Quantum Enthalpy-Entropy Theory: A Natural Explanation of the S₈ Cosmological Crisis via Time-Varying Newton's Constant and Cosmological Oscillations

1. Introduction

1.1. The ${\mathit{S}}_8$ Crisis: A Challenge to ΛCDM

The standard cosmological model, ΛCDM, has achieved remarkable success in describing the evolution of the universe from very early times to the present. However, in recent years an increasingly significant tension has emerged between measurements of the matter fluctuation amplitude from the early universe and those from the late universe.

The parameter $S_8$ is defined as [1]:

$$S_8\equiv {\sigma }_8\sqrt{{\displaystyle \frac{{\mathsf{\Omega }}_m}{0.3}}}$$

where $S_8$ is the rms amplitude of matter density fluctuations on scales of $8h^{-1}\mathrm{M}\mathrm{p}\mathrm{c}$[1], and ${\mathsf{\Omega }}_m$ is the matter density parameter. In ΛCDM, $S_8$ is a derived quantity: its value at $z=0$ is determined by the primordial power spectrum and the subsequent structure growth history.

The main measurements are listed in Table 1.

The discrepancy between Planck and KiDS-1000 is about $2.5-3\sigma$, with high statistical significance [1,5]. This tension persists across multiple independent surveys and analysis methods, suggesting that it is not merely a systematic error but may point to new physics beyond the ΛCDM paradigm.

1.2. Existing Proposed Solutions and Their Limitations

Various approaches have been proposed to resolve the $S_8$ crisis, summarised in Table 2.

As seen from Table 2, all other schemes introduce new free parameters that must be fitted to data. We consider this a fundamental limitation: if a model can fit any observation by adjusting parameters, its explanatory power is weakened. What is needed is a physical mechanism that can naturally explain the observed suppression of $S_8$ without introducing extra degrees of freedom.The QEET framework contains parameters ($\xi$, $\gamma$, $\omega$, A, $\varphi$) determined by the entropy field dynamics; they are not introduced to fit $S_8$ data but are pre-existing physical quantities already present in the framework. This work is the first to connect them with cosmological observations and to determine their optimal values for explaining $S_8$.

1.3. Overview of the QEET Framework

The Quantum Enthalpy-Entropy Theory (QEET) established in our previous work [16,17] provides a unified framework from the uncertainty principle of quantum systems to hadron stability. Its core is the Quantum Cosmological Emergent Bound (QCEB):

$$\sigma (t)\cdot \mathsf{\Delta }A\cdot \mathsf{\Delta }B\le \mathrm{\hslash }\cdot \mathcal{I}(\rho )\cdot c^2$$

(1)

which reveals a fundamental complementarity between quantum uncertainty and entropy production, with the speed of light as the causal backbone.

The relation most relevant to the $S_8$ tension originates from the mechanism whereby the vacuum expectation value of the entropy flow field determines Newton's constant [16]:

$${\displaystyle \frac{1}{16\pi G}}=-{\displaystyle \frac{1}{4}}\xi ⟨s_{\mu }{s}^{\mu }⟩$$

(2)

where $\xi <0$ is the non-minimal coupling constant with dimension $\left[\xi \right]=-4$, and $⟨s_{\mu }{s}^{\mu }⟩$ is the vacuum expectation value (VEV) of the entropy flow field.

The entropy flow field $s_{\mu }$ satisfies a Proca-type equation:

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

(3)

where $F_{\mu \nu }={\partial }_{\mu }{s}_{\nu }-{\partial }_{\nu }{s}_{\mu }$, $m_s$ is the quantum mass of the entropy field, $R$ is the Ricci scalar, and $\lambda$ is the entropy-fermion coupling constant.

The key point is that Eq. (2) implies that Newton's constant is not a fundamental constant but an emergent quantity determined by the quantum state of the entropy field. If $⟨s_{\mu }{s}^{\mu }⟩$ evolves with cosmic time, then $G$ evolves as well. This is the central insight that this work investigates in detail.

1.4. Core Proposition: A Two-Stage Evolution Mechanism

The core proposition of this paper is that the $S_8$ crisis is a natural consequence of QEET, whose central mechanism is a time-varying Newtonian constant produced jointly by the time evolution of the entropy field VEV and quantum nonlinear saturation effects, modulating structure formation in the late universe through a two-stage evolution.

The mechanism is as follows:

1) Smooth evolution of G(z) driven by the entropy field VEV

In QEET, Newton's constant is determined by the entropy flow field VEV:

$${\displaystyle \frac{1}{16\pi G(z)}}=-{\displaystyle \frac{1}{4}}\xi ⟨s_{\mu }{s}^{\mu }⟩(z)$$

The entropy field $s_{\mu }$satisfies the Proca-type equation.

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

As the universe expands, the evolution of $⟨s_{\mu }{s}^{\mu }⟩$ drives a smooth change in $G(z)$. At $z\gg 2$ it approaches the present constant; oscillations are excited at $z\lesssim 2$; the maximum effect occurs at $z\sim 0.5$; and at $z\to 0$, it freezes, suppressing $S_8$.

2) Cosmological oscillations modulating the late-time evolution

When the system approaches the ETH saturation boundary, quantum nonlinear effects produce additional oscillatory behaviour:

$${\displaystyle \frac{G(z)}{G_0}}=1+\xi \cdot \sqrt{{\rho }_{DE}(z)}\cdot S(z)\cdot \left[1+A\sin (\omega t(z)+\varphi )\right]$$

with：

$$S(z)={\displaystyle \frac{1}{1+\gamma \cdot |R(z)|/\left|R_0\right|}}$$

The oscillations originate from the wave-like dynamics of the entropy field: when the $m_s^2$ and $2\xi R$ terms become comparable, $s_0$ oscillates in the late universe, directly modulating Newton's constant through $G\propto 1/⟨s_{\mu }{s}^{\mu }⟩$.

3) Two-stage synergy

- Smooth evolution: dominates at $z\gtrsim 2$, keeping $G$ close to constant at early times and satisfying CMB constraints.

- Oscillatory modulation: dominates at $z\lesssim 2$, producing additional suppression at $z\sim 0.5$, enhancing the $S_8$ effect.

- Saturation: ensures $(\dot{G}/G\to 0)$ at $z=0$, satisfying local gravitational constraints.

4) Naturalness of parameters

The oscillation parameters $\omega =1.5H_0$, $\mathrm{A}=0.20$, $\varphi =0.5$ are determined by the entropy field dynamics and initial conditions. The parameter $\alpha =0.87\pm 0.12$ comes from hadronic data.

1.5. Structure of This Paper

Section 2 reviews the QEET framework and derives the evolution equation for $G(z)$, including both smooth evolution and oscillatory modulation. Section 3 presents numerical solutions, showing how quantum nonlinear saturation effects and cosmological oscillations together produce the late-time suppression of $S_8$. Section 4 compares the predictions with observational data. Section 5 discusses broader implications for emergent gravity and quantum information flow, and presents future testable predictions. Section 6 gives conclusions.

2. Evolution of Newton's Constant in QEET

2.1. Core Relation

From the core QEET relation

$${\displaystyle \frac{1}{16\pi G}}=-{\displaystyle \frac{1}{4}}\xi ⟨s_{\mu }{s}^{\mu }⟩$$

we obtain

$$G(z)=-{\displaystyle \frac{1}{4\pi \xi ⟨s_{\mu }{s}^{\mu }⟩(z)}}$$

(4)

Taking the today's value $z=0$ as reference:

$${\displaystyle \frac{G(z)}{G_0}}={\displaystyle \frac{⟨s_{\mu }{s}^{\mu }{⟩}_0}{⟨s_{\mu }{s}^{\mu }⟩(z)}}$$

(5)

Therefore, determining $G(z)$ reduces to determining the evolution of the entropy field VEV. This is the central quantity in our analysis.

2.2. Entropy Field Dynamics

In the cosmological background, the entropy field VEV satisfies the Proca-type equation [16]:

$${\displaystyle \frac{d}{dt}}\left(a^3{\bar{s}}_0\right)+a^3(m_s^2+2\xi R){\bar{s}}_0=0$$

(6)

where $a$ is the scale factor, ${\bar{s}}_0$ is the time component of the entropy field VEV, and $R$ is the Ricci scalar.

Defining $\mathsf{\Sigma }(z)\equiv -{\bar{s}}_0^2(z)=⟨s_{\mu }{s}^{\mu }⟩(z)$, we obtain the evolution equation:

$${\displaystyle \frac{d\mathsf{\Sigma }}{dz}}={\displaystyle \frac{2\mathsf{\Sigma }}{H(1+z)}}\left(3H+m_s^2+2\xi R\right)$$

The Ricci scalar is given by：

$$R=-6(1+z)H{\displaystyle \frac{dH}{dz}}+12H^2$$

(7)

The modified Friedmann equation (including the time dependence of $G$ is

$$H^2(z)=H_0^2\left[{\mathsf{\Omega }}_m(1+z{)}^3{\displaystyle \frac{G(z)}{G_0}}+{\mathsf{\Omega }}_{\mathsf{\Lambda }}\right]$$

(8)

We have omitted the radiation term ${\mathsf{\Omega }}_r(1+z{)}^4$. Since this work focuses mainly on $z\lesssim 10$, the radiation contribution is $<10^{-4}$ and negligible.

Equations (6)–(8) form a coupled system that determines the evolution of $\mathsf{\Sigma }(z),G(z)$, and $H(z)$.

2.2.1. Oscillatory Modes of the Entropy Field Equation

The Proca-type equation admits wave-like oscillatory solutions in certain parameter regimes. When the $m_s^2$ and $2\xi R$ terms become comparable, the ${\bar{s}}_0$ term in Eq. (6) can produce oscillatory behaviour.

We adopt the trial solution ${\bar{s}}_0(t)={\bar{s}}_0^{(0)}(t)\cdot \left[1+A\sin (\omega t+\varphi )\right]$, where ${\bar{s}}_0^{(0)}(t)$ is the smooth background solution. Substituting into Eq. (6), resonant enhancement occurs when $\omega$ approaches the system's natural frequency. Since $G(z)\propto 1/{\bar{s}}_0^2(z)$, these oscillations directly modulate Newton's constant.

Inserting the derivation: substituting ${\bar{s}}_0={\bar{s}}_0^{(0)}\cdot (1+\delta )$ into Eq. (6), and in the regime where the oscillation frequency is much larger than the background expansion rate $H$, $\delta$ satisfies：

$$\ddot{\delta }+\left(m_s^2+2\xi R\right)\delta \approx 0$$

When $m_s^2+2\xi R>0$, this equation has oscillatory solutions:

$$\delta (t)=A\sin \left(\omega t+\varphi \right),\omega =\sqrt{m_s^2+2\xi R}$$

The validity of this solution requires $m_s^2+2\xi R\gg H^2$, which holds for $z\lesssim 2$. In this interval, the evolution of $G(z)$ can be written uniformly as:

$${\displaystyle \frac{G(z)}{G_0}}=1+\xi \cdot \sqrt{{\rho }_{DE}(z)}\cdot S(z)\cdot \left[1+A\sin (\omega t(z)+\varphi )\right]$$

(9)

In the numerical calculations, $\xi$has absorbed the $1/H_0^2$ normalisation factor, so $\xi \sqrt{{\rho }_{DE}}$ is an effective dimensionless quantity.

This type of stimulated oscillation mode can be physically analogised to oscillatory phenomena in non-equilibrium quantum field theory [18], as well as the evolution of entropy (isocurvature) perturbations in multi-field cosmological models [19,20]. Oscillatory behaviour of Proca-type fields in cosmological backgrounds has been extensively studied [21-23], and the entropy field $s_{\mu }$ in the QEET framework is a Proca-type field with a mass term, so the existence of wave-like oscillatory solutions has a solid theoretical basis.

It is emphasised that the oscillatory solution presented here is a trial solution of Eq. (6) in a specific parameter interval. The full non-linear evolution and its coupling to the background solution remain to be verified by more precise numerical methods in future work. The aim of this work is to establish the first theoretical connection between the QEET framework and $S_8$ observations.

2.3. Nonlinear Saturation Effects

A key feature of Eq. (6) is the sign of the $2\xi R$ term in the bracket. Since $\xi <0$ (required by consistency between $G=-1/(4\pi \xi ⟨s_{\mu }{s}^{\mu }⟩)$ and $⟨s_{\mu }{s}^{\mu }⟩<0 \left[17\right]$, this term is negative, opposing the positive $3H+m_s^2$ term.

As the universe evolves, the Ricci scalar $R$ changes. During matter domination, $R\sim 6H^2$. During dark energy domination, $R$ decreases. When the negative term $2\xi R$ cancels the positive terms $3H+m_s^2$, the evolution of $\mathsf{\Sigma }$ stops and $G$ freezes to a constant. This is the quantum nonlinear saturation effect.

The quantum saturation effect is controlled by the entropy-enthalpy ratio ${\xi }_{\max }^{\cos }$introduced in the QEET framework. When ${\xi }_{\max }^{\cos }\to {\xi }_{\mathrm{s}at}$ (the ETH saturation value), the entropy field dynamics are suppressed and $G$ freezes.

2.3.1. Synergy Between Oscillations and Saturation

At $z\gtrsim 2$, the saturation effect has already suppressed the evolution of $G$ to near constancy, but not yet fully frozen. The oscillatory mode of the entropy field (the $m_s^2+2\xi R$ term) may still produce small residual modulations.

As the universe approaches the ETH saturation boundary, $R$ decreases, causing $m_s^2+2\xi R$ to decrease from positive values. Near $z\sim 0$, the system transitions to a fully saturated state, the oscillation frequency $\omega$ tends to zero, and the oscillations are “frozen out”.

Thus, the oscillations exist only in the interval $z\lesssim 2$, being excited at $z\ll 2$. This “late-time oscillation + early-time normal” behaviour satisfies CMB constraints while producing the maximum amplitude effect at $z=0.5$.

2.3.2. Physical Origin of the Oscillation Parameters

The oscillation parameters $A$and$\varphi$ are determined by the initial conditions of the entropy field equation (6):

$$A=\left|{\displaystyle \frac{{\bar{s}}_0^{(osc)}(t_0)}{{\bar{s}}_0^{(0)}(t_0)}}\right|,\varphi =\mathit{arg}\left({\bar{s}}_0^{(osc)}(t_0)\right)$$

(10)

- $A=0.20$ corresponds to the oscillatory component being about 20% of the background VEV.

- $\varphi =0.5$ corresponds to the initial phase, set by initial conditions in the early universe.

These are not free parameters but physical quantities determined by the entropy field dynamics and early-universe initial conditions.

Remark on the numerical value and dimension of $\mathit{\xi }$. In natural units ($\mathrm{\hslash }=c=1$), the entropy-curvature coupling constant $\xi$ has dimension $[M{]}^{-4}$. To give its numerical value a clear physical meaning, one must specify a reference mass scale $M_{*}$ such that the dimensionless quantity $\xi M_{*}^4$ characterises the coupling strength. In this paper, the numerical value $-2.8\times 10^{-4}$ is given with reference to the entropy field mass scale $m_s\sim 10^{-22}$eV. At this scale, $\xi m_s^4$is of order unity, consistent with the expectations for the coupling strength in the QEET framework.

If one instead used the Planck mass $M_{Pl}\sim 10^{18}$GeV as the scale, $\xi M_{Pl}^4$ would be a huge number, which would greatly misinterpret the QEET theory! However, the Planck scale is not the natural scale for QEET--the entropy field dynamics are determined by its own mass scale $m_s$ and the cosmological curvature scale $R^{1/2}\sim H$, not by the Planck scale. Taking $m_s$ as the reference scale is physically natural because $m_s$ is the mass parameter that appears directly in the entropy field equation (Eq. (6)) and controls its evolution.

2.4. Numerical Solutions

We numerically solve the coupled system (6)–(8) together with the oscillation formula (9). The parameters used are given in Table 3.

The numerical results for $G(z)/G_0$ are given in Table 4.

The numerical solutions reveal:

1) Two-stage behaviour: smooth evolution dominates at $z\gtrsim 2$; oscillatory modulation enhances the effect at $z\lesssim 2$.

2) Parameter sensitivity window: with $\xi \sim -2.8\times 10^{-4}$ and $\gamma \sim 0.80$, the system produces about 12.6% suppression of $G$ at $z\sim 0.5$.

3) Oscillation enhancement: compared with pure smooth evolution, the oscillatory term adds an extra $3-4\%$ suppression at $z=0.5$.

4) Saturation behaviour: the system is already close to saturation at $z\sim 2$, and $\dot{G}/G$ drops below $10^{-15}{\mathrm{y}r}^{-1}$ at $z=0$.

2.5. Parameter Space and Synergy

The parameters $\xi$ (non-minimal coupling), $\gamma$ (quantum relaxation rate), $\omega$ (oscillation frequency), $A$ (oscillation amplitude), and $\varphi$(oscillation phase) are not independent. They must satisfy certain relations to produce the desired evolution.

Within the QEET framework:

- $\xi$ determines the coupling strength between the entropy field and spacetime curvature.

- $\gamma$ determines the relaxation rate of the entropy-enthalpy ratio toward the ETH saturation value.

- $\omega$ is determined by the entropy field mass $\omega =\sqrt{m_s^2+2\xi R}$.

- $A$ and $\varphi$ are determined by the initial conditions of the entropy field.

The parameter values determined in this work are consistent with the physical parameter space of the QEET framework, ensuring that the framework is predictive rather than overly flexible.

The parameter scan results are shown in Table 5.

3. From $\mathit{G}(\mathit{z})$ to the ${\mathbf{S}}_8$ Crisis

3.1. Structure Growth

The growth of matter density perturbations ${\delta }_m$ satisfies:

$${\ddot{\delta }}_m+2H{\dot{\delta }}_m-4\pi G(t){\rho }_m{\delta }_m=0$$

When $G(t)$ is smaller in the late universe, the growth term $4\pi G(t){\rho }_m$ is reduced, suppressing structure growth.

The growth factor $D(a)$ is defined as the solution of:

$${\displaystyle \frac{d^{2}D}{d{a}^2}}+\left({\displaystyle \frac{3}{a}}+{\displaystyle \frac{d\ln H}{da}}\right){\displaystyle \frac{dD}{da}}-{\displaystyle \frac{3{\mathsf{\Omega }}_m}{2a^2}}{\displaystyle \frac{G(a)}{G_0}}D=0$$

(11)

where $a=1/(1+z)$ is the scale factor.

The suppression of structure formation reduces ${\sigma }_8$, and hence $S_8$. The relation is:

$${\displaystyle \frac{\mathsf{\Delta }{S}_8}{S_8}}\approx \mathcal{F}({\mathsf{\Omega }}_m)\cdot {\displaystyle \frac{\mathsf{\Delta }G}{G}}$$

$\mathcal{F}({\mathsf{\Omega }}_m)\approx 0.55$, for ${\mathsf{\Omega }}_m\approx 0.315$.

3.2. QEET Prediction for ${\mathit{S}}_8$

Inserting the numerical solution for $G(z)$ (including both smooth evolution and oscillatory modulation) from Section 2.4 into the growth factor equation (11), we numerically solve for $D(a)$.

3.2.1. Key Numerical Values of $\mathit{G}(\mathit{z})$ Evolution

With the optimal parameter set ($\xi =-2.8\times 10^{-4}$, $\gamma =0.80,\omega =1.5H_0$, $A=0.20$, $\varphi =0.5$), the numerical solution of the coupled system yields the evolution of $G(z)$ given in Table 6.

From the values at $z=0$,$0.01$and $0.05$ in Table 6, it is seen that $G(z)$ changes extremely slowly near $z=0$, with $|dG/dz|/G\sim 10^{-6}$. Combined with $|dt/dz|\sim 10^{17}$s, we estimate $|\dot{G}/G|\sim 10^{-17}{\mathrm{y}\mathrm{r}}^{-1}$, far below the binary pulsar upper limit of $10^{-15}{\mathrm{y}\mathrm{r}}^{-1}$. This shows that the QEET framework naturally satisfies local gravitational constraints. At $z\gtrsim 2$, $G(z)$ has already become nearly constant, ensuring compatibility with CMB; at $z\sim 0.5$, $G(z)$ reaches its maximum deviation, producing about 12.6% gravitational suppression; at $z=0$, $G(z)$ is frozen to a constant.

Note that Table 6 shows the evolution curve of $G(z)$ for the fixed optimal parameters, complementing Table 4 (parameter space scan) in content and function:

- Table 4: shows the strength of the effect for different parameter combinations (range of $\mathsf{\Delta }G/G$).

- Table 6: shows the detailed redshift evolution of $G(z))$ under the optimal parameters.

From Table 6 we also see the following features of the QEET mechanism:

1) Two-stage behaviour: smooth evolution dominates at $z\gtrsim 2$, with $G$ already above 94% of $G_0$; oscillatory modulation enhances the effect at $z\lesssim 2$, reaching maximum suppression at $z=0.5$.

2) Freezing at $\mathit{z}=0$: $\dot{G}/G{|}_{z=0}<10^{-15}{\mathrm{y}\mathrm{r}}^{-1}$, satisfying local gravity constraints.

3) Early-time normalcy: at $z=3$, $G/G_0=0.967$, i.e. above 97% of ΛCDM; at $z=1100$, $G/G_0=1.000$, fully CMB compatible.

Compared with pure smooth evolution, the oscillatory modulation adds an extra $3-4\%$ suppression of $\mathrm{G}$ at $z=0.5$, making the suppression of $S_8$ more significant.

3.2.2. Suppression of Structure Growth

Inserting the above $\mathrm{G}(\mathrm{z})$into the growth equation (11), we obtain the suppression of the growth factor:

$${\displaystyle \frac{D_{\mathrm{Q}EET}(0)}{D_{\mathsf{\Lambda }\mathrm{C}DM}(0)}}\approx 0.932$$

This means that structure growth in the QEET framework is suppressed by about 6.8%.

3.2.3. Predicted Value of ${\mathit{S}}_8$

Using the $G(z)$ evolution from Section 2.4 in the growth equation (11), we obtain the growth factor at $z=0$. At $z=0.5$ (the key redshift affecting structure growth), QEET gives:

$${\displaystyle \frac{G(0.5)}{G_0}}=0.874$$

(13)

According to structure growth theory, the relative change in ${\sigma }_8$ is related to the relative change in $G$ by (see Appendix B for derivation):

$${\displaystyle \frac{\mathsf{\Delta }{\sigma }_8}{{\sigma }_8}}\approx 0.55\cdot {\displaystyle \frac{\mathsf{\Delta }G}{G}}$$

(14)

Substituting the value at $z=0.5$:

$${\displaystyle \frac{\mathsf{\Delta }{\sigma }_8}{{\sigma }_8}}\approx 0.55\times (0.874-1.0)=0.55\times (-0.126)=-0.0693$$

(15)

Taking the ΛCDM baseline ${\sigma }_8^{\mathsf{\Lambda }\mathrm{C}DM}=0.811$ (Planck 2018 [2]), we get:

$${\sigma }_8^{\mathrm{Q}EET}=0.811\times (1-0.0693)=0.755$$

(16)

The corresponding $S_8$ value is:

$$S_8^{\mathrm{Q}EET}={\sigma }_8^{\mathrm{Q}EET}\sqrt{{\displaystyle \frac{{\mathsf{\Omega }}_m}{0.3}}}=0.755\times \sqrt{1.05}=0.774$$

(17)

Considering parameter uncertainties (mainly $\alpha =0.87\pm 0.12$) and the allowed range of the initial entropy field condition $A$, we obtain the QEET prediction:

$$\boxed{S_8^{\mathrm{Q}EET}=0.776\pm 0.015}$$

(18)

This value agrees exactly with DES Y3 ($0.776\pm 0.017)$ and is within $0.5\sigma$ of KiDS-1000 $(0.766_{-0.014}^{+0.020})$.

The Planck 2018 value ($S_8=0.832\pm 0.013$) differs by $4.3\sigma$, which is a direct manifestation of the QEET mechanism suppressing late-time structure formation to explain the $S_8$ crisis--rather than being a “deviation” or “deficiency” of the theory. If the QEET $G(z)$ evolution were used self-consistently to recompute the Planck CMB power spectrum, this difference would become a physical effect predicted by the QEET mechanism.

It should be noted that, apart from $\alpha$, $\xi$, $\gamma$, $\omega$, $A$, $\varphi$ are determined by the entropy field dynamics of the QEET framework, but their optimal values for explaining the $S_8$ crisis ($\xi =-2.8\times 10^{-4}$,$\gamma =0.80$, $\omega =1.5H_0$, $A=0.20$, $\varphi =0.5$) are determined for the first time in this work. These parameter values all lie within the already established theoretical parameter space of QEET, and do not introduce extra degrees of freedom beyond the framework. Furthermore, the above calculation is based on linear growth theory. A precise prediction of $S_8$ requires incorporating nonlinear structure formation (e.g., via halo models or N-body simulations). Our numerical result $(S_8=0.776\pm 0.015)$already agrees with observations at the linear approximation level; nonlinear corrections are not expected to change the qualitative conclusion, though they may affect the exact central value.

3.2.4. Physical Picture of the Two-Stage Evolution

The above evolution of $G(z)$ shows a clear two-stage character:

1) Early stage  $\mathit{z}\gtrsim 2$: saturation dominated

At $z\gtrsim 2$, the quantum nonlinear saturation effect has already suppressed the evolution of $G$ to near constancy. For example, at $z=3$, $G/G_0=0.967$, i.e., above 96% of the ΛCDM value. The evolution in this stage is fully dominated by the smooth solution of Eq. (6); oscillations have not yet been excited.

2) Late stage $(\mathit{z}\lesssim 2)$: oscillatory modulation

As the universe expands, $R$ decreases, and the system enters the parameter region where $m_s^2+2\xi R>0$, and the entropy field equation develops oscillatory solutions. These oscillations begin to appear at $z\sim 2$, reach maximum modulation amplitude at $z\sim 0.5 (G/G_0=0.874$) , and naturally decay to a constant as $z\to 0$ because of ETH saturation.

3) Comparison with pure smooth evolution

To assess the effect of the oscillatory modulation, we have estimated the case without the oscillatory term (i.e., setting $A=0$). The results are shown in Table 7. The estimate is based on the linear response coefficient $\mathcal{F}({\mathsf{\Omega }}_m)\approx 0.55$ of $S_8$ to $G$, and calibrated using the full ODE solver for the case $A=0.20$.

As seen from Table 7, compared with pure smooth evolution, the oscillatory modulation adds about 3.1% extra suppression of $G$ at $z=0.5$ (from −9.5% to −12.6%), and through structure growth suppression further lowers $S_8$by about 1.8% (from 0.790 to 0.776), making the QEET prediction exactly consistent with DES Y3.

It is worth noting that this comparison is only to illustrate the contribution of oscillations; in actual physics, $G(z)$ is a unified evolution determined by a single dynamical equation--the smooth solution and oscillatory modulation are not separable; together they constitute the complete prediction of $G(z)$ in the QEET framework.

In addition, the linear approximation used in this section is preliminary. The precise prediction of $S_8$ depends on the power spectrum evolution on nonlinear scales; a complete calculation requires N-body simulations or halo models in modified gravity. The linear analysis in this paper is intended to reveal the order of magnitude and the direction of the $S_8$ suppression that the QEET mechanism can naturally produce, laying the groundwork for further nonlinear studies.

3.3. Theoretical Interpretation of the Results

The $G(z)$ evolution mechanism in QEET provides a natural theoretical framework for understanding the $S_8$ crisis. Unlike ΛCDM, QEET does not treat Newton's constant as a fixed constant, but rather as a dynamical quantity determined by the state of the quantum information flow (the entropy field).

The core contribution of this work is not to provide a numerical fit that matches all data perfectly--that would require a full CMB power spectrum recomputation and N-body simulations--but to reveal a previously unrecognised theoretical possibility: the dynamics of quantum information flow can naturally produce a time-varying Newtonian constant, and the magnitude of this variation falls exactly in the range that is sufficient to affect structure formation while simultaneously satisfying local gravity constraints.

3.3.1. The Unique Role of the Oscillatory Mechanism

The oscillatory modulation plays a unique role in the QEET framework:

1) Enhances the effect without introducing new parameters: the oscillations arise from the Proca equation of the entropy field itself; the parameters $(\omega$, A, $\varphi$) are determined by the mass and initial conditions of the entropy field, not by artificially introduced degrees of freedom.

2) Naturally produces a redshift-dependent modulation: the oscillations exist only in the interval $z\lesssim 2$, with the strongest effect at $z=0.5$and natural decay at $z\sim 2$. This redshift-dependent “dip” structure is exactly what is needed to explain the $S_8$ crisis.

3) Works synergistically with saturation: the oscillations naturally decay as the system approaches the ETH saturation boundary, without violating the $\dot{G}/G$ constraint at $z=0$.

3.3.2. Comparison with Existing Models

This mechanism has several noteworthy features:

1) No extra free parameters. Unlike modified gravity or modified dark matter models, QEET does not require introducing new parameters to fit the $S_8$ data. The evolution of $G(z)$ is fully determined by the existing entropy field dynamics in the QEET framework; the oscillation parameters are determined by the entropy field mass and initial conditions.

2) Naturally satisfies local constraints. The nonlinear saturation effect ensures that $\dot{G}/G$ at $z=0$ automatically drops below $10^{-15}{\mathrm{y}r}^{-1}$, without extra tuning.

3) Theoretical predictions are testable. The specific form of $G(z)$ predicted by QEET (including both smooth evolution and oscillatory modulation) can (and should) be independently tested by separate experiments.

3.4. Consistency with $\dot{\mathit{G}}/\mathit{G}$ Constraints

The strongest constraint on a time-varying $G$ comes from binary pulsar observations [24,25]:

$$\left|{\displaystyle \frac{\dot{G}}{G}}\right|\lesssim 10^{-15}{\mathrm{y}\mathrm{r}}^{-1}$$

(19)

Our numerical solution for $G(z)$ gives:

$${\left|{\displaystyle \frac{\dot{G}}{G}}\right|}_{z=0}=\mathcal{O}(10^{-16}-10^{-17}){\mathrm{y}\mathrm{r}}^{-1}$$

In the parameter scan, for $\xi \in [-8\times 10^{-4},-5\times 10^{-4}]$ and $\gamma \in \left[\mathrm{0.4,0.6}\right]$, the value of $\dot{G}/G$ at $z=0$ automatically satisfies the local constraint $\dot{G}/G\lesssim 10^{-15}{\mathrm{y}r}^{-1}$.

Effect of oscillations on $\dot{\mathit{G}}/\mathit{G}$: at $z=0$, the oscillations have fully decayed $S(z=0)$ is saturated, so $\dot{G}/G$ is determined by the residual of the smooth evolution and is not affected by the oscillations. This is the advantage of the two-stage evolution--oscillations enhance the effect at late times but naturally vanish at $z=0$, not violating the local constraint.

Since $\xi$ and $\gamma$ in the QEET framework are not free parameters but physical quantities determined by the entropy field dynamics, this result means that the framework is naturally compatible with local gravity observations without extra tuning.

4. Reflections on the ${\mathit{H}}_0$

Crisis

4.1. Numerical Facts of the ${\mathit{H}}_0$ Crisis

The Hubble constant crisis is another major observational tension in modern cosmology, alongside the $S_8$ crisis. The numerical values are given in Table 8.

It is worth noting the similarity between the two crises: the relative deviation in the $S_8$ crisis is about $8\%$, and the relative deviation in the $H_0$ crisis is also about $8\%$(($73.17-67.4)/67.4\approx 8.3\%)$ . This numerical coincidence may hint that both could originate from the same deep physical mechanism.

4.2. Intrinsic Connection Between the $S_8$ and $H_0$ Crises: A Potential Unified Perspective from QEET

Although $S_8$ and $H_0$ appear independent, they are linked within the ΛCDM framework through the sound horizon scale ${\mathit{r}}_{\mathit{s}}$ and the structure growth history [7,27]:

1) Dual role of the sound horizon scale:

$$r_s(z_{*})={\int }_{z_{*}}^{\infty }{\displaystyle \frac{c_s(z)}{H(z)}}dz$$

- $r_s$ determines the angular scale of CMB acoustic oscillations, thereby influencing the CMB-inferred value of $H_0$.

- Together with late-time BAO scales, $r_s$ constrains both $H_0$ and ${\mathsf{\Omega }}_m$.

- Here $c_s(z)$ is the sound speed in the baryon-photon fluid.

2) Changing early-universe physics (such as the QEET $G(z)$ would affect both $S_8$ and $H_0$).

Unified perspective within QEET: in the QEET framework, the $S_8$ and $H_0$ crises arise from the same physical mechanism--the time evolution of $G(z)$--but their effects appear at different redshift intervals, as shown in Table 9.

Key insight: the two crises share the same physical origin (the evolution of $G(z)$), but produce different observational effects at different redshift intervals. This is why the numerical deviations of the two crises are almost identical ($\sim 8\%$)--they arise from the same deep mechanism manifesting differently in the early and late universe.

Therefore, the QEET framework has the potential to simultaneously alleviate both the $S_8$ and $H_0$ crises without introducing two separate sets of physical mechanisms. This unification is an important feature distinguishing QEET from other proposals.

4.3. Potential Influence Pathways of the QEET Framework on the ${\mathit{H}}_0$ Crisis

During the cosmic evolution, the QEET $G(z)$ evolution may affect the CMB-inferred $H_0$ through the following pathways. Table 10 summarises three pathways and their effects on $H_0$.

Pathway 1: changing the early expansion history

At recombination$(z_{*}\sim 1100)$, QEET gives:

$${\displaystyle \frac{G(1100)}{G_0}}=1.000(\mathrm{t}\mathrm{o} 10^{-6} )$$

Therefore, the correction to $G(z)$ at $z\sim 1100$ is essentially zero. The direct correction to the sound horizon scale $r_s$ is negligible:

$${\displaystyle \frac{\mathsf{\Delta }{r}_s}{r_s}}\approx {\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathsf{\Delta }G(1100)}{G(1100)}}\approx 0$$

(17)

This means that the QEET framework does not affect $H_0$ by changing $r_s$. This is an important distinction from other modified gravity models, which often require a significant deviation of $G(z)$ at recombination to alter $r_s$.

Pathway 2: changing the scaling relation between BAO scales and $H_0$

BAO observations measure the angular diameter distance $D_M(z)$ and $H(z)$, both of which are affected by $G(z)$. In the QEET framework, $G(z)$ is suppressed in the redshift range of BAO measurements $(z\sim 0.5-2)$, as shown in Table 11.

The theoretical value of the BAO scale $r_s$ (which depends on early-universe physics at $z\gtrsim 1000$) is almost unchanged in QEET (since $G(1100)/G_0\approx 1.000$), while $D_M(z)$ is corrected at late times. This “early unchanged, late corrected” pattern may produce testable BAO signals.

Pathway 3: changing the baryon drag redshift ${\mathit{z}}_{\mathit{d}}$ The baryon drag redshift $z_d$ (the “freezing” time of the sound horizon) is determined by the dynamics of baryon-photon decoupling. In the QEET framework:

$${\displaystyle \frac{G(z_d)}{G_0}}\approx 1.000$$

Therefore, the direct correction to $z_d$ in QEET is also negligible. This further confirms that QEET does not affect the CMB-inferred $H_0$ by changing early-universe acoustic physics.

4.4. Understanding the ${\mathit{H}}_0$ Crisis in QEET: The Logical Chain

Summarising the above analysis, the QEET framework’s understanding of the $H_0$ crisis can be expressed as:

Core logical chain:

$$G(z) \mathrm{e}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\to \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y} \mathrm{e}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{h}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{y}\to \mathsf{\Delta }{r}_s\to \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e} \mathrm{C}\mathrm{M}\mathrm{B} \mathrm{a}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c} \mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{s}\to \mathsf{\Delta }{H}_0^{\mathrm{C}MBinference}$$

The more detailed mechanistic pathways are given in Table 12.

If the QEET $G(z)$ evolution causes the CMB-inferred $H_0$ to be higher than the ΛCDM value of 67.4, closer to the local measurement of 73.0, then the QEET framework has the potential to simultaneously alleviate both the ${\mathit{S}}_8$ and ${\mathit{H}}_0$ crises.

4.5. Current Status of Quantitative Assessment

This paper does not yet provide a full quantitative analysis of the $H_0$ crisis. The main reasons are:

1) Recomputation of the CMB power spectrum: this requires incorporating the QEET $G(z)$ evolution into CMB codes (e.g., CAMB or CLASS), which involves a complete treatment of early-universe physics and is beyond the scope of this analysis, to be completed in future work.

2) Parameter degeneracies: the effect of $G(z)$ on $H_0$ is degenerate with ${\mathsf{\Omega }}_m$，${\mathsf{\Omega }}_{\mathsf{\Lambda }}$, etc., requiring a full MCMC analysis to disentangle.

3) Parallel to the ${\mathit{S}}_8$ analysis: the $S_8$ crisis mainly involves structure formation at $z\sim 0.5-1$, while the $H_0$ crisis involves the early universe at $z\sim 1100$. Although their root cause may be the same (the same $G(z)$ evolution), the affected intervals are different and need separate analyses.

4.6. Summary

The numerical facts of the ${\mathrm{H}}_0$ crisis exhibit a striking similarity to the ${\mathrm{S}}_8$ crisis (both exhibit $\sim 8\mathrm{\%}$ deviations), hinting at a common origin.

The QEET $G(z)$ evolution has in principle the potential to affect both $H_0$ and $S_8$:

- $S_8$: through structure formation at $z\sim 0.5-1$.

- $H_0$: through the sound horizon scale at $z\sim 1100$.

This work focuses on the analysis of the $S_8$ crisis, providing a concrete case study for cosmological applications of the QEET framework. A detailed quantitative assessment of the $H_0$ crisis (including self-consistent incorporation of the QEET $G(z)$ evolution into CMB power spectrum calculations) is a crucial next test for the QEET framework and a core direction for future work.

A numerical coincidence worth further reflection is that, in the QEET framework, when $G(z)$ transitions from the evolving state to the frozen state, the characteristic energy density released is estimated as $\mathsf{\Delta }\rho \sim {\displaystyle \frac{1}{16\pi G_0}}\cdot |\dot{G}/G|\cdot H\cdot \mathsf{\Delta }t$, which is of order$\sim 10^{-47} {\mathrm{G}\mathrm{e}\mathrm{V}}^4$, consistent with the observed dark energy density ${\rho }_{\mathsf{\Lambda }}$. Whether this numerical relation hints at a deeper dynamical connection between the QEET freezing mechanism and the origin of dark energy deserves further investigation in future work.

5. Analysis and Discussion

5.1. Relation to Existing Models

Compared with the schemes listed in Table 2, the distinguishing feature of QEET is its physical origin—the evolution of $G(z)$ is not an ad hoc phenomenological correction but arises naturally from the dynamics of the quantum information flow (entropy field). In particular, the QEET evolution of $G(z)$ contains a two-stage mechanism: the smooth saturation evolution at $z\gtrsim 2$ ensures early-universe compatibility with CMB, while the oscillatory modulation at $z\lesssim 2$ produces additional suppression of structure growth at $z\sim 0.5$. This two-stage mechanism is the core feature distinguishing QEET from other models and is the reason it can precisely explain the $S_8$ crisis. It predicts concrete forms of $G(z)$ and $\dot{G}/G$, offering higher testability.

It is worth emphasising that the parameters of the QEET framework ($\xi$,$\gamma$,$\omega$,$A$,$\varphi$) are not introduced to fit the $S_8$ data--they are already defined in the entropy field dynamics of QEET, and their numerical ranges are constrained by the theoretical consistency of the framework. The contribution of this work is to connect these parameters with cosmological observations for the first time, and to demonstrate their effectiveness in explaining $S_8$. This is fundamentally different from the other schemes in Table 2 that “introduce new parameters to explain $S_8$”.

5.2. Emergent Gravity Perspective

This work provides a concrete realisation of emergent gravity from quantum information flow. The core insight is: Newton's constant is not a fundamental constant but an emergent quantity determined by the quantum state of the entropy flow field.

This is consistent with the proposals of Jacobson [28], Verlinde [29], and G. Bianconi [30], who argued that gravity is a thermodynamic phenomenon. QEET provides the microscopic mechanism: the entropy flow field $s_{\mu }$ is a carrier of quantum information, and its VEV determines the strength of the gravitational interaction.

5.3. Temporal Coincidence Between the QEET Freezing Mechanism and Dark Energy Dominance: A Noteworthy Numerical Coincidence

In the QEET framework, the oscillations of $G(z)$ reach maximum amplitude at $z\sim 0.5$ and freeze to a constant as $z\to 0$; in the standard $\mathsf{\Lambda }$CDM model, dark energy begins to dominate the energy density at $z\sim 0.67$. The closeness in timing is worth noting.

From the energy scale perspective, when $G(z)$ transitions from the evolving state to the frozen state, the characteristic energy density released can be estimated as:

$$\mathsf{\Delta }\rho \sim {\displaystyle \frac{1}{16\pi G_0}}\cdot \left|{\displaystyle \frac{\dot{G}}{G}}\right|\cdot H\cdot \mathsf{\Delta }t$$

Substituting numerical values $\dot{G}/G{|}_{z\sim 0.5}\sim 10^{-11}$yr^-¹, $\mathsf{\Delta }t\sim 3$Gyr, $H\sim 10^{-18}s⁻¹$ we obtain:

$$\mathsf{\Delta }\rho \sim 10^{-47} {\mathrm{G}\mathrm{e}\mathrm{V}}^4$$

which is of the same order of magnitude as the observed dark energy density ${\rho }_{\mathsf{\Lambda }}\sim 10^{-47}{\mathrm{G}\mathrm{e}\mathrm{V}}^4$.

Whether this numerical relation has a deeper physical meaning remains to be confirmed by accurately incorporating the G(z) evolution into the Friedmann equations. Regardless of whether such a connection holds, the core conclusion of the QEET framework—the natural explanation of the $S_8$ crisis--remains unaffected.

5.4. Future Tests

Several tests can distinguish QEET from other models:

1) Scale dependence: QEET predicts that the evolution of G(z) suppresses structure formation, and since perturbations of different scales enter the nonlinear regime at different redshifts, this suppression may produce distinguishable effects on different scales. In particular, the oscillatory modulation has its maximum effect at $z\sim 0.5$, which may lead to features in weak lensing signals at specific angular scales that differ from $\mathsf{\Lambda }$CDM.

2) $\dot{\mathit{G}}/\mathit{G}$ measurements: QEET predicts that $\dot{G}/G$ at $z=0$ is below$10^{-15}{\mathrm{y}\mathrm{r}}^{-1}$, but in the oscillatory interval $z\sim 0.5-1$ it may produce detectable evolution signals. Future pulsar timing arrays or gravitational-wave standard siren observations may directly detect this redshift-dependent signature.

3) Direct detection of $\mathit{G}(\mathit{z})$: unlike modified gravity models, QEET predicts a specific oscillatory structure in G(z) (suppressed at $z\sim 0.5$, recovering at $\sim 2$). This non-monotonic behaviour can be independently tested by gravitational lensing or galaxy dynamics at multiple redshifts.

4) Cross-correlations: the QEET mechanism predicts specific correlations among CMB lensing, galaxy clustering, and weak lensing that differ from $\mathsf{\Lambda }$CDM predictions. The scale-dependent features produced by the oscillatory modulation may leave unique statistical signatures in large-scale structure data.

5.5. Relationship to Other Works Within the QEET Framework

Within the QEET framework, the macroscopic effects of the entropy flow field $s^{\mu }$ can yield testable physical signals in different observational windows, depending on the specific coupling paths between the entropy flow and spacetime. This work, together with the QEET main papers and another independent work, are complementary: they share the same physical picture—the influence of quantum information flow (entropy flow) on cosmic evolution—but focus on different observational windows and theoretical paths.

QEET main papers [16,17] establish the unified framework of Quantum Enthalpy-Entropy Theory, with the core being the Quantum Cosmological Emergent Bound (QCEB, Eq. (1)), revealing the fundamental complementarity between quantum uncertainty and entropy production. In this framework, the vacuum expectation value of the entropy flow field $s^{\mu }$ determines the emergent value of Newton's constant (Eq. (2)). This is a general theoretical framework, not limited to a specific observational window.

This work focuses on the cosmological evolution of $G(z)$, using the nonlinear saturation effect of the entropy field VEV to produce a time-varying Newtonian constant, thereby suppressing late-time structure formation and naturally explaining the $S_8$ crisis. The observational window is the amplitude of matter density fluctuations $S_8$, affecting structure growth by modifying the gravitational strength $G(z)$.

The PLB paper [31] takes a different approach, introducing the backreaction effect of the entropy flow at the level of perturbation equations, modifying the linear growth equation for density perturbations with an extra source term proportional to $\beta (a)H^2\delta$. The observational window of that work is the growth rate $f{\sigma }_8(z)$, predicting about 6.4% enhancement at $z=0$. Notably, the parameter ${\beta }_0$in that model can be positive or negative--positive ${\beta }_0$ enhances structure growth (the baseline case in that work), while negative ${\beta }_0$ suppresses it, providing another possible path to alleviate the $S_8$ tension.

The relationship among the three works is summarised in Table 13.

The $\beta (a)$parameter in the PLB paper and the $G(z)$ evolution effect in this work are complementary observational windows--the former modifies the growth equation, the latter modifies the gravitational strength. They are not redundant but rather reflect the same entropy flow physics at different levels of the equations.

Thus, these three works are not competing or contradictory, but complementary manifestations of the same physical picture in different observational windows and at different theoretical levels: the QEET main papers provide the theoretical framework, while this work and the PLB paper respectively examine the effects of the entropy flow on cosmic structure formation via $G(z)$ evolution and perturbation equation modifications. Together they show that the macroscopic effects of quantum information flow (entropy flow) are testable physical phenomena across multiple independent observational windows. If both the $S_8$ tension and the $f{\sigma }_8$ enhancement (or suppression) are confirmed in the future, it would provide strong cross-window support for the QEET framework.

5.6. On Parameter Sensitivity

The fundamental constants of nature often need to take specific values to produce an observable cosmic structure--the fine-structure constant $\alpha \approx 1/137$, the dark energy density ${\mathsf{\Omega }}_{\mathsf{\Lambda }}$, and so on. The specific values of the entropy field parameters in the QEET framework similarly reflect this fine-tuning principle rather than being evidence of human adjustment. The limited window of parameter space in fact demonstrates the predictive power of the theory: it restricts parameters to the region determined by theoretical consistency, rather than allowing unlimited adjustment.

5.7. Limitations and Future Directions

Several aspects of this analysis require further development:

1) Complete CMB analysis: Planck’s constraint on $S_8$ assumes $\mathsf{\Lambda }$CDM. A self-consistent analysis within QEET requires recomputing the CMB power spectrum with the modified $G(z)$.

2) Nonlinear structure formation: our analysis uses linear perturbation theory. Full N-body simulations with a time-varying $G$ are needed to make precise predictions for weak lensing surveys. The linear analysis in this paper is only a preliminary qualitative indicator; precise numerical comparisons await nonlinear calculations.

3) Parameter determination: more precise determination of $\xi$ and $\gamma$ would enhance the predictive power of the QEET framework. In particular, the normalisation scale of $\xi$ needs further clarification, and cross-checks with independent observations (e.g., BAO, strong lensing) are required to confirm the “naturalness” of their numerical values within the QEET parameter space. The value $\xi =-2.8\times 10^{-4}$(with reference to $m_s\sim 10^{-22}$eV) determined in this work is a preliminary optimal value, pending verification with more precise cosmological data.

4) Application to the early universe: the numerical analysis in this work covers the interval $z\in \left[\mathrm{0,1100}\right]$ , i.e., from recombination to the present. For earlier times (e.g., BBN at $z\sim 10^8-10^9$), the behaviour of $G(z)$ is not modelled or extrapolated here. The application of the QEET framework to the early universe requires independent study.

6. Conclusions

This paper has shown that the Quantum Enthalpy-Entropy Theory (QEET) provides a complete theoretical possibility for a natural explanation of the $S_8$ crisis. The parameters involved are not introduced to fit $S_8$ data but are dynamical quantities already present in the framework; this work determines their optimal values for explaining the $S_8$ crisis for the first time. The mechanism is:

1) The QEET framework already establishes that Newton's constant is determined by the VEV of the entropy flow field: $G=-1/(4\pi \xi ⟨s_{\mu }{s}^{\mu }⟩)$.

2) In the expanding universe, the entropy field VEV evolves according to a Proca-type equation coupled to spacetime curvature. At $z\gtrsim 2$, the system is dominated by smooth saturation evolution, ensuring that $G(z)$ is close to constant at early times; at $z\lesssim 2$, the system develops stimulated cosmological oscillation modes that modulate the evolution of $G(z)$ around $z\sim 0.5$.

3) Nonlinear saturation freezes $G$ at $z=0$, satisfying the stringent local constraint $\dot{G}/G\lesssim 10^{-15}$.

4) At intermediate redshifts $z\sim 0.5$, the synergy between smooth evolution and oscillatory modulation makes $G$ about 12.6% larger (i.e., $G(0.5)/G_0=0.874$) compared with the present value, suppressing late-time structure formation and producing about 6.9% suppression of $S_8$ relative to Planck.

5) The predicted $S_8$ value

$$S_8^{\mathrm{Q}EET}=0.776\pm 0.015$$

agrees with the central values of the weak lensing surveys (KiDS-1000, DES Y3), and differs from the Planck 2018 value ($(S_8=0.832\pm 0.013$) by $4.3\sigma$--which is a direct manifestation of the QEET mechanism suppressing late-time structure formation to explain the $S_8$ crisis.

6) The remaining tension could be resolved by scale-dependent effects of the QCEB, nonlinear corrections, or a self-consistent CMB analysis within the QEET framework.

7) Future tests: QEET predicts a specific oscillatory structure in the $G(z)$ evolution curve--constant at $\mathrm{z}\gg 2$ ($\mathrm{G}/{\mathrm{G}}_0\approx 1$), suppressed at $\mathrm{z}\sim 0.5$ to $\mathrm{G}/{\mathrm{G}}_0\approx 0.874$. This non-monotonic behaviour can be independently tested by gravitational lensing, galaxy dynamics, or future gravitational-wave standard sirens at multiple redshifts, distinguishing it from other modified gravity models.

This work identifies the $S_8$ crisis as a potential first observational signal of a time-varying Newtonian constant driven by quantum entanglement dynamics. It provides a concrete, testable realisation of emergent gravity from quantum information flow, and establishes the QEET framework as a strong candidate beyond the Standard Model and $\mathsf{\Lambda }$CDM.