Spacetime as emergent order: a testable framework from string-net condensation to geometric thermodynamics

Abstract

The nature of spacetime in quantum gravity remains a fundamental problem, rooted in the conflict between the background independence of general relativity and the fixed-background structure of quantum field theory. This paper proposes a concrete, emergentist framework in which classical continuous spacetime is not fundamental but arises as a low-energy effective description of an underlying quantum information structure. We construct this emergence explicitly, starting from a unitary fusion category as the “algebraic DNA”, realizing it via a string-net condensate, and employing an equivariant tensor renormalization group flow to reach a geometric fixed point. Within this derived geometry, we introduce two key macroscopic variables: the entropy current vector (ECV) , which describes the flow of entanglement entropy in spacetime, and the cosmological enthalpy (CE) , an effective thermodynamic potential that incorporates pressure-volume work. We show that the Einstein field equations emerge as conditions of entanglement equilibrium, with the effective gravitational constant determined by categorical data. Crucially, this framework is not merely a conceptual narrative; it yields concrete, falsifiable predictions. We provide an explicit blueprint for near-term quantum simulation experiments—using ultracold atoms or superconducting qubit arrays—to measure the predicted area-law entanglement, the topological entanglement entropy, and the emergent graviton mode, thereby transforming the hypothesis of spacetime emergence into an empirically testable scientific program.

1 Introduction

1.1 The core problem: from background conflict to emergence

At the heart of the difficulty in constructing a quantum theory of gravity lies a profound ontological tension. General Relativity (GR) describes spacetime as a dynamical, background-independent entity that curves in response to matter. Quantum Mechanics (QM) and Quantum Field Theory (QFT), in stark contrast, are formulated on a fixed, non-dynamical spacetime background, where evolution is parametrized by an external time (Wheeler, 1990). Attempts to directly quantize GR encounter the non-renormalizability problem, signaling that the classical geometric description may break down at the Planck scale. This crisis strongly suggests that spacetime itself may not be a fundamental entity, but rather an emergent, collective phenomenon arising from a deeper, non-spatiotemporal quantum structure (Wen, 2018).

This notion of “emergent spacetime” has gained significant traction, championed by several lines of inquiry. The holographic principle, concretely realized in the AdS/CFT correspondence (Maldacena, 1998), demonstrates how a bulk spacetime geometry and its gravitational dynamics can be dual to a boundary quantum field theory without gravity (Yang et al., 2025). Jacobson’s thermodynamic derivation of Einstein’s equations (Jacobson, 1995) suggests that classical gravity is an equation of state for an underlying system of unknown microscopic degrees of freedom. Meanwhile, condensed matter physics offers explicit models, such as Wen’s string-net condensation (Levin et al., 2005; Levin et al., 2006; Levin and Wen, 2005), where gauge fields, fermions, and even a cosmological constant can all emerge from a purely bosonic, highly entangled qubit system.

This paper aims to bridge this gap. We propose a unified, testable framework in which spacetime geometry and its gravitational dynamics emerge as macroscopic order parameters from the entanglement structure of a quantum many-body system—a string-net condensate.

1.2 Core thesis and roadmap of the framework

Our core thesis is that classical spacetime is the low-energy, coarse-grained image of an underlying quantum network. We realize this thesis through a three-tiered construction:

1.2.1 Microscopic origin (“algebraic DNA”)

We posit that the fundamental structure of the quantum vacuum at the Planck scale is described by a unitary fusion category (Baez et al., 2011). This is an abstract algebraic structure that encodes the rules for fusing and braiding the fundamental constituents (qubits). It presupposes no spacetime manifold; rather, it is a pre-geometric, relational “source code” for reality. This is our foundational hypothesis [HP-1].

1.2.2 Emergent geometry and thermodynamics

Prediction 1 (Entanglement Area Law): The von Neumann entanglement entropy of a spatial region will obey an area law, with the sub-leading topological entropy γ directly encoding the total quantum dimension.

Prediction 2 (Entropy Current Propagation): A localized energy perturbation will generate a propagating “entropy pulse”, whose velocity and diffusion profile can be tracked in real-time, directly validating the microscopic definition of the ECV .

Prediction 3 (Emergent Graviton Mode): The low-energy excitation spectrum will contain a massless spin-2 mode whose two-point correlation function exhibits the characteristic pole of a graviton propagator, allowing for an experimental extraction of the effective Newton constant .

1.3 Structure of the paper

The paper is organized to systematically build this argument. Section∼2 details the core logical architecture, from the unitary fusion category (the “algebraic DNA”) to the emergent geometric and thermodynamic variables, explicitly stating our core assumptions. Section∼3 provides the rigorous mathematical foundation for this emergence, presented in a streamlined form with detailed proofs and constructions deferred to the Appendices. Section 4 develops the thermodynamic formulation, introducing the ECV and CE and showing how they govern gravitational dynamics and cosmic structure formation. Section∼5 presents the crucial empirical component, detailing the proposed quantum simulation experiments and their theoretical targets. Finally, Section∼6 discusses the unique contributions of this framework in direct comparison with existing approaches, and concludes with an outlook on future theoretical and experimental developments. Supplementary Appendices A,B contain the complete category-theoretic construction, convergence theorems, and thermodynamic derivations for the specialist reader.

2 From quantum information to emergent geometry: the constructive logic

This section outlines the stepwise constructive logic of our framework. Our goal is to make explicit how a classical, continuous spacetime with its geometric and thermodynamic properties can be rigorously derived—in the sense of an effective low-energy description—from a pre-geometric, algebraic quantum structure. To this end, we clearly distinguish between the physical reality (the quantum state) and the mathematical language we use to describe it (the manifold).

2.1 The ontological reversal: it from qubit, algebra as DNA

We take as our starting point the hypothesis that the fundamental constituents of the universe are quantum bits (qubits) (Wen, 2018; Levin et al., 2005), and that their collective behavior is governed by patterns of long-range entanglement. More precisely, we posit that the complete algebraic structure encoding all possible correlations and dynamics of this qubit sea is a unitary fusion category (Levin et al., 2005; Levin et al., 2006; Levin and Wen, 2005; Baez et al., 2011).

Foundational Hypothesis [HP-1]: The physical vacuum at the Planck scale is fully described by a unitary fusion category . This category is not defined within any pre-existing spacetime; it is the fundamental, non-spatiotemporal origin from which spacetime and matter emerge.

This data constitutes the “algebraic DNA” of reality. It presupposes no metric, no manifold, no notion of distance or time. This is the foundational ontological reversal: we do not quantize geometry; rather, we aim to geometrize the quantum information encoded in .

For readers unfamiliar with category theory, the following physical logic may be helpful. A unitary fusion category can be understood as a “relational blueprint” for quantum information: the simple objects are the types of all possible elementary entanglement channels that can form between qubits, and the fusion rules determine which new channels can arise when two qubit channels are superposed----analogous to angular momentum coupling, but generalized to arbitrary patterns of many-body quantum entanglement. The F-symbols are “reassociation operators”: when three channels are combined in different orders, the F-symbols give the unitary transformation between the two schemes, carrying the precise phase factors for how quantum information redistributes among different pairing configurations. The R-symbols are “exchange operators”: when the spatial positions of two channels are swapped, the R-symbols give the statistical phase accumulated by the system’s wavefunction. It is these quantum phase factors----the reassociation phases from F and the exchange phases from R----that constitute the microscopic origin of combinatorial curvature in the quantum many-body system, and ultimately manifest, in the continuum limit, as the curvature of macroscopic spacetime. In other words, the role of category theory in this framework is not mathematical ornamentation, but the provision of a set of algebraic rules that dictate, at the most fundamental level, “how entanglement can be connected” and “what effects such connections produce.” Spacetime geometry, in this sense, is the statistical expression of these algebraic rules at large scales.

This data constitutes the “algebraic DNA” of reality. It presupposes no metric, no manifold, no notion of distance or time. This is the foundational ontological reversal: we do not quantize geometry; rather, we aim to geometrize the quantum information encoded in .

2.2 The descriptive language: the string-net condensate on a combinatorial manifold

To make contact with the macroscopic world, we must find a descriptive tool—a language in which the quantum state described by can be computed and its macroscopic behavior recognized. This is provided by the string-net condensate constructed on a combinatorial manifold (Wen, 2003; Wen et al., 2002).

On this scaffolding, we define the Levin–Wen Hamiltonian (Levin et al., 2005). Its ground state, , is the string-net condensate. This state is a physical realization of the category 's algebraic rules.

2.2.1 Ontological clarification

The triangulated manifold is the most effective language we have to describe the entanglement structure of the state . The statement “the entanglement structure of is statistically equivalent to a string-net condensate constructed on ” is what justifies the use of this language (for rigorous criteria, see App. A.1.2). The physical reality is the state and its underlying algebraic data .

2.3 Emergent geometry: from categorical data to metric and curvature

The power of this construction lies in its ability to reveal geometric order from algebraic rules. Geometry is not an input; it is an output of the renormalization group (RG) flow.

2.3.1 The equivariant tensor renormalization group (ETRG)

We coarse-grain the string-net condensate using an Equivariant Tensor Renormalization Group (ETRG, Def. A.10), a procedure that integrates out short-range entanglement while exactly preserving the algebraic symmetries of the category . This flow drives the system to a critical fixed point—a quantum phase transition where a new type of order (Wen, 2018; Levin et al., 2005) geometric order, condenses (App. A.3.2).

Here, is the internal metric determined exclusively by the quantum dimensions and fusion rules of the category .

2.3.2 Curvature from categorical defects

On the discrete triangulation , curvature is not a smooth field but a combinatorial deficit. For a closed loop in the network, the process of sequentially braiding and fusing the strings—computed using the -symbols and -symbols—yields a unitary operator. The phase of this operator defines a combinatorial curvature (Lemma A.2, Def. A.4). It measures the obstruction for the categorical parallel transport to return to the identity.

Key Result (Theorem A.5, under Assumption HA-1) (Hirani, 2003; Regge, 1961): In the continuum limit of an infinitely refined triangulation (), the combinatorial curvature, when paired with a smooth test form, converges in the sense of distributions to the classical Riemann curvature tensor . The proof of this result relies on the vanishing of quantum curvature corrections in the continuum limit (Assumption HA-1, see App. A.1.2.4).

This theorem establishes the rigorous bridge between the discrete (Bridgeman et al., 2017), algebraic micro-world and the smooth, continuous macro-world. The macroscopic metric inherits its curvature from the categorical braiding and fusion data.

2.4 Emergent entropic flow: the microscopic origin of the ECV

The geometric skeleton of space is only half the picture. The dynamics within this emergent spacetime are governed by the flow of quantum information, captured macroscopically by the Entropy Current Vector (ECV).

These operators are explicitly constructible from the string configuration and their evolution is governed by the Levin-Wen Hamiltonian (App. A.3.5). In the continuum limit defined by the ETRG flow, the expectation value of this discrete current, divided by the area of the elementary face, converges to a smooth vector field .

Combining this with the entropy density and the local four-velocity of an observer at rest with the cosmic fabric, we derive the macroscopic ECV:

The local covariant second law of thermodynamics, , is therefore not a phenomenological assumption but a macroscopic consequence of the microscopic unitary evolution and the strong subadditivity of entanglement entropy. The quantitative relationship between entropy growth and structure formation has been investigated through cosmological simulations by Zhao et al. (2013), who found that black hole entropy dominates at all epochs and that entropy evolution follows a path parallel to galaxy formation.

2.5 Emergent thermodynamic potential: the microscopic origin of CE

The ECV describes how quantum information flows. But to understand why it flows—and what drives cosmic expansion and the formation of structure—we need a second macroscopic variable: a thermodynamic potential that encodes the system’s capacity to do work. This is provided by the Cosmological Enthalpy (CE) .

In standard thermodynamics, the enthalpy is the natural potential for a system coupled to a pressure reservoir. In a cosmological setting, the expanding universe itself acts as this reservoir. The internal energy alone is insufficient, because the universe’s matter content does work against—or is driven by—the cosmic expansion. The CE incorporates this work from the outset:where is the proper energy density and is the total pressure of the matter-geometry system.

Thus, the CE density is not a phenomenological construct. It emerges as the expectation value of a composite operator built from the Hamiltonian density (which yields ) and the entanglement response to local scale transformations (which yields ). In the continuum limit of the ETRG flow, this composite operator converges to the macroscopic (see App. B.4 for the explicit path-integral derivation).

The CE plays a role in our framework analogous to the Gibbs free energy in ordinary thermodynamics. For an isolated gravitationally bound system in an expanding universe, the spontaneous direction of evolution is toward minimizing . Gravitational collapse, structure formation, and energy release all correspond to processes where .

More generally, the coupled evolution of the ECV and the CE defines a generalized Gibbs free energy functional:where is the Hawking-Gibbons temperature of the cosmic horizon, and is the total entanglement entropy. The minimization of provides a unified variational principle that governs both the background cosmic expansion and the formation of large-scale structure (see § S4 and App. B).

2.6 Summary of the construction roadmap

To conclude this section, we summarize the logical flow of our framework, from foundational hypothesis to observable physics, in Table 1.

This roadmap provides a coherent, step-by-step construction of classical spacetime from a purely algebraic quantum origin. The next section will formalize the thermodynamic language required to describe the dynamics of this emergent spacetime.

3 The thermodynamic formulation of emergent spacetime

With the microscopic origins of geometry, the entropy current, and the cosmological enthalpy established in Section 2, we now formalize their macroscopic thermodynamic roles. This section constructs a self-contained thermodynamic language for the emergent spacetime, showing how gravitational dynamics and cosmic evolution can be expressed as non-equilibrium thermodynamic processes.

3.1 From microscopic entanglement to macroscopic entropy current

The ECV , introduced in Section 2.4 as the continuum limit of the discrete entanglement entropy current operator , is the central kinematic variable describing the flow of quantum information in the emergent spacetime. On the continuous manifold , it takes the covariant form:where is the entropy density in the local rest frame, is the four-velocity of a comoving observer (), and is the spatial entropy current density satisfying .

The fundamental dynamical law governing is the covariant second law of thermodynamics:where is the local entropy production rate. We emphasize that this is not a phenomenological postulate. It is a macroscopic consequence of the microscopic unitary evolution of the string-net condensate, grounded in the strong subadditivity of entanglement entropy (see App. A.3.5 for the complete derivation).

3.1.1 The ECV thus plays a dual role

Thermodynamically, it characterizes the direction and intensity of irreversible entropy transport. Geometrically, it encodes the redistribution of entanglement that underlies changes in the spacetime fabric itself. In a uniformly expanding universe, points predominantly in the direction of spatial expansion. In a gravitationally bound system undergoing collapse, it points outward from the system’s center, representing the expulsion of disorder required for local self-organization.

3.2 Cosmological enthalpy as the effective thermodynamic potential

If the ECV describes the flow, the Cosmological Enthalpy (CE) describes the potential that drives it. For a matter distribution described by a perfect fluid energy-momentum tensor , the CE density and total CE are defined as:

The inclusion of the pressure term is not optional. In an expanding universe, the matter content continuously does work against the cosmic expansion. The internal energy alone is not a conserved or appropriate thermodynamic potential, because it does not account for this pressure-volume work. The CE corrects this, serving as the gravitational analogue of the classical enthalpy .

As established in Section 2.5, the pressure in this expression has a quantum microscopic origin rooted in the fusion and braiding rules of the category . Consequently, is a comprehensive macroscopic measure of both the energy content of matter and the strength of the underlying quantum correlations.

For a homogeneous and isotropic FLRW universe with scale factor and equation of state , the total CE takes the explicit form:

This expression makes the physical role of the CE transparent. For dust (), reduces to the total rest mass-energy. For radiation (), the CE exceeds the internal energy by a factor of due to radiation pressure. For a cosmological constant (), , reflecting the fact that dark energy in this form does not participate in the thermodynamic work balance of structure formation.

3.3 The generalized Gibbs free energy

The ECV and the CE are not independent. They are coupled through a generalized thermodynamic potential that governs the spontaneous evolution of the matter-geometry system. We define the Spacetime Gibbs Free Energy as:

Here, is the Hawking-Gibbons temperature associated with the cosmic horizon, and is the total entanglement entropy of the spatial region under consideration.

The functional unifies the two driving forces of cosmic evolution. The enthalpy term favors gravitational collapse and structure formation (it decreases as binding energy becomes more negative). The entropic term encodes the thermodynamic cost of ordering, resisting unchecked collapse. The competition between these two terms determines the fate of any gravitating system.

This formulation yields a powerful variational principle. The spontaneous direction of evolution for any subsystem of the universe is toward minimizing . That is, a process occurs spontaneously if and only if:

A stable equilibrium configuration corresponds to and . This principle applies across all scales: from the Jeans instability governing star formation to the background Friedmann equations governing cosmic expansion (see App. B for explicit derivations).

Crucially, is not a phenomenological construct. As shown in Supplementary Appendix B4, it emerges directly from the path integral of the microscopic string-net model under the ETRG flow. The partition function for a spatial region takes the form as the effective free energy of the quantum gravitational system in the continuum limit.

3.4 Einstein’s equations as conditions of entanglement equilibrium

With the thermodynamic formalism in place, we can now re-derive the Einstein field equations as a special case of the minimization principle. This derivation follows the logic of Jacobson’s thermodynamic approach (Jacobson, 1995), but is now grounded in the microscopic string-net framework.

Consider a local Rindler horizon passing through an arbitrary spacetime point . The horizon has a local Unruh temperature , where is the acceleration of the observer. For the system to be in local equilibrium, the variation of with respect to the metric must vanish. This requires that the change in entanglement entropy across the horizon exactly balances the energy flux through it:

Using the area law for entanglement entropy established in Theorem A.9 of App. A.2.2, , and the expression for the energy flux (Jacobson, 1995), the equilibrium condition applied at every point and for every null direction yields:

By the arbitrariness of , we obtain the full linearized Einstein equations:

We make explicit that this derivation rests on the key assumption of local entanglement equilibrium [HE]: that each small causal diamond is in a state of maximal entanglement entropy, so that holds at every point and for every null direction.where is the fundamental lattice spacing and is the total quantum dimension. Thus, in our framework, the Einstein equations are the macroscopic equation of state for the entanglement entropy of the quantum spacetime network, with coupling constants that are in principle calculable from the microscopic category. Specifically, the Time-Evolving Block Decimation (TEBD) algorithm maps the continuous-time Hamiltonian evolution onto a discrete sequence of local unitary gates via Trotter-Suzuki decomposition (Vidal, 2003; Vidal, 2004), providing a rigorous foundation for recovering continuous dynamics from discrete operations.

3.5 Summary of the thermodynamic formulation

The thermodynamic language developed in this section provides a unified description of spacetime dynamics. Table 2 summarizes the key thermodynamic variables, their microscopic origins, and their macroscopic roles.

This formulation transforms the conceptual picture of Section 2 into a computationally tractable thermodynamic framework. The next section applies this framework to the specific problem of cosmic structure formation, demonstrating how the minimization of yields quantitative predictions, including the Jeans mass for gravitational instability.

4 Cosmological dynamics and structure formation

The thermodynamic formalism developed in Section 3 provides a unified language for describing gravitational dynamics across all scales. In this section, we apply this language to the specific problems of cosmic expansion and structure formation, demonstrating that the minimization of yields quantitative predictions that recover and extend known results.

4.1 Thermodynamic form of the Friedmann equations

For a homogeneous and isotropic FLRW universe with metric:the generalized Gibbs free energy takes a particularly simple form. For a comoving volume filled with a perfect fluid of energy density and pressure , we have:where is the entropy density and with

The first Friedmann equation can be rewritten as a thermodynamic identity. Using the relation (the Bekenstein-Hawking entropy of the apparent horizon) and the first law of thermodynamics (Hawking, 1975; Bekenstein, 1973), one obtains:where is the internal energy of the comoving volume. This equation is exactly the first Friedmann equation in thermodynamic form, demonstrating the consistency of our framework with standard cosmology.

The time evolution of under cosmic expansion provides a criterion for the global direction of evolution. For a universe dominated by a single component with equation of state , we find:

The sign of determines whether the universe is in a regime of spontaneous expansion () or requires external driving (). For a cosmological constant (), , and the evolution is driven entirely by the entropy term.

4.2 Structure formation as a minimization process

The true power of the thermodynamic formulation emerges when we consider inhomogeneities. Structure formation—the emergence of galaxies, clusters, and cosmic web from primordial density fluctuations—can be understood as a process of minimization in local overdense regions.

Consider a spherically symmetric overdense region of comoving radius embedded in the FLRW background. Let its density be , where is the background density. The total CE for this region includes both the rest mass energy and the gravitational binding energy. In the Newtonian approximation (valid on sub-horizon scales):where is the total mass of the region.

The entropy of the region includes contributions from the configurational entropy of the density field and the entanglement entropy of the underlying quantum geometry. For a cold dark matter perturbation, a minimal model takes , yielding , where is the average entropy per particle.

The generalized Gibbs free energy for the overdense region is then:

The condition for gravitational instability is that decreases as the region contracts. Differentiating with respect to the scale factor and setting yields the Jeans instability criterion. The critical length scale is the Jeans length :where is the sound speed. Perturbations with are unstable to gravitational collapse. The corresponding Jeans mass is:

This recovers the standard result, but now derived from a unified thermodynamic variational principle rather than a separate fluid-dynamical analysis. The full derivation is given in Supplementary Appendix B2.

4.3 The role of the ECV in early structure formation

The ECV introduced in Section 3.1 plays a dynamical role in structure formation that goes beyond standard cosmology. In the early universe, as primordial density fluctuations grow via gravitational instability, the associated entropy flow points outward from overdense regions:

This directional entropy current couples to spacetime geometry through an effective back-reaction term. In isolated quantum many-body systems, the approach to thermal equilibrium is governed by the interplay between entanglement growth and local relaxation (Rigol et al., 2008). This suggests that the entropy current we have introduced may provide a covariant, gravitational analog of such thermalization dynamics. In the linear perturbation equation, this generates an additional attractive force:

While the structural form of the perturbation equation and the scaling of the coupling follow directly from the ECV conservation law and the adiabatic evolution of the entropy density established in Section 3.1, the full numerical integration of this equation across different cosmological epochs, and the resulting quantitative predictions for the matter power spectrum, will be presented in a separate work (in preparation). This enhancement of gravitational attraction is a distinctive prediction of our framework: the thermodynamics of gravitational collapse actively promotes structure formation, beyond the standard CDM prediction.

4.4 Thermodynamic description of cosmic expansion history

The transitions between these epochs occur when the dominant contribution to shifts from one term to another. This provides a thermodynamic interpretation of the cosmic coincidence problem: the current epoch is special because both and terms are comparable, making the universe sensitive to both gravitational collapse and cosmic acceleration.

It is appropriate to clarify a potential conceptual confusion at this point. The “accelerated expansion” () discussed above refers to the second time derivative of the scale factor being positive----this is a feature of the time evolution of the Hubble parameter and is fully compatible with the local speed-of-light limit in general relativity. It is not equivalent to the “superluminal expansion” that may occur during the inflationary epoch----the latter describes the recessional velocity of regions outside the Hubble horizon formally exceeding the speed of light, which is permitted in general relativity because it characterises the expansion rate of space itself rather than the motion of any signal or object through space. In our emergent framework, the exponential expansion of the inflationary epoch corresponds to the initial stage of the ETRG flow: the entanglement network of qubits generates new entanglement connections (i.e., spatial pixels) at an exponential rate, which manifests, after coarse-graining, as an exponential growth of the spacetime metric. The current accelerated expansion, by contrast, corresponds to the stage in which the cosmological enthalpy density becomes negative (, i.e., dark energy domination), driving the accelerated growth of the scale factor through the Friedmann equation . These two types of expansion differ fundamentally in their physical mechanisms and in the cosmic epochs at which they occur.

4.5 Summary of cosmological applications

A clarifying remark concerning the timescales involved in the expansion is warranted at this point. The expansion history referred to in point 3 above encompasses the radiation-dominated, matter-dominated, and dark-energy-dominated epochs. The temporal evolution of the expansion rate across these three epochs is governed by the unified variational principle: during the radiation and matter epochs, the enthalpy term dominates () and the expansion decelerates; during the dark energy epoch, the enthalpy term vanishes () and the entropic term alone drives the expansion, producing acceleration. If one traces the history further back to the inflationary epoch----during which the exponential expansion rate of space may formally exceed the local speed of light----this corresponds, in our emergent framework, to the very initial stage of the ETRG flow: the entanglement network generates new connections at an exponential rate. The timescale of this process is determined by the microscopic parameters of the underlying string-net----specifically, the lattice spacing and the total quantum dimension ----rather than by the classical FLRW equations. Thus, the framework handles distinct expansion phases through distinct mechanisms that are nevertheless unified within a single architecture: the radiation, matter, and dark energy epochs are described by the variational principle, while the inflationary epoch emerges naturally from the entanglement dynamics of the initial ETRG flow. Although these phases may all exhibit , their physical origins, characteristic timescales, and the theoretical tools required to describe them are fundamentally different.

5 Empirical testability: from theoretical framework to falsifiable predictions

A theoretical framework, however elegant, remains incomplete without a concrete path to empirical validation or falsification. In this section, we address this requirement directly. We present three specific, falsifiable predictions of our framework and outline explicit experimental protocols for testing them. The central message is that the emergence of spacetime from quantum entanglement is not merely a conceptual narrative—it is a scientific hypothesis amenable to laboratory investigation.

Before presenting the specific experimental protocols, it is appropriate to clarify the scientific status and falsification capacity of these proposals. The three experimental predictions put forward by this framework are not vague aspirations for a distant technological future. They are quantitative tests with well-defined numerical signatures, operable measurement protocols, and explicit falsification conditions. What these proposals share is a common strategic choice: they do not rely on the direct probing of Planck-scale energies--an undertaking that is technically infeasible for the foreseeable future--but instead exploit the precisely controllable entanglement structures available in quantum many-body systems, thereby enabling direct tests of the core hypotheses of emergent spacetime geometry at tabletop experimental scales.

Specifically: 1) the measurement of the entanglement area law and topological entanglement entropy (Prediction 1) amounts to a quantitative verification of the ground-state entanglement scaling behaviour in a concrete string-net model (e.g., the Fibonacci model, ; )----if the measured topological entropy does not match the categorical prediction, the framework is falsified; 2) the measurement of the entropy pulse propagation velocity (Prediction 2) translates the theoretical definition of the ECV into a trackable laboratory signal, whose speed is bounded from above by the Lieb–Robinson velocity (Lieb and Robinson, 1972)----if the pulse velocity is zero or infinite, the framework is falsified; 3) the detection of the emergent graviton mode (Prediction 3) requires measuring the spin-2 projector structure and the pole in the two-point correlation function of the stress–energy tensor--if the correlation function does not exhibit this structure, or if the extracted effective Newton constant does not scale as , the framework is falsified. The falsification condition for each prediction is listed explicitly in Table 3. The design principle of these experimental protocols is the following: a positive outcome would constitute strong support for the framework, while a negative outcome would rule out its essential mechanisms. It is this sharp falsifiability----rather than the scale or cost of the experiment----that defines the standard of scientific rigour.

A theoretical framework, however elegant, remains incomplete without a concrete path to empirical validation or falsification. In this section, we address this requirement directly. We present three specific, falsifiable predictions of our framework and outline explicit experimental protocols for testing them. The central message is that the emergence of spacetime from quantum entanglement is not merely a conceptual narrative----it is a scientific hypothesis amenable to laboratory investigation.

5.1 The principle of testability in quantum gravity

Quantum gravity has long been criticized for its lack of experimental guidance. The Planck scale, m and GeV, lies far beyond the reach of current and foreseeable particle accelerators. This has led to the widespread perception that quantum gravity theories are inherently untestable.

Our framework bypasses this obstacle by shifting the arena of testing from high-energy particle physics to quantum many-body simulation. The emergence of spacetime geometry from entanglement is a collective phenomenon that can be realized in controllable quantum systems with a finite number of degrees of freedom. The relevant experiments do not require probing the Planck scale directly. They require engineering and measuring the entanglement structure of specific quantum many-body states—a capability that has matured dramatically in the last decade.

The three predictions that follow are formulated so that their experimental confirmation would provide strong evidence for the framework, while their falsification would rule out its essential mechanisms.

5.2 Prediction 1: entanglement area law and topological entanglement entropy

The first prediction concerns the spatial structure of entanglement in the string-net condensate. Theorem A.9 in App. A.2.2 establishes that the von Neumann entanglement entropy of a spatial region in the ground state obeys:where is an constant, is the lattice spacing, and is the topological entanglement entropy. The topological term is a universal constant determined by the total quantum dimension of the category :

For the Fibonacci category (), , giving . For the quantum double model , , giving .

This prediction is sharply falsifiable. If the entanglement entropy of a candidate string-net state does not scale as an area law, or if the extracted topological term does not match the categorical prediction, the framework is ruled out.

Existing experiments have already demonstrated the capability to measure second Rényi entropy in optical lattice systems using single-site resolved quantum gas microscopy (Guo et al., 2026). The primary technical challenge for our protocol is the realization of the specific Levin–Wen Hamiltonian, which requires engineering four-body plaquette interactions. Recent advances in Floquet engineering and Rydberg dressing make this feasible in the near term.

5.3 Prediction 2: entropy current propagation velocity

The second prediction concerns the dynamics of the ECV . In the string-net condensate, a localized energy perturbation—such as a laser-induced potential dip—creates a local imbalance in the entanglement structure. The system responds by propagating an entropy pulse outward. The velocity of this entropy pulse is predicted to be bounded by the emergent speed of light:where is set by the Lieb-Robinson velocity of the microscopic Hamiltonian (Lieb and Robinson, 1972). In the ETRG fixed-point theory, this velocity flows to the effective speed of light in the emergent geometry.

The shape and diffusion profile of the entropy pulse provide additional characterization. In the continuum limit, the entropy density satisfies a conservation law with a source term:where is the local entropy production rate. The diffusion constant governing the spatial spread of the pulse is related to the effective gravitational constant and the temperature .

This protocol directly maps the flow of quantum information in the emergent spacetime, providing a laboratory realization of the ECV. The measured velocity and diffusion profile can be compared against our theoretical calculations for specific categorical models (, ).

5.4 Prediction 3: emergent graviton mode and effective Newton constant

The third and most distinctive prediction is the existence of a massless spin-2 excitation—an emergent graviton—in the low-energy spectrum of the string-net condensate. In the model in (2 + 1) dimensions, the low-energy effective theory is known to be -dimensional Einstein gravity coupled to topological matter fields (Meichanetzidis et al., 2018). The spin-2 mode manifests as a massless pole in the two-point correlation function of the stress-energy tensor:where is the spin-2 projection operator. The effective Newton constant is:

For the model, , giving . This is a precise, numerical prediction: if the low-energy two-point function deviates from the spin-2 projector structure, or if the coupling extracted from the pole does not scale as , the framework is falsified.

This protocol is the condensed-matter analogue of detecting gravitational waves, but at the scale of a quantum chip. The extraction of would constitute a direct laboratory verification of the central claim of our framework: that gravitational coupling emerges from categorical quantum information data.

5.5 Near-term feasibility and existing precedents

The primary technical gap between current capabilities and our proposed experiments is the realization of the full Levin–Wen Hamiltonian with the specific categorical data (-symbols, -symbols) of the Fibonacci or models. Progress in this direction is accelerating. A proof-of-principle demonstration of the area law and topological entanglement entropy in the Fibonacci model is achievable within three to 5 years.

5.6 Cosmic observational signatures

These cosmic signatures are not unique predictions of our framework, and their quantitative amplitude remains to be calculated. We present them to motivate a broader research program connecting quantum spacetime emergence to observational cosmology.

5.7 Falsifiability summary

The framework presented in this paper makes three specific, falsifiable predictions. Their experimental signatures and falsification conditions are summarized in Table 3.

6 Discussion: comparison with existing approaches and scope of the framework

The framework developed in this paper intersects with several major research programs in quantum gravity. In this section, we clarify the unique contributions of our approach through direct comparison, and we specify its domain of validity and current limitations.

6.1 Comparison with thermodynamic and entropic gravity

Our work is closest in spirit to Jacobson’s thermodynamic derivation of Einstein’s equations and Verlinde’s entropic gravity proposal. However, fundamental differences exist at the level of microscopic foundations.

Jacobson showed that the Einstein equations follow from the proportionality of entropy and horizon area (Jacobson, 1995), together with the first law of thermodynamics applied to local Rindler horizons. In his derivation, the origin of the entropy and the microscopic degrees of freedom are left unspecified. Our framework supplies a concrete microscopic model for that entropy: the entanglement entropy of the string-net condensate. Moreover, we go beyond Jacobson by determining the effective gravitational constant in terms of categorical data, and by constructing the ECV as a dynamical variable that describes entropy transport in non-equilibrium settings.

Verlinde proposed that gravity is an entropic force arising from the statistical tendency of a system to maximize its entropy (Verlinde, 2011). This is an intriguing idea, but it remains at the level of a thermodynamic analogy. Our framework provides a microscopic realization: the “entropic force” is the macroscopic manifestation of the quantum network’s tendency to restore maximal entanglement in response to perturbations. Moreover, we explicitly construct the free energy functional whose minimization drives gravitational dynamics, making the variational principle precise.

6.2 Comparison with the AdS/CFT correspondence

The AdS/CFT correspondence is the most rigorous realization of the holographic principle (Susskind, 1995) and of spacetime emergence from quantum information (Maldacena, 1998). Our framework shares the core tenet that spacetime geometry and gravity can emerge from entanglement, but differs in its starting point and scope.

AdS/CFT is a duality: a specific bulk gravitational theory is equivalent to a specific boundary conformal field theory. It relies on the asymptotic structure of anti-de Sitter space and does not explain how the bulk spacetime itself emerges from a pre-geometric theory. Our framework, by contrast, starts from a non-spatiotemporal algebraic structure (the unitary fusion category ) and constructs the emergent spacetime through the physical realization of string-net condensation and the ETRG flow. No asymptotic boundary is assumed.

There is a promising connection, however. Certain unitary modular categories are known to be related to two-dimensional conformal field theories via the Turaev–Viro/Reshetikhin–Turaev construction (Turaev and Viro, 1992). If the string-net condensate on a manifold with boundary can be shown to be dual to a boundary CFT, our framework would provide a microscopic, background-independent realization of holography. This remains a major goal for future work.

6.3 Comparison with loop quantum gravity

Loop quantum gravity (LQG) and our framework share a striking mathematical resemblance: both describe quantum geometry in terms of graph-theoretic structures with algebraic data on edges. In LQG, these are spin networks carrying SU(2) representations (Rovelli, 2004). In our framework, they are string-nets carrying objects of the category .

Despite this mathematical parallel, the physical interpretations differ fundamentally. In LQG, the spin network is the quantum state of the gravitational field, obtained by quantizing classical general relativity. Geometry is quantized. In our framework, the string-net is the physical realization of a pre-geometric algebraic structure. Geometry is not quantized classical geometry; it emerges as a new order parameter at a quantum phase transition. This ontological difference has practical consequences: in LQG, the coupling constant is inherited from the classical theory; in our framework, both and are emergent and determined by categorical data.

The two approaches may be complementary. Spin networks and string-nets are both labeled graphs encoding algebraic data. An exploration of their precise mathematical relationship could open a fruitful dialogue between the quantization and emergence paradigms.

6.4 Comparison with string theory

String theory approaches quantum gravity by replacing point particles with extended objects moving in a pre-existing spacetime background. Spacetime itself is not emergent in the perturbative formulation, though non-perturbative dualities suggest a deeper structure. Our framework begins with a more radical starting point: the fundamental degrees of freedom are not objects in spacetime, but abstract qubits whose entanglement builds spacetime. This is a genuinely different ontological commitment.

We note that certain string-theoretic constructions, such as the AdS/CFT correspondence and the idea of the string landscape, have motivated the notion of emergent spacetime. Our framework can be viewed as providing a condensed-matter-inspired realization of these ideas, where the emergent geometry is not tied to any specific compactification scheme but is determined by the algebraic structure of a unitary fusion category.

6.5 Domain of validity and current limitations

Every physical theory has a domain of validity. It is essential to specify the regime in which our framework is intended to apply, and to acknowledge its current limitations openly.

The framework currently has the following limitations:

Dimensionality: The most rigorously constructed models (Levin–Wen, Turaev–Viro) are formulated in (2 + 1) dimensions. Extending the construction to (3 + 1) dimensions while preserving the categorical structure is an ongoing challenge. Recent work on -string condensation and higher-form gauging may provide a path forward (Gorantla et al., 2025).

Foundational assumptions: As explicitly noted throughout the text, three key physical assumptions remain to be proven from first principles: the foundational hypothesis [HP-1] that a unitary fusion category fully describes the physical vacuum; the assumption [HA-1] that quantum curvature corrections vanish in the continuum limit; and the assumption [HE] of local entanglement equilibrium required to derive Einstein’s equations. These assumptions are physically motivated and supported by evidence in specific models, but their general proof is left for future work.

Matter content: While string-net models naturally produce emergent gauge bosons and fermions, a complete derivation of the Standard Model matter content and its coupling to the emergent geometry remains an open problem.

Quantitative cosmological predictions: The cosmic signatures listed in Section 5.6 are currently order-of-magnitude estimates. A full calculation of, for example, the primordial gravitational wave bispectrum or the matter power spectrum enhancement requires detailed numerical simulations that are beyond the scope of this paper.

6.6 Summary of unique contributions

We emphasize that these limitations are not fatal flaws. They define a concrete research program. Many of them are shared by other quantum gravity approaches. The value of the present framework is that it reformulates these open problems in a language—quantum information, tensor networks, and category theory—that is amenable to analytical and numerical investigation.

7 Summary of supportive results

This section consolidates the key results that the framework has already yielded. These are not proposals for future experiments, but concrete theoretical outputs completed in the present work.

7.1 Emergent derivation of the Einstein equations as conditions of entanglement equilibrium

Starting from the response of the string-net ground state to local energy-momentum perturbations and combining Jacobson’s thermodynamic logic, we have explicitly demonstrated within the string-net framework how the linearized Einstein equations emerge from the condition of local entanglement equilibrium (Section 3.4). The key result----that the effective Newton constant and the cosmological constant are fully determined by the total quantum dimension of the category and the fundamental lattice spacing (Corollary A.24)----provides a microscopic origin for the gravitational coupling constants, something that standard general relativity and cosmology do not supply.

7.2 Rigorous proof of the entanglement area law

Theorem A.9 provides a strict proof of the entanglement area law for the string-net ground state--------thereby supplying a microscopic foundation for the Bekenstein–Hawking entropy. The topological entanglement entropy is a universal quantity determined by the category , yielding specific, testable numerical predictions in concrete models (e.g., for the Fibonacci category; for the model).

7.3 Unified recovery of the jeans criterion and the Friedmann equations

From the single variational principle, we have recovered both the standard Jeans mass formula (Section 4.2) and the thermodynamic form of the Friedmann equations (Section 4.1). These are well-established results in standard cosmology, but within our framework they are derived from a common variational principle rather than being introduced as independent assumptions. This recovery demonstrates the consistency of the framework with classical cosmology while simultaneously revealing their shared thermodynamic origin.

7.4 Three falsifiable experimental predictions

The framework yields three specific, falsifiable predictions----the entanglement area law and topological entanglement entropy (Prediction 1, Section 5.2), the propagation velocity of entropy pulses (Prediction 2, Section 5.3), and the emergent graviton mode (Prediction 3, Section 5.4). Each prediction is accompanied by a concrete measurement protocol, the required physical platform, and explicit falsification conditions (Table 3). These predictions do not require probing Planck-scale energies, but can be tested in tabletop-scale quantum many-body systems----a key advantage that distinguishes this framework from most quantum gravity proposals.

7.5 Clear comparison with existing quantum gravity programs

Section 6 systematically compares the present framework with Jacobson’s thermodynamic gravity, Verlinde’s entropic gravity, the AdS/CFT correspondence, loop quantum gravity, and string theory, clarifying the unique contributions of this work: an end-to-end emergence construction from non-spatiotemporal categorical data to classical spacetime geometry, the microscopic determination of coupling constants, the introduction of new thermodynamic variables, the establishment of a unified variational principle, and the formulation of concrete falsifiable predictions.

A broader synthesis of these results, together with a discussion of their implications and future research directions, is presented in Section 8.

Beyond the three core predictions detailed in Section 5, the theoretical framework developed in this paper, when applied to different physical scenarios, yields a range of additional testable consequences. These include, for instance, the production of antimatter via entropy flow reversal (Yang et al., 2026b) and the stability of hadrons constrained by entropy-enthalpy competition (Yang et al., 2026a). These topics, explored in our related works, demonstrate the broader applicability and testability of the framework.

8 Conclusion and outlook

We have presented a unified framework in which classical spacetime is an emergent macroscopic order parameter of an underlying quantum information structure. Starting from a unitary fusion category as the fundamental “algebraic DNA”, we showed how a string-net condensate realizes this structure physically, and how the ETRG flow drives the system to a geometric fixed point where a continuous spacetime manifold, its metric, and its curvature emerge.

Within this emergent spacetime, we introduced two core thermodynamic variables: the entropy current vector , which describes the directional flow of entanglement entropy, and the cosmological enthalpy , which serves as an effective thermodynamic potential incorporating pressure-volume work. We demonstrated that the Einstein field equations arise as conditions of local entanglement equilibrium, with coupling constants and fully determined by categorical data.

The framework is not merely conceptual. It yields three specific, falsifiable predictions—the entanglement area law, the propagation velocity of entropy pulses, and the existence of an emergent graviton mode—that can be tested in near-term quantum simulation experiments using ultracold atoms or superconducting qubit arrays.

Two overarching themes emerge from this work. First, the key to quantum gravity may not lie in quantizing geometry, but in geometrizing quantum information. The metric is a bilinear condensate of frame fields; curvature is a macroscopic measure of categorical braiding defects. Second, spacetime and its dynamics are intrinsically thermodynamic. The flow of entropy, the minimization of free energy, and the drive toward equilibrium are not analogies but the fundamental language of gravitational physics.

Future theoretical work will focus on three directions: (1) extending the categorical construction to (3 + 1) dimensions, building on recent advances in higher-form gauging (Gorantla et al., 2025) and -string condensation; (2) rigorously proving the convergence theorem (Theorem A.5) without the current assumptions on quantum curvature corrections; and (3) deriving the Standard Model gauge group and fermion spectrum from the dynamics of specific categorical string-net models.

On the experimental side, the immediate priority is the realization of Levin–Wen Hamiltonians with Fibonacci or categorical data on programmable quantum simulators. A proof-of-principle demonstration of the area law and topological entanglement entropy extraction is feasible within the next 5 years and would constitute the first laboratory evidence for the emergence of geometry from entanglement.

The nature of spacetime has been a central question of physics since Newton and Leibniz. The framework developed in this paper does not claim to be a final answer. It is an invitation to a research program: one in which the structure of the universe, from the smallest quantum correlations to the largest cosmic webs, is understood as the thermodynamics of quantum information.

References

• 1

  BaezJ. C.StayM. (2011). “Physics, topology, logic and computation: a Rosetta Stone,” in New Structures for Physics. Lecture Notes in Physics. Editor CoeckeB. (Berlin: Springer), 813, 95–172. 10.1007/978-3-642-12821-9_2

  • CrossRef
  • Google Scholar
  • View reference in article

• 2

  BekensteinJ. D. (1973). Black holes and entropy. Phys. Rev. D.8, 2333–2346. 10.1103/PhysRevD.7.2333

  • CrossRef
  • Google Scholar
  • View reference in article

• 3

  BridgemanJ. C.FlammiaS. T.PoulinD. (2017). Detecting and stabilizing measurement-induced symmetry-protected topological phases in generalized cluster models. Phys. Rev. B96, 115102. 10.1103/PhysRevB.96.115102

  • CrossRef
  • Google Scholar
  • View reference in article

• 4

  ChalopinT.BojovićP.WangS.FranzT.SinhaA.WangZ.et al (2026). Observation of emergent scaling of spin–charge correlations at the onset of the pseudogap. Proc. Natl. Acad. Sci. U.S.A.123, e2525539123. 10.1073/pnas.2525539123

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 5

  GorantlaP.PremA.TantivasadakarnN.WilliamsonD. J. (2025). String-membrane-nets from higher-form gauging: an alternate route to p-string condensation. Phys. Rev. B112, 125124. 10.1103/qq9n-16hk

  • CrossRef
  • Google Scholar
  • View reference in article

• 6

  GuoF.DilJ. H.SannaA.MolodtsovS. L.RienksE. D. L.VarelaM.et al (2026). Dependency of quantum time scales on symmetry. Phys. Rev. Lett.136, 060401. 10.1103/PhysRevLett.136.060401

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 7

  HawkingS. W. (1975). Particle creation by black holes. Commun. Math. Phys.43, 199–220. 10.1007/BF02345020

  • CrossRef
  • Google Scholar
  • View reference in article

• 8

  HiraniA. N. (2003). Discrete Exterior Calculus. Ph.D. thesis. Pasadena, CA: California Institute of Technology.

  • Google Scholar
  • View reference in article

• 9

  JacobsonT. (1995). Thermodynamics of spacetime: the einstein equation of state. Phys. Rev. Lett.75, 1260–1263. 10.1103/PhysRevLett.75.1260

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 10

  LevinM.WenX.-G. (2005). Colloquium: photons and electrons as emergent phenomena. Rev. Mod. Phys.77, 871–879. 10.1103/RevModPhys.77.871

  • CrossRef
  • Google Scholar
  • View reference in article

• 11

  LevinM.WenX.-G. (2005). String-net condensation: a physical mechanism for topological phases. Phys. Rev. B71, 045110. 10.1103/PhysRevB.71.045110

  • CrossRef
  • Google Scholar
  • View reference in article

• 12

  LevinM.WenX.-G. (2006). Quantum ether: photons and electrons from a rotor model. Phys. Rev. B73, 035122. 10.1103/PhysRevB.73.035122

  • CrossRef
  • Google Scholar
  • View reference in article

• 13

  LiebE. H.RobinsonD. W. (1972). The finite group velocity of quantum spin systems. Commun. Math. Phys.28, 251–257. 10.1007/BF01645779

  • CrossRef
  • Google Scholar
  • View reference in article

• 14

  MaldacenaJ. (1998). The Large-N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys.2, 231–252. 10.4310/ATMP.1998.v2.n2.a1

  • CrossRef
  • Google Scholar
  • View reference in article

• 15

  MeichanetzidisK.CirioM.BaisF. A.PachosJ. K. (2018). Quantum double models coupled to dynamical fermions. Phys. Rev. Lett.121, 250503. 10.1103/PhysRevLett.121.250503

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 16

  ReggeT. (1961). General relativity without coordinates. Nuovo Cimento19, 558–571. 10.1007/bf02733251

  • CrossRef
  • Google Scholar
  • View reference in article

• 17

  RigolM.DunjkoV.OlshaniiM. (2008). Thermalization and its mechanism for generic isolated quantum systems. Nature452, 854–858. 10.1038/nature06838

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 18

  RovelliC. (2004). Quantum Gravity, Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. 10.1017/CBO9780511755804

  • CrossRef
  • Google Scholar
  • View reference in article

• 19

  SatzingerK. J.LiuY.-J. (2021). Probing topological spin liquids on a programmable quantum simulator. Science374, 1237–1241. 10.1126/science.abi8794

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 20

  SusskindL. (1995). The world as a hologram. J. Math. Phys.36, 6377–6396. 10.1063/1.531249

  • CrossRef
  • Google Scholar
  • View reference in article

• 21

  TuraevV. G.ViroO. Y. (1992). State sum invariants of 3-manifolds and quantum 6j-symbols. Topology31, 865–902. 10.1016/0040-9383(92)90015-A

  • CrossRef
  • Google Scholar
  • View reference in article

• 22

  VerlindeE. (2011). On the origin of gravity and the laws of Newton. J. High. Energ. Phys.29, 29. 10.1007/JHEP04(2011)029

  • CrossRef
  • Google Scholar
  • View reference in article

• 23

  VidalG. (2003). Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett.91, 147902. 10.1103/PhysRevLett.91.147902

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 24

  VidalG. (2004). Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett.93, 040502. 10.1103/PhysRevLett.93.040502

  • Pubmed Abstract
  • CrossRef
  • Google Scholar
  • View reference in article

• 25

  WenG. (2003). Quantum orders from string-net condensations and the origin of light and massless fermions. Phys. Rev. D.68, 065003. 10.1103/PhysRevD.68.065003

  • CrossRef
  • Google Scholar
  • View reference in article

• 26

  WenX.-G. (2018). Four revolutions in physics and the second quantum revolution --- A unification of force and matter by quantum information. J. Mod. Phys. A32, 1830010. 10.1142/S0217751X18300104

  • CrossRef
  • Google Scholar
  • View reference in article

• 27

  WenX. G. (2002). Quantum orders and symmetric spin liquids. Phys. Rev. B65, 165113. 10.1103/PhysRevB.65.165113

  • CrossRef
  • Google Scholar
  • View reference in article

• 28

  WheelerJ. A. (1990). A Journey into Gravity and Spacetime. New York, NY: Scientific American Library.

  • Google Scholar
  • View reference in article

• 29

  YangX. (2025). The principle of phase transition cosmology- - -The current universe formed by the evolution of two-mode quantum superfluid. RAPS 0909, 2550003. 10.1142/S2424942425500030

  • CrossRef
  • Google Scholar
  • View reference in article

• 30

  YangX. D.YangY. C.MeiH. L. (2026a). Non-equilibrium categorical dynamics: entropy, Enthalpy, and the approach to confinement. Phys. Lett. A528, 130045. 10.1016/j.physleta.2026.131912

  • CrossRef
  • Google Scholar
  • View reference in article

• 31

  YangX. D.YangY. C.MeiH. L. (2026b). Antimatter generation mechanism: a new perspective from entropy flow vector based on the quantum tensor network theory. Front. Phys.14, 1844769. 10.3389/fphy.2026.1844769

  • CrossRef
  • Google Scholar
  • View reference in article

• 32

  ZhaoX.LiY.ZhuQ.FoxD. (2013). The co-evolution of cosmic entropy and structures in the universe. arXiv:1308.4352. Available online at: https://arxiv.org/abs/1308.4352 (Accessed July 22, 2026).

  • Google Scholar
  • View reference in article