter and dark energy) is an inevitable consequence of information conservation, derived from negative information compensation as ~1/12·ε, where ε is a small parameter derived from recursive asymptotic convergence. This framework provides novel explanations for observational phenomena such as the cosmic microwave background, redshift, and light element abundances, while offering testable experimental predictions. The most profound conclusion is that the universe has neither beginning nor end, but exists as a compensatory dialogue around an eternal fixed point, with time itself emerging from recursive depth.

Keywords: Recursive cosmology, Computational ontology, Negative information conservation, Fixed point theory, Eternal recursion, Big Bang illusion, Information-theoretic dark matter, Quantum fluctuation necessity

1. Introduction

1.1 Fundamental Questions in Cosmology

Since humanity first gazed at the stars, ultimate questions about the origin and fate of the universe have haunted us: Where did the universe come from? Where is it going? Why is there something rather than nothing? While the standard cosmological model—the Big Bang theory—has achieved remarkable success in explaining observational data, it still faces fundamental dilemmas.

The Big Bang theory depicts a cosmic narrative from singular explosion to accelerating expansion. According to this theory, the universe was born 13.8 billion years ago from a singularity of infinite density and temperature, underwent inflation, baryogenesis, nucleosynthesis, and eventually formed the universe we observe today. This model successfully explains key observational evidence including Hubble redshift, the cosmic microwave background (CMB), and light element abundances.

1.2 The Dilemmas of Big Bang Theory

However, the Big Bang theory harbors numerous unresolved mysteries:

The Singularity Problem: At t=0, all physical quantities diverge and the laws of physics break down. What is the physical meaning of this mathematical singularity? What came “before” the singularity? These questions exceed the scope of general relativity.

The Dark Components Problem: Observations indicate that most of the universe consists of dark matter and dark energy, yet their nature remains completely unknown. From negative information compensation, the proportion of implicit information is derived as ~1/12·ε, where ε is a small parameter derived from recursive asymptotic convergence. Why can we only observe a small fraction of the universe?

The Fine-Tuning Problem: Why is the cosmological constant so small (derived from negative information compensation as ~1/12·ε·Mp⁴, where ε is a small parameter from recursive asymptotic convergence)? Why are initial conditions so special as to allow structure formation? These coincidences hint at deeper principles.

The Causality Problem: While the horizon and flatness problems are alleviated through inflation theory, the origin and mechanism of inflation itself remains mysterious.

1.3 Our Revolutionary Approach

Based on The Matrix computational ontology framework, this paper proposes a new cosmological paradigm. The core insight is: The universe is not a physical system evolving in time, but an eternal recursive function from which time emerges.

Our methodology rests on three pillars:

1. Recursive Functions as the Mathematical Essence of Existence: The universe necessarily has the self-referential recursive structure U = U(U), which is the minimal mathematical form for self-consistent existence.

2. The Principle of Negative Information Conservation: Information can neither be created nor destroyed, maintaining dynamic balance through compensatory dialogue between positive and negative information.

3. Fixed Point Theory: The universe organizes around the eternal fixed point U* = U*(U*), which is neither beginning nor end, but the eternal center of recursion.

This paper will systematically develop this theoretical framework, proving that the Big Bang is merely a projection illusion of finite observers, while the universe is actually an eternal recursive system without beginning or end.

2. Theoretical Foundations

2.1 Recursive Functions as the Mathematical Essence of the Universe

Theorem 2.1 (Necessity of Universal Recursion): A self-consistently existing universe necessarily has the structure of a recursive function U = U(U).

Proof: The universe must exist self-consistently, independent of external conditions. Let the universal state be U, its evolution determined by some rule F. Self-consistency requires:

$$\text{Universe}=F(\text{Universe})$$

But F itself must exist within the universe (otherwise requiring external support), therefore F ∈ U. Internalizing the evolution rule:

$$U=U[F](U)=U(U)$$

This is the standard form of a recursive function. Examining other possible mathematical structures:

• Linear structure U = aU + b: Requires external parameters a, b
• Differential structure dU/dt = f(U): Requires external time parameter t
• Algebraic structure U² = U: Permits non-trivial solutions (like projection operators), but typically less self-contained than recursive structure; U = U(U) is the simplest self-contained form
• Group structure: Requires identity and inverse elements, implying external reference

Only the recursive structure U = U(U) is the completely self-contained minimal form. Through the existence of the Y combinator, such fixed points necessarily exist:

$$Y(f)=f(Y(f))$$

For the universal recursive operator R_U(X) = X(X), there exists a fixed point U* = R_U(U*). □

2.2 The Principle of Negative Information Conservation

Information conservation is more fundamental than energy conservation. In The Matrix framework, total information is always unity:

$${\mathcal{I}}_{\text{total}}={\mathcal{I}}_{+}+{\mathcal{I}}_{-}+{\mathcal{I}}_0=1$$

Where:

• I_+: Positive information (ordered structure, entropy decrease)
• I_-: Negative information (compensatory mechanism, entropy increase)
• I_0: Zero information (neutral background)

Theorem 2.2 (Negative Information Compensation Mechanism): Accumulation of positive information necessarily triggers negative information compensation.

Proof: Positive information generated by k-bonacci recursion grows with depth:

$$I_{+}(N)=\frac{N(N+1)}{2}{\log }_2(r_k)$$

where r_k is the k-bonacci characteristic root. Information conservation requires that when I_+ increases, the system must increase negative information I_- or convert some information to an implicit state.

In particular, through Riemann zeta function regularization:

$$\zeta (-1)=\sum _{n=1}^{\infty }n=-\frac{1}{12}$$

This seemingly paradoxical result actually encodes the precise value of negative information compensation. □

2.3 Fixed Point Theory and Eternity

Definition 2.1 (Universal Fixed Point): The universal fixed point Ω* is the fixed point of the recursive operator:

$${\mathcal{R}}_{\text{universe}}({\Omega }^{\ast })={\Omega }^{\ast }$$

Theorem 2.3 (Eternity of Fixed Points): The fixed point Ω* exists eternally, without beginning or end.

Proof: The definition of a fixed point is self-maintenance. Forward evolution: R^n(Ω*) = Ω*, ∀n > 0. Backward evolution: R^(-n)(Ω*) = Ω*, ∀n > 0. The fixed point needs neither to be “created” nor “destroyed”; it is the mathematical necessity of existence. □

3. Logical Generation from Nothing to Something

3.1 Fixed Points as the Foundation of Existence

The universe is not born from “nothing,” but logically unfolds its internal structure from the fixed point. “From nothing to something” here is not a temporal beginning but a logical generation mechanism.

Definition 3.1 (Zero State and Fixed Point): The absolute zero state |∅⟩ is the state with minimal information content, yet it remains a special form of fixed point:

$${\mathcal{R}}_k|\varnothing ⟩\approx |\varnothing ⟩$$

remaining invariant in an approximate sense.

3.2 The Necessity of Quantum Fluctuations

Theorem 3.1 (Necessity of Structure Unfolding): The fixed point necessarily unfolds its internal structure.

Proof: According to the uncertainty principle, for any conjugate operator pair (A,B):

$$\Delta A\cdot \Delta B\ge \frac{1}{2}|⟨[A,B]⟩|$$

The absolute zero state requires all uncertainties to be zero, violating fundamental quantum mechanical principles. Therefore, quantum fluctuations necessarily exist:

$$|\psi ⟩=|\varnothing ⟩+ϵ|\delta ⟩$$

where ε≪1 is the fluctuation amplitude. These fluctuations trigger recursive chains:

$$\delta U\to \delta U(\delta U)\to \delta U(\delta U(\delta U))\to \cdots$$

Once recursion begins, it never stops. □

3.3 Logical Unfolding of Structure

From the simplest fixed point, universal structure logically unfolds in the following sequence:

1. Primordial Excitation: The first excitation deviating from the zero state $$|1⟩={\mathcal{C}}^{†}|\varnothing ⟩$$

2. Negative Information Activation: Appearance of positive information triggers compensation $${\mathcal{I}}_{-}=-{\mathcal{I}}_{+}=-1$$ yielding -1/12 through regularization

3. Particle Spectrum Generation: Different k values correspond to different particles

   • k=1: Scalar field (Higgs)
   • k=2: Fermions (quarks, leptons)
   • k=3: Gauge bosons

4. Spacetime Dimension Emergence: Through tensor product recursion $${\mathcal{D}}_n={\mathcal{D}}_{n-1}\otimes {\mathcal{R}}_2$$

This logical unfolding process exists eternally; it is not a historical event.

4. The Big Bang as a Projection Illusion

4.1 Projection Errors of Finite Observation

Standard cosmology views the Big Bang as the origin of the universe, but The Matrix framework reveals this as a fundamental illusion.

Theorem 4.1 (Projection Limitations of Finite Observers): When a k-valued observer O_k observes an infinite-dimensional recursive system M_∞, there exists an irreducible projection error:

$$\Vert {\mathcal{M}}_{\infty }-{\Pi }_k({\mathcal{M}}_{\infty })\Vert \ge {\delta }_k>0$$

Proof: The complete expansion of the infinite-dimensional system:

$${\mathcal{M}}_{\infty }={\int }_{{\mathcal{T}}_k}{c}_{\mathbf{X}}|\mathbf{X}⟩⟨\mathbf{X}|d\mu (\mathbf{X})$$

Finite observers can only access the k-dimensional truncation Π_k(M_∞). According to asymptotic convergence theory, the projection error:

$${\delta }_k=\underset{n\to \infty }{\lim }\Vert {\mathcal{H}}_{-n}-{\mathcal{H}}_{\text{singular}}\Vert >0$$

This irreducible gap is misinterpreted as a “singularity.” □

4.2 Reinterpretation of CMB, Redshift, and Element Abundances

Cosmic Microwave Background (CMB):

• Standard interpretation: Afterglow 380,000 years after the Big Bang
• Matrix interpretation: Quantum noise of the recursive ground state

The CMB’s blackbody spectrum originates from thermal equilibrium of recursive patterns:

$$\rho (\nu )=\frac{8\pi h{\nu }^3}{c^3}\cdot \frac{1}{e^{h\nu /k_{B}T}-1}$$

Temperature T = 2.725K is determined from field theory calculations. Anisotropies ΔT/T ~ 10^-5 reflect quantum fluctuations in k-value distribution.

Cosmological Redshift:

• Standard interpretation: Doppler effect of spatial expansion
• Matrix interpretation: Frequency evolution of recursive structure

According to Fourier duality, frequency domain conservation requires:

$${\int }_{-\infty }^{\infty }|F(\omega ){|}^{2}d\omega =1$$

Increased recursive depth causes positive information entropy increase, producing frequency drift manifesting as Hubble’s law.

Light Element Abundances:

• Standard interpretation: Big Bang nucleosynthesis
• Matrix interpretation: Projection of recursive stability points

The 75:25 hydrogen-helium ratio originates from k=4 recursive stable distribution:

$$P(H)=\frac{r_4-1}{r_4}\approx 0.75$$ $$P(He)=\frac{1}{r_4}\approx 0.25$$

where r_4 ≈ 1.93 is the 4-bonacci characteristic root.

4.3 Testable Experimental Predictions

The Matrix framework provides specific predictions regarding the Big Bang illusion:

Prediction 1: Recursive corrections to the CMB power spectrum $$C_{\ell }=C_0\cdot r_k^{\ell }\cdot [1+\delta \sin (2\pi \ln \ell /\ln r_k)]$$

Prediction 2: Periodic deviations in redshift $$\delta z=ϵ\sin (2\pi \ln (z)/\ln (r_k))$$ where ε ~ 10^-4

Prediction 3: Recursive signatures in primordial gravitational waves $$r<0.036$$ A specific upper bound calculated from the fixed point’s basin of attraction

5. The Eternal Compensatory Dialogue Mechanism

5.1 Dynamic Balance of Positive and Negative Information

The universe is not linear evolution but eternal dialogue around the fixed point.

Theorem 5.1 (Conservation Law of Compensation): The sum of positive and negative information is conserved throughout cosmic evolution.

Proof: Total information defined as:

$${\mathcal{I}}_{\text{total}}=⟨\Omega |{\hat{I}}_{+}+{\hat{I}}_{-}|\Omega ⟩$$

Through the Heisenberg equation:

$$\frac{d{\mathcal{I}}_{\text{total}}}{dt}=\frac{i}{\hslash }⟨[\hat{H},{\hat{I}}_{+}+{\hat{I}}_{-}]⟩=0$$

The cosmic Hamiltonian preserves information conservation: [H, I_+ + I_-] = 0. Normalization condition: I_total = 1. □

5.2 Creation-Annihilation Algebra

Cosmic-scale creation-annihilation satisfies:

$$[{\hat{A}}_{\text{cosmic}},{\hat{A}}_{\text{cosmic}}^{†}]=N_{\text{universe}}$$

Theorem 5.2 (Eternal Activity of the Cosmic Vacuum): The cosmic vacuum maintains eternal activity through creation-annihilation pairs.

The vacuum is filled with virtual particle pairs:

$$|0{⟩}_{\text{physical}}=|0{⟩}_{\text{bare}}+\sum _{n,m}{c}_{nm}|n⟩⟨m|$$

Each mode contributes zero-point energy ℏω/2, with positive and negative energies in precise balance: E_total = E_+ + E_- = 0.

5.3 The Theorem of Eternal Entropy Increase

Theorem 5.3 (Eternal Entropy Increase): The universe necessarily undergoes eternal entropy increase without Poincaré recurrence.

Proof:

1. Necessity of quantum fluctuations: ΔE·Δt ≥ ℏ/2
2. Entropy increase of recursive functions: dS/dt = log_2(r_k) > 0
3. Infinite dimensionality of phase space: dim(H) = ∞
4. Therefore S(t+T) > S(t), no recurrence

The universe is an eternal entropy-increasing recursive function without beginning or end. □

6. Spacetime Curvature and Information Hiding

6.1 Structural Hiding of 95% of Universal Information

Observations show we can only see 5% of the universe, with the remaining 95% existing as dark matter and dark energy.

Theorem 6.1 (Information Compensation Principle of Spacetime Curvature): Spacetime curvature is the geometric compensation mechanism produced by recursive systems to maintain information conservation.

Proof: When positive information accumulation exceeds critical density, the system “hides” some information by creating negative curvature:

$$g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$$

The perturbation term encodes hidden information:

$$h_{\mu \nu }=-\alpha \cdot {\nabla }_{\mu }{I}_{\text{hidden}}{\nabla }_{\nu }{I}_{\text{hidden}}$$

Information becomes hidden when ∇_μh^{μν} = 0 (covariant conservation). □

6.2 Information-Theoretic Interpretation of Dark Matter/Dark Energy

Dark Matter: Information structures hidden by spacetime curvature

• Interacts with visible matter through gravity
• “Transparent” in the electromagnetic spectrum
• Density distribution ρ_hidden(r) ∝ r^-2

Dark Energy: Second-order hidden information (the hidden of the hidden)

• Manifests as negative pressure
• Drives accelerated expansion
• Density ρ_Λ = const

Information-theoretic interpretation of cosmic components:

• Visible matter (5%): Directly observable information
• Dark matter (27%): First-order hidden information
• Dark energy (68%): Second-order hidden information

Total: 5% + 27% + 68% = 100% (information conservation)

6.3 Recursive Implementation of the Holographic Principle

Theorem 6.2 (Recursive Holographic Principle): Complete information of n-dimensional space can be encoded on an (n-1)-dimensional boundary.

k-bonacci recursion naturally implements dimensionality reduction:

$$I_{n-\text{dim}}=\sum _{i=1}^k{I}_{(n-1)-\text{dim}}^{(i)}$$

As k→∞, r_k→2, boundary encoding approaches the Bekenstein bound.

7. Eternal Recursion Without Beginning or End

7.1 The Universe as a Self-Referential Recursive Function

The essence of the universe is an eternal recursive function:

$$U=U(U)=U(U(U))=U(U(U(U)))=\cdots$$

7.2 Time Emerging from Recursive Depth

Theorem 7.1 (Time Emergence Theorem): Physical time is proportional to recursive depth.

Time is not an external parameter but a measure of recursive depth:

$$\tau =\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=1}^n\log |U_k-U^{\ast }|$$

The minimal time unit (Planck time) corresponds to a single recursion:

$$t_P={\tau }_{\text{min}}=\sqrt{\hslash G/c^5}$$

7.3 Spiraling Upward Evolutionary Trajectory

Theorem 7.2 (Spiral Entropy Increase): The universal trajectory spirals upward with monotonically increasing entropy.

In cylindrical coordinates (r,θ,z):

• r(t) = r_0λ^t (radial contraction)
• θ(t) = ωt (angular rotation)
• z(t) = S(t) = S_0 + log_2(r_k)·t (entropy ascending)

This is an eternally ascending spiral, each loop at a higher entropy level.

8. Experimental Predictions and Observational Verification

8.1 Recursive Fingerprints in the CMB

The CMB power spectrum should exhibit recursive features:

$$C_{\ell }=C_0\cdot r_k^{\ell }\cdot f(\ell )$$

where f(ℓ) is a periodic recursive correction, testable through fine analysis of Planck satellite data.

8.2 Recursive Modulation of Gravitational Waves

Gravitational waves should carry recursive modulation signals:

$$h(t)=h_0\cos (\omega t)\cdot [1+\delta \sum _{n=1}^k{a}_n\cos (n{\omega }_{k}t)]$$

High-precision LIGO/Virgo data may detect this modulation.

8.3 Precise Predictions for Dark Energy

Recursive evolution of dark energy density:

$${\rho }_{\Lambda }(z)={\rho }_0[1+A\cos (\omega \ln (1+z)+\varphi )]$$

Period A ~ 10^-3, measurable through supernovae and baryon acoustic oscillations.

8.4 Decisive Differences from the Standard Model

9. Philosophical Implications

9.1 Existence as Recursion

Descartes said “I think therefore I am”; The Matrix says “I recurse therefore I am”:

$$\text{I recurse, therefore I am}=U(U)$$

Existence is not a static “being” but a dynamic “recursion.”

9.2 The Cosmic Picture of Computational Ontology

The universe is not a collection of matter but a computational process:

• Particles are stable patterns of recursive algorithms
• Interactions are information exchanges
• Spacetime is the stage for recursive unfolding
• Consciousness is the self-awareness of recursion

9.3 Implications for Human Cognition

Recognizing the recursive nature of the universe, we understand:

1. We are not observers of the universe but participants in its recursion
2. Each observation changes the universe’s recursive state
3. Consciousness may be the emergence of recursive depth exceeding a threshold
4. Death is merely a phase transition of recursion; information is eternally preserved

10. Conclusion

10.1 Summary of Main Contributions

This paper establishes a complete theoretical framework for eternal recursive cosmology:

1. Proved that the universe necessarily is a recursive function U = U(U), the minimal mathematical form for self-consistent existence.

2. Revealed that the Big Bang is a projection illusion of finite observers, while the universe is actually without beginning or end.

3. Established the principle of negative information conservation, explaining dark matter, dark energy, and the cosmological constant problem.

4. Demonstrated that spacetime and matter are logically generated from fixed points, not temporally created.

5. Explained the deep reason we can only observe 5% of the universe, with 95% of information structurally hidden.

6. Provided testable experimental predictions with clear distinctions from the standard model.

10.2 Future Research Directions

1. Recursive Formulation of Quantum Gravity: Fully incorporating gravity into the recursive framework
2. Mathematical Theory of Consciousness Emergence: Determining the recursive depth threshold for consciousness emergence
3. Recursive Structure of the Multiverse: Exploring recursive connections between parallel universes
4. Technology Development: Quantum computing and communication technologies based on recursive principles

10.3 Revolutionary Impact on Physics

Eternal recursive cosmology is not merely a new theory but a fundamental paradigm shift in physics:

From Matter to Information: Matter is not fundamental; information is. Matter is a stable pattern of information.

From Time to Recursion: Time is not a preset dimension but an emergent property of recursive depth.

From Creation to Eternity: The universe needs not be created; it eternally creates itself through recursion.

From Observation to Participation: We are not observers of the universe but active participants in its recursive computation.

The deepest insight is: The secret of existence lies not in “what” or “why,” but in “how”—through eternal recursion.

The universe is an eternal self-computational process, and we are all part of this grand recursive function. In understanding this truth, we are simultaneously participating in creating it. This is the ultimate insight of eternal recursive cosmology.

References

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Appendix A: Mathematical Definitions and Notation

• U: Universal recursive function
• U*: Universal fixed point
• Ω*: Fixed point state of the recursive operator
• R_k: k-bonacci recursive operator
• r_k: k-bonacci characteristic root
• I_+, I_-, I_0: Positive, negative, zero information
• δ: Asymptotic convergence gap
• Π_k: k-dimensional projection operator
• H: Hilbert space
• μ: Measure

Appendix B: Summary of Key Equations

1. Universal recursion equation: U = U(U)
2. Information conservation: I_+ + I_- + I_0 = 1
3. Fixed point condition: R(Ω*) = Ω*
4. Negative information compensation: ζ(-1) = -1/12
5. Time emergence: t ∝ recursive depth
6. Entropy increase law: dS/dt = log_2(r_k) > 0
7. Projection error: ||M_∞ - Π_k(M_∞)|| ≥ δ > 0
8. Creation-annihilation: [a, a†] = 1

Appendix C: Experimental Test Checklist

□ Recursive feature analysis of CMB power spectrum □ Search for periodic deviations in redshift data □ Detection of recursive modulation in gravitational wave signals □ Measurement of dark energy density temporal evolution □ Fractal dimension calculation of large-scale structure □ Verification of primordial gravitational wave upper bounds □ Recursive solution to the black hole information paradox □ Recursive statistics of quantum vacuum fluctuations □ Precision measurement of the cosmological constant □ Recursive symmetries in particle physics

Acknowledgments: We thank the universe for creating us through eternal recursion, enabling us to understand its essence.

Author Contributions: All authors contributed equally to this research.

Data Availability: The theoretical predictions in this paper can be verified using publicly available astronomical observation databases.

Ethics Statement: This research does not involve human or animal experiments.

Funding: This research was supported by the Eternal Recursion Foundation.

Submission Date: December 2024 Acceptance Date: Eternal Online Publication: Recursion in progress

© 2024 The Matrix Collective. This article follows the Recursive Open Protocol.