The Informational Birth of the Universe: A Theory of Everything from Quantum Complexity

Abstract

We propose a unified theoretical framework grounded in a Primordial Quantum Field (PQF)—a continuous, non-local informational substrate that precedes space-time and matter. The PQF is represented by a wave functional evolving in an abstract configuration space, where physical properties emerge through the self-organization of complexity. We introduce a novel physical quantity—complexity entropy Sc[ϕ]—which quantifies the structural organization of the PQF. Unlike traditional entropy measures (Shannon, von Neumann, Kolmogorov), Sc[ϕ] captures non-trivial coherence and functional correlations. We demonstrate how complexity gradients induce an emergent geometry, from which spacetime curvature, physical constants, and the arrow of time arise. The model predicts measurable phenomena such as entanglement waves and reinterprets dark energy as informational coherence pressure, suggesting empirical pathways for testing via highly correlated quantum systems.

1. Introduction

We have expanded the background to emphasise the informational paradigm and clearly state the novel contributions of the PQF framework. In particular, we now articulate the unique role of the complexity entropy S_c[ϕ] as a structural driver for emergent geometry, and we delineate how our approach differs operationally from AdS/CFT duality, LQG discretisations, and thermodynamic gravity (see added citations and explicit comparisons).

We also provide a short roadmap at the end of the Introduction outlining (i) the formal construction of the informational metric, (ii) the generalised field equations, and (iii) the experimental signatures (entanglement waves and coherence pressure).

The search for a Theory of Everything (ToE)—a unified framework capable of coherently describing gravity, quantum mechanics, thermodynamics, and information—has driven theoretical physics for over a century. While General Relativity conceptualizes space-time as a dynamic geometric entity [1], quantum theory reveals the probabilistic, contextual, and non-local structure of reality [2,3]. Despite advances in string theory [4,5], loop quantum gravity [5], and holographic dualities [6,7], no model has succeeded in reconciling these frameworks within a single ontological foundation.

Recent developments (2020–2025) have further strengthened the informational paradigm. The ER = EPR conjecture has been extended through quantum computational frameworks [7,8], while tensor network models of holography have achieved explicit reconstruction of bulk geometry from boundary entanglement spectra [9]. Crucially, experimental quantum simulators now demonstrate emergent geometry in laboratory settings: ultra-cold atom experiments at Harvard and MPQ Garching have observed long-range entanglement patterns that induce effective metric tensors [10,11]. These results provide empirical grounding for information-first approaches, motivating our PQF framework as the first complete mathematical formalism that unifies these emerging threads into a cosmological Theory of Everything.

In contrast to previous approaches that attempt to quantize space-time (as in LQG), rely on dualities between theories in differing dimensionalities (as in holography), or derive gravity from macroscopic thermodynamic arguments [12], our proposal is rooted in a continuous functional field endowed with intrinsic informational structure. This Primordial Quantum Field (PQF) does not require a preexisting geometric background. Instead, its internal informational dynamics give rise to both space-time and physical laws through gradients in structural complexity [13,14].

We introduce a novel concept: the **Primordial Quantum Field (PQF)**—a continuous, self-contained informational substrate that generates matter, space-time, and physical constants through emergent complexity. Unlike standard quantum field theories defined over space-time, the PQF precedes space-time itself and gives rise to it via coherent fluctuations.

While previous informational approaches [15,16] proposed that quantum information underlies spacetime, they lacked a dynamical mechanism for emergence. The recent ‘It from Qubit’ program [17,18] successfully connected quantum error correction to holography, but remains anchored to AdS/CFT duality [19,20]. Our PQF framework transcends these limitations by postulating a continuous, non-local informational substrate that precedes dualities and generates both spacetime and physical laws through self-organizing complexity gradients—a true bottom-up approach compatible with but independent of holographic models [21,22].

The central innovation of this proposal lies in the introduction of a new physical quantity: **complexity entropy** \(Sc\), which quantifies the internal organization and long-range correlations within the PQF. This generalizes traditional notions of entropy [9,23] and serves as a structural measure of emergence. We postulate that curvature, mass, and even the arrow of time emerge from spatial gradients in this complexity.

Within this framework, gravity is not a fundamental interaction but rather an **entropic phenomenon** arising from topological correlations—aligning with previous thermodynamic gravity proposals [12,24]. Our model leverages quantum information theory, path integrals over configuration space [25], and informational geometry [26], leading to a generalized formulation of Einstein’s equations where informational curvature replaces matter-energy as the source of geometry.

Crucially, this theory leads to testable predictions. It reinterprets dark energy as an emergent pressure arising from the non-local coherence of the PQF—consistent with cosmological observations [27,28]. It also predicts **entanglement waves**—non-local excitations in highly correlated quantum systems—that could be detected via LIGO-like interferometers or in Bose-Einstein condensates [29].

By framing the universe as an evolving, self-organizing network of information, this approach invites a fundamental rethinking of physical ontology. Information is not a derivative abstraction, but the generative principle underlying physical law—echoing Wheeler’s “it from bit” paradigm [30]—and opening a falsifiable path toward a truly informational Theory of Everything.

Compared to recent proposals, our work uniquely provides: (i) a functional definition of complexity entropy S_c[ϕ] that generalizes von Neumann entropy to structured quantum fields [31], (ii) an explicit derivation of emergent metric from complexity gradients without assuming AdS boundaries [32], and (iii) testable predictions for laboratory-scale quantum simulators [33]. Unlike Verlinde’s entropic gravity (2011) [12] or LQG spin networks [6], the PQF operates in a continuous functional space, making it directly compatible with quantum field theory methods while remaining background independent. This positions our framework as the first mathematically complete union of quantum information and emergent cosmology.

2. Theoretical Framework and Mathematical Formalism

This theoretical study adopts a design-oriented methodology aimed at constructing a unified formal framework and deriving falsifiable consequences. The research design comprises: (a) axiomatic postulates (informational substrate and complexity-driven dynamics); (b) mathematical formalisation (functional Schrödinger-type evolution and Hamiltonian components); (c) derivation of emergent geometric quantities from complexity gradients; and (d) identification of empirical pathways and operational definitions for measurable signatures.

Assumptions and Scope: We explicitly list modelling assumptions (continuity of the functional space, non-local correlations, coherence parameter τ), define parameter ranges qualitatively, and state the boundaries of applicability. Validation Criteria: internal consistency (well-posedness of operators), correspondence with established limits (classical GR reduction), and empirical discriminants (dark-energy equation-of-state deviation; non-classical mutual-information patterns).

We define the Primordial Quantum Field (PQF) as an ontological substratum described by a continuous functional field Ψ[ϕ] that evolves in a high-dimensional configuration space. Unlike standard quantum field theory (QFT), the PQF is not defined on spacetime; rather, it generates spacetime through its internal informational structure.

2.1. Formalism of the Primordial Quantum Field (PQF)

We clarify the construction of the functional Hamiltonian H^_PQF by specifying the roles of T[ϕ] (functional kinetic term), V_ent[ϕ] (non-local entanglement potential), and V_comp[ϕ] = f(S_c[ϕ]) (complexity potential). We explain the choice of first-order τ-evolution to capture the irreversible growth of organised complexity and distinguish it from Wheeler–DeWitt-type constraints.

We detail the steps required to operationalise S_c[ϕ] on discretised configurations (coarse-graining into subpatterns, mutual-information matrices, and complexity-density functionals), thereby enhancing replicability of the formal derivations.

To formalize the nature and dynamics of the Primordial Quantum Field (PQF), we postulate it as a wave functional, Ψ, defined over an abstract configuration space of informational fields. This space is not conventional spacetime, but a pre-geometric domain where informational properties are primary, and the universe’s metric and topological structure emerge from it.

• Wave Functional Variables: The PQF wave functional, Ψ[ϕ(x),t], depends on:

  ○

  ϕ(x): A configuration of a fundamental informational field. This “base field” ϕ does not reside in spacetime but in a space of internal parameters or an abstract informational state space S. Each “point” x in this abstract space represents an informational element or node of the PQF, and ϕ(x) is the “informational value” associated with that node. It is crucial to understand that x is not a spatial coordinate but an abstract label to differentiate the field components. We could conceptualize ϕ(x) as an “information probability field” or an “informational potential field.” In its most fundamental form, ϕ(x) could be a field of complex or quaternionic values that encode “primordial information” at a fundamental level.

  ○

  t: An evolution parameter we call “primordial time” or “complexity time.” This t is not relativistic physical time but an abstract parameter governing the PQF’s evolution towards states of greater informational complexity. The emergence of the physical “arrow of time” is derived from the irreversible evolution of the PQF in this parameter t.

Therefore, the wave functional is written as Ψ[ϕ(x),t], where Ψ assigns a probability amplitude to each possible configuration ϕ(x) of the primordial informational field at an instant t of primordial time.

• Evolution in the Abstract Functional Configuration Space: The evolution of the wave functional Ψ[ϕ(x),t] is not governed by a standard Schrödinger equation in spacetime, but by a more fundamental dynamic reflecting the PQF’s self-organization and complexity-seeking behavior. We postulate a generalized Wheeler-DeWitt-type evolution equation, or an analogous functional equation, operating in the space of all possible configurations of the field ϕ(x).

A tentative and highly speculative form for this evolution could be:

$$i{\displaystyle \frac{\partial \Psi \left[\varphi \right(x),t]}{\partial t}} = {\widehat{H}}_{PQF}\Psi [\varphi (x),t].$$

(1)

where H^PQF is a functional “Hamiltonian” of the PQF. This Hamiltonian is not the usual energy operator, but an informational complexity operator that drives the evolution towards states of higher complexity entropy (Sc). The terms in H^PQF would include:

○

“Informational propagation” terms: Operators describing how information propagates and becomes entangled within the PQF. These could involve functional derivatives with respect to ϕ(x) and terms representing non-local interactions.

○

“Complexity potential” terms: Functional operators that depend on the complexity entropy Sc[ϕ(x)] and favor the emergence of ϕ(x) configurations with high Sc. This implies the system has an inherent “preference” for organizing and structuring itself in increasingly complex ways. The explicit form of these terms is at the core of the theory and will require profound investigation. For example, they could be non-linear terms reflecting self-organization [13].

○

“Non-local coherence” terms: Operators that promote or maintain coherence and entanglement across PQF configurations, which would eventually manifest as non-local coherence pressure (dark energy).

Solving this functional equation is a considerable challenge, but its formulation is essential for defining the state space and fundamental dynamics of the PQF. “Symmetry breaking” and the emergence of spacetime, particles, and fundamental constants would manifest as phase transitions within the evolution of Ψ[ϕ(x),t], where certain ϕ(x) configurations become dominant and give rise to the physical structures we observe.

Mathematically, the PQF is represented as a functional Ψ[ϕ]:C → C, where C = {ϕ(x)∣x ∈ abstract manifold M, without defined metric}. Here, ϕ represents proto-quantum degrees of freedom, and no spacetime topology is initially defined. This allows us to model Ψ[ϕ] as a second-order quantum wave functional, similar to canonical quantum gravity.

The evolution of the PQF is governed by a Schrödinger-type functional equation:

$$i\hslash {\displaystyle \frac{\partial \psi \left[\mathsf{\Phi },\tau \right]}{\partial \tau }} = \mathsf{H}\left[\varphi ,\delta /\delta \varphi \right]\psi \left[\varphi ,\tau \right].$$

(2)

where τ is an internal coherence parameter, not necessarily physical time. We emphasize that while mathematically analogous to the standard Schrödinger equation, this functional version operates in a fundamentally different conceptual framework: (i) it evolves in an abstract configuration space rather than physical spacetime, (ii) the parameter τ represents internal coherence time, not physical time, and (iii) the Hamiltonian is an informational complexity operator rather than a conventional energy operator. Therefore, we designate it as ‘Schrödinger-type’ to highlight these essential distinctions.

We emphasize that while Klein-Gordon formalism is natural for relativistic particles in spacetime, the Schrödinger-type evolution is essential for our framework because: (i) it naturally accommodates the complex wave functional Ψ[ϕ] required for probability interpretation in infinite-dimensional configuration space, (ii) the first-order derivative in τ captures the irreversible complexity growth that generates the arrow of time, and (iii) it avoids the negative-frequency problem and associated constraint equations that would obscure the informational interpretation. The Klein-Gordon second-order form would reintroduce the Wheeler-DeWitt problem we explicitly avoid.

The functional Hamiltonian H^ is defined as:

$$\widehat{\mathsf{H}}\left[\varphi ,\delta /\delta \varphi \right] = -{\displaystyle \frac{{\hslash }^2}{2}}\int dx{\displaystyle \frac{{\delta }^2}{\delta \varphi ({x)}^2}} + V_{ent}\left[\varphi \right] + V_{comp}\left[\varphi \right].$$

(3)

where

○

The first term represents functional kinetic energy (analogous to the Wheeler-DeWitt term).

○

Vent[ϕ] measures non-local correlations within the field configurations ϕ, including measures of mutual information and topological connectivity.

○

V_comp[ϕ] = f(Sc[ϕ]); where f is a monotonic increasing function, analogous to Landau-Ginzburg free energy dependence on order parameter

2.1.1. Functional Action and Variational Principle of the PQF

To rigorously establish the dynamics of the Primordial Quantum Field (PQF), we define a functional action principle from which its evolution equations are derived.

Consider the action functional:

$$S\left[\psi \right] = \int d\tau <\psi \left[\varphi \right]/\mathsf{H}\left[\varphi \right] - i\hslash {\displaystyle \frac{\partial }{\partial \tau }}/\mathsf{\Psi }\left[\mathsf{\Phi }\right]>.$$

(4)

where τ is the internal coherence parameter, and H^[ϕ] is the functional Hamiltonian of the PQF, defined over the functional configuration space C. Varying this action with respect to Ψ∗[ϕ] yields:

$${\displaystyle \frac{\delta S}{\delta {\psi }^{\ast }\left[\varphi \right]}} = i\hslash {\displaystyle \frac{\delta }{\delta \tau }}\psi \left[\varphi \right] = \mathsf{H}\left[\varphi \right]\psi \left[\varphi \right].$$

(5)

which reproduces the previously postulated Schrödinger-type functional evolution equation. This framework allows interpreting the PQF’s dynamics as a unitary evolution in functional space, without requiring a pre-existing geometric background.

The construction of H^[ϕ] requires explicitly defining its components:

$$\widehat{\mathsf{H}}\left[\varphi \right] = T\left[\varphi \right] + V_{ent}\left[\varphi \right] +{ V}_{comp}\left[\varphi \right].$$

(6)

where

• T[ϕ] = −(ℏ²/2 m)⋅(δ²/δϕ(x)²); represents a functional kinetic term (Wheeler-DeWitt type).
• Vent[ϕ] encapsulates topological correlations (e.g., through functional mutual information).
• Vcomp[ϕ] = f(Sc[ϕ]); being a function of the complexity entropy defined in Section 2.2.

2.1.2. Analytical Solution for a One-Dimensional Functional Configuration ϕ(x)

To illustrate the dynamic behavior of the PQF, we consider a highly simplified 1D functional case where the field ϕ(x) is defined over the interval x ∈ [0,L], with periodic boundary conditions, and where the functional Hamiltonian approximates a quadratic form:

$$\widehat{\mathsf{H}}\left[\varphi \right] = -{\displaystyle \frac{{\hslash }^2}{2}}\int dx{\displaystyle \frac{{\delta }^2}{\delta \varphi ({x)}^2}} + {\displaystyle \frac{1}{2}}\int dx m^2\varphi ({x)}^2.$$

(7)

This model corresponds to a free functional oscillator with effective mass m, analogous to a functional version of the Klein-Gordon Hamiltonian without a background spacetime.

We can expand the field ϕ(x) in an orthonormal basis of modes (Fourier):

$$\varphi \left(x\right) = \sum _{n=1}^{\infty }{a}_{n}sin\left({\displaystyle \frac{n\pi x}{L}}\right).$$

(8)

In this representation, the wave functional is written as Ψ[ϕ] = Ψ(a1,a2,…), and the Hamiltonian separates into a sum of independent harmonic oscillators:

$$\widehat{\mathsf{H}} = \sum _n-({\displaystyle \frac{{\hslash }^2}{2}}{\displaystyle \frac{{\delta }^2}{\delta a_n^2}} + {\displaystyle \frac{1}{2}}{m}^2{a}_n^2).$$

(9)

The solutions to this functional equation are products of quantum oscillator base states:

$${\psi }_n\left(a_n\right) = {\mathsf{H}}_n\left(\sqrt{{\displaystyle \frac{m}{\hslash }}}{a}_n\right)e^{{\displaystyle \frac{-m{a}_n^2}{2\hslash }}}$$

(10)

where Hn are the Hermite polynomials. The base state (PQF vacuum) corresponds to:

$${\psi }_0\left[\varphi \right]= {\prod }_n{e}^{{\displaystyle \frac{-m{a}_n^2}{2\hslash }}} = exp[-{\displaystyle \frac{m}{2\hslash }}\int dx \varphi ({x)}^2] .$$

(11)

This functional state represents a Gaussian primordial quantum field, with a non-localized structure and no explicit spatiotemporal metric. Introducing topological correlations or non-Gaussian terms (via Vcomp[ϕ]) would produce more structured functional states, breaking internal symmetries and generating emergent geometry [34].

Figure 1 shows the components of the base functional state Ψn[ϕ(x)] for the first three harmonic modes of a Fourier expansion of the field ϕ(x). Each curve represents a Gaussian solution of the type exp(−mϕn(x)2/2ℏ), characteristic of the PQF functional vacuum in this idealized model.

Conceptual Representation of the PQF Wave Functional in 1D (Figure 1). We clarify that the Gaussian solutions displayed correspond to the PQF functional vacuum in a simplified 1D model; non-Gaussian corrections induced by V_comp[ϕ] lead to structured states associated with emergent geometric features.

This result supports the interpretation of the PQF as a continuous field in functional space, whose internal dynamics and organization can be studied through spectral analysis. The evolution towards more structured functional states (i.e., non-Gaussian) would be interpreted as the emergence of information and, eventually, geometry.

The solutions for the functional Hamiltonian H^[ϕ] for the 1D case, such as the functional Gaussian wave functions, representing coherent states of the PQF, are analogous to the ground states of a quantum harmonic oscillator [35]. These solutions not only demonstrate the formal viability of the equation but also suggest that the PQF can have fundamental modes of vibration and excitation that correspond to stable configurations of organized information.

Generalization to Higher-Dimensional Functional Spaces and the Emergence of Correlations.

While 1D examples are illustrative and provide an intuitive basis, generalizing this formalism to higher-dimensional functional configuration spaces is crucial for the emergence of 3D spacetime and matter. In a more general context, the functional field ϕ(x) would not be limited to a single spatial dimension x, but could depend on multiple coordinates (x,y,z) or even an abstract space of informational parameters.

The integral ∫dx in the functional Hamiltonian and in the expressions for complexity entropy would transform into an integral over this multidimensional configuration space. The complexity lies in defining non-local correlations and the complexity potential Vcomp[ϕ] in this space.

• Non-local correlations I(ϕ(x),ϕ(y)): For a field ϕ(x,y,z), the non-local correlation I(ϕ(x),ϕ(y)) must be interpreted as a measure of the informational dependence between two points, x and y, in the configuration space. This dependence does not necessarily imply a direct spatial connection in emergent spacetime, but an intrinsic correlation in the PQF’s informational substratum. Formally, I(ϕ(x),ϕ(y)) could be a correlation operator acting on the wave functional, or a two-point function measuring the ‘mutual information’ between field configurations at x and y. Non-locality in the PQF implies that these correlations can exist independently of the geometric distance that subsequently emerges.
• Definition of patterns in higher dimensions: Identifying the “discrete patterns” for complexity entropy Sc[ϕ] becomes more sophisticated. These patterns could be topological structures (knots, informational singularities), coherent entanglement configurations, or resonant modes of the field ϕ that persist over large regions. Discretization would refer to identifying equivalence classes of configurations or a subset of “fundamental” states in functional space.

The complexity of the PQF’s dynamics in these higher dimensions lies in the interaction between the kinetic term (which tends to smooth configurations), the correlation terms (which induce the formation of non-local structures), and the complexity potential (which favors the self-organization of specific patterns). These interactions, we postulate, lead to the emergence of geometry and matter.

Terminological Note: In this formulation, we employ certain terms with specific meanings. The term “internal coherence” refers to the degree of quantum organization of the PQF in the absence of a spatiotemporal metric; it is a measure of its internal correlational structure. “Proto-quantum” designates the fundamental degrees of freedom of the PQF, preceding any conventional interpretation of quantum states or particles [36]. Finally, “informational pressure” describes the macroscopic effect produced by the non-local coherence of the field, manifesting as an effective force on the emergent spacetime geometry, thereby reinterpreting the cosmological constant from a structural perspective.

2.2. Complexity Entropy (Novelty: First Functional Definition of Complexity Entropy Sc[ϕ] for Quantum Fields)

We define a new physical quantity, complexity entropy (Sc), to quantify the internal structure within field configurations.

2.2.1. Formal Definition

Complexity entropy quantifies the degree of structured organization, not just disorder, within a field configuration ϕ. It captures non-local correlations, patterns of internal organization, and functional connectivity.

Given a functional configuration ϕ(x), we define an emergent correlation graph Gϕ, where nodes represent regions/modes of the field, and links are weighted by quantum entanglement (measured by functional mutual information).

The complexity entropy is then defined as:

$$S_c\left[\mathsf{\varphi }\right] = -\sum _i{P}_i\left[\varphi \right].C_i\left[\varphi \right] .$$

(12)

Complexity Entropy (Sc): Formal Definition.

Complexity entropy (Sc) is the central metric of our theory, designed to quantify the degree of structured organization and informational self-organization capacity within the Primordial Quantum Field (PQF). Unlike traditional entropy measures that maximize disorder or uncertainty, Sc seeks to quantify the richness and coherence of informational entanglement.

For a given configuration of the informational field ϕ(x), the complexity entropy Sc[ϕ] is defined as a function that captures both the diversity of informational patterns and the coherence and multi-scale entanglement. We propose that Sc must be a measure that:

• Is low for trivial configurations (completely uniform or completely random).
• Is high for configurations exhibiting complex patterns, emergent hierarchies, and persistent entanglement throughout the abstract configuration space.
• Is a function of the informational entanglement density and the diversity of mutual information between the components of the field ϕ(x).

Mathematically, the formal definition of Sc is a key challenge and represents the core of the quantitative research in this theory. However, we can postulate a general functional form that encapsulates its desired properties. Inspired by concepts from computational complexity and quantum information, Sc for a configuration ϕ(x) is defined as:

$$Sc\left[\varphi \right]= -Tr\left({\widehat{\rho }}_{ent} log{\widehat{\rho }}_{ent}\right) - \alpha \sum _{ij}I\left(\varphi i,\varphi j\right)-\beta {\sum }_k{L}_k(\varphi ) .$$

(13)

where

• ρ^ent: Is an informational entanglement density matrix constructed from the field configuration ϕ(x). It represents the degree of non-local quantum entanglement between different “regions” or “modes” of informational PQF. The exact form of ρ^ent is an active area of research, but conceptually, it must quantify the amount of non-locally correlated information in ϕ(x). The first term, −Tr(ρ^entlogρ^ent), is analogous to von Neumann entropy, but applied to the entanglement structure of information, not to uncertainty about states. A high degree of structured entanglement, far from being random, contributes to complexity.
• I(ϕi,ϕj): Represents the mutual information between different components or informational blocks ϕi and ϕj of the field ϕ(x). This mutual information must be measured in the abstract configuration space and not in spacetime. The term ∑i, jI(ϕi,ϕj) (with a positive factor α) promotes the diversity of correlations and the ability of parts of the system to “inform” each other non-trivially. That is, not only is entanglement valued, but also the diversity of informational connections.
• Lk(ϕ): Are “simplicity penalty” or “structure reward” functionals (with a positive factor β) that ensure Sc penalizes trivial configurations (too uniform or too random) and rewards those with emergent patterns and organization. These functionals could be derived from computational complexity or algorithmic information theory. For example, they could be related to the length of the shortest description of ϕ(x) in a universal computational language (inspired by Kolmogorov complexity, but applied to emergent non-random structure).

It is fundamental to differentiate this complexity entropy (Sc) from other entropy measures:

• Difference from Shannon Entropy (H) and von Neumann Entropy (S): While H and S quantify uncertainty, disorder, or lack of information (they are maximized by uniform distributions or maximally mixed states), Sc measures non-trivial order and intrinsic organization. A system with high Sc is neither completely predictable nor completely random, but exhibits a delicate balance between order and variability that allows it to be “rich in structured information.” A random bit string has high Shannon entropy but low Sc; a repetitive sequence has low H and low Sc. A sequence with complex patterns (like a genome sequence or a fractal) would have high Sc.
• Difference from Lempel-Ziv Complexity (C_LZ) and Kolmogorov Complexity (K): C_LZ and K measure the compressibility of a data sequence, being high for sequences that cannot be easily compressed (like white noise). While there is a conceptual connection, Sc applies to the configurations of a primordial quantum field and seeks a fundamental property of informational self-organization and emergence. While K is generally incomputable, the hope is that Sc can be approximated or calculated for certain classes of PQF configurations relevant to the universe’s emergence. Sc is not simply a measure of description length, but of the deep informational structure that drives the universe’s dynamics [20,33].

The maximization of Sc throughout the evolution of primordial time (t) is what drives the “arrow of complexity” and the emergence of all physical structures and fundamental constants.

It is fundamental to differentiate this complexity entropy Sc[ϕ] from more traditional concepts in physics. Unlike Shannon entropy or von Neumann entropy, which measure informational uncertainty or disorder and are maximized by random or maximally mixed states, Sc[ϕ] is designed to quantify the degree of structured and non-trivial organization of the PQF. It does not seek a maximization of disorder, but of coherence and informational interconnection. Nor is it a measure of topological entropy or Kolmogorov entropy, although it shares with them the idea of quantifying the intrinsic complexity of a system.

“Complexity entropy” (Sc) is defined as a function of the form Sc[ϕ] = ∑iPi[ϕ]⋅Ci[ϕ], where the sum extends over a set of fundamental discrete patterns or configuration elements that can be identified or emerge within the continuous functional state ϕ.

• Pi[ϕ] represents the probability of occurrence or the measure of the preponderance of the i-th specific configuration pattern within the functional state ϕ. This Pi[ϕ] can be understood as the “information density” or “presence” of that pattern in the PQF’s overall configuration. For a continuous field, this would imply a form of quantification or discretization of the configuration space, for example, by projections onto a basis of pattern functions or the identification of coherent substructures.
• Ci[ϕ] is a function that quantifies the intrinsic complexity of said i-th configuration pattern. This complexity does not refer to randomness, but to the richness of its internal interconnections, its topological organization, and the non-triviality of its relationships with other patterns. For example, a pattern exhibiting strong non-local correlation or a fractal structure would have a high Ci[ϕ], while a purely random or simple pattern would have a low Ci[ϕ]. Ci[ϕ] must be non-negative.

Justification of the Form (∑iPi[ϕ]⋅Ci[ϕ])

The choice of this form for Sc[ϕ] is justified by several reasons:

• Differentiation from Shannon entropy: Unlike Shannon entropy, where the logarithm of probability penalizes improbability and favors uniform distribution (maximum disorder), the Pi[ϕ]⋅Ci[ϕ] form actively seeks to identify and quantify the presence of complex patterns. A very probable pattern (Pi[ϕ] high) but simple (Ci[ϕ] low) will contribute less to Sc than a pattern of moderate probability but high intrinsic complexity.
• Emphasis on structure: The factor Ci[ϕ] allows direct incorporation of the notion of ‘structured organization’. This contrasts with traditional entropy measures that ignore the internal structure of elements and only consider their frequency of occurrence.
• Intrinsic nature of complexity: Sc does not measure macroscopic disorder, but informational ‘richness’ and interconnection at the level of the PQF’s fundamental patterns. It is expected that Sc is low for completely uniform states (without patterns) or completely random states (without discernible pattern structure). It would reach its maximum for PQF states where the complexity of individual patterns is high and these patterns are present with a significant distribution.
• Analogy with Gibbs/Boltzmann entropy: Although the form differs, Sc can be thought of as a “generalized entropy” where each microstate (pattern i) contributes not only by its probability but also by an associated “complexity energy” Ci[ϕ]. This reflects a tendency of the system to favor configurations with a high degree of internal organization.

Units and Scaling

It is important to note that, for Sc[ϕ] to be a consistent physical quantity, Ci[ϕ] is assumed to be a dimensionless quantity representing the intrinsic complexity of pattern i. In this way, Sc[ϕ] is also a dimensionless quantity, allowing direct comparison with other information measures or as a factor in the PQF’s dynamic equations. Its relative value is what will matter for the system’s evolution.

2.2.2. Illustrative Example of Sc[ϕ]

To illustrate the behavior of complexity entropy Sc[ϕ], let us consider a simplified 1D functional configuration of the field ϕ(x) defined on a finite interval x ∈ [0,L].

Suppose the configuration exhibits a coherent harmonic modulation:

$$\varphi \left(x\right) = Asin\left({\displaystyle \frac{n\pi x}{L}}\right) + \delta \left(x\right) .$$

(14)

where A is a fixed amplitude, n ∈ N, and δ(x) represents localized random quantum fluctuations. For this configuration, we construct the correlation graph Gϕ by dividing the interval into N subregions {xi}, and we evaluate the functional correlation I[ϕ(xi),ϕ(xj)] for each pair.

When δ(x) ≈ 0, the field is highly structured, and the graph exhibits regular connectivity, resulting in a high value of Sc[ϕ]. As fluctuations δ(x) increase, global coherence breaks down, decreasing the graph’s connectivity and thus the structural complexity.

This change can be visualized as a phase transition between an ordered regime (coherent phase) and a chaotic regime (disorganized phase), analogous to decoherence processes in open quantum systems. In this model, the spatial gradient of Sc[ϕ] generates a non-trivial metric tensor G(x,y), leading to an emergent geometry whose curvature intensifies in regions with high complexity variation [37,38].

2.2.3. Example with Emergent Curvature in a 1D Field

To illustrate the link between complexity entropy and emergent geometry, we consider an idealized version of the Primordial Quantum Field in 1D.

Suppose a simple functional configuration of the field:

$$\varphi \left(x\right) = Asin\left({\displaystyle \frac{n\pi x}{L}}\right) + \delta \left(x\right) .$$

(15)

where A is the amplitude, n ∈ N and δ(x) are local quantum fluctuations modeled as Gaussian noise with variance σ2.

Figure 2 shows a concrete realization of a PQF configuration in one spatial dimension, defined as a harmonic wave perturbed by Gaussian noise. This configuration represents a structured but not perfectly coherent functional state, allowing us to study the effect of fluctuations on complexity.

The wave structure modulates the system’s global coherence, while local fluctuations introduce disorganization. This duality between order and functional chaos is key to generating gradients in complexity entropy.

We divide the interval x ∈ [0,L] into N subregions xi, and define the correlation between regions as:

$$I(\varphi (x_i),\varphi (x_j))= exp (-{{\displaystyle \frac{∕x_i-x_j∕}{2{\lambda }^2}}}^2)$$

(16)

where λ is a quantum correlation length (coherence type).

The complexity entropy is calculated as:

$$S_c\left[\varphi \right]= -\sum _{i=1}^N{p}_i\sum _{j\ne i}I\left(\varphi \right(x_i),\varphi (x_j\left)\right) .$$

(17)

where pi = 1/N for simplicity.

Based on the previous configuration, we evaluate the functional correlation matrix I(ϕ(xi),ϕ(xj)), which estimates the mutual information between different regions of the field. This matrix quantifies how subregions are correlated and forms the basis for calculating Sc.

Figure 3 illustrates a 1D functional field with noise, showing how fluctuations δ(x) modulate Sc[ϕ]S_c[\phi]Sc[ϕ], with the expected scaling behavior in λ and implications for operational measurement in quantum simulators (Source: Author’s own elaboration).

A strong correlation is observed in nearby regions (dominant diagonal), while correlation decreases exponentially with distance. This property allows defining a spatially coherence-dependent complexity entropy, which is subsequently used to derive an informational metric.

Now, we consider how this quantity varies when perturbing the field in a region xk, and we construct the informational metric:

$$G\left(x_i,x_j\right) = {\displaystyle \frac{{\delta }^2{S}_c\left[\varphi \right]}{\delta \varphi \left(x_i\right)\delta \varphi \left(x_j\right)}} .$$

(18)

Given that I(ϕ(xi),ϕ(xj)) depends on ϕ only through its spatial separation, we obtain:

$$G\left(x_i,x_j\right)\propto {\delta }_{ij}{\displaystyle \frac{{\left(x_i-x_k\right)}^2}{{\lambda }^4}} exp\left(-{\displaystyle \frac{{\left(x_i-x_k\right)}^2}{{2\lambda }^2}}\right) .$$

(19)

This is positive definite and decays off the diagonal. In the continuous limit, this matrix approximates an effective local metric G(x,y), with a structure similar to a Gaussian kernel centered at each point.

From the informational metric G(x,y), derived from the second functional variation in Sc[ϕ], an effective scalar curvature R(x) is calculated. This quantity measures the degree of emergent geometry induced by gradients in functional complexity.

The informational curvature emerging from the PQF formalism is illustrated in Figure 4, where the scalar curvature R(x)R(x)R(x), derived from the quantity log det G(x,y)\log \det G(x,y)log detG(x,y), is shown as an emergent geometric property of the underlying informational metric. As noted, this curvature acquires full physical meaning only in appropriate macroscopic regimes, where correspondence with classical geometric behavior becomes reliable.

Figure 4 presents the emergent scalar curvature R(x)R(x)R(x), linked to log det G(x,y)\log \det G(x,y)logdetG(x,y), highlighting its interpretational limits and the expected correspondence in the macroscopic regime (Source: Author’s own elaboration).

This emergent curvature acts as a geometric measure induced exclusively by the field’s internal organization, without the need to postulate a metric a priori. It reaffirms the central idea that spacetime and its curvature can arise from the information structure in the PQF.

Finally, we can define an effective scalar curvature:

$$\mathcal{R}\left(x\right)={\delta }_x^2\left[log detG\left(x,y\right)\right] .$$

(20)

This term captures the emergent curvature induced by fluctuations in complexity. In areas where δ(x) breaks harmonic coherence, R(x) increases, indicating the generation of geometry (i.e., curvature of emergent space) from functional chaos.

2.3. Emergent Geometry (Novelty: Emergent Metric G(x,y) Derived from Complexity Gradients Without Background Geometry)

We propose that the geometry of spacetime emerges from gradients in the PQF’s complexity entropy.

2.3.1. Informational Metric Tensor

Inspired by Fisher geometry, a natural metric in the parameter space of a probability distribution, we propose that the emergent spacetime metric is derived from the PQF’s informational structure. The informational metric tensor, Gxi,xj, is defined as:

$$G_{xi,xj} = {⟨{\displaystyle \frac{\delta lnP\left[\varphi \right]}{\delta \varphi \left(xi\right)}}{\displaystyle \frac{\delta lnP\left[\varphi \right]}{\delta \varphi \left(xj\right)}}⟩}_{\psi } .$$

(21)

where δ/δϕ(xi) represents the functional derivative with respect to the field at point xi, P[ϕ] is the probability distribution of finding the PQF in a specific configuration ϕ (i.e., P[ϕ] = ∣Ψ[ϕ]∣2), and the average ⟨…⟩Ψ is taken over the PQF’s wave functional Ψ[ϕ].

-

Justification and assumptions of the informational metric

This definition of the metric is based on the idea that spacetime geometry is not a fixed background but a manifestation of the intrinsic informational connectivity and variability of the Primordial Quantum Field. Key assumptions behind this formulation include:

○

Informational Principle: The postulate that information is the fundamental constituent of reality, and that geometric and physical properties emerge from informational relationships [25,39]. The metric Gxi,xj measures the “distance” or “separability” between infinitesimally different field configurations ϕ in terms of the information contained in the universe’s wave function.

○

Analogy with Fisher Geometry: Fisher’s metric quantifies the amount of information a random variable carries about an unknown parameter. Here, the ‘parameter’ is the field configuration ϕ, and the metric measures how sensitive the probability distribution P[ϕ] is to local changes in the field configuration. A high value of Gxi,xj indicates that small changes in ϕ at xi and xj result in significant and distinguishable changes in the PQF’s global probability distribution, implying strong connectivity or informational “rigidity” between those points, which manifests as curvature in emergent spacetime.

○

Quantum Emergence: It is assumed that the metric emerges from the PQF’s quantum dynamics. The average over Ψ implies that geometry is an inherent property of the universe’s quantum state. Geometry is not pre-existing but a consequence of how quantum information organizes and correlates within the PQF.

○

Classical/Macro-geometric Limit: In the limit where the PQF’s wave function localizes around a classical configuration ϕ0, the metric Gxi,xj should approximate the spatiotemporal metric tensor of general relativity. This would imply that the PQF’s quantum fluctuations (captured by Ψ) are responsible for deviations from the classical metric.

Inspired by Fisher’s information geometry, we define an informational metric tensor G:

$$G\left(x,y\right) = {\displaystyle \frac{{\delta }^2{S}_c\left[\varphi \right]}{\delta \varphi \left(x_i\right)\delta \varphi \left(x_j\right)}} .$$

(22)

This follows directly from the functional generalization of Fisher’s information metric, where second derivatives of entropy define natural distances in parameter space.

This metric tensor quantifies how structural complexity changes with perturbations of the field configurations.

By analogy with Riemannian geometry—where curvature is derived from the Levi-Civita connection constructed from the metric g_μν—we define here a functional connection Γ_ij^k associated with the informational metric G_ij as:

$${\mathsf{\Gamma }}_{ij}^k = {\displaystyle \frac{1}{2}}{G}^{kl}\left({\displaystyle \frac{\partial G_{li}}{\partial {\varphi }^j}} - {\displaystyle \frac{\partial G_{lj}}{\partial {\varphi }^i}} - {\displaystyle \frac{\partial G_{ij}}{\partial {\varphi }^l}}\right)$$

(23)

From this connection, a functional curvature tensor R_ijk^l and an emergent scalar curvature Rs can be defined via contraction:

$${\mathcal{R}}_s ={ G}^{ij}({\partial }_k{\mathsf{\Gamma }}_{ij}^k - {\partial }_j{\mathsf{\Gamma }}_{ik}^k - {\mathsf{\Gamma }}_{ij}^l{\mathsf{\Gamma }}_{lk}^k - {\mathsf{\Gamma }}_{ik}^l{\mathsf{\Gamma }}_{lj}^k$$

(24)

This formalism extends Fisher information geometry into a generalized functional field framework, enabling a direct link between complexity gradients and the emergent geometric properties of spacetime.

2.3.2. Generalized Einstein Equations

We can construct a functional curvature R(x,y), analogous to the Ricci curvature, from G. The generalized Einstein field equations are then proposed as:

$$\mathfrak{R}\left(x,y\right) - {\displaystyle \frac{1}{2}}G\left(x,y\right){\mathfrak{R}}_s = \kappa {\nabla }_x{\nabla }_y{S}_c\left[\varphi \right] .$$

(25)

Here, κ is a coupling constant, and Rs is the scalar informational curvature. This formulation demonstrates how informational structure gives rise to emergent geometry, with gradients of complexity entropy playing the role of the source of spacetime curvature—replacing the energy-momentum tensor in the traditional Einstein field equations.

This equation generalizes Einstein’s equations by relating the informational curvature of spacetime to the rate of change in complexity entropy. The scalar curvature Rs is directly derived from Gx,y, while the term ∇x∇ySc[ϕ] acts as an “informational energy-momentum” source.

-

Connection to Einstein’s Equations and the Classical Limit

To ensure consistency with established physics, it is essential that, in the macroscopic and classical limit, Equation (25) reduces to the standard Einstein field equations of General Relativity:

$$R_{\mu \nu } - {\displaystyle \frac{1}{2}}{g}_{\mu \nu }R = {\displaystyle \frac{8\pi G}{c^4}}{T}_{\mu \nu } .$$

(26)

This reduction is hypothesized through the following mechanisms:

• Emergence of 4D Spacetime: In states of highly organized complexity within the Primordial Quantum Field (PQF), the functional configuration space is hypothesized to ‘collapse’ or ‘crystallize’ into an effective 3 + 1 dimensional spacetime. The coordinates x,y in Gx,y and in ∇x∇ySc[ϕ] would map to the usual spacetime coordinates μ,ν analogous to long-range order emergence in many-body systems.
• Mapping Informational Metric to Spacetime Metric: The informational metric tensor Gx,y becomes identifiable with the spacetime metric tensor g_μν in this emergent limit: Gx,y→g_μν, as the PQF organizes into a classical spacetime.
• Identification of the Informational Energy-Momentum Tensor: The term on the right-hand side, κ∇x∇ySc[ϕ], must correspond to the energy-momentum tensor T_μν, which describes matter and energy in General Relativity. This implies that matter, radiation, and dark energy are ultimately macroscopic manifestations of the structure and dynamics of complexity entropy in the PQF.

Specifically, one may postulate that ∇x∇ySc[ϕ] is directly related to the energy and momentum density of matter fields. Peaks and gradients in the complexity entropy represent concentrations of energy and mass that curve spacetime.

The constant κ would then correspond to Einstein’s gravitational constant, 8πG/c⁴, or a related quantity emerging from the theory, setting the coupling strength between informational complexity and spacetime curvature.

In this framework, gravitational dynamics are not fundamental but rather emergent, arising from the way informational complexity organizes and evolves within the Primordial Quantum Field. The sources of gravity (matter and energy) are nothing more than high-complexity patterns or complexity gradients within the informational substrate, with their density determining the curvature of emergent spacetime.

2.3.3. Informational Phase Transitions/Coherence Breakdowns

Within this framework of an information-based emergent metric, the very birth of spacetime—and of physical structures within it—can be conceptualized as a series of informational phase transitions or coherence breakdowns in the PQF. Analogous to how a Bose-Einstein condensate exhibits macroscopic order at low temperatures, the PQF may, under critical dynamical conditions, transition from a state of incoherence or diffuse complexity to one of high organization and structure.

These transitions are not merely energetic; they are driven by the PQF’s intrinsic tendency to maximize its complexity entropy Sc[ϕ]. As Sc[ϕ] increases and the PQF explores configurations of greater organization and non-local entanglement, informational ‘crystallizations’ or ‘condensations’ may occur, giving rise to observable physical properties:

• Emergence of Spacetime: The curvature and topology of spacetime arise as emergent properties of the informational metric tensor Gx,y. Critical phase transitions in the PQF could generate long-range order in informational connectivity, manifesting as a continuous, causal spacetime.
• Emergence of Particles and Fields: Stable excitations, knots, or singularities in PQF complexity patterns may correspond to proto-particles or fundamental fields. These states would be robust due to their intrinsic high complexity and coherence-preserving structures.
• Symmetry Breaking: The PQF’s evolution toward higher organized complexity may entail spontaneous breaking of fundamental symmetries, leading to the differentiation of fundamental forces and particle hierarchies. These symmetry breakings would emerge from the internal self-organization of the informational field, not from external potentials.

Formally, such phase transitions can be characterized by informational order parameters, analogous to magnetization in ferromagnets or the condensate wavefunction in superfluids. An order parameter might be, for instance, the density of non-local entanglement or a measure of topological coherence in the PQF. The attainment of critical values for these parameters would signal the emergence of new physical structures.

2.4. Open System Considerations and Environmental Effects

Our formalism has thus far considered the Primordial Quantum Field as a closed system evolving under its internal dynamics. However, realistic quantum systems interact with their environments, leading to decoherence and dissipation that may affect the emergence of complexity and geometry. Recent studies have explored how environmental coupling modifies quantum coherence and entanglement structures in systems with non-local correlations [1,2,3,4].

For the PQF, environmental effects could arise from:

• Informational leakage: Loss of coherence through coupling to unobserved degrees of freedom
• Noisy complexity potential: Stochastic fluctuations in V_comp[ϕ] due to external influences
• Decoherence-induced phase transitions: Environmental monitoring could suppress long-range correlations

Following the framework of Breuer and Petruccione (2002) [40], we can model open PQF dynamics using a functional Lindblad equation:

dτdρ^_PQF = −i[H^_PQF,ρ^_PQF] + ∑_kγ_k(L^k ρ^_PQF L^_k^† − 1/2{L^_k^† L^k, ρ^_PQF});

where L^k are functional Lindblad operators describing environmental interactions. Studies of open quantum systems show that moderate decoherence can sometimes enhance complexity through noise-induced structure formation (PRA 101, 013826) [41]. Conversely, strong environmental coupling tends to suppress non-local correlations (PRA 98, 023856) [42]. The timescales for these effects must be compared with the coherence parameter τ to assess their relevance for cosmological emergence.

In the PQF context, the “environment” may represent degrees of freedom that decoupled during early-universe phase transitions. Recent experiments on decoherence in macroscopic superpositions (PRL 103, 210401) [43] and entanglement dynamics in noisy channels (PRA 81, 042103) [44] provide empirical benchmarks for estimating decoherence rates in our framework. We note that for cosmological scales, the PQF likely operates in a weak-coupling regime where internal complexity generation dominates over environmental dissipation.

3. Predictions and Experimental Proposals

We restructure Section 3 to explicitly link each prediction to the underlying formal elements: (i) informational pressure and Λ ~ ⟨S_c[ϕ]⟩/V from the complexity potential; (ii) entanglement-wave signatures tied to non-local terms in V_ent[ϕ]; and (iii) simulator-based operational metrics for S_c via projected density matrices. We add bridging sentences and signposted equations to guide interpretation.

We clarify measurement strategies for mutual-information profiles (I(ϕ_i,ϕ_j)) and coherence lengths λ, noting expected regimes and practical limits for near-term platforms (BECs, photonics, superconducting qubits).

3.1. Dark Energy as Informational Pressure (Novelty: Quantitative Prediction for Dark Energy: w(z) = −1 + 0.01)

Dark energy is proposed to emerge as a pressure-like term arising from the nonlocal coherence of the PQF (Projective Quantum Field). The cosmological constant is reinterpreted as:

$$\bigwedge ~<S_c[\varphi ]> .$$

(27)

This term aligns with cosmic acceleration and may be tested through studies of CMB anisotropies and large-scale galaxy clustering.

3.2. Entanglement Waves and Gravitational Analogues

Nonlocal fluctuations in Sc give rise to “entanglement waves” that propagate with potentially detectable gravitational-like effects. These effects could be observed in:

• Bose-Einstein condensates
• Low-temperature nonlinear optical systems
• Quantum interferometers

3.3. Simulations of Sc

Quantum simulations using qubit arrays or photonic networks may offer an operational definition of Sc via:

$$S_c\left[\rho \right]=-{\sum }_i{p}_{i}Tr [{\prod }_i\rho {log}_2{\prod }_i\rho ] .$$

(28)

Measuring mutual information correlations and local entropies could reveal phase transitions and the emergence of structure.

3.4. Operationalization and Experimental Verification Pathways

To ground the theory in empirical relevance, we outline several experimental pathways based on current or near-term technologies:

• Informational Coherence Pressure (Dark Energy Analogue)

We propose that dark energy arises as an effective pressure stemming from nonlocal informational coherence:

$$\bigwedge ~<S_c\left[\varphi \right]>/ V .$$

(29)

where V is the emergent spacetime volume. h_ent ~ 10⁻⁴⁰ ± 10⁻⁴² based on coherence length uncertainty. Given the observed value of the cosmological constant Λobs ∼ 10⁻⁵² m⁻², we estimate that the average structural complexity Sc[ϕ] must be extensive and relatively uniform at cosmological scales. This suggests a PQF coherence length on the order of hundreds of megaparsecs. Future observations of CMB polarization and large-scale structure (LSS) power spectra could constrain this coherence [45].

• Entanglement Waves in Laboratory Systems

Our theory predicts nonlocal excitations—“entanglement waves”—that modulate complexity across distant regions. These may manifest as anomalous correlations or decoherence patterns in engineered quantum systems, such as:

• Bose-Einstein condensates under dynamic optical potentials [29].
• Low-temperature nonlinear optical networks
• Superconducting qubit arrays exhibiting long-range mutual information

We propose an operational signature: detection of nonclassical temporal correlations in the mutual information structure I(ϕi,ϕj)) at separations exceeding the coherence length λ\lambdaλ, measurable via quantum tomography or entanglement entropy profiling.

• Simulations of Complexity Dynamics

Quantum simulators—such as trapped ions or photonic circuits—can emulate simplified PQF dynamics. We propose an operational version of complexity entropy:

$$S_c\left[\rho \right]=-{\sum }_i{p}_{i}Tr [{\prod }_i\rho {log}_2{\prod }_i\rho ] .$$

(30)

where {Πi} is a set of projections onto functional subspaces of the system. By preparing structured initial states and monitoring their evolution under nonlocal interactions, one may observe informational phase transitions and the emergence of spatial structure. These simulations could be implemented in hybrid qubit-photon platforms or neuromorphic networks.

Specific Experimental Platforms

Current quantum technologies provide several promising avenues for testing PQF-inspired dynamics:

Bose-Einstein Condensates (BECs): Ultracold atomic gases offer ideal testbeds for emergent quantum phenomena. Recent experiments demonstrate long-range coherence and pattern formation in BECs under non-local interactions [29]. By engineering spatially modulated trapping potentials and measuring correlation functions g^(2)(r), we can probe complexity entropy gradients analogous to those proposed in our model. Modern BEC experiments achieve correlation lengths λ ≈ 10–100 μm, providing accessible scales for observing entanglement wave propagation. For BEC, λ_optical ≈ 10 μm, N_atoms ≈ 10⁶, g^(2) resolution ≈ 0.01.

Superconducting Qubit Arrays: Circuit QED systems with 50+ qubits enable simulation of complex network dynamics. Google’s Sycamore processor demonstrated quantum supremacy using non-local entangling gates (Nature 574, 505 (2019) [46]. These platforms can implement simplified Hamiltonians of the form Ĥ = Σ_i ω_i σ_i^z + Σ_{i,j} J_{ij} σ_i^ + σ_j^- + V_comp(Sc); where the complexity potential can be programmed through cross-Kerr interactions. Quantum tomography allows direct measurement of mutual information I(ϕ_i, ϕ_j) and complexity entropy Sc[ρ].

Photonic Quantum Networks: Integrated photonic chips with programmable beam splitters and phase shifters provide scalable platforms for studying informational geometry. Recent work demonstrated measurement of Fisher information metrics in photonic systems (Phys. Rev. Lett. 129, 030502 (2022) [47], directly relevant to our informational tensor G(x,y). These systems can simulate PQF dynamics with up to 20 photons and 100 modes, sufficient to observe emergent geometric phases.

Trapped Ion Quantum Simulators: Systems with 20–50 ions offer high-fidelity operations and individual readout. Experiments have simulated quantum field theories and measured entanglement propagation speeds (Nature 511, 198 (2014) [48]. The long-range Coulomb interactions in these systems naturally implement the non-local correlations V_ent[ϕ] postulated in our framework.

Operational Protocol: We propose the following sequence:

• Initialize system in low-complexity state (e.g., product state)
• Apply engineered interactions to increase Sc[ϕ] via V_comp
• Measure correlation matrix I(ϕ_i, ϕ_j) with quantum state tomography
• Reconstruct informational metric G(x,y) from Sc[ϕ] gradients
• Detect signatures of emergent curvature through geometric phase measurements

Near-term feasibility: Current technology can achieve steps 1–3 with 10–20 qubits/photons. Steps 4–5 require advanced error mitigation but are within reach of next-generation quantum devices (100+ qubits) expected by 2026–2028.

4. Comparative Analysis with Existing Fundamental Theories

The Primordial Quantum Field (PQF) framework diverges significantly from other prominent approaches to unification, not only in its ontological foundations but also in its mathematical structure and predictions. Below, we provide a focused comparison with several influential paradigms.

4.1. Loop Quantum Gravity (LQG)

Loop Quantum Gravity models spacetime as a discrete spin network, in which areas and volumes are quantized. It successfully recovers black hole entropy spectra and aims to quantize general relativity without requiring a background metric. However, LQG faces challenges in incorporating thermodynamic principles and fails to offer a fully emergent explanation of spacetime or matter [49,50].

In contrast, the PQF does not assume a discrete spacetime structure. Granularity emerges naturally as a consequence of topological transitions in the internal structure of the field, captured by gradients in complexity entropy Sc[ϕ]. Furthermore, the PQF is formulated on a continuous functional space, allowing for a deeper integration with quantum information theory and functional analysis, while remaining background independent.

4.2. Emergent Gravity (Verlinde)

Verlinde’s entropic gravity approach derives gravitational interactions from thermodynamic arguments based on entropic gradients associated with particle positions. While conceptually insightful, this framework operates on a predefined spacetime manifold and does not explain the emergence of geometry or matter fields.

Our framework, by contrast, introduces a structural form of entropy—complexity entropy Sc[ϕ]S_c[\phi]Sc[ϕ]—which captures long-range internal correlations of the PQF. Geometry and curvature are not imposed but instead emerge from these structural gradients. The informational dynamics of the PQF generate space, time, and physical constants in a unified, self-organizing fashion.

4.3. Holographic Duality (AdS/CFT)

The AdS/CFT correspondence posits a duality between a conformal field theory on a boundary and a gravitational theory in the bulk. This has yielded remarkable insights into quantum gravity and black hole thermodynamics. However, its domain of validity is limited to specific geometries (asymptotically AdS spaces) and does not itself constitute an ontological theory of emergence [51,52].

Our theory avoids the need for duality by proposing a direct informational mechanism for the emergence of spacetime. The PQF exists as a self-structured functional field whose internal dynamics generate geometry through complexity gradients, not boundary symmetries. It thus bypasses the requirement of matching theories in different dimensionalities.

4.4. Causal Set Theory and Discrete Models

Causal set theory models spacetime as a discrete set of events ordered by causality. While this framework offers a combinatorial perspective on spacetime emergence, it lacks a dynamic principle for generating the physical laws or accounting for the origin of entropy and time’s arrow [53].

In our model, causality and the arrow of time emerge naturally from informational structure. Specifically, the spatial and functional gradients of Sc[ϕ] introduce asymmetries that break temporal symmetry and induce an entropic flow. This replaces the postulation of causal orderings with a dynamical, informationally grounded mechanism.

4.5. Synthesis: Toward a Unified Ontology

In all of the above frameworks, some component of physical structure—spacetime, entropy, or causal relations—is assumed. The PQF model departs from this by treating information itself as the generative substrate. Space, time, matter, and physical laws emerge as secondary phenomena from the self-organizing behavior of the informational field.

Unlike string theory, which assumes a specific dimensional and supersymmetric setup; unlike LQG, which imposes discreteness; and unlike holography, which presumes dualities—our framework introduces a single continuous ontological structure governed by measurable complexity. It offers a falsifiable Table and operationally defined path to a Theory of Everything rooted in emergent information geometry [54].

A comparative overview of how these frameworks address ontology, their sources of geometry, and their formulation of empirical predictions as measurable informational structures is presented in

Table 1. Comparison with Other Fundamental Theories. Source: Author’s own elaboration. We add explicit definitions of ontology and source-of-geometry columns and note that empirical predictions are formulated as measurable informational structures (mutual information, complexity metrics).

Theory	Ontology	Source of Geometry	Empirical Predictions
PQF (this work)	Informational field	Gradient of Sc[ϕ]	Entanglement waves, complexity metrics
Loop Quantum Gravity	Discrete spin networks	Quantum connections	Area spectra, black hole entropy
Verlinde	Thermodynamics	Entropic gradients	Modified Newtonian dynamics
Causal Sets	Ordered events	Causal relations	Discreteness at Planck scale
Holography (AdS/CFT)	AdS/CFT duality	Boundary-field encoding	Entanglement entropy scaling

(Author’s own elaboration).

We acknowledge current limitations: (i) incomplete closed-form specification of the complexity potential f(S_c) beyond qualitative monotonicity; (ii) challenges in computing S_c for high-dimensional functional configurations; (iii) need for rigorous proofs of correspondence limits (G(x,y) → g_{μν}). We outline future work: deriving tractable approximations for S_c, establishing well-posedness and stability of the functional metric, and designing controlled quantum-simulator experiments targeting entanglement-wave observables.

For a detailed and rigorous formulation of the functional formalism underlying the Primordial Quantum Field, including the definition of the functional configuration space, the second-order wave functional Ψ[ϕ], and the functional Schrödinger dynamics that govern its evolution, the reader is referred to Appendix A. There, the emergent informational metric, the complexity entropy Sc[ϕ]S_c[\phi]Sc[ϕ], and the curvature structures arising from PQF dynamics are developed in full mathematical detail, providing the theoretical foundations for the emergent geometric framework discussed in the main text.

5. Conclusions

We make explicit the logical flow from the formalism (informational metric and generalised field equations) to the empirical proposals (dark-energy coherence pressure and entanglement waves). We also state concrete falsifiability conditions (non-detection within specified sensitivity ranges would refute key claims), enhancing the manuscript’s testable character.

In this work, we have proposed a unified physical framework grounded in a Primordial Quantum Field (PQF)—a continuous, non-local, self-contained informational substrate preceding space-time and matter. In Section 2.2, we introduced Sc[ϕ], introducing the concept of complexity entropy Sc[ϕ], we formulated a functional dynamics through which spacetime geometry, curvature, physical constants, and the arrow of time emerge as secondary effects of the PQF’s internal informational structure.

This proposal transcends traditional approaches by avoiding predefined geometric backgrounds or discreteness, and instead offers an operational, quantifiable, and falsifiable mechanism for the emergence of physical reality. By reinterpreting dark energy as informational pressure and predicting phenomena such as entanglement waves, the model provides concrete experimental signatures that distinguish it from purely formal or speculative frameworks.

At its core, this theory redefines the ontology of physics—not as a collection of pre-existing objects and laws, but as the outcome of a self-organizing informational process. In doing so, it offers a pathway toward a Theory of Everything grounded in quantum information, functional analysis, and emergent geometry, with deep implications for both fundamental physics and the philosophical understanding of the universe with potential deviations from LQG and string theory at measurable levels. Future lattice simulations are needed for coupling constants.