Thermodynamic Emergence of Quantum Behavior from Recursively Coupled Energy Structure

1. Introduction

Unifying gravity with quantum physics remains one of the central challenges in theoretical physics. While general relativity (GR) passes all tests at macroscopic scales, its merger with quantum theory remains elusive. String theory, though extensively developed, fails to yield a unique low-energy limit, limiting its predictive power [1,2,3]. Loop quantum gravity (LQG) offers a background-independent quantisation of gravity and predicts discrete spectra for geometric operators [4,5]. Yet, it struggles to incorporate standard model fields and to recover classical GR in the semiclassical limit. An alternative class of models modifies gravity or introduces new dynamical fields. Scalar-tensor theories, including quintessence, invoke a dynamical scalar field to explain late-time cosmic acceleration [6,7], offering an alternative to a cosmological constant and addressing fine-tuning and coincidence problems [8]. Many modified gravity models, such as $f\left(R\right)$ and Horndeski theories, can be recast as scalar-tensor frameworks with non-minimal couplings [9]. Within quantum gravity, two complementary programs stand out. The asymptotic-safety scenario posits a non-perturbative UV fixed point that makes gravity fundamentally renormalizable [10,11]. Renormalisation-group flows drive gravitational couplings to finite values at Planckian scales, while IR fixed points can trigger spontaneous scale symmetry breaking, generating particle masses and dynamically selecting fundamental scales [12,13]. A concrete realisation is the cosmon framework, where a time-dependent scalar field remains scale-invariant at both fixed points but breaks symmetry in between accounting for inflation and dark energy without fine-tuning or extra fields [14,15]. Second, several approaches treat gravity as an emergent rather than fundamental interaction. Entropic and holographic scenarios derive Newtonian and Einsteinian dynamics from thermodynamic or information-theoretic properties of microscopic degrees of freedom, linking quantum information to geometry [16,17,18,19]. Recently, a consistent framework coupling classical gravity to quantum matter was proposed, avoiding the need to quantize the metric [20]. The theory is linear, completely positive, and trace-preserving, reproducing Einstein’s equations in the classical limit. Crucially, fundamental stochasticity in both geometry and matter allows the model to evade no-go theorems while naturally inducing decoherence without invoking measurement postulates. In a related framework, ’t Hooft models systems via discrete ontological states evolving under a unitary matrix, with quantum mechanics emerging as a statistical description and the wave function reflecting epistemic, not ontological, information [21]. A multiscale framework by Wetterich links particle mass generation and cosmic evolution to a time-varying scalar field, thereby avoiding an initial singularity [12]. Similarly, Verlinde’s entropic approach reframes gravity as an information-based ordering principle rather than a fundamental force [16]. While these frameworks offer theoretical insights, none has yet received decisive experimental support.

Rather than relying on a specific microscopic origin such as entropic, holographic, or informational mechanisms [12,16,17,21] the present model offers a structural unification of these perspectives through a dynamic, energy-based interpretation in which collective gravitational behavior arises organically from recursive multiscale organisation. Crucially, the local internal energy at each scale is defined as a functional of the internal energies of its subsystems, establishing a fully nested energy hierarchy. This recursive structure enables natural coupling across all physical scales and provides a mechanism for emergent quantum behavior, arising from structurally organized multiscale interactions governed by cosmological dynamics. The multiscale model thus invites for a reconsideration of foundational assumptions from a dynamic, network-based viewpoint.

The importance of multiscale modeling has long been recognized in other fields of science, notably in chemistry, where the 2013 Nobel Prize was awarded to Martin Karplus, Michael Levitt and Arieh Warshel for pioneering methods that couple quantum and classical regimes to simulate complex systems efficiently [22,23,24,25]. Building on this principle, the present framework extends the concept of cross-scale coupling beyond molecular systems by establishing recursive multiscale interaction as a structural foundation of physical organisation. This formulation introduces a dynamical paradigm in which Newtonian mechanics emerges from recursively structured multiscale couplings, rather than being postulated a priori. Gravitational theory remains operationally valid at macroscopic scales, yet its source terms are recast as effective expressions of deeply nested energy architectures across physical hierarchies.

Modeling decisions, such as which scales to resolve and with what precision, depend on the application. Quantum mechanical, mesoscopic, and classical descriptions may coexist across spatial regions without conceptual conflict, as long as coupling remains consistent. From this broader perspective, chemical multiscale modeling can be viewed as a mature, domain-specific realisation of a more general paradigm: one that integrates macroscopic behavior with microscopic structure through coherent scale-bridging strategies. Building on this idea, we now construct the model from a thermodynamic viewpoint, beginning with a classical analogy and postponing field-theoretic or continuum assumptions.

2. The Recursive Thermodynamic Model

Before the mathematical formulation is introduced, it is essential to clarify the terminology and conceptual framing employed in this work. Several expressions used throughout—such as “nested structure”, “non-equilibrium hierarchy”, and “thermodynamic emergence”—have specific, technical meanings within the context of statistical mechanics and systems theory.

The term nested structure refers to a hierarchically organized sequence of thermodynamic subsystems, each embedded within a larger one, and each contributing to the energetic and entropic balance of the next higher level. Such systems are known from climate models, biological metabolism, and multiscale modeling in statistical physics. In this framework, a quantum-scale system is assumed to be embedded within a structured thermodynamic background composed of non-equilibrium processes acting on multiple time and length scales.

Structured substrate here denotes this surrounding thermodynamic environment, whose influence cannot be fully resolved at the scale of the quantum system itself. This unresolved statistical interaction gives rise to apparent indeterminacy, which we interpret as epistemic uncertainty projected into Hilbert space formalism.

The notion of thermodynamic emergence is not meant in a metaphysical or ontological sense, but rather refers to a shift in the effective system description resulting from coarse-graining over unresolved energy exchanges. This concept is aligned with Jaynes’ view of statistical mechanics as inference theory and with interpretations of the quantum wavefunction as a knowledge-based statistical tool.

In particular, we do not assume or invoke any gravitational, cosmological, or superdeterministic mechanisms. The formalism remains fully compatible with conventional quantum theory, but explores the possibility of reinterpreting the wavefunction as a statistical projection arising from a deeply coupled thermodynamic embedding.

The proposed approach is based on two simple postulates:

Assumption A1. 

The universe is regarded as an isolated thermodynamic system ${\mathcal{S}}_0$ with no other systems of equal rank. Its total energy equals its internal energy $U_0$, as kinetic energy is initially set to zero. It remains unchanged in time

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}U=:{\dot{U}}_0=0.$$

(1)

This is interpreted as the minimum requirement for the universe.

Assumption A2. 

All astronomical systems, macroscopic matter, and composite particles except elementary particles possess a thermodynamic internal energy $U_{‌}$, in its simplest form defined as

$$U_{=}\sum _{\alpha }{E}_{\alpha }=\sum _{\alpha }{U}_{\alpha }^{\mathrm{loc}}+{\mathcal{H}}_{\alpha },$$

(2)

where $U_{\alpha }^{\mathrm{loc}}$ is the local internal energy and ${\mathcal{H}}_{\alpha }=K_{\alpha }+P_{\alpha }$ the Hamilton function, with $K_{\alpha }$ denoting kinetic and $P_{\alpha }$ potential contributions of the sublevels of U. Each $U_{\alpha }^{\mathrm{loc}}$ acts as a functional of the family of all recursively nested energy contributions of its subsystems, in the same manner as $U_{‌}$.

Importantly, the potential energy terms $P_{\alpha }$ must be understood as referring to the interactions between the systems included within the subsystem α

In this formulation, the energies are treated as globally integrated quantities but may be understood as arising from spatial integration over corresponding field variables. A local, differential representation of the first law, as common in continuum thermodynamics, is not used here. When point-like particles are considered, the continuum assumption may be partially dropped, and expressions known from classical mechanics are used in place of volumetric integrals.

Remark 1. [regarding Assumption A2:] More formally, U is defined as a recursively nested family of energy contributions ${\left\{E_{\alpha }\right\}}_{\alpha }$ where each element $E_{\alpha }$ depends on a local internal energy $U_{\alpha }^{\mathrm{loc}}$, itself determined by a subordinate family ${\left\{E_{\alpha \beta }\right\}}_{\beta }$ at the next sublevel

$$\begin{array}{c}\hfill {\left\{E_{\alpha }\left(U_{\alpha }^{\mathrm{loc}}\left({\left\{E_{\alpha \beta }\right\}}_{\beta }\right)\right)\right\}}_{\alpha }.\end{array}$$

(3)

Thereby, a fully nested nonlinear hierarchical structure is established, where $\alpha \in \{1,\dots ,A\}\subset \mathbb{N},A\in \mathbb{N},$ denotes the contributions at the first hierarchical sublevel (comprising A subsystems), and $\alpha \beta$, with $\beta \in \mathbb{N}$, refers to the contributions at the sublevel of subsystem α. This can be quantitatively expressed for the first two scales:

$$\begin{array}{c}\hfill U_{=}\sum _{\alpha =1}^A{E}_{\alpha }\left(U_{\alpha }^{\mathrm{loc}}\left({\left\{E_{\alpha \beta }\right\}}_{\beta }\right)\right),\mathrm{where}{E}_{\alpha }=U_{\alpha }^{\mathrm{loc}}\left({\left\{E_{\alpha \beta }\right\}}_{\beta }\right)+{\mathcal{H}}_{\alpha }.\end{array}$$

(4)

This procedure can straightforwardly be generalized to more than two hierarchical scales by considering

$$\begin{array}{c}\hfill {\left\{E_{\alpha }\left(U_{\alpha }^{\mathrm{loc}}\left({\left\{E_{\alpha \beta }\left(U_{\alpha \beta }^{\mathrm{loc}}\left({\left\{E_{\alpha \beta \gamma }(\dots )\right\}}_{\gamma }\right)\right)\right\}}_{\beta }\right)\right)\right\}}_{\alpha }.\end{array}$$

(5)

In general, each local internal energy $U_{\alpha \beta \dots \xi }$ is defined analogously in terms of its own subsystems,

$$U_{\alpha \beta \dots \xi }=\sum _{\zeta =1}^{Z\left(\xi \right)}{E}_{\alpha \beta \dots \xi \zeta },$$

thereby enabling unrestricted hierarchical recursion and capturing the multiscale nesting of energy contributions.

The second postulate serves as the foundational hypothesis of this framework implying that each energy contribution on one level acts as an emergent base for further decompositions, establishing nonlinear, multi-scale couplings from cosmic to quantum domains.

2.1. Internal Energy of the Universe

Based on the two postulates, the internal energy of the universe $U_0$ is given depending on multiple scales

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_0\stackrel{!}{=}\mathrm{const}.& =\sum _{\alpha =1}^{J_0}{E}_{\alpha }=\sum _{\alpha =1}^J{\mathcal{H}}_{\alpha }+U_{\alpha }^{\mathrm{loc}}\hfill \\ \hfill & =\sum _{\alpha =1}^{J_0}\left({\mathcal{H}}_{\alpha }+\sum _{\beta =1}^{J\left(\alpha \right)}\left({\mathcal{H}}_{\alpha \beta }+\sum _{\gamma =1}^{J\left(\beta \right)}\left({\mathcal{H}}_{\alpha \beta \gamma }+\dots +U_{\alpha \beta \gamma \dots \zeta }^{\mathrm{loc}}\right)\right)\right)),\hfill \end{array}\end{array}$$

(6)

abbreviated as

$$\begin{array}{c}\hfill \sum _{\begin{array}{c}\alpha ,\beta ,\gamma ,\dots ,\zeta \end{array}}^{J\left(0,\alpha ,\beta ,\gamma ,...,\zeta \right)}{\mathcal{H}}_{\alpha \beta \gamma ...\zeta }+U_{\alpha \beta \gamma ...\zeta }^{\mathrm{loc}}=:{\mathcal{A}}_{\begin{array}{c}\alpha ,\beta ,\gamma ,\dots ,\zeta \end{array}},\end{array}$$

(7)

where $\mathcal{A}$ is introduced as an energy-operator, with the indices as information about the scales are to be considered. For the universe, taking all scales into account, there is a further definition

$$\begin{array}{c}\hfill {\mathcal{A}}_{\begin{array}{c}\alpha ,\beta ,\gamma ,\zeta ,\dots ,\xi \end{array}}=:{\mathcal{A}}_{\square }=U_0\stackrel{!}{=}\mathrm{const}..\end{array}$$

(8)

2.2. Further Consequences Resulting from the Recursive Model

Each subsystem obeys a local form of the first law

$$\begin{array}{c}\hfill \dot{E}=\dot{U}+\dot{K}+\dot{P}=\dot{W}+\dot{Q},\end{array}$$

(9)

yet remains embedded in a globally coupled thermodynamic structure, as given in Equation (8). Consequently, no subsystem evolves independently: cosmic-scale variations such as orbital motion, solar output, or gravitational shifts propagate downward and modulate internal energy states across scales.

This has direct thermodynamic implications. According to the Clausius-Duhem inequality of rational thermodynamics [26]:

$$\begin{array}{c}\hfill T{\dot{s}}^{\mathrm{irr}}=T\dot{s}-{\mathcal{P}}^{‌}+\dot{u}+{\displaystyle \frac{1}{T}}\overrightarrow{q}·\overrightarrow{\nabla }T\ge 0,\end{array}$$

(10)

the irreversible part of the entropy rate is directly proportional to the rate of internal energy accumulation. Here, ${\dot{s}}^{\mathrm{irr}}$ denotes the irreversible part of the entropy rate, $\dot{s}$ the total entropy rate, $\mathcal{P}$ the power density, $\dot{u}$ the rate of internal energy accumulation, $\overrightarrow{q}$ the heat flux vector, and $\overrightarrow{\nabla }T$ the temperature gradient. All quantities are defined per unit volume as common in continuum formulation of balance equations e.g., $U={\int }_{\Omega }u\mathrm{d}V$. This obviously excludes the possibility of total equilibrium states if u exhibits the nested, dynamically modulated structure of postulate 2. Non-equilibrium behavior is then not incidental but driven by persistent energy redistribution from cosmic dynamics.

In local formulations, global couplings remain implicitly embedded, much like boundary conditions in classical mechanics encode global constraints into local motion or how path integrals embed the full history of all paths into the current amplitude. Effective parameters such as mass, energy or charge must then absorb multiscale influences by construction. Effective physical parameters such as mass and inertia, respectively, electric charge or energy must then, by construction, absorb and reflect multiscale influences from these nested structures. In particular, experimentally accessible ratios such as the mass-to-charge ratio $m/q$ may serve as sensitive probes for underlying hierarchical effects. The resulting structure acts as an informational mediator across scales, much like spacetime in general relativity encodes and transmits the mutual dependence between energy distributions and geometric curvature

This closure renders gauge freedom superfluous, not as a contradiction to field theory, but as a consequence of full resolution. Gauge invariance thus reflects incomplete modeling, not fundamental symmetry.

3. Emergence of Newtonian Laws from Recursive Energy Structures

In this framework, Newton’s axioms emerge as specific limiting cases. The first law follows by considering a universe containing only a single system with ${\dot{U}}_0={\dot{E}}_1=0$ (point mass). From this, the classical formulation of inertia naturally arises

$$\begin{array}{c}\hfill {\dot{K}}_1=\dot{\overrightarrow{p}}·\dot{\overrightarrow{x}}=0\Rightarrow \dot{\overrightarrow{p}}=0\forall \dot{x}\ne 0,\end{array}$$

(11)

where $\dot{\overrightarrow{x}}$ and $\dot{\overrightarrow{p}}$ are referring to the vectors of velocity and linear momentum. Without the minimum requirement for the universe, the result is not tenable without detailed statements about the work performed on the system and the transferred heat. If we furthermore consider the case of two point-mass systems in the universe from the perspective of classical physics, the well-known relationships emerge:

$$\begin{array}{c}\hfill {\overrightarrow{F}}_1=-{\overrightarrow{F}}_2\mathrm{and}{m}_i{\ddot{\overrightarrow{x}}}_i={\overrightarrow{F}}_i^{\mathrm{G}}\forall {\dot{x}}_i\mathrm{with}i=1,2.\end{array}$$

(12)

These relations follow entirely without invoking Newton’s axioms or assuming energy conservation. Instead, they result directly from the minimum condition for the universe (Postulate 1), together with Postulate 2, via:

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\mathcal{A}}_{\square }=\sum _{\alpha =1}^2{\dot{K}}_{\alpha }+{\dot{P}}_{\alpha }={\dot{\overrightarrow{x}}}_1\left(m_1{\ddot{\overrightarrow{x}}}_1-G{\displaystyle \frac{m_1{m}_2}{r^2}}\right)+{\dot{\overrightarrow{x}}}_2\left(m_2{\ddot{\overrightarrow{x}}}_2+G{\displaystyle \frac{m_1{m}_2}{r^2}}\right)\stackrel{!}{=}0.\end{array}$$

(13)

This holds for all velocities. Here $r=\left|\right|{\overrightarrow{x}}_1-{\overrightarrow{x}}_2\left|\right|$ and $F^{\mathrm{G}}$ is the Newtonian gravitational force. If the above is interpreted as a derivation by prefixing postulates 1 and 2, then the foundational role typically ascribed to these equations becomes questionable. This demonstrates that Newton’s second and third laws arise as consistency conditions of recursive multiscale couplings, rather than as independent postulates. This equation no longer applies if the system is not a point mass and therefore has a non-zero internal energy.

4. Hierarchical Energy Coupling and Local Observables

4.1. Thermodynamic Fluctuations Across Scales

Illustrating structural interdependence across scales, we consider a molecule on Earth $E_{\mathrm{E}1}^{\mathrm{M}}$ as subsystem in the multiscale structure of the planet. The first law of thermodynamics in conjunction with Earth’s internal energy, according to postulate 2, provides

$$\begin{array}{c}\hfill {\dot{E}}_{\mathrm{E}1}^{\mathrm{M}}={\dot{W}}_{\mathrm{E}1}^{\mathrm{M}}+{\dot{Q}}_{\mathrm{E}1}^{\mathrm{M}}={\dot{U}}^{\mathrm{E}}-\left(\sum _{\beta =2}^{J_E}{\dot{U}}_{\mathrm{E}\beta }+{\dot{\mathcal{H}}}_{\mathrm{E}\beta }\right)=:{\dot{U}}^{\mathrm{E}}-{\dot{U}}_{\mathrm{E}}^{\mathrm{residual}}.\end{array}$$

(14)

The indices M and E indicate the contributions of the molecule and Earth. Obviously, the total energy of the molecule $E_{\mathrm{E}1}^{\mathrm{M}}$ changes in the same way as the difference $U_{\mathrm{E}}-U_{\mathrm{E}}^{\mathrm{residual}}$, e.g., when the Earth adopts a different orbit and thereby alters its energy level, or when $U_{\mathrm{E}}$ and $U_{\mathrm{E}}^{\mathrm{residual}}$ become identical or both vanish. Due to strong couplings, $U_{\mathrm{E}}$ and $U_{\mathrm{E}}^{\mathrm{residual}}$ cannot be treated as independent. Temporarily relaxing this constraint allows hypothetical scenarios involving changes in the velocity of the considered material subsystem. Large masses and high velocities significantly affect this balance, forcing the system to reduce its energy and, in extreme cases, perturbing the orbital dynamics of the carrier system. Such recursive feedback propagates across structurally coupled systems, both laterally and vertically, triggering scale-spanning energy reconfigurations throughout the nested hierarchy. This conceptual exploration underscores that local energy fluctuations cannot remain isolated within the total system and illustrates why the universe, given its vast energy content, large masses, and high velocities on average exhibits remarkable large-scale stability. The macroscopic influence on the microscale becomes particularly visible when ${\dot{U}}_{\mathrm{E}}$ is substituted in Equation (14) via the local first law, applied to the Earth system

$$\begin{array}{c}\hfill {\dot{E}}_{\mathrm{E}1}^{\mathrm{M}}={\dot{W}}_{\mathrm{E}1}^{\mathrm{M}}+{\dot{Q}}_{\mathrm{E}1}^{\mathrm{M}}={\dot{W}}_{\mathrm{E}}+{\dot{Q}}_{\mathrm{E}}-{\dot{\mathcal{H}}}_{\mathrm{E}}-{\dot{U}}_{\mathrm{E}}^{\mathrm{residual}}.\end{array}$$

(15)

It becomes obvious that the external energies of the Earth as well as the work performed on the Earth and the heat flux across Earth’s surface have an influence on the behavior of all matter and its subsystems.

4.1.1. Brownian Dynamics Under the Influence of Cosmological Fluctuations

Moreover, Equations (14) and (15) suggest that classical Brownian motion may not be purely stochastic in origin, but rather a local manifestation of globally structured fluctuations projected onto subsystems. This becomes evident when recast in the form of a Langevin-type equation:

$$\begin{array}{c}\hfill m\ddot{x}={\displaystyle \frac{1}{\dot{x}}}\left({\dot{W}}_{\mathrm{E}}+{\dot{Q}}_{\mathrm{E}}-{\dot{\mathcal{H}}}_{\mathrm{E}}-{\dot{U}}_{\mathrm{E}}^{\mathrm{residual}}-{\dot{P}}_{\mathrm{E}1}^{\mathrm{M}}-{\dot{U}}_{\mathrm{E}1}^{\mathrm{M}}\right).\end{array}$$

(16)

Whereas conventional models like the Langevin approach treat friction and noise as externally imposed stochastic inputs, this framework derives both as intrinsic manifestations of recursive multiscale energy exchange. This perspective generalizes the fluctuation-dissipation relation and reveals how nested thermodynamic structures systematically constrain emergent motion.

4.2. Cosmic Effects on Quantum Systems

For a refined understanding of quantum behaviour we start with examining Earth’s internal energy, which, when resolved to quantum scales following postulate 2, reveals the following structure:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill U^{\mathrm{E}}=\sum _{\alpha =1}^J\left({\mathcal{H}}_{\alpha }^{\mathrm{E}}+\sum _{\beta =1}^{J\left(\alpha \right)}\left({\mathcal{H}}_{\alpha \beta }^{\mathrm{E}}+\sum _{\gamma =1}^{J\left(\beta \right)}\left({\mathcal{H}}_{\alpha \beta \gamma }^{\mathrm{E}}+\sum _{\gamma =1}^{J\left(\gamma \right)}\left({\mathcal{H}}_{\alpha \beta \gamma \delta }^{\mathrm{E}}+U_{\alpha \beta \gamma \delta }^{\mathrm{E},\mathrm{loc}}\right)\right)\right)\right)).\end{array}\end{array}$$

(17)

The electron of interest, e.g., in quantum level 1111, is removed from its interacting background and its energy is expressed in isolated form as

$$\begin{array}{c}\hfill U_{1111}^{\mathrm{E}}+{\mathcal{H}}_{1111}^{\mathrm{E}}=E_{1111}^{\mathrm{E}}.\end{array}$$

(18)

Neglecting spin and exchange terms for simplification, the Schroedinger equation follows as

$$\begin{array}{c}\hfill {\mathcal{H}}_{1111}^{\mathrm{E}}\varphi =E_{1111}^{\mathrm{E}}\varphi .\end{array}$$

(19)

Naturally, ${\mathcal{H}}_{‌}$ here becomes an operator since the kinetic term in Equation (19) retains its standard quantum-mechanical form, as it is unaffected by the interpretational shift introduced in this framework. The Schroedinger equation (19) describes an electron in an effectively isolated state, neglecting explicit multiscale interactions with its cosmological environment. Due to the principled inaccessibility of fully resolving all multiscale interactions in Equation (17), the wave function $\varphi$ in Equation (19) must be regarded not as a fundamental ontological entity, but as an effective emergent representation of unresolved recursive multiscale couplings within a hierarchically structured universe. These unresolved couplings are implicitly encoded in the wave function. This perspective interprets quantum behavior including superposition, interference, and apparent indeterminacy as emerging from coarse-grained constraints and global boundary conditions rather than intrinsic randomness or isolation. The Schrödinger equation thus captures not an isolated microstate, but a projection of nonlocally correlated, scale-transcending dynamics into a computationally tractable form. This is in agreement with [21], where quantum mechanics emerges only as an effective statistical description and the wave function encodes epistemic rather than ontological information. In the case of a multi-electron system, explicit interactions are often introduced retrospectively. However, these methods mainly consider interactions within the system and do not take into account the higher astronomical scales that would trivially be included in our multiscale model, see Appendix A.

4.3. Zero-Point Energy as an Emergent Property of Recursive Thermodynamic Partitioning

The Hamilton function of an ideal/monoatomic gas is given by the following equation

$$\begin{array}{c}\hfill {\mathcal{H}}_{\alpha }={\displaystyle \frac{3}{2}}N{k}_{\mathrm{B}}T+P.\end{array}$$

(20)

Here, N is the particle number of the considered system, and $k_{\mathrm{B}}$ is the Boltzmann constant. If the temperature is set to zero, the Hamilton function vanishes, but the total energy does not disappear as long as the substructures $\beta$ possess quantum mechanical degrees of freedom ${\displaystyle \frac{1}{2}}\hslash {\omega }_{\alpha \beta }>0$ per mode

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E_{\alpha }(T=0)=U_{\alpha }+0+P_{\alpha }& =\sum _{\beta }{U}_{\alpha \beta }+{\mathcal{H}}_{\alpha \beta }+P_{\alpha }\hfill \\ \hfill & =\sum _{\beta }{\displaystyle \frac{1}{2}}\hslash {\omega }_{\alpha \beta }+{\mathcal{P}}_{\alpha \beta }+P_{\alpha }\ge 0.\hfill \end{array}\end{array}$$

(21)

This equation is embedded within the multiscale model and must also satisfy the equations governing the higher-order systems even if $P_{\alpha }$ and $P_{\alpha \beta }$ are zero; thus, it is evident that it cannot vanish. Only if there were no further cosmic energy changes, implying an absolutely static universe. Consequently, within this recursive multiscale framework, a non-thermal energy remains that is not attributable to statistical thermodynamics, but emerges from the embedded energetic dependencies between structural levels. While this residual energy structurally resembles quantum mechanical zero-point energy, it originates from classical, deterministic nesting rather than quantum uncertainty.

4.4. Remarks on the Uncertainty Principle of Quantum Physics

Remarkably, this framework suggests that quantum uncertainty is not a manifestation of intrinsic randomness, but an emergent consequence of constrained energy nesting across hierarchically structured physical scales. The Heisenberg uncertainty principle thereby arises from residual structural constraints imposed by unresolved subscale interactions, rather than from fundamental indeterminacy.

Heisenberg’s measurement-based relation can thus be interpreted as a special case of a broader principle: any localization attempt perturbs a subsystem’s energetic embedding within the recursive multiscale hierarchy, inducing a redistribution of constrained degrees of freedom. This redistribution manifests as effective uncertainty. Without such hierarchical modulations, neither quantum fluctuations nor stable bound states would be sustainable.

In the case of structureless particles such as electrons, recursive feedback does not originate from internal substructure, but from coupling with thermodynamically organized modes in the surrounding environment. Even under nominal isolation, such particles remain energetically embedded within their contextual hierarchy—rendering apparent randomness a macroscopic expression of unresolved energetic entanglement rather than intrinsic quantum noise.

5. Constraining the Energy-Momentum Tensor through Recursive Thermodynamic Structure

Having established the recursive thermodynamic framework we now turn to its implications for spacetime geometry. Jacobson famously showed that Einstein’s equations can be derived as an equation of state by applying the Clausius relation to local Rindler horizons [18]. Lemaitre interpreted the cosmological constant as a form of vacuum energy by comparing geometric terms to a perfect-fluid energy-momentum tensor [27]. This suggest that gravity may emerge from underlying thermodynamic structure. While Einstein’s equations relate geometry to energy-momentum, they do not specify the internal origin of $T_{\mu \nu }$. Rather than postulating fields or fluids, our framework implies in accordance with [18] that $T_{\mu \nu }$ emerges from recursively nested internal energies:

$$T_{\mu \nu }^{\mathrm{eff}}={\mathcal{T}}_{\mu \nu }\left[U^{\mathrm{loc}}\left(\left\{E_{\alpha \beta \dots \gamma }\right\}\right)\right].$$

Rather than starting from a predefined spacetime geometry, this framework suggests a reversal of the conventional approach: the energy-momentum tensor $T_{\mu \nu }$, derived from recursive thermodynamic structure, serves as the primary object. Geometry then emerges by solving the Einstein field equations with prescribed $T_{\mu \nu }$. In this view, spacetime curvature is not fundamental, but constrained by the nested energy architecture of matter.

6. Conclusions and Discussion

With this thermodynamically consistent, recursive multiscale model, quantum phenomena such as the wave function and zero-point energy, but as well as Brownian motion are no longer considered fundamental, but emerge from energy exchanges across hierarchically nested physical systems. The second postulate serves as the foundational hypothesis of this framework, implying that each energy contribution on one level acts as the emergent base for further decompositions. This establishes nonlinear, cross-scale couplings that span from cosmological to quantum domains. Unlike traditional models that invoke distinct fundamental interactions at each scale, the present framework postulates a unified recursive energy structure, governed by the first law of continuum thermodynamics across scale transitions. Ultimately, the wave function is reinterpreted as an effective descriptor of unresolved multiscale couplings. It no longer describes a static microstate but encodes the system’s current energetic embedding within the multiscale hierarchy. This perspective redefines locality as a scale-relative concept and opens paths to experimental access of global energetic correlations—for instance, via high-precision interferometry sensitive to cosmological background modulations , see Appendix B. Within this structure, true equilibrium states are fundamentally inaccessible, as the dynamics of the cosmic scale permanently perturb any lower-scale configuration. Thus, local equilibria can only exist transiently, stabilized by continuous energetic compensation across scales. This implies that inertia is not static, but subject to cyclic adjustment through interscale energetic feedback. Consequently, geometry itself may not be a fundamental input but a derived construct determined from the energy-momentum tensor. A full derivation of the metric remains open, but the structure suggests that the geometry is constrained by the recursive multiscale model. The model generalizes and embeds approaches such as Wetterich’s variable-mass cosmology and Verlinde’s entropic gravity into a unified energetic architecture grounded in recursive thermodynamic structure. In Wetterich’s scenario, cosmological redshift arises from increasing particle masses rather than metric expansion. This concept is extended here by permitting energy or mass modulations on one hierarchical level to induce compensatory redistributions on others, consistent with global conservation constraints. These interscale adjustments impose nonlocal consistency conditions, which may manifest as quantum uncertainty, relativistic inertia modulation, or effective gravitational interaction.