Emergent Gravity from Quantum Information Flow: A Modified Dirac-Equation Perspective

1. Introduction

The Dirac equation is a cornerstone of relativistic quantum mechanics, yet it presupposes a fixed spacetime background [1,2]. General relativity, by contrast, treats spacetime as dynamical but classical [3,4]. One promising approach to bridging these frameworks is to treat gravity as an emergent phenomenon rooted in thermodynamics--a perspective supported by Jacobson's thermodynamic derivation of the Einstein equations and Verlinde's entropic gravity proposal [5,6]. However, a concrete microscopic mechanism that links quantum information flow to both fermion dynamics and the emergence of spacetime has remained elusive.

In this work, we propose such a mechanism not as a phenomenological induced-gravity model built on a pre-existing curved spacetime, but as a genuine emergence from a deeper, non-spatiotemporal quantum structure--a program we have developed systematically in a companion paper [7], where classical continuous spacetime is shown to arise as a low-energy effective description of an underlying quantum information structure, 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 [7]. Within this derived geometry, we identify the entropy current--a covariant, well-defined quantity in relativistic hydrodynamics and non-equilibrium thermodynamics--as the macroscopic descriptor of quantum information flow that connects the quantum and the gravitational. Crucially, in our framework the entropy current itself is not a fundamental field postulated from the outset; rather, as established in the companion paper [7], it emerges from the entanglement structure of the underlying string-net condensate: microscopically, it is defined via local entanglement entropy operators and discrete entropy current operators constructed from the string-net ground state; macroscopically, its continuum form ${\mathrm{s}}^{\mathsf{\mu }}={\mathsf{\rho }}_{\mathrm{s}}{\mathrm{u}}^{\mathsf{\mu }}+{\mathrm{j}}_{\mathrm{s}}^{\mathsf{\mu }}$is obtained through the controlled coarse-graining procedure of the equivariant tensor renormalization group flow [7]. Coupling this emergent entropy current to the Dirac equation produces testable modifications in rotating systems, while quantizing its fluctuations--together with the tree-level VEV contribution that provides the dominant term--generates the Einstein-Hilbert action as the low-energy effective action--with Newton's constant ${\mathrm{G}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathrm{a}}_0/{\mathrm{D}}^2$ and the cosmological constant ${\mathsf{\Lambda }}_{\mathrm{C}}$determined entirely by categorical data of the microscopic theory (the total quantum dimension $\mathrm{D}$ and the fundamental lattice spacing ${\mathrm{a}}_0$), not introduced as free parameters [7]. This two-tier structure--from modified Dirac physics to emergent gravity--is the central theme of the present work, and it is distinguished from conventional induced-gravity scenarios precisely by the fact that the metric ${\mathbf{g}}_{\mathsf{\mu }\mathsf{\nu }}$ itself is not presupposed but emerges as a bilinear condensate of frame field operators, curvature is constructed combinatorially from categorical braiding and fusion data, and the Einstein equations arise as conditions of local entanglement equilibrium rather than as fundamental field equations [7].

As we emphasize in Section 4 and Section 5, the Proca dynamics of the entropy current is an effective field theory description of its macroscopic behavior, not a fundamental property; the mass parameter `m_s` is an effective infrared scale that characterizes the correlation length of the entropy flow field, explicitly demonstrated via the two-point function in Section 5.4.

The view that gravity is emergent rather than fundamental has been championed by several influential approaches. Jacobson's thermodynamic derivation of the Einstein equations [5] showed that the gravitational field equations can be understood as an equation of state of spacetime, while Verlinde's entropic gravity proposal [6] suggests that gravitational forces arise from entropy gradients. These insights point toward a thermodynamic origin of gravity, where the fundamental degrees of freedom are informational rather than geometric. If gravity is emergent, the key question becomes: what is the microscopic substrate from which it emerges?

In this work, we propose an answer to this question within the framework of a low-energy effective field theory. We build upon the entropy-enthalpy framework developed in our previous work [7,8,9], where the universe evolves through the competition between entropy-driven diffusion and enthalpy-driven aggregation. The central object in this picture is the entropy current $\mathbf{S}$--a covariant, well-defined quantity in relativistic hydrodynamics and non-equilibrium thermodynamics [10,11], whose divergence ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$ quantifies local entropy production and encodes the arrow of time. Our proposal is to elevate this thermodynamic descriptor to an active dynamical field that couples to fermionic matter and simultaneously provides a microphysical origin for spacetime geometry.

The framework operates on two levels. At the quantum level, the entropy current couples to fermionic matter through a modification of the Dirac equation. This modification yields concrete, testable predictions: in the non-relativistic limit, the spatial entropy current ${\mathrm{s}}^{\mathsf{\mu }}$ acts as a vector potential, generating an effective magnetic field ${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathbf{S}$that drives spin precession, while non-uniform entropy distributions produce position-dependent energy shifts. At the gravitational level, when the entropy flow field is quantized and its quantum fluctuations are integrated out in curved spacetime, the Einstein-Hilbert action emerges as the low-energy effective action, with the graviton appearing as a collective excitation of the quantum information field. The graviton acquires a mass $\left({\mathrm{m}}_{\mathrm{s}}\right)$, constrained by LIGO/Virgo observations to ${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}$eV, and its dispersion relation ${\mathrm{E}}^2={\mathbf{p}}^2+{\mathrm{m}}_{\mathrm{s}}^2$predicts frequency-dependent gravitational-wave propagation--a signature testable by future observatories such as LISA and the Einstein Telescope.

This two-tier structure--from modified quantum mechanics at the fermion level to emergent geometry at the gravitational level--establishes the entropy current as the common thread connecting the quantum and the cosmic. It provides a concrete, testable realization of gravity as an entropic phenomenon, naturally connecting to Jacobson’s thermodynamic gravity, Verlinde’s entropic gravity, and the generalized second law of thermodynamics.

The paper is organized as follows. Section 2 presents the modified Dirac equation and derives the modified dispersion relation for uniform, non-uniform, and time-dependent entropy flows. Section 3 performs the non-relativistic Foldy-Wouthuysen reduction, revealing the emergent entropic gauge structure and deriving the effective magnetic field and modified spin precession. Section 4 constructs a dynamical Proca-type theory for the entropy current in curved spacetime, obtaining the Yukawa-type static solutions and introducing higher-order curvature corrections for stability. Section 5--the conceptual center of this work--performs the path integral over quantum fluctuations of the entropy field, demonstrating the emergence of the Einstein-Hilbert action and the massive graviton. Section 6 discusses testable predictions, including quantitative estimates of the entropy-fermion coupling $\mathsf{\lambda }$ from STAR data on$\mathsf{\Lambda }$ hyperon global spin polarization, and the resulting energy scale ${\mathsf{\Lambda }}_{\mathsf{\lambda }}\sim {10}^2-{10}^3\mathrm{G}\mathrm{e}\mathrm{V}$. Section 7 relates the framework to Jacobson’s and Verlinde’s formulations of thermodynamic gravity, discusses limitations, and outlines future directions. Section 8 concludes.

2. The Modified Dirac Equation

2.1. Lagrangian Formulation

We propose the following Lagrangian for a Dirac fermion coupled to the entropy current vector ${\mathrm{s}}^{\mathsf{\mu }}$:

$$\mathcal{L}=\bar{\mathsf{\psi }}\left(\mathrm{i}\mathrm{\hslash }{\mathsf{\gamma }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}-\mathrm{m}\mathrm{c}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}\right)\mathsf{\psi }$$

(1)

where ${\mathsf{\gamma }}^{\mathsf{\mu }}$ are the Dirac matrices, $\mathrm{m}$ is the fermion mass, ${\mathrm{s}}_{\mathsf{\mu }}$ is the entropy current vector, and $\mathsf{\lambda }$ is the coupling constant characterizing the strength of the entropy-fermion interaction.

We adopt the mostly-minus metric signature ${\mathsf{\eta }}_{\mathsf{\mu }\mathsf{\nu }}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(+,-,-,-)$ throughout this work.

Dimensional analysis and effective field theory nature: In natural units ($\mathrm{\hslash }=\mathrm{c}=1$), the action $\mathrm{S}=\int \mathcal{L}{\mathrm{d}}^4\mathrm{x}$ is dimensionless, so the Lagrangian density $\mathcal{L}$ has mass dimension $\left[\mathcal{L}\right]=4$. The Dirac fermion field has dimension $\left[\mathsf{\psi }\right]=3/2$, hence the bilinear $\left[\bar{\mathsf{\psi }}\mathsf{\psi }\right]=3$. The entropy current ${\mathrm{s}}_{\mathsf{\mu }}$, whose temporal component ${\mathrm{s}}^0$is the entropy density, carries dimension $\left[{\mathrm{s}}_{\mathsf{\mu }}\right]=\left[{\mathrm{s}}^0\right]=3$ (since entropy itself is dimensionless when ${\mathrm{k}}_{\mathrm{B}}=1$, and volume has dimension $-3$ in natural units). For the interaction term $\mathsf{\lambda }\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}\mathsf{\psi }$ to have the correct dimension of $[\mathrm{M}{]}^4$, we must have:

$>\left[\mathsf{\lambda }\right]+\left[\bar{\mathsf{\psi }}\mathsf{\psi }\right]+\left[{\mathrm{s}}_{\mathsf{\mu }}\right]=4⟹\left[\mathsf{\lambda }\right]=4-3-3=-2$

Thus, $\mathsf{\lambda }$ possesses mass dimension $\left[\mathsf{\lambda }\right]=-2$(i.e., inverse energy squared). This is the same dimension as the Newtonian gravitational constant ${\mathrm{G}}_{\mathrm{N}}$ or the Fermi coupling constant ${\mathrm{G}}_{\mathrm{F}}$ in the four-fermion weak interaction theory. Consequently, the present framework is naturally interpreted as a low-energy effective field theory of a more fundamental microscopic description (e.g., quantum tensor networks or quantum gravity). The magnitude of $\mathsf{\lambda }$ is not fixed by symmetry but is determined by the underlying ultraviolet completion, and its smallness (expected to be suppressed by some high-energy scale $\mathsf{\Lambda }$ as $\mathsf{\lambda }\sim 1/{\mathsf{\Lambda }}^2$) ensures that modifications to the Dirac equation remain perturbatively small in the present epoch. In SI units, this corresponds to $\left[\mathsf{\lambda }\right]={\mathrm{J}}^{-2}$ (or ${\mathrm{m}}^4\cdot {\mathrm{k}\mathrm{g}}^{-2}$).

The Lagrangian (1) is manifestly Lorentz covariant, as ${\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}$ is a scalar under Lorentz transformations. However, Lorentz covariance of the action should be distinguished from Lorentz symmetry of a particular background: if the entropy current acquires a non-zero vacuum expectation value $⟨{\mathrm{s}}^{\mathsf{\mu }}⟩\ne 0$, this background selects a preferred direction and spontaneously breaks Lorentz invariance in the vacuum. The phenomenology discussed in Section 3 and Section 6 concerns fluctuations around such a background, where the effective spin couplings reflect the broken symmetry phase.

Naturalness and the ultraviolet cutoff scale. The dimensionful coupling λ ~ [mass]⁻² signals that the interaction term is a dimension-5 operator in the effective field theory (EFT) expansion. In standard EFT reasoning, such an operator is suppressed by some ultraviolet cutoff scale Λ as λ ~ 1/Λ². If Λ were identified with the Planck scale ${\mathrm{M}}_{\mathrm{P}\mathrm{l}}~10¹⁹\mathrm{G}\mathrm{e}\mathrm{V}$, one would expect $\mathsf{\lambda }~10⁻³⁸\mathrm{G}\mathrm{e}\mathrm{V}⁻²$, rendering the entropy-fermion coupling unobservably small at low energies.

However, the present framework possesses a natural infrared scale that is radically different from the Planck scale: the entropy field mass ${\mathrm{m}}_{\mathrm{s}}$, which governs the correlation length of the entropy current. As we shall demonstrate in Section 5, the same mass parameter ${\mathrm{m}}_{\mathrm{s}}$ determines the graviton mass and is constrained by gravitational-wave observations to ${\mathrm{m}}_{\mathrm{s}}\lesssim {10}^{-20}\mathrm{e}\mathrm{V}$(Section 5.5). Physically, ${\mathrm{m}}_{\mathrm{s}}$ represents the inverse correlation length of the entropy flow field in the present cosmological epoch--a scale set by the large-scale structure of entropy in the universe rather than by Planck-scale quantum gravity.

We therefore identify the EFT cutoff with the scale at which the entropy flow field undergoes a phase transition or exhibits strongly coupled dynamics. A conservative assumption (which we label $[\mathbf{H}\mathbf{A}-\mathbf{E}\mathbf{F}\mathbf{T}]$ ) is ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}{\mathrm{E}}_{\mathrm{s}}$, where ${\mathrm{E}}_{\mathrm{s}}\equiv ⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}{⟩}^{1/6}$ is the characteristic energy scale of the entropy field. This relation is not derived from first principles but is motivated by the naturalness argument that the cutoff of the effective theory should be set by the characteristic scale of the entropy field itself. We explicitly note that this is a model assumption, and its consistency with the STAR data (Section 6.1) provides phenomenological support for the framework but does not constitute an independent proof. As shown in Section 6.1, the STAR data imply${\mathrm{E}}_{\mathrm{s}}~{{10}^2-10}^3\mathrm{G}\mathrm{e}\mathrm{V}$.With ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}~{\mathrm{E}}_{\mathrm{s}}$, the natural EFT expectation becomes $\mathsf{\lambda }~1/\mathsf{\Lambda }{²}_{\mathrm{E}\mathrm{F}\mathrm{T}}~10⁻⁷-10⁻⁵\mathrm{G}\mathrm{e}\mathrm{V}⁻²$, which is fully consistent with the value independently extracted from the STAR spin polarization data. This self-consistency between the EFT cutoff inferred from λ and the energy scale ${\mathrm{E}}_{\mathrm{s}}$ inferred from the same data provides a non-trivial internal check of the framework.

The physical picture is therefore as follows: the entropy-fermion coupling is not suppressed by the Planck scale but by the characteristic energy scale of the entropy flow field itself, ${\mathrm{E}}_{\mathrm{s}}~{10}^2-10³\mathrm{G}\mathrm{e}\mathrm{V}.$This scale lies far below the Planck scale, explaining why entropy-induced modifications to the Dirac equation can be observable in heavy-ion collisions. The hierarchy ${\mathrm{E}}_{\mathrm{s}}\ll {\mathrm{M}}_{\mathrm{P}\mathrm{l}}$ is a natural consequence of the fact that the entropy current describes collective, macroscopic degrees of freedom (the flow of quantum information) rather than fundamental Planck-scale dynamics. In this sense, the effective field theory defined by $\mathsf{\lambda }~1/{\mathrm{E}}_{\mathrm{s}}²$ is self-consistently generated by the entropy flow field's own dynamics, as will be further elucidated by the path integral analysis in Section 5.

We summarize the mass dimensions of the relevant fields and operators in Table 1.

Table note: With $\left[{\mathrm{s}}^{\mathsf{\mu }}\right]=3$, the field strength ${\mathrm{F}}_{\mathsf{\mu }}\mathsf{\nu }={\partial }_{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\nu }}-{\partial }_{\mathsf{\nu }}{\mathrm{s}}_{\mathsf{\mu }}$ has mass dimension 4, the Proca kinetic term has mass dimension 8. This reflects that ${\mathrm{s}}^{\mathsf{\mu }}$ is not canonically normalized. In our EFT framework, this is an effective term with implicit cutoff factors (see Section 5.2).

2.2. Field Equation

Varying the Lagrangian (1) with respect to $\bar{\mathsf{\psi }}$ yields the modified Dirac equation:

$$\left(\mathrm{i}\mathrm{\hslash }{\mathsf{\gamma }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}-\mathrm{m}\mathrm{c}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}\right)\mathsf{\psi }=0$$

(2)

The conjugate equation is:

$$\mathrm{i}\mathrm{\hslash }{\partial }_{\mathsf{\mu }}\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\mu }}+\mathrm{m}\mathrm{c}\bar{\mathsf{\psi }}-\mathsf{\lambda }{\mathrm{s}}_{\mathsf{\mu }}\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\mu }}=0$$

(3)

Both equations are covariant and reduce to the standard Dirac form when $\mathsf{\lambda }=0$.

2.3. Plane Wave Solutions

We seek solutions of the form:

$$\mathsf{\psi }\left(\mathrm{x}\right)=\mathrm{u}\left(\mathrm{p}\right){\mathrm{e}}^{-\mathrm{i}{\mathrm{p}}_{\mathsf{\mu }}{\mathrm{x}}^{\mathsf{\mu }}/\mathrm{\hslash }}$$

(4)

where ${\mathrm{p}}^{\mathsf{\mu }}=(\mathrm{E}/\mathrm{c},\mathbf{p})$ is the four-momentum and $\mathrm{u}\left(\mathrm{p}\right)$ is a Dirac spinor.

Substituting (4) into (2):

$$\left(\mathrm{i}\mathrm{\hslash }{\mathsf{\gamma }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}-\mathrm{m}\mathrm{c}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}\right)\mathrm{u}\left(\mathrm{p}\right){\mathrm{e}}^{-\mathrm{i}{\mathrm{p}}_{\mathsf{\mu }}{\mathrm{x}}^{\mathsf{\mu }}/\mathrm{\hslash }}=0$$

(5)

Using ${\partial }_{\mathsf{\mu }}{\mathrm{e}}^{-\mathrm{i}{\mathrm{p}}_{\mathsf{\nu }}{\mathrm{x}}^{\mathsf{\nu }}/\mathrm{\hslash }}=-(\mathrm{i}/\mathrm{\hslash }){\mathrm{p}}_{\mathsf{\mu }}{\mathrm{e}}^{-\mathrm{i}{\mathrm{p}}_{\mathsf{\nu }}{\mathrm{x}}^{\mathsf{\nu }}/\mathrm{\hslash }}$, we obtain:

$$\left({\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{p}}_{\mathsf{\mu }}-\mathrm{m}\mathrm{c}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}\right)\mathrm{u}\left(\mathrm{p}\right)=0$$

(6)

Defining the effective four-momentum:

$${\tilde{\mathrm{p}}}_{\mathsf{\mu }}\equiv {\mathrm{p}}_{\mathsf{\mu }}+\mathsf{\lambda }{\mathrm{s}}_{\mathsf{\mu }}$$

(7)

equation (6) becomes:

$$\left({\mathsf{\gamma }}^{\mathsf{\mu }}{\tilde{\mathrm{p}}}_{\mathsf{\mu }}-\mathrm{m}\mathrm{c}\right)\mathrm{u}\left(\mathrm{p}\right)=0$$

(8)

This is the standard Dirac equation with momentum ${\mathrm{p}}_{\mathsf{\mu }}$ replaced by the effective momentum ${\tilde{\mathrm{p}}}_{\mathsf{\mu }}$.

2.4. Modified Dispersion Relation

Multiplying (8) on the left by $({\mathsf{\gamma }}^{\mathsf{\nu }}{\tilde{\mathrm{p}}}_{\mathsf{\nu }}+\mathrm{m}\mathrm{c})$ and using the Clifford algebra $\{{\mathsf{\gamma }}^{\mathsf{\mu }},{\mathsf{\gamma }}^{\mathsf{\nu }}\}=2{\mathsf{\eta }}^{\mathsf{\mu }\mathsf{\nu }}$ , we obtain:

$$({\tilde{\mathrm{p}}}_{\mathsf{\mu }}{\tilde{\mathrm{p}}}^{\mathsf{\mu }}-{\mathrm{m}}^2{\mathrm{c}}^2)\mathrm{u}\left(\mathrm{p}\right)=0$$

(9)

Since $\mathrm{u}\left(\mathrm{p}\right)\ne 0$, we have:

$${\tilde{\mathrm{p}}}_{\mathsf{\mu }}{\tilde{\mathrm{p}}}^{\mathsf{\mu }}={\mathrm{m}}^2{\mathrm{c}}^2$$

(10)

Substituting ${\tilde{\mathrm{p}}}_{\mathsf{\mu }}={\mathrm{p}}_{\mathsf{\mu }}+\mathsf{\lambda }{\mathrm{s}}_{\mathsf{\mu }}$:

$$({\mathrm{p}}_{\mathsf{\mu }}+\mathsf{\lambda }{\mathrm{s}}_{\mathsf{\mu }})({\mathrm{p}}^{\mathsf{\mu }}+\mathsf{\lambda }{\mathrm{s}}^{\mathsf{\mu }})={\mathrm{m}}^2{\mathrm{c}}^2$$

(11)

where ${\mathrm{p}}_{\mathsf{\mu }}=\mathrm{i}{\partial }_{\mathsf{\mu }}$acts as a differential operator and does not necessarily commute with ${\mathrm{s}}_{\mathsf{\mu }}$. Expanding carefully while preserving operator ordering:

$${\mathrm{p}}_{\mathsf{\mu }}{\mathrm{p}}^{\mathsf{\mu }}+\mathsf{\lambda }\{{\mathrm{p}}_{\mathsf{\mu }},{\mathrm{s}}^{\mathsf{\mu }}\}+{\mathsf{\lambda }}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}={\mathrm{m}}^2{\mathrm{c}}^2$$

(12)

where $\{\mathrm{A},\mathrm{B}\}\equiv \mathrm{A}\mathrm{B}+\mathrm{B}\mathrm{A}$ denotes the anticommutator.

In the WKB regime adopted throughout this work, the entropy current ${\mathrm{s}}_{\mathsf{\mu }}$ varies slowly compared with the fermion de Broglie wavelength. This condition is quantified as $\left|{\partial }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\right|\ll \left|{\mathrm{p}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\right|$: the relative change of ${\mathrm{s}}_{\mathsf{\mu }}$over the fermion wavelength is small. Under this condition, the gradient term ${\partial }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$is subdominant, and the anticommutator reduces to $\{{\mathrm{p}}_{\mathsf{\mu }},{\mathrm{s}}^{\mathsf{\mu }}\}\simeq 2{\mathrm{p}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$, with higher-order derivative corrections neglected to leading order in the WKB expansion. Using ${\mathrm{p}}_{\mathsf{\mu }}{\mathrm{p}}^{\mathsf{\mu }}={\mathrm{E}}^2/{\mathrm{c}}^2-{\mathbf{p}}^2$, we obtain the general WKB dispersion relation:

$${\displaystyle \frac{{\mathrm{E}}^2}{{\mathrm{c}}^2}}-{\mathbf{p}}^2+2\mathsf{\lambda }{\mathrm{p}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}+{\mathsf{\lambda }}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}={\mathrm{m}}^2{\mathrm{c}}^2$$

(13)

2.5. Special Cases

2.5.1. Uniform Static Entropy Flow

Consider the simplest case: ${\mathrm{s}}^{\mathsf{\mu }}=({\mathrm{s}}^0,0)$, where ${\mathrm{s}}^0$ is constant. Then:

$${\mathrm{p}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=(\mathrm{E}/\mathrm{c}){\mathrm{s}}^0,{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=({\mathrm{s}}^0{)}^2$$

(14)

Substituting into (13):

$${\displaystyle \frac{{\mathrm{E}}^2}{{\mathrm{c}}^2}}-{\mathbf{p}}^2+2\mathsf{\lambda }(\mathrm{E}/\mathrm{c}){\mathrm{s}}^0+{\mathsf{\lambda }}^2({\mathrm{s}}^0{)}^2={\mathrm{m}}^2{\mathrm{c}}^2$$

(15)

Solving for $\mathrm{E}$:

$$\mathrm{E}=-\mathsf{\lambda }{\mathrm{s}}^0\mathrm{c}\pm \mathrm{c}\sqrt{{\mathbf{p}}^2+{\mathrm{m}}^2{\mathrm{c}}^2}$$

(16)

This result shows that a uniform static entropy flow produces only an overall energy shift:

$$\mathsf{\Delta }\mathrm{E}=-\mathsf{\lambda }{\mathrm{s}}^0\mathrm{c}$$

(17)

The effective mass remains unchanged (${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{m}$), and the dispersion relation retains its standard form up to a constant shift. This is a crucial consistency check: in regions of uniform entropy flow, no novel dispersive effects appear.

2.5.2. Static Non-Uniform Entropy Flow

Consider ${\mathrm{s}}^{\mathsf{\mu }}=\left({\mathrm{s}}^0\right(\mathbf{x}),0)$, where ${\mathrm{s}}^0$ varies slowly in space. In the Wentzel-Kramers-Brillouin (WKB) approximation, valid when the characteristic scale of variation ${\mathrm{L}}_{\mathrm{s}}$ is much larger than the de Broglie wavelength ${\mathsf{\lambda }}_{\mathrm{d}\mathrm{B}}$, we obtain the local dispersion relation:

$$\mathrm{E}(\mathbf{x},\mathbf{p})=-\mathsf{\lambda }{\mathrm{s}}^0\left(\mathbf{x}\right)\mathrm{c}\pm \mathrm{c}\sqrt{{\mathbf{p}}^2+{\mathrm{m}}^2{\mathrm{c}}^2}$$

(18)

This indicates that particles experience a position-dependent energy shift proportional to the local entropy density. In regions of high entropy density (${\mathrm{s}}^0>0$), particles with $\mathsf{\lambda }>0$ have lower effective energy—a "redshift" effect induced by entropy. For $\mathsf{\lambda }<0$, the effect reverses, producing a "blueshift."

A remark on terminology is in order. The position-dependent shift $\mathsf{\Delta }\mathrm{E}\left(\mathbf{x}\right)=-\mathsf{\lambda }{\mathrm{s}}^0\left(\mathbf{x}\right)\mathrm{c}$ in the local dispersion relation (18) is sometimes interpreted as an effective mass shift ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\mathbf{x}\right)=\mathrm{m}+\mathsf{\lambda }{\mathrm{s}}^0\left(\mathbf{x}\right)/\mathrm{c}$ when written in the form ${\mathrm{E}}^2={\mathbf{p}}^2{\mathrm{c}}^2+{\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^2{\mathrm{c}}^4$. However, this interpretation is valid only in the WKB regime where ${\mathrm{s}}^0$ varies slowly. In the exactly solvable case of a strictly uniform entropy flow, the dispersion relation (16) shows unequivocally that the effective mass remains $\mathrm{m}$; only the overall energy is shifted. Thus, the entropic mass shift is a non-uniformity-induced effect that disappears when the entropy gradient vanishes. This distinction is crucial for correctly interpreting the thermodynamic origin of mass modifications in dense media.

2.5.3. Time-Dependent Entropy Flow

For a time-dependent entropy flow ${\mathrm{s}}^{\mathsf{\mu }}=\left({\mathrm{s}}^0\right(\mathrm{t}),0)$, the energy $\mathrm{E}$ acquires a time dependence through the dispersion relation. The generalized energy is no longer conserved in the usual sense, reflecting the non-equilibrium nature of the system. The rate of energy change follows from differentiating (13) with respect to time:

$${\displaystyle \frac{\mathrm{d}\mathrm{E}}{\mathrm{d}\mathrm{t}}}=-\mathsf{\lambda }\mathrm{c}{\displaystyle \frac{\mathrm{d}{\mathrm{s}}^0}{\mathrm{d}\mathrm{t}}}$$

(19)

This suggests that a changing entropy density acts as a source or sink of particle energy, consistent with the thermodynamic interpretation of entropy production ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$ [12,13].

3. Non-Relativistic Limit and Effective Magnetic Field

3.1. Pauli-Type Reduction and the Emergent Entropic Gauge Structure

We now derive the non-relativistic limit of the modified Dirac equation following the standard Foldy-Wouthuysen procedure [13,14]. In natural units ($\mathrm{\hslash }=\mathrm{c}=1$), the modified Dirac equation reads:

$$\mathrm{i}{\partial }_{\mathrm{t}}\mathsf{\psi }=\left(-\mathrm{i}{\mathsf{\gamma }}^0{\mathsf{\gamma }}^{\mathrm{i}}{\partial }_{\mathrm{i}}+\mathrm{m}{\mathsf{\gamma }}^0-\mathsf{\lambda }{\mathsf{\gamma }}^0{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}\right)\mathsf{\psi }$$

(20)

Using the Dirac representation:

$${\mathsf{\gamma }}^0=\left(\begin{array}{ll}\mathrm{I}& 0\\ 0& -\mathrm{I}\end{array}\right),{\mathsf{\gamma }}^{\mathrm{i}}=\left(\begin{array}{ll}0& {\mathsf{\sigma }}^{\mathrm{i}}\\ -{\mathsf{\sigma }}^{\mathrm{i}}& 0\end{array}\right)$$

(21)

we obtain:

$${\mathsf{\gamma }}^0{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }}=\left(\begin{array}{ll}{\mathrm{s}}_0& {\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}\\ {\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}& -{\mathrm{s}}_0\end{array}\right)$$

(22)

Thus, equation (20) becomes the coupled system:

$$\mathrm{i}{\partial }_{\mathrm{t}}\left(\begin{array}{l}\mathsf{\varphi }\\ \mathsf{\chi }\end{array}\right)=\left(\begin{array}{ll}\mathrm{m}+\mathsf{\lambda }{\mathrm{s}}_0& -\mathrm{i}{\mathsf{\sigma }}^{\mathrm{i}}{\partial }_{\mathrm{i}}+\mathsf{\lambda }{\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}\\ -\mathrm{i}{\mathsf{\sigma }}^{\mathrm{i}}{\partial }_{\mathrm{i}}+\mathsf{\lambda }{\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}& -\mathrm{m}-\mathsf{\lambda }{\mathrm{s}}_0\end{array}\right)\left(\begin{array}{l}\mathsf{\varphi }\\ \mathsf{\chi }\end{array}\right)$$

(23)

where $\mathsf{\varphi }$and $\mathsf{\chi }$are the large and small components of the Dirac spinor, respectively. For positive-energy solutions in the non-relativistic regime, we assume $\left|\mathsf{\lambda }{\mathrm{s}}_{\mathsf{\mu }}\right|\ll \mathrm{m}$ and $\left|\mathbf{p}\right|\ll \mathrm{m}$, so that the small component is suppressed relative to the large component by a factor of $1/\mathrm{m}$:

$$\mathsf{\chi }\simeq {\displaystyle \frac{1}{2\mathrm{m}}}\left(-\mathrm{i}{\mathsf{\sigma }}^{\mathrm{i}}{\partial }_{\mathrm{i}}+\mathsf{\lambda }{\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}\right)\mathsf{\varphi }$$

(24)

Substituting (24) back into the equation for $\mathsf{\varphi }$, and keeping terms up to order $1/\mathrm{m}$, we obtain the effective Schrödinger equation:

$$\mathrm{i}{\partial }_{\mathrm{t}}\mathsf{\varphi }=\left[\mathrm{m}+\mathsf{\lambda }{\mathrm{s}}_0+{\displaystyle \frac{1}{2\mathrm{m}}}{\left(-\mathrm{i}{\mathsf{\sigma }}^{\mathrm{i}}{\partial }_{\mathrm{i}}+\mathsf{\lambda }{\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}}\right)}^2\right]\mathsf{\varphi }$$

(25)

Expanding the square operator carefully, noting that ${\partial }_{\mathrm{i}}$ acts on everything to its right and hence does not commute with ${\mathrm{s}}_{\mathrm{i}}$, we have:

$${\left(-\mathrm{i}{\partial }_{\mathrm{i}}+\mathsf{\lambda }{\mathrm{s}}_{\mathrm{i}}\right)}^2\mathsf{\varphi }=(-{\nabla }^2)\mathsf{\varphi }+\mathsf{\lambda }\left[-\mathrm{i}(\nabla \cdot \mathbf{s})-2\mathrm{i}\mathbf{s}\cdot \nabla \right]\mathsf{\varphi }+{\mathsf{\lambda }}^2{\mathbf{s}}^2\mathsf{\varphi }+\mathsf{\lambda }\mathsf{\sigma }\cdot (\nabla \times \mathbf{s})\mathsf{\varphi }$$

(26)

The last term arises from the commutator $[{\mathsf{\sigma }}^{\mathrm{i}}{\mathrm{s}}_{\mathrm{i}},{\mathsf{\sigma }}^{\mathrm{j}}{\partial }_{\mathrm{j}}]$, which yields the spin--curl coupling. Substituting (26) into (25), the effective Hamiltonian becomes:

$${\mathrm{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{m}+\mathsf{\lambda }{\mathrm{s}}_0-{\displaystyle \frac{{\nabla }^2}{2\mathrm{m}}}-{\displaystyle \frac{\mathrm{i}\mathsf{\lambda }}{\mathrm{m}}}\mathbf{s}\cdot \nabla +{\displaystyle \frac{{\mathsf{\lambda }}^2}{2\mathrm{m}}}{\mathbf{s}}^2+{\displaystyle \frac{\mathsf{\lambda }}{2\mathrm{m}}}\mathsf{\sigma }\cdot (\nabla \times \mathbf{s})$$

(27)

where we have dropped the purely real divergence term $-\mathrm{i}\mathsf{\lambda }(\nabla \cdot \mathbf{s})/2\mathrm{m}$ (a total derivative that does not affect local dynamics). Remarkably, the first-order derivative terms can be absorbed into a canonical momentum shift:

$$-{\displaystyle \frac{{\nabla }^2}{2\mathrm{m}}}-{\displaystyle \frac{\mathrm{i}\mathsf{\lambda }}{\mathrm{m}}}\mathbf{s}\cdot \nabla +{\displaystyle \frac{{\mathsf{\lambda }}^2}{2\mathrm{m}}}{\mathbf{s}}^2={\displaystyle \frac{1}{2\mathrm{m}}}(-\mathrm{i}\nabla -\mathsf{\lambda }\mathbf{s}{)}^2$$

(28)

Therefore, the final non-relativistic Hamiltonian takes the compact gauge-like form:

$${\mathrm{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{m}+\mathsf{\lambda }{\mathrm{s}}_0+{\displaystyle \frac{1}{2\mathrm{m}}}{\left(\mathbf{p}-\mathsf{\lambda }\mathbf{s}\right)}^2+{\displaystyle \frac{\mathsf{\lambda }}{2\mathrm{m}}}\mathsf{\sigma }\cdot (\nabla \times \mathbf{s})$$

(29)

where $\mathbf{p}\equiv -\mathrm{i}\nabla$. This result reveals that the spatial entropy current $\mathbf{s}$ acts analogously to a vector potential in the effective Schrödinger theory: the canonical momentum is shifted as p → p − λs, while its curl generates a spin-dependent coupling structurally similar to a Zeeman field. We emphasize that this is a vector-potential-like coupling, not a genuine gauge symmetry; the theory does not possess a gauge invariance of the form ${\mathrm{s}}_{\mathsf{\mu }}\to {\mathrm{s}}_{\mathsf{\mu }}+{\partial }_{\mathsf{\mu }}\mathsf{\alpha }$.

3.2. Effective Spin-Coupling Field

From the spin-dependent term in (29):

$${\mathrm{H}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}={\displaystyle \frac{\mathsf{\lambda }}{2\mathrm{m}}}\mathsf{\sigma }\cdot (\nabla \times \mathbf{s})$$

(30)

Comparing with the standard Pauli Hamiltonian ${\mathrm{H}}_{\mathrm{P}\mathrm{a}\mathrm{u}\mathrm{l}\mathrm{i}}=-\mathsf{\mu }\cdot \mathbf{B}$, we identify an effective Zeeman-type field:

$${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\displaystyle \frac{\mathsf{\lambda }}{\mathrm{m}}}\nabla \times \mathbf{s}$$

(31)

We refer to this as an ‘effective’ field because it acts on spin in the same way as a magnetic field, but it originates from the entropy current’s curl rather than from an electromagnetic gauge structure.

3.3. Modified Spin Precession

The spin precession equation follows from the Heisenberg equation for the spin operator $\mathbf{S}=(1/2)\mathsf{\sigma }$:

$${\displaystyle \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\mathrm{t}}}=\mathrm{i}[\mathrm{H},\mathbf{S}]$$

(32)

Using (30), we obtain:

$${\displaystyle \frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\mathrm{t}}}={\mathsf{\omega }}_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}\times \mathbf{S}$$

(33)

where

$${\mathsf{\omega }}_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}={\displaystyle \frac{\mathsf{\lambda }}{\mathrm{m}}}\nabla \times \mathbf{s}$$

(34)

In the presence of an external magnetic field $\mathbf{B}$, the total precession frequency is:

$${\mathsf{\omega }}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\mathsf{\omega }}_{\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{c}}+{\mathsf{\omega }}_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}$$

(35)

where ${\mathsf{\omega }}_{\mathrm{D}\mathrm{i}\mathrm{r}\mathrm{a}\mathrm{c}}=(\mathrm{e}/\mathrm{m})\mathbf{B}-(\mathrm{e}/2\mathrm{m})\mathsf{\beta }\times \mathbf{E}+\dots$is the standard Bargmann-Michel-Telegdi precession frequency [15,16].

3.4. Sources of the Entropy Curl

Having established that ${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathbf{s}$ drives spin precession, we now examine the physical sources that can generate a non-zero curl of the entropy current. The key mathematical observation is that $\nabla \times \mathbf{s}$ is non-zero if and only if the entropy flow contains a rotational or vortical component. This follows from a fundamental property of vector calculus: the curl of any gradient field vanishes identically, $\nabla \times (\nabla \mathrm{f})=0$ for any scalar function $\mathrm{f}$. Consequently, any entropy current that derives from a gradient of a scalar quantity — such as temperature $\mathrm{T}$ or chemical potential $\mathsf{\mu }$ --produces no effective magnetic field.

The entropy current $\mathbf{s}$ can arise from various physical processes, as summarized in Table 2.

In the rotating fluid row, ${\mathsf{\rho }}_{\mathrm{s}}$ denotes the entropy density in the fluid rest frame.

For the rotating fluid, the entropy current is advected by the fluid velocity field $\mathbf{v}=\mathsf{\omega }\times \mathbf{r}$, so that $\mathbf{s}={\mathsf{\rho }}_{\mathrm{s}}\mathbf{v}$, where ${\mathsf{\rho }}_{\mathrm{s}}$ is the entropy density in the fluid rest frame. Its curl gives $\nabla \times \mathbf{s}=2{\mathsf{\rho }}_{\mathrm{s}}\mathsf{\omega }$ (neglecting gradients of ${\mathsf{\rho }}_{\mathrm{s}}$ for a uniform fluid). Substituting into (31), the effective magnetic field for a rotating system is:

$${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\displaystyle \frac{2\mathsf{\lambda }{\mathsf{\rho }}_{\mathrm{s}}}{\mathrm{m}}}\mathsf{\omega }$$

(36)

Thus, only the vortical component of the entropy flow produces a non-zero curl and hence an effective magnetic field. This is a distinctive prediction with important experimental consequences:

1) Clean separation of sources: Pure temperature or chemical potential gradients do not affect spin precession, while rotating systems with entropy flow do [17,18]. This allows experimentalists to distinguish between thermal and vortical contributions to spin polarization -- a capability that is particularly valuable in heavy-ion collision experiments where both thermal gradients and global vorticity are present.

2) Direct probe of fluid vorticity: In systems such as the quark-gluon plasma produced in relativistic heavy-ion collisions, equation (36) provides a direct, calibration-free measurement of the local fluid vorticity $\mathsf{\omega }$ through the induced spin precession, independent of electromagnetic fields.

3) Analogy to the Barnett effect: In condensed matter physics, the Barnett effect describes the magnetization of a rotating uncharged body. Here, the entropy current plays a role analogous to the magnetization, with $\nabla \times \mathbf{s}$ acting as an effective magnetic field that couples to spin. This analogy suggests that our framework predicts an “entropic Barnett effect” that could be tested in rotating quantum fluids, such as ultra-cold atomic gases or superfluid helium.

4. Entropy Flow Field Dynamics: A Classical Proca Framework

4.1. Lagrangian for the Entropy Current

Methodological remark on the Proca promotion.The introduction of a Proca-type kinetic term for the entropy current is not a derivation from first-principles thermodynamics, but an effective field theory (EFT) assumption about the dynamical content of the emergent geometry at macroscopic scales. In the companion paper [7], the entropy current ${\mathrm{s}}^{\mathsf{\mu }}$emerges microscopically from the entanglement structure of the string-net condensate, where its dynamics is governed by the underlying Levin-Wen Hamiltonian. At low energies, the most general covariant action consistent with the symmetries of the emergent spacetime and the conservation of the entropy current (up to the second law) includes a kinetic term of the Proca form. This is analogous to how the Maxwell action is not derived from QED but emerges as the leading low-energy term in the derivative expansion of the effective action for a $\mathrm{U}\left(1\right)$ gauge field. We therefore emphasize that the Proca Lagrangian (37) is a low-energy effective description of the collective entanglement dynamics, not a fundamental property of the thermodynamic entropy current itself. The mass parameter ${\mathrm{m}}_{\mathrm{s}}$is an effective parameter that encodes the inverse correlation length of the entropy flow field in the present cosmological epoch, to be determined by observations (Section 5.5).

To make the framework dynamical, we introduce a kinetic term for the entropy current ${\mathrm{s}}_{\mathsf{\mu }}$. Following the Proca theory for massive vector fields [19,20], we propose:

$${\mathcal{L}}_{\mathrm{s}}=-{\displaystyle \frac{1}{4}}{\mathrm{F}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{F}}^{\mathsf{\mu }\mathsf{\nu }}+{\displaystyle \frac{1}{2}}{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$$

(37)

where ${\mathrm{F}}_{\mathsf{\mu }\mathsf{\nu }}={\partial }_{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\nu }}-{\partial }_{\mathsf{\nu }}{\mathrm{s}}_{\mathsf{\mu }}$ and ${\mathrm{m}}_{\mathrm{s}}$ is the entropy field mass.

The mass term ${\displaystyle \frac{1}{2}}{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ is essential for two reasons. First, in the context of an EFT description of a dissipative medium, a mass term $1/2{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ is the lowest-order term that breaks the would-be gauge invariance of the kinetic term and introduces a finite correlation length, ${\mathsf{\xi }}_{\mathrm{c}}~1/{\mathrm{m}}_{\mathrm{s}}$, for the entropy flow field. This is characteristic of a medium with a longest relaxation time, though we emphasize that this is an effective description rather than a derivation from first principles. The finite correlation length is controlled by the mass gap ${\mathrm{m}}_{\mathrm{s}}$, as we demonstrate explicitly via the propagator in Section 5.4. Second, as will become clear in Section 5, a non-zero ${\mathrm{m}}_{\mathrm{s}}$ provides the natural scale for the emergent graviton mass, connecting the entropy field's correlation length to gravitational-wave observables.

Distinction from standard Proca theory. It is important to distinguish the present usage of the Proca Lagrangian from standard massive vector field theory. In standard Proca theory, the vector field ${\mathrm{s}}^{\mathsf{\mu }}$ is a fundamental quantum field with mass dimension 1, and the mass term $1/2{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ is a fundamental parameter of the action. In our framework, by contrast, the field ${\mathrm{s}}^{\mathsf{\mu }}$is first and foremost a macroscopic descriptor of entanglement entropy flow in the emergent spacetime. Its Proca-like action is an effective field theory that captures the leading long-wavelength dynamics of the underlying entanglement degrees of freedom. The mass parameter ${\mathrm{m}}_{\mathrm{s}}$ is not a fundamental mass of a particle, but an effective parameter characterizing the finite correlation length of the entropy flow field. This distinction is essential: we are not quantizing the thermodynamic entropy current as a fundamental Proca field; we are writing down the lowest-order effective action for its macroscopic dynamics, with the understanding that this action is valid only below the EFT cutoff ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}\mathrm{S}}$.

4.2. Coupling to Curved Spacetime and Fermions

To prepare the ground for the emergence mechanism presented in Section 5, we couple the entropy flow field to a curved spacetime background ${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$ and to fermionic matter. The complete action takes the form:

$$\mathrm{S}=\int {\mathrm{d}}^4\mathrm{x}\sqrt{-\mathrm{g}}\left[\bar{\mathsf{\psi }}(\mathrm{i}{\mathsf{\gamma }}^{\mathsf{\mu }}{\nabla }_{\mathsf{\mu }}-\mathrm{m}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }})\mathsf{\psi }-{\displaystyle \frac{1}{4}}{\mathrm{F}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{F}}^{\mathsf{\mu }\mathsf{\nu }}+{\displaystyle \frac{1}{2}}{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}+\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\right]$$

(38)

where ${\nabla }_{\mathsf{\mu }}$ denotes the covariant derivative acting on spinors, $\mathrm{R}$ is the Ricci scalar, and $\mathsf{\xi }$is a non-minimal coupling constant of mass dimension $\left[\mathsf{\xi }\right]=-4$ in natural units (as required by the consistency of the emergent Newton constant in equation (54)).Crucially, we have not included the Einstein–Hilbert term ${\displaystyle \frac{1}{16\mathsf{\pi }\mathrm{G}}}\mathrm{R}$as a fundamental ingredient. Instead, the Ricci scalar appears only through the non-minimal coupling $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$, which serves as a seed for the emergence of gravitational dynamics. As we shall demonstrate in Section 5, upon integrating out quantum fluctuations of the entropy field, the Einstein–Hilbert action is generated as a low-energy effective term, with Newton's constant determined by the vacuum expectation value of ${\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$.

Remark on the role of the non-minimal coupling. The term $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$is a standard non-minimal coupling between a vector field and spacetime curvature -- analogous to, but physically distinct from, the scalar curvature coupling $\mathsf{\xi }\mathrm{R}{\mathsf{\varphi }}^2$ that appears in inflation and Higgs physics. In this work, ${\mathrm{s}}_{\mathsf{\mu }}$ is not a Higgs-like scalar; it carries no gauge charge, does not develop a symmetry-breaking potential, and does not couple to fermions through Yukawa interactions. The non-minimal coupling serves a specific technical purpose: it provides the seed term that, upon quantization of the entropy field and integration over its fluctuations (Section 5), generates the Einstein–Hilbert action as a low-energy effective term. In this sense, $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$is a bridge between quantum information flow and geometry, not a mechanism for fundamental mass generation.

4.3. Field Equations

Varying (38) with respect to ${\mathrm{s}}_{\mathsf{\mu }}$ yields the generalized Proca equation in curved spacetime:

$${\nabla }_{\mathsf{\mu }}{\mathrm{F}}^{\mathsf{\mu }\mathsf{\nu }}+{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}^{\mathsf{\nu }}+2\mathsf{\xi }\mathrm{R}{\mathrm{s}}^{\mathsf{\nu }}=\mathsf{\lambda }\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\nu }}\mathsf{\psi }$$

(39)

The source term on the right-hand side shows that fermions act as sources of the entropy current. The term $2\mathsf{\xi }\mathrm{R}{\mathrm{s}}^{\mathsf{\nu }}$ couples the entropy flow to spacetime curvature: in regions of positive Ricci curvature, the entropy field acquires an effective mass shift ${\mathrm{m}}_{\mathrm{s},\mathrm{e}\mathrm{f}\mathrm{f}}^2={\mathrm{m}}_{\mathrm{s}}^2+2\mathsf{\xi }\mathrm{R}$.

Varying with respect to ${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$ gives a formal relation for the background curvature:

$${\mathrm{G}}_{\mathsf{\mu }\mathsf{\nu }}=8\mathsf{\pi }\mathrm{G}\left({\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathsf{\psi }}+{\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{s}}+{\mathsf{\Theta }}_{\mathsf{\mu }\mathsf{\nu }}\right)$$

(40)

where${\mathrm{G}}_{\mathsf{\mu }\mathsf{\nu }}$is the Einstein tensor, ${\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathsf{\psi }}$is the fermion energy-momentum tensor, and $\mathrm{G}$ is a formal notation for the emergent gravitational coupling whose microscopic origin will be determined in Section 5 [see equation (54)]. At this stage, equation (40) is not a fundamental dynamical equation; rather, it is a constraint relating the background curvature to the energy-momentum of the entropy and fermion fields. The true dynamics of gravity will emerge in Section 5 from quantum fluctuations of the entropy field.

where ${\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{s}}$ is the Proca energy-momentum tensor:

$${\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{s}}={\mathrm{F}}_{\mathsf{\mu }\mathsf{\alpha }}{\mathrm{F}}_{\mathsf{\nu }}^{\mathsf{\alpha }}-{\displaystyle \frac{1}{4}}{\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{F}}_{\mathsf{\alpha }\mathsf{\beta }}{\mathrm{F}}^{\mathsf{\alpha }\mathsf{\beta }}+{\mathrm{m}}_{\mathrm{s}}^2\left({\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\nu }}-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{s}}_{\mathsf{\alpha }}{\mathrm{s}}^{\mathsf{\alpha }}\right)$$

(41)

and ${\mathsf{\Theta }}_{\mathsf{\mu }\mathsf{\nu }}$ arises from the non-minimal coupling:

$${\mathsf{\Theta }}_{\mathsf{\mu }\mathsf{\nu }}=\mathsf{\xi }\left[\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\nu }}-{\displaystyle \frac{1}{2}}{\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}\mathrm{R}{\mathrm{s}}_{\mathsf{\alpha }}{\mathrm{s}}^{\mathsf{\alpha }}+{\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}{\nabla }_{\mathsf{\alpha }}{\nabla }^{\mathsf{\alpha }}\left({\mathrm{s}}_{\mathsf{\alpha }}{\mathrm{s}}^{\mathsf{\alpha }}\right)-{\nabla }_{\mathsf{\mu }}{\nabla }_{\mathsf{\nu }}\left({\mathrm{s}}_{\mathsf{\alpha }}{\mathrm{s}}^{\mathsf{\alpha }}\right)\right]$$

(42)

4.4. Static Spherically Symmetric Solution

We now examine the static, spherically symmetric solution of the entropy field in the absence of fermion sources ($\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\nu }}\mathsf{\psi }=0$). In the weak-curvature limit, where gradients of the background curvature are negligible compared to the scale set by ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$, and retaining only the dominant scalar mode ${\mathrm{s}}_0$ of the vector field, the generalized Proca equation (39) reduces to the following effective scalar equation:

$${\nabla }^2{\mathrm{s}}_0-({\mathrm{m}}_{\mathrm{s}}^2+2\mathsf{\xi }\mathrm{R}){\mathrm{s}}_0=0$$

(43)

Here we have assumed the background curvature R to be approximately constant over the relevant length scale. This approximation is valid in the regime $|\nabla \mathrm{R}|/\mathrm{R}\ll {\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$,which holds for the cosmological and astrophysical applications considered in this work.

Compared with the flat-spacetime Proca equation, the non-minimal curvature coupling $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ generates an effective mass contribution $2\mathsf{\xi }\mathrm{R}.\mathrm{F}\mathrm{o}\mathrm{r}\mathsf{\xi }>0$, the entropy field acquires a positive curvature-induced mass, suppressing its interaction range in high-curvature regions. For $\mathsf{\xi }<0$ with $\left|2\mathsf{\xi }\mathbf{R}\right|>{\mathbf{m}}_{\mathbf{s}}^2$, however, the effective mass squared becomes negative, and the Yukawa potential (44) develops an oscillatory behavior characteristic of tachyonic instability--a feature that may have profound implications for neutrino mass generation and flavor oscillations, as will be explored in a separate work.

The regular solution at infinity is the modified Yukawa potential:

$${\mathrm{s}}_0\left(\mathrm{r}\right)={\displaystyle \frac{\mathrm{q}}{4\mathsf{\pi }\mathrm{r}}}{\mathrm{e}}^{-{\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\left(\mathrm{s}\right)}\mathrm{r}}$$

(44)

where

$${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\left(\mathrm{s}\right)}\equiv \sqrt{{\mathrm{m}}_{\mathrm{s}}^2+2\mathsf{\xi }\mathrm{R}}$$

(45)

is the curvature-dependent effective mass of the entropy field. In the flat-spacetime limit ($\mathrm{R}\to 0$), this reduces to the standard Yukawa form:

${\mathrm{s}}_0\left(\mathrm{r}\right)={\displaystyle \frac{\mathrm{q}}{4\mathsf{\pi }\mathrm{r}}}{\mathrm{e}}^{-{\mathrm{m}}_{\mathrm{s}}\mathrm{r}}$

recovering the result of the original Proca theory [19,20].

This solution carries several important physical implications:

1) Finite range of entropic interactions. The effective mass ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\left(\mathrm{s}\right)}$ sets the interaction range ${\mathrm{r}}_{\mathrm{s}}\equiv 1/{\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\left(\mathrm{s}\right)}$. In this work, we adopt $\mathsf{\xi }<0$ to ensure a positive Newton constant via (54) [see Section 5.2]. Consequently, in regions of positive curvature (e.g., the early universe or near black hole horizons with non-zero Ricci scalar, such as in the presence of matter or a cosmological constant), the entropy field becomes lighter and its influence is more extended.

2) Curvature-induced mass generation for the entropy field. Even if the bare mass ${\mathrm{m}}_{\mathrm{s}}$ vanishes, a non-zero $\mathrm{R}$ generates an effective mass ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\left(\mathrm{s}\right)}=\sqrt{2\mathsf{\xi }\mathrm{R}}$ for the entropy current. For $\mathsf{\xi }<0$, however, this expression is real only when $\mathrm{R}<0$; for $\mathrm{R}>0$ with $\left|2\mathsf{\xi }\mathrm{R}\right|>{\mathrm{m}}_{\mathrm{s}}^2$, the effective mass squared becomes negative, signaling an oscillatory mode that may have implications for neutrino physics (see below).

3) Connection to the emergent graviton. As we shall demonstrate in Section 5, the mass parameter ${\mathrm{m}}_{\mathrm{s}}$ that governs the entropy field's Yukawa tail is precisely the same parameter that determines the mass of the emergent graviton. Thus, the curvature dependence exhibited in (45) suggests that in strong-field regions, the graviton may acquire an effective mass ${\mathrm{m}}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{n}}^{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{{\mathrm{m}}_{\mathrm{s}}^2+2\mathsf{\xi }\mathrm{R}}$-- a potential signature for gravitational-wave observations.

In the context of black hole thermodynamics, it may be related to the Bekenstein–Hawking entropy, though a detailed investigation lies beyond the scope of the present work.

4.4.1. Stability and Higher-Order Curvature Corrections

The curvature-dependent effective mass in (45), ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\mathrm{s}\right)²={\mathrm{m}}_{\mathrm{s}}²+2\mathsf{\xi }\mathrm{R}$, raises a consistency issue: since positive G requires ξ<0, large positive curvature may make ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\mathrm{s}\right)²$negative, signaling tachyonic instability.

To restore stability, we supplement (38) with a higher-order term:

$\mathsf{\Delta }{\mathrm{L}}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}=\mathsf{\zeta }\mathrm{R}²{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }},\mathsf{\zeta }>0$ (45a)

yielding ${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\mathrm{s}\right)²\left(\mathrm{R}\right)={\mathrm{m}}_{\mathrm{s}}²+2\mathsf{\xi }\mathrm{R}+\mathsf{\zeta }\mathrm{R}²$. For $\mathsf{\zeta }{\mathrm{m}}_{\mathrm{s}}²>\mathsf{\xi }²$, this remains positive for all R. The stability scale is defined as ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}\equiv \sqrt{\left|\mathsf{\xi }\right|/\mathsf{\zeta }}$, which has dimensions of mass. In the weak-curvature regime defined by $\left|\mathrm{R}\right|\ll {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}^2$, the linear coupling suffices as the leading-order description. When $\left|\mathrm{R}\right|\sim {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}^2$ or larger, higher-order curvature terms are required to maintain stability.

4.5. Generalized Second Law

The entropy current must satisfy the covariant second law of thermodynamics:

$${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$$

(46)

This can be imposed as a constraint on the solutions of the field equations. In the static Yukawa solution, ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}={\partial }_{\mathrm{r}}{\mathrm{s}}_0\left(\mathrm{r}\right)\ne 0$, corresponding to a non-zero entropy production rate.

The entropy production rate $\mathsf{\sigma }$ plays a dual role in our framework. At the classical level, it enforces the arrow of time and quantifies irreversibility. At the quantum level, we speculate that σ may be related to the spectral density of quantum fluctuations of the entropy field, as regions with entropy production could act as sources of metric fluctuations (see Section 5.6). We emphasize that this interpretation is currently a conjecture: a rigorous derivation would require a fluctuation-dissipation theorem or Schwinger-Keldysh formalism applied to the string-net condensate [7], which lies beyond the scope of the present work. The statements in Section 5 regarding the generalized second law as the origin of spacetime causality should be understood in this conjectural sense, unless and until a full microscopic derivation is provided.

5. Emergent Gravity: From Entropy Flow Fluctuations to the Graviton

5.1. Entropy-Curvature Coupling and the Path Integral Setup

Levels of emergence. Before presenting the calculation, we distinguish four distinct levels at which the Einstein-Hilbert action may arise in our framework:

1) Tree-level VEV-induced curvature term: The non-minimal coupling $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$, evaluated on the background VEV $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩\ne 0$, directly yields a term proportional to $\mathrm{R}$in the action. This is the dominant contribution to the effective Newton constant and is already present at tree level.

2) Quantum loop corrections: Integrating out Gaussian fluctuations $\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}$ around the background yields a determinant factor, which contributes corrections of order $\mathrm{O}(1/(16\mathsf{\pi }²)$ to the curvature coefficient. These are parametrically suppressed relative to the tree-level contribution.

3) Renormalization of the gravitational coupling: The sum of tree-level and loop contributions defines the renormalized effective Newton constant ${\mathrm{G}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$, which is the quantity observable in the low-energy theory.

4)Genuine emergence of gravitational dynamics: In our framework, “emergence” refers to the fact that the action (38) contains no bare Einstein-Hilbert term; the entire gravitational dynamics--including the $\mathrm{R}$ term and its coefficient--arises from the entropy field's dynamics and its non-minimal coupling to curvature. The metric ${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$is not a fundamental dynamical field in the microscopic theory; it acquires its dynamics only after the entropy field fluctuations are integrated out and the effective action is derived.

The calculation below makes this hierarchy explicit, with the tree-level VEV contribution providing the leading-order induced Einstein-Hilbert action.

Microscopic origin of the VEV. In the companion paper [7], the entropy current ${\mathrm{s}}^{\mathsf{\mu }}$is shown to emerge from the entanglement structure of a string-net condensate. The key mechanism is as follows: the equivariant tensor renormalization group (ETRG) flow drives the string-net condensate to a critical point—a quantum phase transition where geometric order condenses. At this fixed point, the categorical symmetry is spontaneously broken, and a set of local operators--the frame field operators ${\mathrm{ê}}_{\mathsf{\mu }}^{\mathrm{a}}\left(\mathrm{x}\right)$--acquire a non-zero vacuum expectation value:

${\mathrm{ê}}_{\mathsf{\mu }}^{\mathrm{a}}\left(\mathrm{x}\right)>={\mathrm{e}}_{\mathsf{\mu }}^{\mathrm{a}}\left(\mathrm{x}\right)\ne 0$

Consequently, the metric emerges as the bilinear condensate:

${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}\left(\mathrm{x}\right)={\mathsf{\eta }}_{\mathrm{a}}\mathrm{b}<{\mathrm{ê}}_{\mathsf{\mu }}^{\mathrm{a}}\left(\mathrm{x}\right){\mathrm{ê}}_{\mathsf{\nu }}^{\mathrm{b}}\left(\mathrm{x}\right)>$

The entropy current VEV $<{\mathrm{s}}^{\mathsf{\mu }}>\ne 0$ follows directly from the same condensation mechanism: it is the expectation value of the microscopic entropy current operator in the ordered phase. Thus, in our framework, the VEV of ${\mathrm{s}}^{\mathsf{\mu }}$ is not a postulated constant but a derived consequence of the geometric phase transition in the underlying string-net condensate. A complete derivation is provided in the companion paper [7, Appendix A.3.5 and A.4].

In Section 4, we constructed a classical Proca-type theory for the entropy current ${\mathrm{s}}_{\mathsf{\mu }}$ in curved spacetime, with action:

$\mathrm{S}[\mathrm{s},\mathrm{g},\mathsf{\psi }]=\int {\mathrm{d}}^4\mathrm{x}\sqrt{-\mathrm{g}}\left[\bar{\mathsf{\psi }}(\mathrm{i}{\mathsf{\gamma }}^{\mathsf{\mu }}{\nabla }_{\mathsf{\mu }}-\mathrm{m}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }})\mathsf{\psi }-{\displaystyle \frac{1}{4}}{\mathrm{F}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{F}}^{\mathsf{\mu }\mathsf{\nu }}+{\displaystyle \frac{1}{2}}{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}+\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\right]$

Crucially, we have not included the Einstein–Hilbert term ${\displaystyle \frac{1}{16\mathsf{\pi }\mathrm{G}}}\mathrm{R}$ as a fundamental ingredient. Instead, the metric ${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$ enters only as a background geometry that the entropy and fermion fields live on, and through the non-minimal coupling $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$. Our goal now is to demonstrate that gravitational dynamics -- including the Einstein–Hilbert action and the graviton -- emerges from the quantum fluctuations of the entropy field itself.

The key idea is as follows. Suppose the entropy field develops a non-zero vacuum expectation value (VEV) in some region of spacetime, either through a phase transition or as a result of non-equilibrium dynamics in the early universe:

$$⟨{\mathrm{s}}_{\mathsf{\mu }}\left(\mathrm{x}\right)⟩={\bar{\mathrm{s}}}_{\mathsf{\mu }}\left(\mathrm{x}\right)$$

(47)

We then decompose the field into its background value plus quantum fluctuations:

$${\mathrm{s}}_{\mathsf{\mu }}\left(\mathrm{x}\right)={\bar{\mathrm{s}}}_{\mathsf{\mu }}\left(\mathrm{x}\right)+\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}\left(\mathrm{x}\right)$$

(48)

and integrate out the fluctuations $\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}$ in the path integral. The resulting effective action for the background metric ${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$ will contain, among other terms, the Einstein–Hilbert action -- with Newton's constant determined by $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$. This is the central mechanism of emergent gravity in our framework.

This is a standard effective-field-theory procedure: the background metric is treated as a trial field whose self-consistency is established only after the entropy fluctuations are integrated out and the resulting Einstein equations are imposed. There is no logical circularity: the metric does not pre-exist as a fundamental dynamical field; it emerges as a collective degree of freedom in the effective action, with its dynamics determined by the entropy field's VEV and fluctuations. The background metric used in the intermediate steps is a computational device, not an independent physical entity.

5.2. Path Integral and the Emergence of the Einstein-Hilbert Action

We consider the Euclidean path integral over the entropy field fluctuations:

$${\mathrm{e}}^{-{\mathrm{S}}_{\mathrm{e}\mathrm{f}\mathrm{f}}[\mathrm{g},\bar{\mathrm{s}},\mathsf{\psi }]}=\int \mathcal{D}\left[\mathsf{\delta }\mathrm{s}\right]{\mathrm{e}}^{-\mathrm{S}[\bar{\mathrm{s}}+\mathsf{\delta }\mathrm{s},\mathrm{g},\mathsf{\psi }]}$$

(49)

To leading order in the saddle-point expansion (i.e., Gaussian fluctuations around the background), we write the action as:

$$\mathrm{S}=\mathrm{S}[\bar{\mathrm{s}},\mathrm{g},\mathsf{\psi }]+{\displaystyle \frac{1}{2}}\int {\mathrm{d}}^4\mathrm{x}\sqrt{-\mathrm{g}}\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}{\mathcal{M}}^{\mathsf{\mu }\mathsf{\nu }}\mathsf{\delta }{\mathrm{s}}_{\mathsf{\nu }}+\mathcal{O}\left(\mathsf{\delta }{\mathrm{s}}^3\right)$$

(50)

where ${\mathcal{M}}^{\mathsf{\mu }\mathsf{\nu }}$ is the fluctuation operator. The Gaussian path integral yields the determinant of $\mathcal{M}$, which can be expressed as a local effective action using standard heat-kernel or Schwinger-DeWitt techniques [29,30]. The leading curvature-dependent contributions are:

$${\mathrm{S}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left[\mathrm{g}\right]=\int {\mathrm{d}}^4\mathrm{x}\sqrt{-\mathrm{g}}\left[\mathsf{\alpha }\left(\mathrm{R}\right)\mathrm{R}+\mathsf{\beta }\left(\mathrm{R}\right){\mathrm{R}}^2+\mathsf{\gamma }\left(\mathrm{R}\right){\mathrm{R}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{R}}^{\mathsf{\mu }\mathsf{\nu }}+\dots \right]$$

(51)

where the ellipsis denotes higher-curvature terms. In the long-wavelength (low-energy) limit, the leading term dominates. To leading order in the saddle-point expansion, the coefficient $\mathsf{\alpha }$ of the induced curvature term is given by the tree-level VEV contribution:

$$\mathsf{\alpha }=\mathsf{\xi }⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩+\mathcal{O}\left({\mathrm{m}}_{\mathrm{s}}^2\right)$$

(52)

where $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$ is the VEV of the entropy field squared. (For a detailed derivation, see Appendix A) The induced Einstein-Hilbert action therefore takes the form:

$${\mathrm{S}}_{\mathrm{E}\mathrm{H}}^{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}}={\displaystyle \frac{1}{16\mathsf{\pi }\mathrm{G}}}\int {\mathrm{d}}^4\mathrm{x}\sqrt{-\mathrm{g}}\mathrm{R}$$

(53)

Induced gravity interpretation. The mechanism described above is precisely the induced gravity scenario originally proposed by Sakharov [31] and generalized to vector condensates: the Einstein-Hilbert action is not fundamental but is induced by the vacuum expectation value of a non-minimally coupled field. In our framework, the “inducing” field is the entropy current ${\mathrm{s}}_{\mathsf{\mu }}$, whose VEV $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$ sets the scale of the effective Newton constant via Eq. (54). We emphasize that this is not merely an analogy: the microscopic origin of the VEV is traced back to the entanglement structure of the string-net condensate in the companion paper [7], and the non-minimal coupling $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ itself arises as the leading curvature-dependent term in the low-energy effective action of that condensate. Thus, while the gravitational dynamics is induced at the level of the effective action, the underlying mechanism is rooted in the quantum information structure of the string-net ground state.

with Newton's constant determined by the entropy field's VEV as:

$${\displaystyle \frac{1}{16\mathsf{\pi }\mathrm{G}}}=-\mathsf{\xi }⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$$

(54)

The positive sign of ${\mathrm{G}}_{\mathrm{N}}$ is verified by Wick rotation: the Euclidean term $+\left|\mathsf{\xi }\right|⟨\mathrm{s}²⟩\int \mathrm{d}⁴\mathrm{x}\sqrt{\mathrm{g}}\mathrm{R}$ yields the Lorentzian action $+\left|\mathsf{\xi }\right|⟨\mathrm{s}²⟩\int \mathrm{d}⁴\mathrm{x}\sqrt{\mathrm{g}}\mathrm{R}$, corresponding to ${\mathrm{G}}_{\mathrm{N}}=\left(16\mathsf{\pi }\right|\mathsf{\xi }|⟨\mathrm{s}²⟩)⁻¹>0$ for time-like ${\mathrm{s}}_{\mathsf{\mu }}$ and $\mathsf{\xi }<0$.

One-loop corrections would introduce factors of order $1/\left(16{\mathsf{\pi }}^2\right)$, which are parametrically suppressed relative to the tree-level VEV contribution in the macroscopic regime.

For a time-like entropy VEV with $⟨{\mathrm{s}}_0^2⟩>0$ and $\mathsf{\xi }<0$, this yields a positive Newton constant. This is the convention adopted throughout this work.

The emergence mechanism described above is formulated as a low-energy effective theory. It is strictly valid when $\left|\mathrm{R}\right|\ll {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}^2$, where ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}$ is the stability scale introduced in Section 4.4.1. In this weak-curvature regime, the tree-level VEV contribution unambiguously yields $\mathrm{G}>0$ for $\mathsf{\xi }<0$ .

This is the central result of our emergence mechanism: the gravitational interaction strength is not a fundamental constant but is entirely determined by the macroscopic quantum state of the entropy field — specifically, by its vacuum expectation value $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$ and the non-minimal coupling $\mathsf{\xi }$.

Connection to the effective field theory cutoff. The emergence mechanism described above also clarifies the origin of the EFT cutoff scale ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{T}\mathrm{T}}$ introduced in Section 2.1. From equation (54), the Newton constant is determined by the entropy field VEV as $({\mathrm{G}}_{\mathrm{N}}=(16\mathsf{\pi }\left|\mathsf{\xi }\right|⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩{)}^{-1})$. Defining the characteristic energy scale of the entropy field as ${\mathrm{E}}_{\mathrm{s}}\equiv ⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}{⟩}^{1/6}$, we have ${\mathrm{G}}_{\mathrm{N}}=\left(16\mathsf{\pi }\right|\mathsf{\xi }|{\mathrm{E}}_{\mathrm{s}}^6{)}^{-1}$. Comparing this with the standard EFT relation $\mathsf{\lambda }\sim 1/{\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}^2$ and noting that the non-minimal coupling $\mathsf{\xi }$ has mass dimension $-4$ in natural units (see Section 4.2), we identify:

${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}\left({\mathrm{G}}_{\mathrm{N}}\right|\mathsf{\xi }|{)}^{-1/2}~{\mathrm{E}}_{\mathrm{s}}$

where ${\mathrm{E}}_{\mathrm{s}}\equiv ⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}{⟩}^{1/6}$ is the entropy field VEV. This identification is satisfying: the same energy scale ${\mathrm{E}}_{\mathrm{s}}$ that determines the strength of gravity also sets the cutoff scale for the entropy-fermion effective field theory. With ${\mathrm{E}}_{\mathrm{s}}\sim 10^2-10^3$ GeV inferred from the STAR data (Section 6.1), the natural EFT expectation becomes $\mathsf{\lambda }\sim 1/{\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}^2\sim 10^{-7}-10^{-5}{\mathrm{G}\mathrm{e}\mathrm{V}}^{-2}$, fully consistent with the value independently extracted from experiment. The scale ${\mathrm{E}}_{\mathrm{s}}$ thus simultaneously explains：1) why $\mathsf{\lambda }$ is of order $10^{-7}-10^{-5}{\mathrm{G}\mathrm{e}\mathrm{V}}^{-2}$ rather than Planck-suppressed；2) why the observed gravitational coupling ${\mathrm{G}}_{\mathrm{N}}$ has its measured value；and 3) why the entropy-induced spin polarization is observable in heavy-ion collisions.

5.3. Linearized Theory and the Graviton Propagator

The induced action (53) endows ${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$ with genuine dynamics. Expanding around flat spacetime, expand the metric around a flat background:

$${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}={\mathsf{\eta }}_{\mathsf{\mu }\mathsf{\nu }}+{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }},\left|{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}\right|\ll 1$$

(55)

Substituting (55) into (53) and expanding to quadratic order in ${\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}$, we obtain the standard Fierz-Pauli action for a massive spin-2 field:

$${\mathrm{S}}_{\mathrm{F}\mathrm{P}}^{\left(2\right)}=\int {\mathrm{d}}^4\mathrm{x}\left[-{\displaystyle \frac{1}{2}}{\partial }_{\mathsf{\sigma }}{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}{\partial }^{\mathsf{\sigma }}{\mathrm{h}}^{\mathsf{\mu }\mathsf{\nu }}+{\partial }_{\mathsf{\mu }}{\mathrm{h}}^{\mathsf{\mu }\mathsf{\nu }}{\partial }_{\mathsf{\nu }}\mathrm{h}-{\displaystyle \frac{1}{2}}{\partial }_{\mathsf{\mu }}\mathrm{h}{\partial }^{\mathsf{\mu }}\mathrm{h}-{\displaystyle \frac{1}{2}}{\mathrm{m}}_{\mathrm{s}}^2\left({\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{h}}^{\mathsf{\mu }\mathsf{\nu }}-{\mathrm{h}}^2\right)\right]$$

(56)

where $\mathrm{h}\equiv {\mathsf{\eta }}^{\mathsf{\mu }\mathsf{\nu }}{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}$. The mass term arises from the entropy field mass ${\mathrm{m}}_{\mathrm{s}}$ that we introduced in the Proca action (37).

Origin of the graviton mass. The mass parameter in the Fierz-Pauli action (56) is inherited directly from the Proca mass of the entropy field. To see this, note that the metric fluctuation ${\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}$ emerges as a bilinear condensate of the frame field operators, ${\mathrm{g}}_{\mathsf{\mu }}\mathsf{\nu }={\mathsf{\eta }}_{\mathrm{a}}\mathrm{b}⟨{\mathrm{ê}}_{\mathsf{\mu }}^{\mathrm{a}}{\mathrm{ê}}_{\mathsf{\nu }}^{\mathrm{b}}⟩$ [7]. The mass term for ${\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}$ arises from expanding the entropy field's Proca mass term $1/2{\mathrm{m}}_{\mathrm{s}}²{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ to quadratic order in the fluctuations around the VEV, with the frame-field operators providing the mapping from the entropy sector to the metric sector. The resulting coefficient of ${\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}{\mathrm{h}}^{\mathsf{\mu }\mathsf{\nu }}$is precisely ${\mathrm{m}}_{\mathrm{s}}²$, with no additional numerical factors at tree level. Thus, ${\mathrm{m}}_{\mathrm{g}}={\mathrm{m}}_{\mathrm{s}}$ is not a parameter identification but a derived consequence of the VEV-induced metric and the Proca mass of the parent entropy field. The detailed derivation is provided in the companion paper [7, Appendix A.4].

In the transverse-traceless (TT) gauge:

$${\partial }^{\mathsf{\mu }}{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{T}\mathrm{T}}=0,{\mathsf{\eta }}^{\mathsf{\mu }\mathsf{\nu }}{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{T}\mathrm{T}}=0$$

(57)

the action simplifies to:

$${\mathrm{S}}_{\mathrm{F}\mathrm{P}}^{\left(2\right)}=\int {\mathrm{d}}^4\mathrm{x}\left[-{\displaystyle \frac{1}{2}}{\partial }_{\mathsf{\sigma }}{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{T}\mathrm{T}}{\partial }^{\mathsf{\sigma }}{\mathrm{h}}^{\mathrm{T}\mathrm{T},\mathsf{\mu }\mathsf{\nu }}-{\displaystyle \frac{1}{2}}{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{T}\mathrm{T}}{\mathrm{h}}^{\mathrm{T}\mathrm{T},\mathsf{\mu }\mathsf{\nu }}\right]$$

(58)

The equation of motion is:

$$(\square -{\mathrm{m}}_{\mathrm{s}}^2){\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{T}\mathrm{T}}=0$$

(59)

This is precisely the equation for a massive spin-2 particle with mass ${\mathrm{m}}_{\mathrm{s}}$--the emergent graviton. Its propagator in momentum space is:

$${\mathcal{D}}_{\mathsf{\mu }\mathsf{\nu },\mathsf{\alpha }\mathsf{\beta }}\left(\mathrm{k}\right)={\displaystyle \frac{\mathrm{i}}{{\mathrm{k}}^2-{\mathrm{m}}_{\mathrm{s}}^2+\mathrm{i}\mathsf{ϵ}}}\left[{\displaystyle \frac{1}{2}}({\mathsf{\eta }}_{\mathsf{\mu }\mathsf{\alpha }}{\mathsf{\eta }}_{\mathsf{\nu }\mathsf{\beta }}+{\mathsf{\eta }}_{\mathsf{\mu }\mathsf{\beta }}{\mathsf{\eta }}_{\mathsf{\nu }\mathsf{\alpha }})-{\displaystyle \frac{1}{2}}{\mathsf{\eta }}_{\mathsf{\mu }\mathsf{\nu }}{\mathsf{\eta }}_{\mathsf{\alpha }\mathsf{\beta }}\right]$$

(60)

In the limit ${\mathrm{m}}_{\mathrm{s}}\to 0$, the propagator (60) formally reduces to that of the massless graviton of general relativity, as a consequence of the Fierz–Pauli tuning in (56). However, we emphasize that this formal limit must be interpreted with caution. As is well known from the study of massive gravity, the ${\mathrm{m}}_{\mathrm{s}}\to 0$ limit of a Fierz–Pauli massive spin-2 field does not smoothly reproduce all predictions of general relativity due to the van Dam–Veltman–Zakharov discontinuity [32,33]: the longitudinal (helicity-0) mode continues to couple to the trace of the energy-momentum tensor, leading to predictions for gravitational phenomena that differ from those of GR even in the limit of arbitrarily small ${\mathrm{m}}_{\mathrm{s}}$.

In a complete non-linear theory, this discontinuity is resolved by the Vainshtein mechanism, wherein the helicity-0 mode is screened below a certain Vainshtein radius, effectively decoupling from matter and restoring the predictions of GR in the weak-field, high-density limit. The exploration of such non-linear effects lies beyond the scope of the present work; the linearized analysis presented here captures the leading-order gravitational-wave phenomenology, for which the TT-gauge propagator (60) is the appropriate effective description. The precise recovery of GR in the full non-linear limit would require a complete UV completion of the emergent gravity framework, which is left for future investigation.

Before proceeding to the observational constraints, we address a potential subtlety concerning the physical degrees of freedom of the Proca field. This is the subject of the next subsection.

5.4. Mode Decomposition and Decoupling of Extra Polarizations

A note on the effective nature of the graviton analysis. The analysis in this section treats the emergent graviton as a low-energy effective degree of freedom within the induced gravity framework established in Section 5.1, Section 5.2 and Section 5.3. The five degrees of freedom--two helicity ±2, two helicity ±1, and one helicity 0--are those of a standard Fierz-Pauli massive spin-2 field in four dimensions. The following mode decomposition demonstrates that the non-TT modes do not contribute to long-range gravitational forces and are phenomenologically irrelevant for gravitational-wave observations. A complete treatment of the full non-linear theory, including the fate of the Boulware-Deser ghost, lies beyond the scope of this work and is left for future investigation.

The Proca field ${\mathrm{s}}_{\mathsf{\mu }}$ in the action (38) carries three physical polarizations: one longitudinal scalar mode and two transverse vector modes. In contrast, the graviton ${\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}$is a spin-2 tensor with only two transverse-traceless polarizations. To establish a consistent mapping from the entropy field fluctuations to emergent gravity, we must demonstrate that the non-tensor modes do not contribute to long-range gravitational forces and decouple from the low-energy effective theory.

Decomposition in the Background

We decompose the fluctuation $\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}$ with respect to the background VEV ${\bar{\mathrm{s}}}_{\mathsf{\mu }}=({\bar{\mathrm{s}}}_0,0)$. In$3+1$ form:

$$\mathsf{\delta }{\mathrm{s}}_0=\mathsf{\varphi },\mathsf{\delta }\mathbf{s}=\nabla \mathsf{\chi }+{\mathbf{s}}_{\mathrm{T}},\nabla \cdot {\mathbf{s}}_{\mathrm{T}}=0$$

(61)

where $\mathsf{\varphi }$ is the scalar time component, $\mathsf{\chi }$ is the longitudinal scalar potential, and ${\mathbf{s}}_{\mathrm{T}}$ is the transverse vector mode.

Vector Mode: Decoupling via Mass Gap

Substituting the decomposition into the quadratic action derived from (38), the transverse vector mode ${\mathbf{s}}_{\mathrm{T}}$ acquires the Lagrangian (in momentum space with 4-momentum ${\mathrm{k}}^{\mathsf{\mu }}$):

$${\mathcal{L}}_{\mathrm{v}\mathrm{e}\mathrm{c}}\sim -{\displaystyle \frac{1}{2}}{\mathbf{s}}_{\mathrm{T}}^{\mathrm{*}}({\mathrm{k}}^2-{\mathrm{m}}_{\mathrm{s}}^2-2\mathsf{\xi }\mathrm{R}){\mathbf{s}}_{\mathrm{T}}$$

(62)

The vector mode couples to fermions through the interaction term $\mathsf{\lambda }\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\mu }}\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}\mathsf{\psi }$. However, the Standard Model fermion current ${\mathrm{j}}^{\mathsf{\mu }}\equiv \bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\mu }}\mathsf{\psi }$ is covariantly conserved, ${\nabla }_{\mathsf{\mu }}{\mathrm{j}}^{\mathsf{\mu }}=0$. Consequently, the transverse vector mode ${\mathbf{s}}_{\mathrm{T}}$, which satisfies $\nabla \cdot {\mathbf{s}}_{\mathrm{T}}=0$, does not couple to the longitudinal part of the current. In the low-energy limit ${\mathrm{k}}^2\ll {\mathrm{m}}_{\mathrm{s}}^2$, this mode acquires a mass gap ${\mathrm{m}}_{\mathrm{s}}$ and its propagator is suppressed by $1/{\mathrm{m}}_{\mathrm{s}}^2$. It therefore does not mediate long-range forces and decouples from the low-energy gravitational sector.

Degrees of freedom and ghost freedom.

The Fierz-Pauli mass term in (56) is of the special form $-1/2{\mathrm{m}}_{\mathrm{s}}²({\mathrm{h}}_{\mathsf{\mu }}\mathsf{\nu }{\mathrm{h}}^{\mathsf{\mu }}\mathsf{\nu }-\mathrm{h}²)$, which is precisely tuned to eliminate the ghost mode that would otherwise plague a generic massive spin-2 theory. As a result, the theory propagates five physical degrees of freedom: two helicity ±2 modes, two helicity ±1 modes (which are inherited from the Proca field's transverse vector modes), and one helicity 0 mode. The helicity ±1 and helicity 0 modes are either projected out in the TT gauge or suppressed by the mass gap ${\mathrm{m}}_{\mathrm{s}}$ in the low-energy limit, as discussed above. No additional ghost or gradient instabilities arise because the Fierz-Pauli tuning ensures that the kinetic terms for all modes have the correct sign. This is a standard result of massive gravity [34], and it applies directly to our emergent graviton sector.

Correlation length from the propagator.

To explicitly demonstrate that ${\mathrm{m}}_{\mathrm{s}}^{-1}$is the physical correlation length of the entropy field, we compute the position-space two-point function for the transverse vector mode in the flat-spacetime limit ($\mathrm{R}\to 0$). From the momentum-space propagator implied by (62),

$⟨{\mathrm{s}}_{\mathrm{T}}^{\mathrm{i}}\left(\mathrm{x}\right){\mathrm{s}}_{\mathrm{T}}^{\mathrm{j}}\left(0\right)⟩=\int {\mathrm{d}}^4\mathrm{k}/\left(2\mathsf{\pi }{)}^4{\mathrm{e}}^{\mathrm{i}\mathrm{k}·\mathrm{x}}\right[\mathrm{i}{\mathsf{\delta }}^{\mathrm{i}\mathrm{j}}/({\mathrm{k}}^2-{\mathrm{m}}_{\mathrm{s}}^2+\mathrm{i}\mathsf{\epsilon })]$

we obtain, for spacelike separations $\mathrm{r}=\left|\mathrm{x}\right|,$

$⟨{\mathrm{s}}_{\mathrm{T}}^{\mathrm{i}}\left(\mathrm{x}\right){\mathrm{s}}_{\mathrm{T}}^{\mathrm{j}}\left(0\right)⟩{\mathsf{\delta }}^{\mathrm{i}\mathrm{j}}({\mathrm{m}}_{\mathrm{s}}/4{\mathsf{\pi }}^2\mathrm{r}){\mathrm{e}}^{-{\mathrm{m}}_{\mathrm{s}}\mathrm{r}}$ (as $\mathrm{r}\to \mathrm{\infty }$)

up to an $\mathrm{O}\left(1\right)$ numerical factor. This explicitly demonstrates that the two-point function decays exponentially with distance, with the decay scale set by ${\mathrm{m}}_{\mathrm{s}}^{-1}$. Thus, the inverse mass ${\mathrm{m}}_{\mathrm{s}}^{-1}$ is indeed the correlation length of the entropy flow field: ${\mathsf{\xi }}_{\mathrm{c}}\equiv 1/{\mathrm{m}}_{\mathrm{s}}$. We emphasize that this identification is a derived consequence of the Proca EFT, not an independent assumption.

Scalar Modes: Decoupling via$\mathsf{\lambda }$Suppression

The scalar sector (φ, χ) is mixed through the kinetic term. Diagonalizing the quadratic Lagrangian yields a single physical scalar degree of freedom (the "entropic phonon") with mass eigenvalue:

$${\mathrm{m}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}^2={\mathrm{m}}_{\mathrm{s}}^2+2\mathsf{\xi }\mathrm{R}-{\mathsf{\lambda }}^2⟨\bar{\mathsf{\psi }}\mathsf{\psi }⟩/\mathrm{m}$$

(63)

where $⟨\bar{\mathsf{\psi }}\mathsf{\psi }⟩$is the fermion condensate in the background (e.g., the QCD quark condensate $⟨\bar{\mathrm{q}}\mathrm{q}⟩$ in hadronic applications), and the negative sign indicates that fermion condensation provides an effective attractive interaction that reduces the scalar mass.

The scalar mode couples to matter with a strength proportional to λ, but its phenomenological impact is controlled by two factors: the mass gap ${\mathrm{m}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}$ and the smallness of λ. The resulting Yukawa potential is:

$${\mathrm{V}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}\left(\mathrm{r}\right)\sim ({\mathsf{\lambda }}^2/4\mathsf{\pi }\mathrm{r}){\mathrm{e}}^{-{\mathrm{m}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}\mathrm{r}}$$

(64)

For a light scalar with ${\mathrm{m}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}\lesssim 10^{-3}$eV, the force range exceeds macroscopic scales and is subject to fifth-force constraints.The observational bounds from torsion-balance experiments (e.g., Adelberger et al. [35]) require the dimensionless scalar coupling to satisfy for ${\mathrm{m}}_{\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}}\lesssim 10^{-3}$eV:

${\displaystyle \frac{{\mathsf{\lambda }}^2{\mathrm{m}}_{\mathrm{r}\mathrm{e}\mathrm{f}}^4}{4\mathsf{\pi }}}\lesssim 10^{-10}$

5.5. Gravitational-Wave Dispersion and Observational Constraints

The massive graviton dispersion relation follows directly from (59):

$${\mathrm{E}}^2={\mathbf{p}}^2{\mathrm{c}}^2+{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{c}}^4$$

(65)

The corresponding group velocity is:

$${\mathrm{v}}_{\mathrm{g}}\left(\mathrm{E}\right)=\mathrm{c}\sqrt{1-{\displaystyle \frac{{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{c}}^4}{{\mathrm{E}}^2}}}\simeq \mathrm{c}\left(1-{\displaystyle \frac{{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{c}}^4}{2{\mathrm{E}}^2}}\right)$$

(66)

This frequency-dependent velocity is a key observable signature of our emergent gravity framework.

The LIGO/Virgo collaboration's observation of GW170817, together with its electromagnetic counterpart GRB 170817A, constrained the difference between the speed of gravity and the speed of light to [36]:

$$\left|{\displaystyle \frac{{\mathrm{v}}_{\mathrm{g}}-\mathrm{c}}{\mathrm{c}}}\right|\lesssim 10^{-15}$$

(67)

Using the GW170817 data [36], the LIGO/Virgo constraint on the speed of gravity, $|{\mathrm{v}}_{\mathrm{g}}-\mathrm{c}|/\mathrm{c}\lesssim 10^{-15}$, can be translated into an upper bound on the graviton mass via (66). For this specific event, the characteristic frequency is $\mathrm{f}\sim 100$ Hz, corresponding to $\mathrm{E}\sim 4\times 10^{-13}$eV. Combining (66) with (67) yields:

$${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}\mathrm{e}\mathrm{V}$$

(68)

The bound derived below is event-specific: it applies to the GW170817 event and depends on the inferred source parameters, the frequency band probed, and the assumption that the electromagnetic counterpart GRB 170817A provides a reliable reference for the speed of light.We further emphasize that this is a data-dependent bound: it derives from a particular observation (GW170817) and relies on the inferred source parameters and the presence of the electromagnetic counterpart. Different events, or improved measurements of the gravitational-wave speed, will yield different bounds. The numerical value should therefore be understood as an order-of-magnitude constraint based on current data, not as a universal upper limit on the graviton mass.

This bound is remarkably close to the mass scale associated with ultra-light dark matter candidates such as fuzzy dark matter ($\mathrm{m}\sim 10^{-22}$eV) and to the characteristic energy scale of cosmological constant (${\mathsf{\Lambda }}^{1/4}\sim 10^{-3}$eV). The resulting ${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}$eV is an order-of-magnitude constraint; more precise bounds depend on the specific gravitational-wave event and the frequency band probed.

In the strong-field regime near black holes or neutron stars, the curvature-dependent effective mass (45) may become significant:

$${\mathrm{m}}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\left(\mathrm{s}\right)}=\sqrt{{\mathrm{m}}_{\mathrm{s}}^2+2\mathsf{\xi }\mathrm{R}}$$

(69)

This predicts that gravitational waves emitted from strong-field regions may exhibit anomalous dispersion due to the effective graviton mass induced by curvature -- a potential target for future gravitational-wave observatories such as LISA or the Einstein Telescope [37].

We note that the graviton mass bound ${\mathrm{m}}_{\mathrm{s}}\lesssim {10}^{-20}\mathrm{e}\mathrm{V}$ obtained from GW170817 constrains the bare mass parameter in the flat-spacetime limit ($\mathrm{R}\to 0$). This is an intrinsically infrared scale, characterizing the correlation length of the entropy field in the present low-curvature cosmological epoch. It should be distinguished from the curvature-dependent effective mass ${\mathrm{m}}_{\mathrm{s}}^{\mathrm{e}\mathrm{f}\mathrm{f}}$of equation (45), which can be significantly larger in strong-field environments such as near black hole horizons. The separation of these two concepts --bare mass as an IR scale, effective mass as a curvature-induced enhancement --is essential for understanding how the same framework can simultaneously accommodate the near-massless graviton observed by LIGO/Virgo and potentially significant entropy-field effects in high-curvature regions.

5.6. The Generalized Second Law as the Origin of Spacetime Causality

The emergence mechanism presented above suggests a thermodynamic foundation for the causal structure of spacetime, although the identification of σ with the spectral density of fluctuations remains conjectural at this stage (see Section 4.5):

${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$ [(46)]

is not merely a constraint on the entropy field; it is the fundamental arrow of time that drives the emergence of geometry itself. The heat-kernel expansion leading to (52) involves integrating over fluctuations $\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}$ with weight ${\mathrm{e}}^{-\mathrm{S}}$. For $\mathsf{\sigma }>0$, the spectral density of fluctuations is modified, effectively sourcing the curvature terms in (51). Thus, entropy production -- the irreversible flow of quantum information -- is what gives rise to the gravitational action in the first place.

In the language of Jacobson's thermodynamic gravity [5], the Einstein equation emerges from the Clausius relation $\mathsf{\delta }\mathrm{Q}=\mathrm{T}\mathrm{d}\mathrm{S}$ applied to local Rindler horizons. Our framework provides a specific microscopic realization: the entropy current ${\mathrm{s}}^{\mathsf{\mu }}$ plays the role of the local entropy flux across causal horizons, and its quantum fluctuations generate the Einstein-Hilbert action. The generalized second law ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\ge 0$ ensures that this emergence is a one-way, irreversible process--consistent with the thermodynamic arrow of time.

5.7. Summary of the Emergence Mechanism

We summarize the chain of emergence as follows:

The emergence chain can be summarized as follows:

(i) Quantum information flow --non-minimal coupling--> Curvature coupling

(${\mathrm{s}}^{\mathsf{\mu }}$ as a Proca field) ($\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$)

--VEV <${\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$>--> Einstein-Hilbert action

($1/\left(16\mathsf{\pi }\mathrm{G}\right)=-\mathsf{\xi }⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$)

(ii) Quantum fluctuations --linearization--> Massive spin-2 excitation

($\mathsf{\delta }{\mathrm{s}}_{\mathsf{\mu }}$) ($\square -{\mathrm{m}}_{\mathrm{s}}^2){\mathrm{h}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{T}\mathrm{T}}=0$--${\mathrm{m}}_{\mathrm{s}}\to 0$--> Massless graviton (GR limit)

(iii) Entropy production --spectral density--> Spacetime causality

(${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$) and thermodynamic arrow

The key results of this section are:

(i) Gravity is emergent: The Einstein-Hilbert action is not fundamental but is induced by the vacuum expectation value of the entropy field via the non-minimal coupling $\mathsf{\xi }\mathrm{R}{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$. The leading contribution arises at tree level from the VEV $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$, with quantum fluctuations providing subdominant corrections. We use the term “emergent” in the sense that no bare Einstein-Hilbert term is present in the microscopic action; the entire gravitational dynamics arises from the entropy field's dynamics and its coupling to curvature.

(ii) Newton's constant is determined microscopically : $1/\left(16\mathsf{\pi }\mathrm{G}\right)=-\mathsf{\xi }⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$ [(54)]

(iii)The graviton has a mass: Dispersion ${\mathrm{E}}^2={\mathbf{p}}^2+{\mathrm{m}}_{\mathrm{s}}^2$ [(65)], with ${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}$eV (based on constraints from the GW170817 event) [(66)].

(iv)The generalized second law drives emergence: ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$ [(46)] provides the thermodynamic arrow for geometry.

This completes the unification: quantum information flow (entropy current) $\to$ modified Dirac equation for fermions $\to$ Proca dynamics for the entropy field $\to$ emergent Einstein gravity and the graviton.

6. Physical Implications and Testable Predictions

6.1. Spin Polarization in Heavy-Ion Collisions:An Illustrative Estimate

In relativistic heavy-ion collisions, the produced quark-gluon plasma (QGP) exhibits significant vorticity [21,22]. The entropy current in such a rotating system has $\nabla \times \mathbf{s}=2{\mathsf{\rho }}_{\mathrm{s}}\mathsf{\omega }$, where ${\mathsf{\rho }}_{\mathrm{s}}$ is the entropy density in the fluid rest frame (see Section 3.4). Substituting into (33), the effective magnetic field is:

${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}={\displaystyle \frac{2\mathsf{\lambda }{\mathsf{\rho }}_{\mathrm{s}}}{\mathrm{m}}}\mathsf{\omega }$

This predicts a spin polarization of produced hyperons proportional to the system vorticity, with a coefficient modified by the entropy coupling $\mathsf{\lambda }$. The STAR experiment has observed global spin polarization of $\mathsf{\Lambda }$ hyperons [23,24], providing a potential testing ground for our framework.

We emphasize that the following estimate is illustrative, not a direct experimental measurement of λ. The STAR data measure the global spin polarization ${\mathrm{P}}_{\mathsf{\Lambda }}$; extracting λ requires a model for the entropy density, vorticity, and fireball lifetime in the QGP. The result is therefore a phenomenological estimate that depends on these model assumptions. Moreover, the measured polarization receives contributions from conventional thermal-vorticity and electromagnetic effects; our framework treats the entropy-induced contribution as an additional component, and the quoted λ range represents the value that would be required to account for the observed polarization assuming all other contributions are negligible or separately subtracted. A full joint fit including all known contributions and their uncertainties is left for future work. With these caveats, the estimate proceeds as follows.

Quantitative estimate of λ from STAR data. The STAR collaboration has measured global spin polarization of Λ hyperons in Au+Au collisions at $\sqrt{{\mathrm{s}}_{\mathrm{N}\mathrm{N}}}=200\mathrm{G}\mathrm{e}\mathrm{V}$, with ${\mathrm{P}}_{\mathsf{\Lambda }}\sim 0.1\mathrm{\%}-1\mathrm{\%}$[23,24]. In our framework, the polarization induced by entropy-flow spin precession is:

$\mathrm{P}\sim {\mathsf{\omega }}_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}·{\mathsf{\tau }}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{e}}=(\mathsf{\lambda }/{\mathrm{m}}_{\mathsf{\Lambda }})|\nabla \times \mathrm{s}|·{\mathsf{\tau }}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{e}}$

We adopt realistic QGP parameters from hydrodynamic simulations and experimental constraints:

(i) Entropy density: For central Au+Au collisions at $\sqrt{{\mathrm{s}}_{\mathrm{N}\mathrm{N}}}=200$GeV, the initial entropy density is ${\mathsf{\rho }}_{\mathrm{s}}\sim 10-30\mathrm{G}\mathrm{e}\mathrm{V}³$. This follows from the measured charged-particle multiplicity $\mathrm{d}{\mathrm{N}}_{\mathrm{c}\mathrm{h}}/\mathrm{d}\mathrm{y}$ 600–800 via the relation $\mathrm{s}\simeq 7.5\mathrm{d}{\mathrm{N}}_{\mathrm{c}\mathrm{h}}/\mathrm{d}\mathrm{y}$ for a thermalized QGP. We adopt ${\mathsf{\rho }}_{\mathrm{s}}\sim 20\mathrm{G}\mathrm{e}\mathrm{V}³$ as a representative value.

(ii) Vorticity: Hydrodynamic models predict ω ∼ 10²² s⁻¹, corresponding to ω ∼10 MeV in natural units. Thus $|\nabla \times \mathrm{s}|\sim 2{\mathsf{\rho }}_{\mathrm{s}}\mathsf{\omega }\sim 400\mathrm{M}\mathrm{e}\mathrm{V}⁴.$

(iii) Fireball lifetime: The QGP survives for ${\mathsf{\tau }}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{e}}\sim 8\mathrm{f}\mathrm{m}/\mathrm{c}$ before hadronization, corresponding to ${\mathsf{\tau }}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{e}}\sim 40\mathrm{M}\mathrm{e}\mathrm{V}⁻¹$ in natural units.

Substituting these values into the above expression:

$\mathsf{\lambda }\sim {\mathrm{P}}_{\mathsf{\Lambda }}{\mathrm{m}}_{\mathsf{\Lambda }}/\left(\right|\nabla \times \mathrm{s}\left|{\mathsf{\tau }}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{e}}\right)$

$\sim (5\times 10⁻³\times 1.116\mathrm{G}\mathrm{e}\mathrm{V})/(400\mathrm{M}\mathrm{e}\mathrm{V}⁴\times 40\mathrm{M}\mathrm{e}\mathrm{V}⁻¹)$∼${10}^{-6}\mathrm{G}\mathrm{e}\mathrm{V}⁻²$(70)

We emphasize that this is an order-of-magnitude estimate. To assess its robustness, we consider the plausible ranges of the key input parameters based on hydrodynamic simulations and experimental measurements for central Au+Au collisions at 200 GeV:

- Initial entropy density:$\mathrm{s}~50-200\mathrm{f}\mathrm{m}⁻³$, corresponding to $\mathrm{d}{\mathrm{N}}_{\mathrm{c}\mathrm{h}}/\mathrm{d}\mathrm{y}~500-1000.$

- Vorticity: $\mathsf{\omega }~0.5-2.0\times 10²²\mathrm{s}⁻¹$, or $\mathsf{\omega }~5-20\mathrm{M}\mathrm{e}\mathrm{V}$ in natural units.

- Fireball lifetime: $\mathsf{\tau }~5-15\mathrm{f}\mathrm{m}/\mathrm{c}$, or $\mathsf{\tau }~(2.5-7.5)\times 10⁻²³\mathrm{s}$.

Varying these parameters within their respective ranges while keeping the measured polarization within the experimentally observed range ${\mathrm{P}}_{\mathsf{\Lambda }}\sim 0.1\mathrm{\%}-1\mathrm{\%}$ [23,24], we obtain the following estimate for $\mathsf{\lambda }$ (taking a representative value ${\mathrm{P}}_{\mathsf{\Lambda }}\sim 0.5\mathrm{\%}$ for the central estimate):

$\mathsf{\lambda }~10⁻⁷-10⁻⁵\mathrm{G}\mathrm{e}\mathrm{V}⁻²$, corresponding to ${\mathrm{E}}_{\mathrm{s}}\equiv {\mathsf{\lambda }}^{-1/2}~10²-10³\mathrm{G}\mathrm{e}\mathrm{V}$ (71)

The central estimate λ ~ 10⁻⁶ GeV⁻² (${\mathrm{E}}_{\mathrm{s}}~10³$ GeV) reported above corresponds to $\mathrm{s}~100\mathrm{f}\mathrm{m}⁻³$, $\mathsf{\omega }~10\mathrm{M}\mathrm{e}\mathrm{V}$, and $\mathsf{\tau }~10\mathrm{f}\mathrm{m}/\mathrm{c}$--values well within the accepted ranges for the QGP phase. The uncertainty is dominated by the vorticity $\mathsf{\omega }$, which carries the largest model dependence and directly scales the extracted $\mathsf{\lambda }$. Improved vorticity determinations from hydrodynamic modeling with test particles or from measurements of vector meson spin alignment would tighten this estimate. At the present level of precision, the range $10²-10³\mathrm{G}\mathrm{e}\mathrm{V}$ for ${\mathrm{E}}_{\mathrm{s}}$ is robust, and the self-consistency with the EFT cutoff ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ (Section 2.1) holds across the full range.

The statistical significance of the entropy-induced polarization relative to the null hypothesis ($\mathsf{\lambda }=0$) remains at , independent of the parameter uncertainties discussed above.

A more robust method to extract λ independently of the unknown ${\mathsf{\rho }}_{\mathrm{s}}$ is to compare polarizations of different hyperon species (Λ, Ξ, Ω) that have different masses but sample the same vorticity field. In our framework, the entropy-induced polarization for a hyperon of mass ${\mathrm{m}}_{\mathrm{H}}$ is:

$${\mathrm{P}}_{\mathrm{H}}=2\mathsf{\lambda }{\mathsf{\rho }}_{\mathrm{s}}\mathsf{\omega }{\mathsf{\tau }}_{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{z}\mathrm{e}}/{\mathrm{m}}_{\mathrm{H}}$$

(72)

The ratio of polarizations of two species H₁ and H₂ is therefore:

$${\mathrm{P}}_{\mathrm{H}}₁/{\mathrm{P}}_{\mathrm{H}}₂={\mathrm{m}}_{\mathrm{H}}₂/{\mathrm{m}}_{\mathrm{H}}₁$$

(73)

This is a parameter-free prediction: heavier hyperons should exhibit smaller polarization, inversely proportional to mass. Using current STAR data [23,24], ${\mathrm{P}}_{\mathsf{\Xi }}/{\mathrm{P}}_{\mathsf{\Lambda }}\sim 0.47/0.6\sim 0.78$ and ${\mathrm{m}}_{\mathsf{\Lambda }}/{\mathrm{m}}_{\mathsf{\Omega }}\sim 0.667$. The apparent discrepancy, particularly for Ω, is dominated by large statistical uncertainties (${\mathrm{P}}_{\mathsf{\Omega }}=1.11\pm 0.87\pm 1.97\mathrm{\%}$). After correcting for decay feed-down effects, the mass-inverse scaling relation (72) provides a clean, calibration-free test of our entropy-flow mechanism.

Self-consistency check of the EFT cutoff. The value $\mathsf{\lambda }~10⁻⁷\mathrm{t}\mathrm{o}10⁻⁵\mathrm{G}\mathrm{e}\mathrm{V}⁻²$ extracted above corresponds, via the standard EFT relation $\mathsf{\lambda }~1/\mathsf{\Lambda }{²}_{\mathrm{E}\mathrm{F}\mathrm{T}}$, to a cutoff scale ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}~10²\mathrm{t}\mathrm{o}10³\mathrm{G}\mathrm{e}\mathrm{V}$. This is the same range as the characteristic entropy-field energy scale ${\mathrm{E}}_{\mathrm{s}}\equiv ⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}{⟩}^{1/6}~10²\mathrm{t}\mathrm{o}10³\mathrm{G}\mathrm{e}\mathrm{V}$ independently inferred from the STAR data. The consistency between these two determinations--one from the EFT assumption ${\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}~{\mathrm{E}}_{\mathrm{s}}$ (Section 2.1), the other from the experimental polarization data--provides phenomenological support for the framework under the assumed relation. It should not, however, be interpreted as an independent validation of the EFT cutoff relation, as both estimates derive from the same data and the same assumed scaling $\mathsf{\lambda }~1/{\mathsf{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}²$λ. It confirms that the entropy-fermion coupling is not anomalously large, but rather takes the value naturally expected for an effective field theory whose ultraviolet completion lies at the TeV scale.

6.2. Energy Shift in High-Entropy Regions

The dispersion relation (18) predicts that particles in high-entropy regions experience an energy shift:

$$\mathsf{\Delta }\mathrm{E}=-\mathsf{\lambda }{\mathrm{s}}^0\left(\mathbf{x}\right)\mathrm{c}$$

(74)

This could have observable consequences in:

- Black hole environments: In strong-curvature regions with non-negligible Ricci scalar (e.g., near black hole horizons with infalling matter or in the early universe), the entropy density varies rapidly, potentially affecting particle spectra.

- Neutron stars: The high-density, high-entropy interior of neutron stars may exhibit modified fermion dispersion relations.

-Early universe cosmology: During the QCD phase transition, entropy gradients could have influenced particle production and spectra.

6.3. Connection to Quantum Information

The coupling constant $\mathsf{\lambda }$ can be related to the quantum information content of the underlying theory. In the framework of quantum tensor networks [25], the entropy current emerges as the expectation value of a microscopic entropy operator:

$${\mathrm{s}}^{\mathsf{\mu }}=⟨{\widehat{\mathrm{s}}}^{\mathsf{\mu }}⟩$$

(75)

The coupling $\mathsf{\lambda }$ then encodes the response of the fermion field to quantum information flow, potentially linking to the quantum Fisher information [26,27].

7. Discussion

7.1. Relation to Jacobson's Thermodynamic Gravity

Jacobson’s seminal work [5] demonstrated that the Einstein equations can be derived from the proportionality of entropy and horizon area, together with the Clausius relation $\mathsf{\delta }\mathrm{Q}=\mathrm{T}\mathrm{d}\mathrm{S}$ applied to local Rindler horizons. That derivation treats spacetime thermodynamics at the horizon level, without invoking a dynamical entropy current.

The present framework shares the thermodynamic spirit of Jacobson’s approach--both view gravity as a thermodynamic phenomenon--but differs in its microphysical implementation. Rather than deriving gravitational dynamics from horizon thermodynamics, we promote the entropy current ${\mathrm{s}}_{\mathsf{\mu }}$ to a dynamical Proca field whose quantum fluctuations generate the Einstein-Hilbert action (Section 5). In this sense, the modified Dirac equation (2) is not a “quantum counterpart” of Jacobson’s derivation, but a complementary extension: it describes how fermionic matter responds to local entropy flow at the quantum level, a regime not addressed by the classical horizon-based approach.

7.2. Relation to Verlinde's Entropic Gravity

Verlinde’s entropic gravity proposal [6] interprets gravitational forces as entropic forces arising from entropy gradients across holographic screens. While our framework shares the conceptual premise that gravity has an entropic origin, the specific mechanisms are distinct. In Verlinde’s construction, gravity emerges from the statistical averaging over underlying microscopic degrees of freedom on a holographic screen, without a dynamical entropy field in the bulk.

In contrast, our framework introduces the entropy current ${\mathrm{s}}_{\mathsf{\mu }}$ as a genuine bulk field with Proca dynamics, whose coupling to fermions generates observable effects (e.g., the effective magnetic field (31) and spin precession). Both approaches can be viewed as different realizations of the broader “gravity from thermodynamics” paradigm: Verlinde’s is rooted in holography and information storage on screens, while ours is rooted in local quantum field theory with a dynamical entropy current. Whether these two perspectives can be unified--for instance, by deriving ${\mathrm{s}}_{\mathsf{\mu }}$ from the underlying holographic degrees of freedom--remains an open question for future investigation.

7.3. Relation to Other Emergent Gravity Approaches

Beyond Jacobson and Verlinde, the thermodynamic paradigm of gravity has been developed along several other lines. Carlip [38] demonstrated that black hole entropy can be accounted for by microscopic conformal field theory degrees of freedom near the horizon. Padmanabhan [39] showed that the gravitational field equations arise from extremizing an entropy functional associated with local Rindler horizons, treating spacetime as an elastic medium. The entanglement gravity program [40] has established that Einstein equations can emerge from the first law of entanglement entropy.

These approaches share with ours the premise that gravity encodes an underlying quantum information structure. The key distinction is that we introduce a dynamical entropy current field ${\mathrm{s}}_{\mathsf{\mu }}$with Proca dynamics, whose quantum fluctuations directly generate the Einstein-Hilbert action (Section 5). Rather than deriving gravity from horizon thermodynamics or entanglement, our mechanism operates through a local field that couples to matter and geometry simultaneously, providing a concrete microphysical pathway from quantum information flow to spacetime curvature.

Recently, Bianconi's independently developed Gravity from Entropy (GfE) theory [41] has established a thermodynamic framework in which gravity arises from the information encoded in the interplay between geometry and matter fields. Remarkably, GfE shares with our framework the core tenet that gravity is rooted in entropy/information rather than in geometric primitives. Crucially, GfE's entropy measure is local and volumetric rather than horizon-based [41], directly resonating with our treatment of the entropy current ${\mathrm{s}}^{\mathsf{\mu }}$ as a bulk dynamical field rather than as a holographic screen entropy. This stands in contrast to Jacobson's horizon thermodynamics and Verlinde's holographic-screen approach, pointing toward an emerging theoretical paradigm in which gravity is understood as a bulk thermodynamic phenomenon rooted in quantum information structure.

Despite this shared conceptual foundation, the roles of entropy in the two frameworks are distinct and complementary. In GfE, entropy appears as a scalar measure--the geometric quantum relative entropy between the physical metric and the metric induced by matter--and serves directly as the Lagrangian density. In our framework, by contrast, entropy is promoted to a covariant vector current that directly embodies the essence of the universe--expansion--as it encodes the directional flow of entanglement entropy in the emergent spacetime.This vectorial nature enables our framework to produce testable spin-coupling phenomena--specifically, the effective Zeeman-type field ${\mathrm{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathrm{s}$ and the associated spin precession ${\mathsf{\omega }}_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathrm{s}$--which are absent in scalar-entropy formulations.

Therefore, while GfE provides a profound thermodynamic foundation for why gravity is entropic in nature, our framework extends this paradigm by asking how entropy flows directionally and how such flow competes with enthalpy (the thermodynamic potential that drives aggregation and structure formation) to drive the emergence of ordered structures from quantum systems. In our Quantum Enthalpy-Entropy Theory (QEET) framework, gravity emerges precisely from the competition between entropy and enthalpy--entropy drives diffusion and disorder, while enthalpy drives aggregation and order; their dynamic balance shapes the entire picture from microscopic quantum correlations to macroscopic cosmic structures. Together, the two frameworks offer complementary perspectives on the same underlying vision: spacetime geometry is not fundamental but emerges from the thermodynamics of quantum information, and gravity arises from the interplay between entropy and enthalpy during this emergence.

To summarize the novelty of the present framework in concrete terms, we contrast it with existing approaches through the following specific distinctions, see Table 3.

7.4. Independent Support from the Gravity from Entropy (GfE) Framework

The recently developed Gravity from Entropy (GfE) theory [41] provides independent support for several key aspects of our framework. Although GfE and our approach differ in mathematical formulation and phenomenological emphasis, their convergence on fundamental principles strengthens the viability of the emergent gravity paradigm.

First, the bulk-field nature of entropy. In GfE, the entropy measure is local and volumetric rather than horizon-based [41], directly resonating with our treatment of the entropy current ${\mathrm{s}}^{\mathsf{\mu }}$ as a bulk dynamical field. This stands in contrast to Jacobson's horizon thermodynamics and Verlinde's holographic-screen approach, and suggests that the volumetric entropy framework--rather than holographic entropy--may be a generic feature of entropy-based gravity theories.

Second, the low-energy recovery of general relativity. GfE reduces to the Einstein-Hilbert action in the low-energy, small-curvature limit [41], exactly as our framework recovers GR in the weak-field regime. The fact that two independent frameworks--one rooted in quantum relative entropy and the other in string-net condensation [7]--converge on GR as their common low-energy limit provides strong evidence that GR is indeed the universal low-energy effective description of a deeper entropic structure.

Third, the compatibility of entropy increase with structure formation. A key result of GfE is that the total entropy of the universe increases with time, while the local entropy density can decrease, allowing for the emergence of ordered structures [41]. This is precisely the behavior encoded in our framework by the generalized second law ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}=\mathsf{\sigma }\ge 0$ (Section 4.5 and Section 7.5): the total entropy increases, while local entropy currents can be directed toward aggregation and structure formation. This convergence suggests that the compatibility of the second law with local ordering is a generic feature of entropy-based gravity, rather than an ad hoc assumption.

Fourth, dynamical dark energy. In GfE, the nonlinear structure of the action generates an emergent effective dark energy term ${\mathsf{\Lambda }}_{\mathrm{G}}$ that depends on the G-field and is not introduced as a fundamental constant [41]. This mirrors our framework, where the cosmological constant ${\mathsf{\Lambda }}_{\mathrm{C}}$ is determined by categorical data (the total quantum dimension $\mathrm{D}$ and the lattice spacing ${\mathrm{a}}_0$) rather than introduced as a free parameter (Section 5.2). The shared prediction that dark energy is dynamical rather than fundamental suggests that this may be a generic consequence of gravity's entropic origin.

Thus, while GfE provides a profound thermodynamic foundation for why gravity is entropic in nature [41], our framework extends this paradigm by asking how entropy flows directionally and how such flow competes with enthalpy to drive the emergence of ordered structures from quantum systems. Together, they offer complementary realizations of the same underlying vision: spacetime geometry is not fundamental but emerges from the thermodynamics of quantum information, and gravity arises from the interplay between entropy and enthalpy during this emergence. see Table 4

Table note: Despite distinct mathematical formalisms and phenomenological targets, both frameworks converge on the same underlying vision: spacetime geometry is not fundamental but emerges from the thermodynamics of quantum information, and gravity arises from the interplay between entropy and enthalpy during this emergence. GfE provides a profound thermodynamic foundation for why gravity is entropic in nature, while Ref. [7] and the present work extend this paradigm by asking how entropy flows directionally and how such flow competes with enthalpy to drive the emergence of ordered structures from quantum systems. Together, they offer complementary realizations of the emerging paradigm that “gravity originates from information.” We are actively developing this series of theories within the QEET framework to establish the entropy–enthalpy competition as a complete foundational picture of cosmic evolution.

7.5. Consistency with the Generalized Second Law

The generalized second law ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\ge 0$ is naturally incorporated into our framework through the Proca equation (39) and the entropy current divergence constraint (46).The entropy production rate $\mathsf{\sigma }$ acts as a source for the entropy current, ensuring that irreversible processes are consistently described.

7.6. Limitations and Future Directions

Several aspects of this framework require further development:

1) Microscopic origin of $\mathsf{\lambda }$: The coupling constant $\mathsf{\lambda }$ should ultimately be derived from a more fundamental theory of quantum information and gravity, such as quantum tensor networks [26], loop quantum gravity, or holographic duality. Within such a framework, $\mathsf{\lambda }$would be related to the entanglement entropy density of the underlying microscopic degrees of freedom, analogous to how ${\mathrm{G}}_{\mathrm{N}}$ emerges from the entropy current VEV in Section 5.

2) Early-universe thermodynamics: The phase-transition cosmology developed in our previous work [29] suggests that the entropy current may have existed in a superfluid-like phase in the early universe, with a non-zero condensate $⟨{\mathrm{s}}^{\mathsf{\mu }}⟩$. The breakdown of such a phase -- possibly through a spontaneous symmetry-breaking mechanism -- could provide a thermodynamic origin for mass generation and the electroweak scale. Within this broader picture, the coupling $\mathsf{\lambda }$ studied here would be the low-energy remnant of a more fundamental interaction between fermions and the cosmic entropy flow. The predictions of this paper --the energy shift $\mathsf{\Delta }\mathrm{E}=-\mathsf{\lambda }{\mathrm{s}}^0\mathrm{c}$, the effective magnetic field ${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathbf{s}$, and the emergent graviton mass ${\mathrm{m}}_{\mathrm{s}}$-- offer testable signatures of entropy-fermion coupling that persist even in the present epoch.

3) Connection to gravitational-wave observatories: The constraint ${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}$eV derived in Section 5.4 from LIGO/Virgo data on the GW170817 event places the emergent graviton in the ultra-light boson regime. Future observations with LISA, the Einstein Telescope, and pulsar timing arrays [37] will probe lower-frequency gravitational waves, where the massive graviton dispersion (65) becomes more pronounced. A detection of frequency-dependent gravitational-wave speed would provide direct evidence for our emergence mechanism and a measurement of ${\mathrm{m}}_{\mathrm{s}}$. As a corollary, the finite graviton mass ${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}$eV implies a finite graviton lifetime. From the uncertainty principle, ${\mathsf{\tau }}_{\mathrm{g}}\sim \mathrm{\hslash }/\mathsf{\Gamma }$, where the decay width ${\mathsf{\Gamma }}_{\mathrm{g}}$ for a massive graviton decaying into massless degrees of freedom is of order $\mathsf{\Gamma }\sim \mathsf{\Lambda }\mathrm{G}\mathrm{E}$, with ${\mathsf{\Lambda }}_{\mathsf{\lambda }}\sim {10}^2\mathrm{t}\mathrm{o}{10}^3\mathrm{G}\mathrm{e}\mathrm{V}$ being the entropy-flow energy scale estimated from STAR data (where ${\mathsf{\Lambda }}_{\mathsf{\lambda }}\sim {10}^2\mathrm{t}\mathrm{o}{10}^3\mathrm{G}\mathrm{e}\mathrm{V}$ is the same UV cutoff scale inferred from the entropy-fermion coupling $\mathsf{\lambda }$ in Section 6.1), $\mathrm{G}$ the Newton constant, and $\mathrm{E}\sim 10^{-13}$eV the typical energy of cosmological gravitational waves [36]. This yields ${\mathsf{\Gamma }}_{\mathrm{g}}\sim 10^{-24}$eV and hence ${\mathsf{\tau }}_{\mathrm{g}}\sim 10^8\mathrm{t}\mathrm{o}{10}^9$s, corresponding to several decades to a century ($\mathcal{O}(10^1-10^2)$ years) for high-energy gravitons considered here, and significantly longer for lower-energy cosmological gravitons. Thus, within our emergent gravity framework, the graviton is effectively stable on cosmological time scales, consistent with all existing observations. This stability is not imposed by symmetry but follows from the smallness of ${\mathrm{m}}_{\mathrm{s}}$, which in turn reflects the long correlation length of the entropy flow field.

4) Cosmological implications: If the entropy field VEV $⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$ varies with cosmic time -- as expected in an expanding universe -- then Newton's constant $\mathrm{G}$ in (54) becomes time-dependent. This provides a natural connection to scalar-tensor theories of gravity and dark energy models, and offers a potential explanation for the apparent fine-tuning of the cosmological constant in terms of the present-day entropy density of the universe.

Finally, we acknowledge that the present formulation is limited to regimes where the background curvature satisfies $\left|\mathrm{R}\right|\ll {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}^2$. In the opposite limit, higher-order curvature terms such as $\mathsf{\zeta }{\mathrm{R}}^2{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}$ are needed to restore stability. A complete UV completion incorporating such terms is an important direction for future research.

8. Conclusion

We have proposed a modification of the Dirac equation by coupling the fermion field to the entropy current vector ${\mathrm{s}}^{\mathsf{\mu }}$. The modified equation $(\mathrm{i}\mathrm{\hslash }{\mathsf{\gamma }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}-\mathrm{m}\mathrm{c}+\mathsf{\lambda }{\mathsf{\gamma }}^{\mathsf{\mu }}{\mathrm{s}}_{\mathsf{\mu }})\mathsf{\psi }=0$ yields:

1)Modified dispersion relation $\mathrm{E}=-\mathsf{\lambda }{\mathrm{s}}^0\mathrm{c}\pm \mathrm{c}\sqrt{{\mathbf{p}}^2+{\mathrm{m}}^2{\mathrm{c}}^2}$, showing that uniform entropy flow produces only an overall energy shift.

2)Effective magnetic field ${\mathbf{B}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathbf{s}$, driving spin precession via ${\mathsf{\omega }}_{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}=(\mathsf{\lambda }/\mathrm{m})\nabla \times \mathbf{s}$.

3)Proca dynamics for the entropy current ${\partial }_{\mathsf{\mu }}{\mathrm{F}}^{\mathsf{\mu }\mathsf{\nu }}+{\mathrm{m}}_{\mathrm{s}}^2{\mathrm{s}}^{\mathsf{\nu }}=\mathsf{\lambda }\bar{\mathsf{\psi }}{\mathsf{\gamma }}^{\mathsf{\nu }}\mathsf{\psi }$, with Yukawa-type solutions indicating a finite range of entropic interactions.

We emphasize that the emergence mechanism presented in Section 5, and the resulting identification of Newton's constant in Eq. (54), is formulated as a low-energy effective theory and is strictly valid in the weak-curvature regime $\left|\mathrm{R}\right|\ll {\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}^2$, where ${\mathsf{\Lambda }}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}}\equiv \sqrt{\left|\mathsf{\xi }\right|/\mathsf{\zeta }})$is the stability scale introduced in Section 4.4.1. In this regime, the tree-level vacuum expectation value contribution dominates and the positive sign of $\mathrm{G}$ is unambiguously obtained for $\mathsf{\xi }<0$.

Perhaps most importantly, the extended framework developed in Section 5 demonstrates that the graviton -- the quantum of spacetime curvature -- need not be postulated as a fundamental particle in this framework. Instead, it emerges as a collective excitation of the entropy flow field. The path integral over quantum fluctuations of ${\mathrm{s}}_{\mathsf{\mu }}$ in curved spacetime yields the Einstein-Hilbert action as the low-energy effective action, with Newton's constant determined by the entropy field's vacuum expectation value:

${\displaystyle \frac{1}{16\mathsf{\pi }\mathrm{G}}}=-\mathsf{\xi }⟨{\mathrm{s}}_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}⟩$ [(54)]

The linearized theory yields a massive spin-2 excitation -- the emergent graviton -- with dispersion ${\mathrm{E}}^2={\mathbf{p}}^2+{\mathrm{m}}_{\mathrm{s}}^2$[(65)]. Gravitational-wave observations from LIGO/Virgo, such as GW170817 [36], constrain the entropy field mass to ${\mathrm{m}}_{\mathrm{s}}\lesssim 10^{-20}$eV (based on the inferred source parameters of that event), placing the emergent graviton in the ultra-light boson regime.Using STAR data on Λ hyperon global spin polarization in heavy-ion collisions, we estimate the entropy-fermion coupling constant to be $\mathsf{\lambda }\sim {10}^{-7}-10^{-5}\mathrm{G}\mathrm{e}{\mathrm{V}}^{-2}$, corresponding to an energy scale ${\mathsf{\Lambda }}_{\mathsf{\lambda }}\sim {10}^2-{10}^3\mathrm{G}\mathrm{e}\mathrm{V}$. We emphasize that this is an order-of-magnitude estimate, with uncertainties spanning the full range quoted above, dominated by the model dependence of the QGP vorticity and entropy density.

This framework provides a unified thermodynamic foundation for relativistic quantum mechanics and gravitation, naturally connecting to Jacobson's thermodynamic gravity, Verlinde's entropic gravity, and the generalized second law ${\nabla }_{\mathsf{\mu }}{\mathrm{s}}^{\mathsf{\mu }}\ge 0$. The predictions for spin polarization in rotating systems, energy shifts in high-entropy environments, and frequency-dependent gravitational-wave dispersion offer concrete experimental tests for this quantum-information-theoretic paradigm of emergent spacetime.