# A DIALECTICAL MATERIALIST ANALYSIS ON PATH DEPENDENCE, IRRE-VERSIBILITY AND TELEOLOGY IN PHYSICAL SYSTEMS

José Mauricio Gómez

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José Mauricio Gómez. A DIALECTICAL MATERIALIST ANALYSIS ON PATH DEPENDENCE, IRRE-VERSIBILITY AND TELEOLOGY IN PHYSICAL SYSTEMS. 2025. hal-04948205

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# A DIALECTICAL MATERIALIST ANALYSIS ON PATH DEPENDENCE, IRREVERSIBILITY AND TELEOLOGY IN PHYSICAL SYSTEMS

José Mauricio Gómez Julián

jose.gomezjulian@ucr.ac.cr, josemgomez@protonmail.ch

## Abstract

This research addresses Loschmidt's paradox—the conflict between microscopic reversibility and macroscopic irreversibility—through dialectical materialism, suggesting that the contradiction stems from theoretical modeling limitations rather than nature itself. We demonstrate how physical systems inherently evolve toward maximum relative stability within material constraints.

Analyzing quantum phenomena reveals probabilistic interpretations as epistemological rather than ontological limits. Path dependence emerges as more fundamental than temporality, with time serving as the structural framework for recording material transformations. This perspective reframes the least action and ground state principles as expressions of matter's inherent teleological nature.

This synthesis reconciles determinism and finality by demonstrating how system histories shape stability-oriented development, offering testable predictions. We propose incorporating historicity and teleology as fundamental features to advance physical theory and bridge quantum descriptions with other domains.

## 1. INTRODUCTION

Loschmidt's paradox serves as a particularly suitable starting point for an in-depth analysis of irreversibility and, from there, for studying path dependence and teleology in physical systems. This paradox is especially relevant because it reveals a fundamental tension between our theoretical models and observable reality—a tension that arises not from a true contradiction in nature but from inherent limitations in our way of conceptualizing physical phenomena.

As will be argued throughout this investigation, this apparent paradox allows us to identify the need to reformulate our understanding of physical systems to organically incorporate both irreversibility and path dependence, recognizing that these are not anomalies to be explained but fundamental characteristics of material reality. This perspective, grounded in dialectical materialism, also enables us to establish deep connections between the irreversibility observed in physical systems, the teleological nature of their transformations, and the universal tendency toward states of maximum relative stability possible given the material constraints of each system.

The paradox in question arises from an apparent contradiction between two levels of physical reality:

1. *Microscopic (fundamental) level:* The physical laws governing the behavior of individual particles are time-reversible. This means that if we were to record a video of particles interacting and play it backward, that inverse sequence would also be physically valid and would obey the same laws.

2. *Macroscopic (observable) level:* However, in our everyday experience, processes have a clear temporal direction. For example, a broken egg does not “unbreak”. coffee with milk does not spontaneously separate, and thermal energy always flows from regions of higher
temperature to those of lower temperature, which entails a constant increase in entropy (second law of thermodynamics).

Thus, the question posed by the paradox is as follows: How can irreversible macroscopic behavior emerge from perfectly reversible microscopic laws? In other words, if each individual interaction between particles is reversible and if macroscopic objects are composed of these particles, why do we never see, for example, a broken egg "rewind" and reassemble itself? More formally, why do we experience a unidirectional arrow of time?

This means that the laws of physics imply the potential reversibility of thermodynamic states, but in observable physical reality, this potentiality never materializes. This paradox challenges our understanding of the relationship between the microscopic and macroscopic factors, the nature of time and causality, and the profound meaning of irreversibility in physics.

## 2. PHILOSOPHICAL FRAMEWORK: THE DIALECTICAL MATERIALIST APPROACH

### 2.1. Background

Heisenberg's uncertainty principle led, from the dialectical materialist point of view, to several interpretations of macroscopic irreversibility: as a limitation in accessing microscopic information, as an entropy's connection to information loss, or as a reflection of our increasing ignorance of microscopic states. Although all these arguments are partially true, the key insight is that these interpretations reveal a deeper conceptual tension in how we understand the relationships between the different levels of physical reality.

To resolve this tension, we must first understand science itself. As Rosental & Iudin (1971, p. 65) define it, science represents a historically developed system of knowledge that continuously validates itself through social practice. Its power lies not merely in observation but also in the discovery of universal laws beneath apparent chaos, progressing from simple cause-effect relationships toward increasingly fundamental principles of existence and thought.

This definition suggests that science's ultimate goal extends beyond mere description—it aims to construct theories that provide the most accurate possible representation of material reality, constrained only by our current knowledge level. This framework helps us understand why the paradox of irreversibility reflects not error in our current explanations but rather limitations in our conceptual approach to multilevel physical phenomena.

Here, a carefully compressed version that maintains the logical flow and key criticism while improving clarity is used:

Loschmidt's paradox suggests that we need to fundamentally rethink our approach to macroscopic laws, incorporating irreversibility as an explicit feature rather than treating it as paradoxical when theory diverges from observable reality. Recent attempts to resolve this through Laplace's demon and Landauer's principle (Binder, 2023, p. 32) illustrate a deeper problem in our theoretical approach: using idealized constructs to explain why reality deviates from idealization is inherently circular—equivalent to using fiction to explain why reality does not conform to fiction.

Moreover, invoking Landauer's principle merely shifts the paradox into information theory without addressing the fundamental question of reality's preferred directional nature. The solution,
therefore, lies not in constructing more sophisticated idealizations but in developing theories that take observed irreversibility as their starting point.

While current macroscopic laws, such as thermodynamics' second law, incorporate irreversibility statistically, we must ask a more fundamental question: should our basic theories themselves reflect the irreversibility we observe in nature? This shifts our inquiry from questioning reality's apparent irreversibility to examining our theoretical frameworks' limitations.

This critique does not reject probabilistic tools in science—they remain valuable for handling computational complexity and managing information in complex dynamical systems. However, we must distinguish between probability's epistemological validity as a tool and claims about reality's inherently probabilistic nature. Mistaking our probabilistic models (the map) for physical reality (the territory) can constrain scientific development by making us settle for statistical explanations rather than seeking deeper causal understanding: “When the terrain disagrees with the map, trust the terrain”, says a Swiss Army proverb. Considering weather prediction, its unpredictability does not mean that weather is inherently random but rather that it is a deterministic system of such complexity that probabilistic models serve as our best approximation.

This distinction becomes crucial in quantum mechanics, where Heisenberg's uncertainty principle represents an ontological fact about reality, not merely an epistemological limitation such as our probabilistic models. This reveals a new kind of determinism—not the classical variety promising perfect prediction but rather the recognition that phenomena develop through interconnected chains of events constrained by dialectical relationships such as position-momentum complementarity. This framework requires us to develop theories that organically incorporate complexity and uncertainty as fundamental features of reality, not as limitations to overcome.

## 2.2. Necessity, Causality and Probability Theory

As Blokhintsev (1968, p. 14) noted, classical mechanics' assumption of uniquely determining future states from initial conditions excludes randomness, yet uncertainty from initial data grows over time until prediction becomes meaningless. This necessitates both conceptual and instrumental probabilistic considerations in physics. While Kolmogorov's axioms established a mathematical foundation linking probability theory to measure theory, in dialectical materialism, randomness serves as an epistemological category reflecting our measurement and conceptual limitations in tracking causal chains rather than negating ontological determinism. It functions as a tool for discovering regularity in mass events, where apparent chance can be progressively integrated into broader deterministic frameworks without reverting to mechanical determinism.

*Necessity*, as defined by Rosental & Iudin (1971, p. 334), represents what must occur under given conditions, whereas *chance* emerges from external phenomena acting upon the system under study. Dialectical materialism uniquely bridges these concepts, rejecting both subjective idealism's reduction to pure chance and Laplacian determinism's complete denial of it. Probability theory then emerges as the systematic study of mass chance events, where stable experimental patterns reveal themselves through statistical properties when similar conditions are reproduced (Rosental & Iudin, 1971, pp. 454-455).

## 2.3. On the Teleology In General

In analyzing teleology within mechanics, we must first address Hegel's materialist question regarding whether practical ends reveal a mechanism's truth or vice versa. Hegel (1976, p. 647) defines teleology as self-determination and establishes truth as the criterion for resolving this relationship. Just as the objective world presents incomplete, phenomenal aspects of truth (Hegel,
1976, p. 647), understanding the partial requires grasping the absolute—which conditions itself. Our analysis therefore examines cutting-edge research across diverse phenomena to demonstrate how systems universally tend toward maximum achievable stability within their available resources.

This universal tendency toward stability manifests across multiple scales. Alexander et al. (2021, pp. 5--71) demonstrate how the universe itself exhibits this characteristic through its capacity to 'learn' and optimize its laws to maintain structure and complexity. At the molecular level, Oparin (2021, pp. 46-48, 52-53) shows how living systems achieve dynamic stability through coordinated chemical reactions, which was experimentally verified by Matsuo & Kurihara (2021, pp. 1-2) in their work with coacervates. Haldane (2022, pp. 4, 10-11) complements this by identifying the critical transition between living and inert matter.

Stability manifests in genetic systems, where DNA serves as a stable template for transcription (Nature, 2014), and in biological populations, where selective forces operate in multiple stable equilibria (Lewontin, 1977, p. 283). This extends to more fundamental levels, as demonstrated by Shepherd et al. (2020, p. 1761) on DNA structural motifs and Teves & Henikoff's (2014, p. 215) observations on DNA torsion-mediated stability.

However, when systems cannot achieve or maintain maximum relative stability, they undergo qualitative transformations. This occurs across various well-known scales: stars exhaust their nuclear fuel to become white dwarfs or explode as supernova, quantum systems experience structural changes under modified external parameters, autocatalytic chemical systems generate new patterns when reactants deplete, and the graphite-to-diamond transformation under pressure, among other changes, exemplifies such fundamental structural changes.

This pattern extends to biological systems, where Gould's punctuated equilibria show periods of stasis followed by rapid speciation, and to ecological systems, where ecosystem tipping points mark abrupt transitions in structure and function. Similar transformations appear in neuroplasticity, where neural networks reorganize in response to injury or learning, and even in social systems, where revolutions mark structural transitions when existing relations of production become incompatible with developing productive forces.

However, what happens if it is not feasible for the system to achieve the maximum possible relative stability or if its fundamental structure does not remain invariant in the long term even upon achieving it?

When systems cannot achieve or maintain maximum relative stability, they undergo qualitative transformations across multiple scales of reality. In physical systems, this manifests when stars exhaust their nuclear fuel, transforming into white dwarfs, neutron stars, black holes, or exploding as supernovae to seed new stellar formations. Similar structural transformations occur in quantum systems under modified external parameters and in chemical systems when reactants are depleted, as evidenced by the pressure-induced transformation of graphite to diamond.

Biological systems demonstrate analogous behavior through Gould's punctuated equilibria, where species undergo rapid changes after periods of stasis, and through more common phenomena such as insect metamorphosis. At the ecosystem level, these transformations appear as tipping points—sudden changes in structure and function when environmental thresholds are exceeded.
This pattern extends to increasingly complex systems: neuronal networks reorganize through plasticity in response to injuries or learning; social systems undergo revolutions when production relations become incompatible with productive forces; and even computational systems exhibit phase transitions in optimization problems and qualitative changes in AI behavior upon reaching complexity thresholds.

Systems that deplete their resources undergo qualitative transformations, either adopting new fundamental structures to resume stability-seeking or becoming components of other systems. This observation helps us understand the relationship between teleology and mechanisms. According to Hegel (1976, pp. 647--648), teleology implies self-determination, whereas mechanisms represent pure external determination.

Mechanism, Hegel (1976, p. 649) argues, represents necessity abstracted from concrete content, accessible primarily through intellect's conceptual framework. Teleology, conversely, depends on concrete characteristics and seeks maximum relative stability through both intrinsic determination (essence) and extrinsic determination (accidental qualities). While teleology's general form represents an infinite process toward intrinsic purpose, its extrinsic manifestations remain finite (Hegel, 1976, p. 649).

German Classical Idealism established that intermediate (extrinsic) teleology subordinates to ultimate (intrinsic) teleology, a principle that modern research continues to validate. At the cosmic scale, Barrow & Tipler (1986, pp. 123--205) suggest that the universe's laws are finely tuned for intelligent life, whereas Smolin (1997, pp. 257--323) proposes that universes evolve to optimize black hole production—both representing intrinsic cosmic teleology.

In biological systems, Wilson & Wilson (2007, pp. 339-342) demonstrate how individual bacterial objectives become subordinate to organism-level functioning, paralleling how colony members' goals serve the colony's adaptive needs. This multilevel organization principle extends to neuroscience, where Friston's (2010, pp. 127--128) free energy principle shows how immediate neural processes serve long-term predictive optimization.

Similar patterns emerge in ecology, where Odum (1969, p. 262) shows that ecosystem changes aim toward climax states, and Bak et al. (1987, p. 381) demonstrate complex systems' evolution toward critical states. This hierarchical organization also appears in artificial systems, with reinforcement learning subordinating immediate rewards to long-term optimization (Sutton & Barto, 2018, pp. 57-59), and in psychology, where Ryan & Deci (2000, pp. 68-73) show that extrinsic motivations serve intrinsic needs.

Even in evolutionary game theory, Smith & Price (1973) demonstrate how short-term competitive strategies ultimately serve long-term evolutionary stability, illustrating how immediate objectives are consistently subordinate to fundamental tendencies (i.e., long-term tendencies) across complex systems.

Following Hegel (1976, p. 649), while both intermediate teleology and mechanisms arise from the same concrete phenomena, their relationships with this concrete content differ fundamentally. For mechanism, concrete content remains accidental, whereas for teleology, even intermediate teleology, it forms an essential part of its nature. This closer connection to concrete reality makes teleology, even in its extrinsic form, a more fundamental truth than a mechanism.
This principle strengthens with intrinsic teleology, which encompasses both concrete particularity and unlimited developmental potential. However, unlike Hegel's idealistic framework, we ground this understanding in material reality. Modern research supports this materialist interpretation: Vopson's work (2019, pp. 1-3; 2021, p. 3; 2022, pp. 3-4), building on Landauer and Shannon's foundations, demonstrates how matter's various states translate into physical information.

The materialist justification for the subordinating mechanism to teleology lies in understanding development's end not as an ideal state but as the necessary result of material contradictions. Recent research has reinforced this view: Terekhovich (2016, p. 2) shows how particles select trajectories on the basis of final states, whereas Alexander et al. (2021, pp. 1, 5--71) demonstrate the universe's tendency to evolve toward maximum long-term stability. Thus, teleology emerges not as a spiritual guide but as an objective characteristic of material processes, where purpose arises from the inherent contradictions of matter.

### 3. QUANTUM SYSTEMS AND PATH DEPENDENCE

#### 3.1. The Schrödinger Equation and the Probabilistic Interpretation of Quantum Mechanics from the Dialectical Materialist Perspective

The time-independent Schrödinger equation is generally defined for a three-dimensional state space as

$$ \dot{H} \psi(x, y, z) = E \psi(x, y, z), $$

The Concrete Universal (which until now had existed only in the history of philosophy in the Hegelian system) must be reinterpreted through dialectical materialism as objective laws and tendencies manifesting through material reality's concrete processes—not as external impositions but as expressions of matter's internal contradictions.

Barrow & Tipler (1986, p. 11) document teleology's historical success in scientific advancement through several key examples: Fermat's correct prediction of light propagation laws, Chamberlain's anticipation of solar nuclear fusion, Boltzmann's explanation of time's direction and thermodynamics, and Hilbert's derivation of Einstein's field equations. However, they also present two significant criticisms that warrant examination.

The first criticism (Barrow & Tipler, 1986, p. 123) involves Monod's Objectivity Postulate, suggesting that teleology's inherent subjectivity precludes scientific objectivity. This objection fails to recognize that materialist teleology (even Hegel is an objectivist from his idealist system), unlike its idealist predecessors, defines purpose not through intellect or will but through inherent material tendencies observable in all systems.

1. Probability in quantum mechanics is determined by performing multiple measurements on identical systems prepared in the same state, which is equivalent to the proposition of calculating probability "experimentally".

2. The stability of the results as the number of experiments increases is related to the "law of large numbers", which was explicitly referenced in our approach.

3. The expression $|\psi(x, y, z)|^2$, which represents the probability density of finding the particle at a specific position, is congruent with the proposition that probability is a "stable magnitude" that describes the behavior of random events en masse.

4. The squares of the quantities of the electric moment matrix elements, which represent transition probabilities between states, fit this probabilistic interpretation, where transitions are the "random events" whose probabilities are being calculated.
This probabilistic interpretation of the Schrödinger equation aligns with our proposed view of dialectical materialism, where chance does not negate determinism but rather reflects different limitations in the knowledge of all causal chains and/or in existing measurement capability.

In the quantum context, the wave function $\psi$ provides a deterministic description of system evolution, whereas its probabilistic interpretation reflects our limitations in precisely predicting individual measurement outcomes. This assertion is fundamentally supported by the following considerations:

1. The Schrödinger equation (in both time-dependent and time-independent forms) describes the evolution of the quantum state in a completely deterministic manner. This means that:

1.1. Given an initial state $\psi(x, y, z, t_0)$, the Schrödinger equation uniquely determines the state $\psi(x, y, z, t)$ for any subsequent time $t$.

1.2. There is no randomness in this evolution since if we know the initial state and the system's Hamiltonian precisely, we can predict the future state with certainty. This determinism is similar to that of classical equations of motion but operates on wave functions rather than particle positions and velocities.

2. Despite the deterministic evolution of $\psi$, the standard interpretation of quantum mechanics establishes that $|\psi(x, y, z, t)|^2$ represents the probability density of finding the particle at a specific position when performing a measurement. This reflects our limitations in several aspects:

2.1. Heisenberg's uncertainty principle implies that we cannot simultaneously know all the properties of a quantum system with arbitrary precision, indicating the impossibility of knowing the complete state.

2.2. The measurement process inevitably disturbs the system, causing the wave function to “collapse” into an eigenstate of the measured observable, which is linked to the very nature of the measurement process.

2.3. Before measurement, the system is theoretically in a superposition of all possible states, and the theory can predict only the probabilities of obtaining each result.

3. This duality between determinism and probability aligns appropriately with our presented view of dialectical materialism:

3.1. The determinism of the Schrödinger equation reflects the idea that everything is subject to law in a fundamental sense.

3.2. The probabilistic interpretation represents our limitations, both at the conceptual level and in measurement instruments, of all causal chains operating in a phenomenon.

3.3. Probability in quantum mechanics acts as an epistemological resource to find the trace of regularity in massive random events.

4. This interpretation is justified by numerous experiments showing the following:

4.1. Quantum systems evolve coherently and predictably when undisturbed (demonstrating the deterministic aspect).

4.2. Individual measurements apparently produce random results, but the statistics of many measurements follow the probabilistic predictions of quantum theory.
As verified from the above exposition, our approach captures the essence of how quantum mechanics reconciles a deterministic mathematical formalism (the Schrödinger equation) with a physical reality that seems inherently probabilistic at the level of individual measurements.

From this, it follows that since the quantum superposition principle states that all possible theoretical states are simultaneously feasible before measurement, while Heisenberg's uncertainty principle states that there exists a reciprocal exclusion relationship between the precision with which a particle's velocity and position can be measured simultaneously, these two principles, although they may appear similar at the observational level, are essentially different in conceptual terms: the quantum superposition principle is the result of the epistemological resource consisting of the probabilistic interpretation of the Schrödinger equation (not because the interpretation of said equation is an epistemological resource—in fact, it is the only way to interpret said equation—but because probabilities themselves are an epistemological resource and not an ontological matter), whereas Heisenberg's uncertainty principle is an ontological matter (i.e., it reflects in the world of concepts the very structure of objective reality).

## 3.2. On the Double-Slit Experiment

To illustrate how the probabilistic interpretation of the Schrödinger equation manifests in physical reality, let us consider the famous double-slit experiment, which clearly demonstrates the wave–particle dual nature of matter and light. The experiment can be summarized as follows:

1. A beam of particles (electrons, photons, etc.) is fired toward a screen with two narrow slits.

2. Behind the slits, a detection screen is placed to record where the particles impact.

3. If the particles behaved only as classical objects, we would expect to see two bands of impact corresponding to the two slits.

4. However, what is observed is an interference pattern, similar to what would be produced if they were waves¹.

The standard interpretation of these results can be summarized as follows:

1. The wave function $\psi$ of each particle passes through both slits simultaneously. The wave function $\psi$ represents the complete quantum state of a particle. When this particle encounters a double-slit system, the wave function evolves according to the Schrödinger equation in a way that is mathematically equivalent to the propagation of a wave through both slits simultaneously. This simultaneity should not be understood in the classical sense of a “divided particle” but rather as a manifestation of the quantum superposition principle. The wave function, in this context, represents a probability amplitude that extends in space and evolves in time according to the system’s Hamiltonian, which includes the boundary conditions imposed by the slits. This coherent evolution of the wave function through both apertures is what gives rise to the wave behavior observed in the interference pattern, even when individual particles are sent.

2. The probability amplitude at each point on the detection screen is calculated through the coherent sum of the wave function contributions passing through each slit. Mathematically, this is expressed as $\psi_{total} = \psi_1 + \psi_2$, where $\psi_1$ and $\psi_2$ are the wave functions associated with passing through the first and second slits, respectively. This sum of complex amplitudes is fundamental to understanding quantum interference, as it allows the relative phases of the wave functions to contribute to the final result. Importantly, this sum occurs at the level of amplitudes (quantified via complex numbers) and not probabilities (which are quantified via real numbers), which allows for the possibility of destructive interference, which would not be possible if we simply summed classical probabilities.

<footnote>1</footnote>
3. The probability of detecting a particle at a given point is proportional to the square of the modulus of the sum of the amplitudes $\psi |v|^2$. This is known as Born's rule and is a fundamental postulate of quantum mechanics. This rule states that the probability of finding a particle at a particular position is proportional to the square of the wave function's modulus at that point: $P(x) \propto |\psi(x)|^2$. In the context of the double-slit experiment, $P(x) \propto \int d^2x \psi_1(x)\psi_2(x) = |\psi_1(x)|^2 + |\psi_2(x)|^2 + 2\Re[\psi_1(x)\psi_2(x)]$, where the last term represents quantum interference. This interference term can be positive or negative, explaining the appearance of maxima and minima in the observed pattern. Importantly, this "squaring" process occurs only at the moment of measurement, and it connects the mathematical formalism of the wave function with experimentally observable probabilities.

4. This process produces the observed interference pattern. This pattern, present in the double-slit experiment, is a macroscopic manifestation of the microscopic quantum processes described above. Each point on the detection screen has a probability of being impacted such that it oscillates spatially because of the interference term in the probability equation. As many individual detections accumulate, a pattern emerges that reflects these probability variations. This pattern is identical to what would be observed with classical waves but arises from the wave nature of the quantum wave function and the measurement process that collapses this wave function. The gradual appearance of this pattern, even when particles are sent one by one, is a powerful demonstration of the probabilistic and wave nature of quantum mechanics and challenges our classical notions of locality and well-defined trajectories.

Building from the concepts presented earlier in section 3.1, it is possible to interpret the results of the double-slit experiment in the following way:

1. The wave-particle duality revealed by the experiment demonstrates the unity and struggle of opposites, a fundamental principle of dialectical materialism. The particle is neither an exclusively wave nor an exclusively particle but rather an entity that manifests properties of both according to the concrete conditions of the experiment.

2. The interference pattern (necessity) emerges from apparently random individual impacts (chance). This illustrates our assertion that chance is only the complement and form of manifestation of necessity.

3. There are epistemological limitations, as we cannot predict where each individual particle will impact, reflecting the previously noted limitations both at the conceptual level and in measurement instruments. However, we can predict with great precision the statistical pattern that will emerge after many impacts.

4. The transition from corpuscular behavior (in emission and detection) to wave behavior (in propagation) should be understood as a manifestation of the law of transformation of quantitative into qualitative changes in the nature of the quantum entity. This is justified because the quantum entity begins with apparently corpuscular behavior since it behaves like a localized particle during emission, with a defined position in space. This is the initial quality of the system. As the quantum entity propagates from the source toward the slits, it experiences an increase in its degree of spatial delocalization. This is the key quantitative change. We can quantify this delocalization in terms of the spatial extension of the wave function $\psi(x,t)$. Initially, the wave function is highly localized, represented by a narrow wave packet. As time passes and the entity propagates, this wave packet expands spatially because of quantum dispersion. The expansion rate depends on the particle's mass and is governed by the Schrödinger equation

$$i\hbar \frac{\partial \psi}{\partial t} = -(\hbar^2/2m) \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi, \text{ where } \hbar \text{ is the reduced Planck constant that is constant with respect to Planck reducida, } m \text{ is the particle mass, and } V(x) \text{ is the potential. The qualitative leap occurs when the spatial extension of the wave function becomes}$$
comparable to or greater than the separation between the two slits. At this point, the quantum entity acquires sufficient spatial coherence to “sense” both slits simultaneously. Once this critical threshold is reached and exceeded, a new quality emerges: wave behavior. The quantum entity can no longer be adequately described as a localized particle but must be treated as a probability wave extending in space. This wave behavior manifests in the quantum entity’s ability to interfere with itself, producing the characteristic interference pattern on the detection screen. The new quality is characterized by the following:

4.1. The entity exists in a superposition of quantum states, passing through both slits simultaneously.

4.2. The ability to exhibit constructive and destructive interference.

4.3. The impossibility of assigning a defined trajectory to the entity, also known as *nonlocality*.

5. The experiment demonstrated the profound interconnection between the quantum entity under study and the measurement instrument, revealing how, in the quantum world, we cannot completely separate the observer from the observed entity. This is congruent with the *Law of Universal Connection and Reciprocal Action*. This is justified since the law in question establishes that no phenomenon in reality can be understood if it is taken outside its connections with surrounding phenomena. In the context of the double-slit experiment, this manifests in the impossibility of separating the quantum particle from the measurement apparatus. The measurement itself is an integral part of the observed phenomenon. It is also verified, in congruence with the referred law, that the totality is more than the mere sum or aggregation of its parts, since the measurement result emerges from the non-linear interaction between both and cannot be reduced to the individual properties of each.

This experiment not only validates the probabilistic interpretation of the Schrödinger equation but also demonstrates how the principles of dialectical materialism manifest in quantum phenomena. The apparent contradiction between wave and particle nature is resolved in a higher synthesis that recognizes both aspects as complementary manifestations of quantum reality in particular, as well as objective reality in general.

**3.3. Why Does the Arrow of Time Have a “Preferred Direction”?**

**3.3.1. *The Proposal***

If we accept that reality exhibits both fundamental determinism (although our understanding requires probabilistic tools) and path dependence, we must address a crucial question: What explains the observed unidirectionality of time—the arrow that points invariably from past to future?

Thus, this section must begin by stating what is understood in the dialectical system (here, the materialist and idealist visions converge) in space time (Hegel, 2005, p. 320):

> Place, being thus the posited identity of space and time, is also primarily that posited *contradiction* wherein space and time each exist on their own. Place is spatial *singularity* and, therefore, indifferent singularity; and it is place only insofar as it is *now spatial*, [that is,] as time, such that place, being immediately indifferent to itself as *this place*, being external to itself, is the negation of itself and [is] *another place*. This *perishing* and *regenerating* of space in time and of time in space, such that time posits itself spatially as *place* and this indifferent spatiality
also posits itself *temporally* in an immediate manner, is *movement*. This becom-
ing, however, is equally the coincidence of its contradiction, *the immediate, iden-
tical and existing* unity of space and time: [it is] *matter*.

This philosophically revolutionary vision was proposed by almost a century as one of the core
contributions of Einstein's theory: the organic union between space and time. From a dialectical
materialist perspective, the unidirectionality of time's arrow philosophically denotes (i.e., in its
conceptual essence) that phenomena are path-dependent (i.e., dependent on their historical tra-
jectory), which is a characteristic that theoretically (and practically) defines complex systems
and is fundamental to the dialectical analysis of reality's phenomena (from both materialist and
idealist perspectives). From this path dependence, one could argue that irreversibility emerges,
along with the impossibility of, for example, time travel to the past.

Thus, a time's arrow is not merely a "direction" but rather the manifestation that history (and the context in which history occurs) matters; future states depend not only on current conditions but also on the entire historical-contextual trajectory that led to them. From this perspective, irreversibility would not be an "anomaly" to explain but rather an inherent characteristic of complexity, since complex systems are defined as fundamentally historical-contextual, meaning that such systems cannot be understood without their evolutionary trajectory rooted in their concrete reality, i.e., in their concrete material conditions of existence.

Revisiting Loschmidt's paradox, what exists is not truly a paradox but rather a tension between a
theoretical model (reversible, ahistorical) and reality (irreversible, historical), with an apparent
paradox arising from the attempt to apply an inadequate model to understanding reality. There-
fore, it is not that reality "contradicts" theory but rather that theory is a limited abstraction of re-
ality, creating tension when attempting to make reality fit the model instead of developing mod-
els (at the fundamental level, beyond thermodynamic and statistical models) that capture reality's
historical-contextual nature. This highlights the need for a different approach to theoretical
physics, one that must incorporate historicity as a fundamental principle, recognizing that path
dependence is an ontological characteristic of reality — that is, one that does not treat irre-
versibility as something to explain from the current fundamental perspective of physical models
but rather as something to incorporate into such a perspective.

What we propose here is that the problem cannot lie in reality "not being well-behaved" but in
our theoretical models at the fundamental level that attempt to "erase" historicity and in our ten-
dency to see it as "paradoxical" when reality does not conform to these ahistorical models.

This fundamental reformulation of the physical world's conception could somehow pave the way
for a quantum theory of gravity because it would allow the formulation of macro laws more con-
gruent with the probabilistic approach to micro phenomena, as these macro laws would start
from complex relationships that could be modeled only probabilistically, allowing methodologi-
cal and epistemological standardization, even though ontologically, they would not be identical
phenomena.

However, some equivalence must exist between both worlds, an equivalence that connects them.
For example, the holographic duality principle suggests that gravity theory and particle theory
are equivalent: what happens mathematically in gravity theory also happens in particle theory,
and vice versa. However, regardless of whether this approach is naive (Levins, 1993, pp. 547-
549), since it is based on mathematics (on something almost purely theoretical) instead of on the
philosophical conceptualization of observable and logically grounded relationships (a complex
binomial between theory and practice, which is known in dialectial materialism as *praxis*), it is
correct in its proposal that there must exist some kind of fundamental equivalence between the
macro and micro.
This raises the following question: does path dependence require a notion of “universal history”? If so, general relativity would suffer a hard blow (philosophically speaking), since in such a theoretical framework, there exists no absolute time, and path dependence, a priori, would seem more consistent and intuitive when conceived on the basis of absolute time. Moreover, quantum theory works with a notion of time more similar to the classical one than the relativistic one (in the time-dependent version of the Schrödinger equation).

In general, the consistency between different inertial reference frames, despite the absence of absolute time, is achieved through mathematical transformations that preserve the causal structure of spacetime because the spacetime interval between events is invariant, geodesics (paths of minimum action) are objective, causality is preserved (events that are causally connected in one frame are connected in all frames), and Einstein's field equations are covariant.

Of course, there is the alternative of reformulating path dependence to ensure that history is relative to the observer, with path dependence being covariant (i.e., physical laws maintain their form under coordinate transformations). The covariance of path dependence means that causal relationships between events are preserved across all reference frames, the sequence of dependencies is maintained even though each observer's “proper” time is different, and the structure of causal relationships is invariant, although their specific temporal manifestation varies. This would mean that although each observer would have their own causal sequence, the different histories would be consistent with each other.

Path dependence could be considered more fundamental than time itself, with causality and history as core concepts. This perspective suggests an emergent “privileged” time arising from path dependence, analogous to how the Cosmic Microwave Background (CMB) establishes a preferred reference frame. The CMB, which emerged approximately 380,000 years after the large Bang, presents a nearly uniform temperature (2.7 K) across all directions. It defines a unique state of cosmic rest—observers stationary relative to the CMB detect identical radiation temperatures omnidirectionally. Furthermore, the CMB's last scattering surface marks a distinct universal moment, establishing a form of cosmic simultaneity observable throughout space.

If we accept that reality exhibits both fundamental determinism (although our understanding requires probabilistic tools) and path dependence, we must address a crucial question: What explains the observed unidirectionality of time—the arrow that points invariably from past to future?

In this way, the preferential frame of the CMB does not contradict special or general relativity since the laws of physics remain the same in all reference frames. However, the CMB provides a natural and physically significant framework for describing the evolution of the universe on a large scale.

Now, what would be the materialist foundation of time and history conceived in terms of the reference frame? On the one hand, the fact that time measurement is relative to the observer's reference frame (as established by relativity) does not mean that time itself is subjective but rather that its measurable manifestation varies according to objective physical conditions (velocity, gravitational field, etc.). On the other hand, while the perception of history may vary between observers, the material sequence of events is objective—that is, matter transformations occur independently of who observes them. Furthermore, causal relationships are objective, although their specific temporal manifestation varies according to the reference frame, implying that the causal structure of the universe is observer independent. Finally, CMB provides a privileged reference frame that emerges from the objective material conditions of the universe, not from a subjective choice. Thus, this establishes a key distinction between objective material reality (which includes the real sequence of transformations) and the different ways of measuring and observing reality (which may vary according to the reference frame).
However, to operationalize the above, we must unambiguously define both what we understand by time and what we understand by history, without resorting to time to define history, since we have established that history (path dependence) is a more fundamental concept than time itself.

Let history be defined as the order of events on the basis of the emergence of physical interac-
tions, that is, on the basis of the order of appearance of relationships between the components of
the analyzed physical reality. Under this definition, history is the emergent order of material in-
teractions on the basis of objective relationships and transformations, where the potential for de-
velopment is an intrinsic characteristic of matter and order emerges from the material necessity
of development itself.

The evolution of matter is fundamentally driven by its ultimate teleological aim: the achievement of increasingly stable states within the unified spatiotemporal structure. Stability emerges as the primary directive of material systems, not as a passive outcome but as an active, purposeful process inherent to the spatiotemporal fabric. For example, cyclic cosmological models can be understood as manifestations of this teleological drive, where the universe exhausts and potentially restarts (Andrei et al., 2022, p. 2), preserving the laws most conducive to achieving maximum relative stability with available resources (Alexander, et al., 2021, pp. 1, 5-71). This stability is 'relative' in that it exists within a dynamic framework of constant transformation yet maintains a sustainable long-term equilibrium that does not deplete the system's fundamental resources or threaten its existential continuity.

In the scenario described above, we can conceive of space-time as a unit that emerges simultane-
ously as a necessary manifestation of path dependence. Fundamental material relations require
both a spatial dimension to manifest and a temporal dimension to organize and develop. Thus,
space and time would be coemergent and indissoluble aspects from the very beginning, arising as
the ‘fabric’ necessary for matter to develop its potentialities: matter tells space how to curve, and
space tells matter how to move.

The four fundamental forces are not merely independent entities but are intrinsically manifested through and within the unified spatiotemporal structure from the very inception of existence. This fundamental unity means that these forces—gravitational, electromagnetic, strong nuclear, and weak nuclear—do not emerge sequentially but are simultaneous expressions of a foundational spatiotemporal fabric. By understanding these forces as integrated manifestations rather than discrete phenomena, we reframe our philosophical approach to comprehending reality as an inherently interconnected and coemergent system.

This simultaneous emergence creates the conditions for both the material registration of changes
and the preservation of historical trajectories, which leads us to examine how systems maintain
and express their evolutionary path.

However, irreversibility is self-grounded in this scheme because it starts from path dependence
(wecall it ‘strong path dependence’ as a principle of existence as a totality, although in its
components—in different systems—there may exist weak or more flexible dependence), which
implies irreversibility by construction (by how we have defined it here, because we precisely
start from the observable fact of irreversibility in material reality). Therefore, irreversibility is
not a consequence but rather a fundamental principle of reality.

To understand the above, an analogy with mathematics is useful: to the best of our knowledge,
there is no proof that natural numbers exist; rather, their existence is taken as a starting point to
prove everything else. This shows that there will always be “loose ends”, reflecting the imperfec-
tion of our knowledge. However, the key to philosophical and scientific thinking lies in knowing
which loose ends are valid to maintain (while seeking ways to “tie them up”), which are not, and
why.
In this case, we maintain this “loose end” (starting from irreversibility) because when analyzing the state of the art across different branches of science, we observe that, ultimately, facts are irreversible: to know the truth of the relative, one must know the absolute—that is, the unconditioned—that has only itself as its own condition. In other words, one must see “the big picture”. Therefore, it is valid to start from the irreversibility of processes, just as valid as assuming that natural numbers exist since we have built our entire society (whose nature is also material and objective) using them as an omnipresent reference. Irreversibility is an ontological principle, not a local/partial consequence.

From this perspective, time emerges as a necessity (of systems in general and the cosmological system in particular) to keep a record of events to be stored in the system’s “memory” (which establishes a bridge with information theory), with path dependence being the structuring principle; time is thus a structure that records the accumulation of changes occurring in space. This approach is valid for various reasons. First, it is not circular because it does not derive irreversibility but takes it as a fundamental empirical datum (equivalent to how mathematics considers natural numbers). Second, it is materialist because it starts from objective material conditions and necessary material relationships.

Space-time, which is the foundation for recording and preserving changes, is not only an epistemological resource but also an ontological fact. This understanding raises important questions about the nature of systems and their evolution. Why do systems need to maintain a record of their transformations, and how does this historical dimension shape their development? These questions must be addressed while recognizing strong path dependence as a fundamental principle of reality. Furthermore, how does the space-time structure enable the preservation and accumulation of changes from the very first moment of existence? The key lies in understanding how the simultaneous emergence of spatial and temporal dimensions creates the necessary conditions for systems to maintain their evolutionary trajectory.

In general, the answers to the previous questions stem from the understanding that space-time not only enables an informational record but also constitutes the material structure through which each state leaves its imprint on the next state—that is, it provides the fundamental framework in which each state contains/preserves something from the previous state. The necessity for space-time existence as a unified structure lies in the fact that matter cannot “disappear” without leaving a trace within both spatial and temporal dimensions, meaning that transformations always conserve/modify something previous, as there is no creation *ex nihilo* in nature.

The need for memory would thus arise because systems are not just their current state but also their history of transformation. System memory is not an acquired characteristic but rather an inherent property of the unified spatiotemporal structure. Every system's state is simultaneously a point of manifestation and a repository of transformational information. This memory is not a historical record but a living, dynamic aspect of the spatiotemporal fabric, where each configuration carries the imprint of previous interactions and potential future states.

Strong path dependence would imply that each state carries within itself the marks of its development, not as a “memory” but as an objective material structure. From the very first moment of existence, accumulation and development have both spatial and temporal characteristics, with both aspects being necessary and indissoluble for the manifestation of increasing complexity. There is no moment of ‘purely’ spatial accumulation, as the very notion of accumulation and development implies both spatiality and temporality.

Thus, space-time, as a unified structure, expresses the impossibility of completely ‘erasing’ the past, being the fundamental framework that enables the ordering and preservation of changes. Rather than emerging as a solution to complexity, space-time is the primordial structure that makes possible the manifestation and development of material relationships. In this sense, strong
path dependence as an axiom implies not only that order exists but also that this order must manifest itself materially through both spatial and temporal dimensions simultaneously. Consequently, space-time constitutes the material manifestation of this fundamental order, providing the necessary dimensions for reality to develop and preserve its evolutionary trajectory.

The philosophical perspective aligns seamlessly with Einstein's general relativity, which fundamentally treats space and time as integrated, dynamic entities. In this framework, spacetime is not a passive backdrop but an active participant in physical phenomena. Gravitational interactions, mass-energy distributions, and temporal flows are understood as interconnected manifestations of this unified structure, reinforcing our conceptual approach of coemergence and fundamental interconnectedness.

The integration of these perspectives offers a comprehensive approach to understanding reality, where mathematical abstraction, physical theory, and philosophical inquiry converge to reveal the profound unity underlying our complex universe.

Our approach maintains existing theories' predictive power and experimental data consistency while providing a more fundamental framework that could resolve current paradoxes.

Taking the time-dependent Schrödinger equation as our starting point,

$$ \dot{\hbar} \frac{\partial \psi}{\partial t} = \hat{H} \psi $$

We reformulate it on the basis of path dependency, where history is defined by physical interaction ordering:

$$ \dot{\hbar} \frac{\partial \psi}{\partial I} = \hat{H}(I) \psi. $$

Here, $I$ represents an interaction index ordered according to their emergence in the physical interaction chain. $\hat{H}(I)$ denotes a history-dependent Hamiltonian operator incorporating all prior interactions that have brought the system to its current state, which can be defined as:

$$ \hat{H}(I) = \hat{H}_0 + \int_0^1 K(I, I') \psi(I') dI', $$

where $\hat{H}_0$ represents the system's base Hamiltonian and where $K(I, I')$ is a kernel describing how previous interactions ($I'$) influence the current interaction ($I$). This integral sums all historical contributions, capturing how each new interaction is determined by the complete chain of prior interactions, not merely the system's 'instantaneous' state. The kernel can be more specified by

$$ K(I, I') = \alpha \exp(-\beta |d(I, I')|) \times P(I, I'), $$

where $\alpha$ is a coupling constant that determines the strength of the historical interaction, $\beta$ is a decay factor that controls how fast the influence of past interactions decreases, and $P(I, I')$ is a projection operator that ensures causality ($P(I, I')=0$ = if $I'$ is not in the causal past of $I$).

The transformation previously performed on the Schrödinger equation is justified by

$$ \frac{\partial \psi}{\partial t} \rightarrow \frac{\partial \psi}{\partial I} \times \left( \frac{\partial I}{\partial t} \right), $$
where $\partial I/\partial t$ represents the accumulation rate of interactions, which can be related to the system complexity:

$$ \partial I / \partial t = \gamma \sqrt{C |I|}, $$

where $\gamma$ is a constant that relates the rate of accumulation of interactions to the complexity of the system. Defining a metric in the interaction space is crucial for the integral's validity. Without a well-defined metric, the integral would attempt to sum incomparable quantities. The integral element is meaningful only if we can define infinitesimal intervals in the interaction space. A proposed metric takes the form:

$$ d(I_1, I_2) = \sqrt{\Delta C^2 + \Delta E^2}, $$

where $\Delta C$ measures the complexity/interaction number change between $I_1$ and $I_2$, while $\Delta E$ measures the energy/information change involved in the interactions. Our metric's key topological properties comprise three fundamental characteristics: continuity (establishing a complete, separable Hausdorff space), local compactness (ensuring finite neighborhoods of related interactions), and path-connectedness (guaranteeing continuous trajectories between interactions).

These properties emerge from deep physical requirements. Completeness ensures well-defined limits for all Cauchy sequences, which are physically translated to definite outcomes for all physical processes. Separability enables approximation of any interaction through countable sequences of simpler interactions, ensuring computational tractability and experimental verifiability. Path-connectedness reflects physical processes' causal continuity, preventing discontinuous system evolution.

To illustrate the necessity of these properties, consider completeness: its absence would permit nonconvergent interaction sequences, violating fundamental physical principles such as energy conservation. The practical implications manifest in three key areas: experimental verifiability through finite measurement sequences (from completeness and separability), continuous causal evolution preserving interaction-space locality (from path-connectedness), and finite relevant interactions at any scale (from local compactness).

These properties enable critical mathematical structures: the Borel measure definition for meaningful integration, differential equation solution existence, and homology theory application for understanding the global interaction structure. Together, they form a coherent framework where the mathematical structure aligns with physical requirements, enabling both theoretical consistency and experimental verification.

This metric ensures a well-defined integral by providing clear continuity, preserving causality (causally connected interactions are 'closer'), and enabling measure definition in the interaction space. Here, $\Delta C$ is defined by

$$ \Delta C = \ln(N_2/N_1) + \sum_{i=1}^{n} (w_i * |I_1 - I_2|), $$

where $N_2/N_1$ is the ratio between the number of interactions of states 2 and 1, $I_i$ represents the intensity of each type of interaction, $w_i$ represents the weights that reflect the relative importance of different types of interactions, and $\ln$ guarantees that the metric is additive for sequential changes. The units and normalization of the metric components are defined by

$$ d(I_1, I_2) = \sqrt{\Delta C/C_0^2 + \Delta E/E_0^2}, $$
where $C_0$ is the complexity fundamental constant (e.g., a qubit complexity) and where $E_0$ is the energy fundamental constant (e.g., the Planck constant). The metric properties of the proposed distance, which are nonnegativity $d(I_1, I_2) \ge 0$, symmetry $d(I_1, I_2) = d(I_2, I_1)$, and triangle inequality $d(I_1, I_3) \le d(I_1, I_2) + d(I_2, I_3)$, can be proven as follows:

$$ \sqrt{\left(\frac{\Delta C_{1,3}}{C_0}\right)^2 + \left(\frac{\Delta E_{1,3}}{E_0}\right)^2} \le \sqrt{\left(\frac{\Delta C_{1,2}}{C_0}\right)^2 + \left(\frac{\Delta E_{1,2}}{E_0}\right)^2} + \sqrt{\left(\frac{\Delta C_{2,3}}{C_0}\right)^2 + \left(\frac{\Delta E_{2,3}}{E_0}\right)^2}. $$

This interaction space metric, when ‘projected’ onto emergent spacetime, should reproduce the Minkowski metric as an effective approximation, analogous to how Newtonian mechanics emerges from quantum mechanics at macroscopic scales. This will be further explored in section 3.3.2.

This framework ensures that interaction ordering is consistent with causality. With respect to Einstein’s “spooky action at distance,” our theoretical framework offers a novel interpretation: rather than viewing entanglement as a distant action, we can understand it as entangled particles sharing a common interaction history. This shifts from particles “communicating” instantaneously to both being part of the same path-dependent structure.

When generating entangled particles, we do not create separate objects requiring mysterious communication but rather a single interaction structure manifesting at distinct spatial locations. Our defined metric captures this by assigning zero distance to entangled particles in terms of interaction history (despite spatial separation), whereas measurement does not “cause” a change in the other particle but completes an interaction structure potentially existing in their shared history.

This does not solve Einstein’s concern traditionally but reframes it: “spooky action at a distance” loses its mysterious nature when space and time emerge from a more fundamental interaction and dependency structure. This interpretation parallels Bohm’s quantum mechanics interpretation with its *implicate order* underlying the *explicate order* we observe, although our path-dependency approach provides a more concrete mathematical structure for understanding this emergence.

The emergence of conventional time from this formulation raises a key consideration: time may emerge as an effective parameter that orders interactions consistent with causality

$$ t = F \left[ \int_{0}^{1} C(I') dI' \right], $$

where $C(I')$ represents interaction complexity or “density” and where $F$ maps this interaction accumulation to conventional time experience. $F$ must satisfy three fundamental properties:

1. Causality preservation: For interaction A enabling B, $F$ must assign A an earlier temporal value.

2. Irreversibility consistency: $F$ must monotonically increase, reflecting unidirectional interaction accumulation.

3. Simultaneous reflection: Different observers may experience different temporal orderings of causally disconnected events.
The complexity measure $C(I)$ likely correlates with system entropy, suggesting fundamental connections between time emergence, information accumulation, and thermodynamics' second law. This temporal reformulation as an emergent structure reframes quantum measurement as establishing new interactions contributing to emergent temporal structure, explaining apparent wave function “collapse” at specific moments. Thus, “collapse” becomes not a mysterious instantaneous event but rather the integration of new interactions into the system’s path-dependent structure.

Bell's theorems deserve special mention. Bell's inequalities demonstrated that any theory preserving local realism contradicts quantum mechanical predictions, which have been experimentally verified. Our model offers an alternative perspective on this dilemma by proposing a more fundamental structure where spacetime emerges from interaction relationships rather than considering locality versus nonlocality in conventional spacetime.

In Bell's framework, 'nonlocal' refers to effects apparently violating locality in emergent spacetime. However, in our model, entangled particles share a common history and remain 'close' according to our metric in the interaction space, despite apparent spatial separation. This framework preserves both realism (interactions and their histories remain objective) and fundamental-level locality (influences propagate continuously in interaction space) while explaining apparent nonlocality in emergent spacetime.

This parallels how two points may appear distant on folded paper yet proximate when flattened; our interaction space metric 'flattens' the structure, revealing connections obscured in emergent spacetime. Rather than circumventing Bell's inequalities, this reinterprets them: correlations inexplicable through conventional spacetime's local realist theories become comprehensible when considering locality in fundamental interaction space.

In our model, decoherence represents a deterministic process governed by underlying interaction structures, expressed as:

$$Decoherence Rate = \gamma \int K(I, I')C(I')dI',$$

This formulation predicts correlated decoherence rates in systems with similar interaction histories. The model correlates the entanglement strength with the interaction space distance, which is quantified as:

$$E(\rho) = E_0 \exp(-\alpha d(I_1, I_2)),$$

where $E(\rho)$ denotes the entanglement measure (von Neumann entropy), $E_0$ represents the normalization constant, and $d(I_1, I_2)$ defines our interaction-space distance metric.

For double-slit interference experiments, the model predicts history-dependent interference patterns, modifying the wave function:

$$\psi_{\text{modified}} = \psi_{\text{standard}} \times F\left[\int K(I, I') \psi(I') dI'\right].$$

Additionally, systems with divergent interaction histories should exhibit temporal flow nonuniformities, with the temporal evolution rate varying as follows:

$$dt/dI = \tau_0(1 + \beta \int C(I') dI'),$$

where $\tau_0$ denotes a constant and where $\beta$ represents the cumulative history influence. These predictions, while requiring precise measurements and controlled experimental conditions, remain experimentally verifiable.
### 3.3.2. Examples of Proposal Equivalence with Other Theoretical Frameworks

#### 3.3.2.1. Classical Regime

Our metric's general form can be expressed as:

$$d_{interactions} = \sqrt{[(\Delta C/C_0)^2 + (\Delta E/E_0)^2 + g(I_1, I_2)]},$$

where $g(I_1, I_2)$ captures the causal structure and quantum correlations, which play crucial roles in regime transitions.

In the classical regime (low-energy, macroscopic systems), the metric reduces to Euclidean form through:

$$\begin{aligned} \Delta C/C_0 &\rightarrow \sqrt{(\Delta x^2 + \Delta y^2 + \Delta z^2)/L_0} \\ \Delta E/E_0 &\rightarrow V(r)/V_0 \\ g(I_1, I_2) &\rightarrow 0, \end{aligned}$$

where $L_0$ and $V_0$ represent the characteristic length and energy scales, respectively.

#### 3.3.2.2. Relativistic Regime

For the relativistic regime (high-velocity limit), the metric reproduces the Minkowski interval via:

$$\begin{aligned} \Delta C/C_0 &\rightarrow \sqrt{(\Delta x^2 + \Delta y^2 + \Delta z^2)/L_0} \\ \Delta E/E_0 &\rightarrow c^2 \Delta t^2/T_0^2 \\ g(I_1, I_2) &\rightarrow -2(\Delta E/E_0)(\Delta C/C_0). \end{aligned}$$

#### 3.3.2.3. Quantum Regime

In the quantum regime, where probabilistic interactions and entanglement dominate:

$$\begin{aligned} \Delta C/C_0 &\rightarrow -\ln|\langle \psi_1 \vee \psi_2 \rangle| \\ \Delta E/E_0 &\rightarrow \hbar(E_1 - E_2)/E_0 \\ g(I_1, I_2) &\rightarrow S(\rho_{1,2}) \end{aligned}$$

where $S(\rho_{1,2})$ represents the entanglement entropy.

These transitions occur smoothly within our unified framework, with each regime emerging naturally under relevant conditions. For example, decoherence manifests when $g(I_1, I_2)$'s entanglement term vanishes through environmental interactions. The metric maintains essential topological properties while enabling different physical structures to emerge across energy and complexity scales.

### 3.3.3. Outlining experimental protocols

#### 3.3.3.1. Decoherence studies

Using trapped ions or superconductor qubits:
1. Prepare *N* identical quantum system pairs.

2. For each pair *i*:

- Apply the controlled interaction sequence $S_i$.

- Measure decoherence rate $D_i$.

3. Correlate $S_i$ complexity with $D_i$.

Keywords: quantum state tomography, operation logging, coherence interferometry. Prediction: correlated decoherence rates beyond standard models.

### 3.3.3.2. Entanglement Analysis

Using parametric downconversion photons:

1. Generate entangled photon pairs.

2. Group division:

- Control: direct measurement.

- Experimental: varied interaction histories.

3. Per group: Apply interactions and measure entanglement.

4. Comparison against exponential prediction.

### 3.3.3.3. Quantum interference (modified double-slit setup)

1. Single-particle source.

2. Modifiable double-slit system.

3. Interaction control elements:

- Variable electromagnetic (EM) fields.

- Adjustable optics.

4. High-resolution detection

Focus: detecting history-dependent interference pattern deviations.

### 3.3.3.4. Time Emergence

Using ultraprecise atomic clocks:

1. Prepare identical atomic clocks.

2. Various interaction histories are applied.

3. The temporal evolution rates are compared.

4. Correlate history complexity with time rates.

### 3.3.3.5. Some Technical Challenges

- Noise Control:

- Quantum isolation.

- Millikelvin temperature control.
- EM shielding.

* Measurement Precision:

- Single-photon detection.

- Ultrafast measurements.

- High-fidelity quantum tomography.

* Variable Control:

- Ultrahigh vacuum.

- Precise EM field control.

- Thermal stabilization

In general, the success criteria to validate or refute this proposal are multiple repetitions, rigorous variable control, sophisticated statistical analysis, and systematic effects distinguishing it from standard theories.

# 4. TELEOLOGY IN PHYSICAL SYSTEMS

## 4.1. Teleological Interpretation of the Least Action Principle

The principle of least action reveals teleology's role in quantum mechanics: the system's final state determines its trajectory (Terekhovich, 2016, p. 2), demonstrating the mechanism's subordination to purpose. As Feynman (2005, p. 9) explains, mechanical systems exhibit an action function that assigns values to possible paths, reaching an extremum for physically realized trajectories. This probabilistic aspect serves as an epistemological tool rather than reflecting ontological indeterminacy, paralleling our broader understanding of probability in quantum mechanics.

This extremum is not always a minimum; rather, the system selects the path that provides sufficient energy for its teleological purpose of achieving stability. This stability represents a fundamental characteristic of matter itself, as Solovej (2011, p. 169) noted, explaining both macroscopic object persistence and the ground state tendency of quantum systems. Feynman (2005, p. 9) further demonstrated that the specific form of the action functional depends on concrete mechanical problems, refuting Kant's notion of a priori universal laws.

The principle thus becomes a tendential law, similar to Marx's falling rate of profit logarithm: the system seeks a maximum energy economy while fulfilling its purpose, subject to various constraining factors. This makes action not an absolute rule but a rule that acts under restrictions (i.e., a tendency), manifesting most clearly in quantum systems where trajectories generally, though not invariably, follow paths of minimal action (Feynman, 2017, p. 105).

## 4.2. Teleological Interpretation of the Tendency of Particles to the Ground State

Quantum systems reveal a fascinating interplay between unpredictability and purpose. As Cubitt, Perez-Garcia, & Wolf (2015, pp. 207--208) demonstrated, we cannot predict in advance the energy needed for a quantum system to jump from its ground state to its first excited state, even
with perfect microscopic knowledge. This limitation highlights how purely mechanical explanations fall short in understanding the development of physical systems.

However, this unpredictability coexists with a profound organizing principle. Bruice (2016, p. 6) describes matter's natural tendency toward its ground state, which dialectical materialism interprets as expressing matter's fundamental need for stabilization. This is not mere metaphor—electrons consistently “prefer” the lowest available energy levels, demonstrating how quantum mechanisms serve teleological purposes.

This theoretical insight has practical applications: Oriols & Mompart (2019, p. xxii) show how we can prepare wave functions by exploiting systems' tendency toward ground states, even though we cannot directly measure these states in single experiments. This practical success reinforces how understanding teleology—matter's inherent drive toward stability—helps us grasp the true nature of physical mechanisms.

## 5. CONCLUSIONS

Research has demonstrated that physical systems exhibit a universal tendency toward maximum possible relative stability within their material constraints. This tendency is not an external imposition but rather an inherent characteristic of matter, manifesting across scales from quantum particles to macroscopic systems. The presented evidence consistently supports this conclusion through both theoretical analysis and the proposed experimental frameworks.

Loschmidt's apparent paradox, rather than representing a fundamental contradiction in nature, reveals the limitations of our theoretical models when attempting to explain observed irreversibility through purely mechanical terms. Our framework resolves this tension by establishing path dependence as a more fundamental principle than time itself, with temporal flow emerging as the necessary structure for recording material transformations. This reformulation suggests that irreversibility should be incorporated as a foundational principle rather than treated as an anomaly requiring explanation.

The probabilistic interpretation of quantum mechanics, analyzed from this perspective, emerges not as a description of inherent indeterminacy but as a methodological resource reflecting practical limitations in measurement and prediction capabilities. This interpretation allows the reconciliation of fundamental determinism with observed probabilistic behavior while providing experimentally testable predictions about decoherence rates, entanglement strength, and interference patterns modified by interaction histories.

Our reinterpretation of the principle of least action reveals how physical systems optimize their trajectories not only for energy efficiency but also in service of their fundamental tendency toward stability. This optimization demonstrates how mechanical aspects are subordinate to teleological purposes, explaining why absolute minimum energy paths are not always selected. Similarly, quantum systems' tendency toward ground states manifests this inherent drive toward maximum achievable stability.

The synthesis of path dependence and teleology provides a unifying framework for understanding the development of physical systems. This framework explains how a system's history actively shapes its evolution toward more stable states, resolving the apparent contradiction between determinism and finality. Our metric formulation in interaction space, which reproduces known physical theories as limiting cases while predicting novel effects, offers a mathematical structure for this philosophical insight.
The experimental protocols we have outlined, particularly in studying decoherence, entangle-
ment, and modified quantum interference, provide concrete ways to test these theoretical predic-
tions. While technically challenging, these experiments could validate or refute specific aspects
of our framework, particularly regarding the relationship between interaction histories and quan-
tum behavior.

Ultimately, this research suggests that advancing physical theory requires explicitly incorporat-
ing both historicity and teleology as fundamental rather than emergent features. This integration
promises not only to resolve existing paradoxes but also to provide new perspectives for under-
standing complex phenomena across all scales of reality, potentially offering insights toward a
unified description of quantum and other domains.

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