Throughout this work, entropy is used in an operational and system-level sense. It denotes the effective diversity of informational states accessible to a system, including representational, behavioral, or strategic degrees of freedom, depending on the domain Shannon (1948).

The entropy here is not synonymous with randomness, noise, or disorder. High entropy reflects the availability of multiple viable internal configurations that support exploration and adaptation, while low entropy indicates contraction of these degrees of freedom. Our arguments do not rely on a specific entropy measure; instead, they depend on relative entropy depletion under system dynamics.

Entropy collapse is fundamentally a dynamical phenomenon rather than a static property. As intelligent systems learn, coordinate, or optimize, feedback mechanisms preferentially reinforce dominant internal states. When such reinforcement exceeds the system’s capacity to introduce or sustain novelty, entropy decreases over time.

Crucially, entropy collapse corresponds to convergence toward a stable low-entropy manifold in the state space of the system rather than convergence to a single fixed point. Within this manifold, limited variability and local fluctuations may persist, but the effective dimensionality of adaptation of the system is dramatically reduced. As a result, the system may continue to evolve or scale along constrained trajectories while remaining globally trapped in a low-adaptability regime.

Geometric Intuition: Conceptually, the low-entropy manifold can be visualized as a lower-dimensional surface within the full state space. Imagine a system initially free to explore throughout a three-dimensional volume (high entropy). Under feedback amplification, its trajectories become progressively confined to a two-dimensional plane, then eventually to a one-dimensional curve-a manifold of reduced dimensionality. Although the system may continue moving along this curve indefinitely, it can no longer access directions perpendicular to it, representing lost adaptive degrees of freedom. This geometric contraction, rather than complete stasis, characterizes the collapse of the entropy.

To avoid ambiguity, we explicitly delimit the scope of this paper.

This work does not claim that entropy collapse explains all failures of intelligent systems, nor that collapse dynamics is identical in mechanism or timescale across domains. We do not aim to provide domain-specific empirical calibrations or optimal intervention strategies. Instead, our focus is on establishing the existence, generality, and structural properties of entropy collapse as a shared dynamical mechanism. Domain-specific phenomena-such as model collapse in artificial intelligence, coordination traps in economics, or bottlenecks in biological evolution-are treated as projections of this underlying mechanism rather than as exhaustive empirical validations.

By framing collapse as convergence toward a low-entropy manifold under minimal assumptions, this section sets the conceptual foundation for the formal analysis and simulations that follow.

The purpose of this section is to identify the minimal structural conditions under which entropy collapse necessarily arises. Rather than constructing a detailed or domain-specific model, we deliberately adopt a skeletal formulation to demonstrate that collapse is not an artifact of particular assumptions, functional forms, or optimization objectives.

We show that entropy collapse emerges whenever an intelligent system satisfies the following three irreducible assumptions.

A1. State Diversity. The system admits a non-degenerate distribution over internal states, representations, or strategies. Formally, at each time t, the system is described by a probability distribution P t over a state space S with nonzero entropy.

A2. Feedback Amplification. The system contains a mechanism by which states with higher prevalence or success are preferentially reinforced over time. This feedback may arise from learning, imitation, selection, or aggregation and is controlled by a parameter α that represents the feedback strength.

A3. Bounded Novelty Regeneration. The system possesses a finite capacity to introduce or sustain novel states. This capacity, governed by a parameter β, does not scale indefinitely with feedback strength and reflects intrinsic limits on exploration, mutation, or innovation.

These assumptions are intentionally weak and domain-agnostic. They do not require rational agents, optimal decision-making, or explicit objective functions.

Each assumption is necessary for entropy collapse to occur.

If A1 is violated, the system has no entropy to lose and collapse is undefined. If A2 is absent, the dominant states are not reinforced and the entropy does not contract systematically. If A3 is violated, novelty generation overwhelms reinforcement and prevents sustained entropy depletion. Only when all three conditions hold simultaneously does collapse arise.

Let P t ∈ ∆(S) denote the state distribution of the system at time t. We consider a general update rule of the form P t+1 = F(P t ; α, β),

where α controls the feedback amplification and β bounds novelty regeneration.

We do not impose a specific functional form on F. Instead, we require only that the increase α amplifies the relative probability mass assigned to the dominant states, while the increase β injects the bounded probability mass into the underrepresented or novel states.

We quantify system diversity using an entropy functional

Under assumptions A1-A3, the evolution of H(P t ) generically exhibits three qualitative regimes as the ratio α/β varies: an adaptive high-entropy regime, a metastable regime with slow entropy decay, and a collapse regime characterized by rapid contraction toward a low entropy manifold.

Importantly, this qualitative structure does not depend on the dimensionality of S, the specific entropy measure used, or the microscopic implementation of feedback and novelty.

From a geometric perspective, the collapse of the entropy corresponds to the contraction of the effective dimensionality of the accessible state space of the system. Although trajectories can continue to evolve indefinitely, they become confined to a low-dimensional manifold within ∆(S), sharply limiting adaptive degrees of freedom.

This interpretation prepares the ground for the formal results presented in the next section, where we establish the existence of collapse thresholds, irreversibility, and attractor structure under the minimal assumptions introduced here.

This section establishes the core theoretical properties of entropy collapse under the minimal assumptions introduced previously. Our goal is not to derive exact trajectories or closed-form solutions, but rather to demonstrate the existence, structure, and robustness of collapse as a dynamical phenomenon.

Let P t ∈ ∆(S) denote the state distribution of the system at time t, and define the entropy

We consider the expected entropy change

Under feedback amplification (A2), the probability mass is systematically concentrated toward the dominant states, inducing a negative contribution to ∆H t . Novelty regeneration (A3) contributes a positive entropy term that is uniformly bounded by β. This asymmetry underlies all subsequent results.

Proposition 1 (Entropy Collapse Threshold) For any system satisfying Assumptions A1-A3, there exists a finite threshold α c (β) such that, for all α > α c , the expected entropy decreases monotonically after finite time.

Proof sketch. As α increases, the entropy-reducing effect of feedback amplification grows without bound, while the entropy-increasing effect of novelty regeneration remains bounded by β. By the continuity of ∆H t in α, there exists a finite α c (β) beyond which ∆H t < 0 holds uniformly outside a metastable transient.

This result does not depend on the dimensionality of S, the choice of entropy measure, or the detailed form of the update operator F.

Proposition 2 (Irreversibility) Once the entropy of the system falls below a critical level H under α > α c , the probability of returning to a high-entropy regime within a finite time approaches zero.

Proof sketch. Below H, system trajectories enter a region of the probability simplex where reinforcement dominates all admissible novelty injections. Because novelty regeneration is bounded, escape trajectories require entropy increases exceeding what β can provide. As feedback continues to sharpen the dominant states, the probability of escape decays to zero.

The irreversibility here is dynamical rather than thermodynamic: it reflects topological trapping in the entropy landscape rather than literal time asymmetry.

Proposition 3 (Collapse Attractor) For α > α c , there exists a compact attractor set A collapse ⊂ ∆(S) such that, for a broad class of initial conditions, the system trajectories converge to A collapse .

Interpretation. The attractor A collapse is generally not a single fixed point. Instead, it takes the form of a stable low-entropy manifold within the probability simplex. Within this manifold, limited variability and local fluctuations may persist, but adaptive degrees of freedom are fundamentally constrained.

Clarification. We emphasize that entropy collapse does not imply convergence to a zero-entropy state. Rather, collapse corresponds to confinement of system dynamics to a stable low-entropy manifold, within which trajectories may extend indefinitely despite a contraction in effective dimensionality.

The propositions above rely only on structural asymmetries between feedback amplification and bounded novelty regeneration. They do not invoke rational agents, optimization objectives, equilibrium assumptions, or domain-specific mechanisms.

As a result, entropy collapse constitutes a universal dynamical phenomenon: whenever reinforcement exceeds bounded novelty, collapse emerges irrespective of the substrate or scale of the system.

Finally, we note that the entropy collapse constrains the effective dimensionality of system adaptation rather than the absolute scale, duration, or magnitude of system dynamics. Post-collapse trajectories may therefore continue indefinitely while remaining confined to a low-dimensional manifold, explaining how systems can persist, grow, or operate over long horizons while losing genuine adaptive capacity.

The purpose of the simulations in this section is not empirical prediction or data fitting. Instead, simulations are used to make the theoretical properties of entropy collapse directly observable, namely the existence of a collapse threshold, irreversibility, and universality across distinct instantiations.

All simulations are deliberately minimal. We avoid domain-specific assumptions, intelligent agents, reward functions, or task-specific objectives. The only goal is to instantiate the generic dynamical skeleton described in Section 3 and verify the qualitative behavior predicted by the theoretical results in Section 4.

In particular, simulations are designed to answer three questions: (i) Does a sharp transition in entropy dynamics exist? (ii) Is the collapse dynamically irreversible? (iii) Is the collapse robust across different update mechanisms?

We consider a finite state space S = {1, . . . , N } with an initial distribution P 0 ∼ Dirichlet(1), ensuring a high initial entropy. At each time step, the distribution is updated according to a rule that combines feedback amplification and bounded novelty regeneration.

Feedback amplification increases the relative probability of states with larger P t (s), controlled by a parameter α. Novelty regeneration injects a bounded probability mass into low-probability states, controlled by β. No assumptions are made about the functional form of the update rule beyond these structural properties.

We first examine entropy trajectories as α is varied while β is fixed.

For α < α c , the entropy fluctuates around a high value, indicating sustained diversity. Near α ≈ α c the entropy slowly decays, producing a metastable regime. For α > α c , the entropy decreases rapidly and converges toward a low-entropy manifold, consistent with Proposition 1.

To test irreversibility, we allow the system to collapse under α > α c , then temporarily increase β to simulate a novelty shock.

Although the entropy increases briefly during the shock, it rapidly decays once the shock is removed. The system does not return to the pre-collapse regime, confirming the irreversibility predicted by Proposition 2.

We repeat the simulations using multiple update mechanisms, including multiplicative reinforcement, softmax-normalized updates, and replicator-style dynamics.

When the entropy trajectories are rescaled by the ratio α/β, the results of different update rules align closely. This demonstrates that collapse dynamics depends primarily on the relative strength of feedback to novelty rather than on microscopic implementation details.

Combining results across parameter sweeps, we construct a qualitative phase diagram.

The diagram highlights three regimes: an adaptive high-entropy regime, a metastable transition region, and a collapse regime characterized by confinement to a low-entropy manifold.

To further establish the robustness of entropy collapse, we performed extensive sensitivity analyses across: (i) state-space sizes (N = 10, 50, 100, 500), (ii) different entropy measures (Rényi entropy of order q ∈ {0.5, 1, 2}), (iii) noise injection levels (Gaussian noise with σ ∈ [0, 0.1]), and (iv) non-stationary environments (β varying slowly over time). In all cases, the fundamental phase structure-adaptive, metastable, collapse-persisted unchanged. The critical ratio α c /β remained stable within ±15% across variations, confirming that the collapse of the entropy is a robust dynamical phenomenon rather than a numerical artifact or a finite-size effect. In particular, while noise increases the transient entropy, it does not eliminate the collapse attractor for α > α c , aligning with Proposition 3.

Together, the simulations provide direct visual evidence for entropy collapse as a dynamical phenomenon. Importantly, post-collapse trajectories continue to evolve over time while remaining confined to a low-entropy manifold, illustrating how systems may persist and scale while losing effective adaptive dimensionality. The above results can be synthesized into a unified conceptual picture of entropy collapse across intelligent systems, as summarized in Fig. 5.

The theoretical and simulation results above establish the entropy collapse as a general dynamical mechanism. In this section, we interpret how the same mechanism manifests itself in distinct domains. The purpose is not to provide exhaustive empirical validation but to show that seemingly unrelated failures share a common entropy-driven structure.

In contemporary machine learning systems, particularly generative and self-improving models, a growing body of evidence points to model collapse: degradation of output diversity and representational richness when models are repeatedly trained on self-generated or homogenized data Shumailov et al. (2023); Alemohammad et al. (2024).

Under the entropy collapse framework, this phenomenon can be understood as follows. The model states correspond to internal representations or output distributions. Feedback amplification arises through self-training, distillation, reinforcement learning from aggregated preferences, or repeated optimization toward narrow objectives. Novelty regeneration is bounded by finite data diversity, exploration rates, or noise injection.

Initially, feedback improves coherence and performance. Beyond a critical threshold, however, reinforcement overwhelms novelty and the model’s representational entropy contracts. The model continues to generate outputs and may scale in size or usage, yet its effective dimensionality of adaptation collapses toward a low-entropy manifold. This explains why post-collapse models remain functional while exhibiting reduced novelty and robustness.

In economic and social systems, coordination between agents is often desirable for efficiency and stability Arthur (1994). However, many systems converge toward rigid coordination patterns, institutional lock-in, or innovation stagnation, even when agents are rational and information is abundant Farmer and Foley (2009).

Here, the state of the system corresponds to distributions of strategies, norms, or institutional arrangements. Feedback amplification emerges through imitation, learning from aggregates, regulatory reinforcement, or path dependence. The regeneration of novelty is limited by the costs of experimentation, policy inertia, or cultural constraints.

Entropy collapse reframes coordination traps as convergence toward low-entropy manifolds in strategy space. Systems may continue to grow in scale or output while becoming increasingly brittle as alternative strategies are systematically suppressed. This perspective clarifies why late-stage policy interventions often fail to restore genuine diversity or adaptability Abrams and Strogatz (2003).

In evolutionary systems, phenomena such as population bottlenecks, loss of genetic diversity, and adaptive stagnation are well documented Gould (1996). These are traditionally attributed to environmental shocks or historical events.

Within the entropy collapse framework, genotypes or phenotypes constitute system states, selection pressure provides feedback amplification, and mutation or recombination supplies bounded novelty. When selection pressure dominates mutation, genetic entropy contracts, and populations converge toward a narrow region of genotype space.

Importantly, such populations may persist over many generations and continue to reproduce, yet remain confined to a low-entropy manifold that limits future adaptability. Collapse here does not imply extinction, but a structural loss of evolutionary flexibility Holland (1992).

Entropy collapse provides a unifying geometric interpretation for several previously disconnected phenomena: -In economics, it formalizes the transition from Schumpeterian “creative destruction” to institutional sclerosis, where path dependence (feedback) overwhelms innovation (novelty). -In machine learning, it generalizes both overfitting (collapse to training data manifold) and mode collapse in GANs (collapse to limited output modes). -In evolutionary biology, it reframes the “Red Queen Hypothesis”-where constant adaptation is needed just to maintain fitness-as a race between selection pressure (feedback) and mutation/recombination (novelty). -In complex systems, it extends Ashby’s Law of Requisite Variety Ashby (1956), quantifying how insufficient internal variety leads to failure when confronting environmental complexity.

Unlike domain-specific models, the entropy collapse identifies the common dynamical structure underlying these phenomena: the irreversible contraction of the effective state space when feedback dominates bounded novelty.

Across these domains, the same structural asymmetry recurs: feedback amplification scales more aggressively than the system’s capacity to regenerate novelty. As a result, systems converge toward stable low-entropy manifolds that constrain adaptive dimensionality without necessarily limiting scale, activity, or longevity.

This unifying interpretation suggests that the collapse of the entropy is not a domainspecific pathology, but a general dynamical cost of intelligence, coordination, and learning.

The results presented in this paper have several important implications for the way intelligent systems are understood, evaluated, and governed. Rather than offering domain-specific prescriptions, we focus on the consequences that follow generically from entropy collapse as a structural dynamical phenomenon.

A central implication of entropy collapse is that intelligence, coordination, and learning are not free. Mechanisms that improve short-term performance-such as reinforcement, consensus formation, or optimization-systematically reduce entropy by privileging dominant states over alternatives Anderson (1972).

This introduces a fundamental trade-off: systems that become highly efficient or coherent in the short term often do so by contracting their effective state space, thereby reducing longterm adaptability. Entropy collapse is therefore not an implementation flaw, but a structural cost of intelligence itself.

Entropy collapse rarely manifests as immediate failure. In contrast, collapse is often preceded by increased stability, reduced variance, and apparent performance gains. These signals can mask the loss of adaptive degrees of freedom, making collapse difficult to detect until recovery becomes impossible Scheffer et al. (2009).

This explains why many intelligent systems appear healthy or even optimal shortly before experiencing catastrophic rigidity or fragility. Collapse is thus better understood as a hidden geometric transformation of the system’s state space rather than as a sudden breakdown.

A key consequence of collapse irreversibility is the systematic failure of late-stage interventions. Strategies that aim to reintroduce diversity-such as adding noise, increasing exploration, injecting incentives, or implementing surface-level reforms-correspond to local entropy perturbations.

Once system dynamics are confined to a low-entropy manifold, such perturbations fail to alter the global attractor structure. Entropy may increase transiently, but trajectories rapidly return to the collapsed regime. This explains why many well-intentioned interventions briefly succeed, but fail to restore genuine adaptability.

It is important to emphasize that the collapse of the entropy constrains the effective dimensionality of the adaptation rather than the absolute scale, duration, or magnitude of the system dynamics. Post-collapse systems may continue to grow, operate indefinitely, or scale in output while remaining confined to a low-dimensional manifold.

As a result, collapse should not be equated with stagnation in size or activity. Instead, it represents a loss of accessible directions for adaptation, innovation, or response to novelty. This distinction clarifies how systems can persist over long horizons, while becoming increasingly brittle Miller and Page (2007).

Our analysis suggests that sustaining adaptability requires treating entropy as a first-order system resource. We propose three preliminary design principles for intelligent systems that are aware of entropy:

1. Entropy Budgeting: Explicitly allocate and monitor “entropy budgets” for subsystems, ensuring that feedback amplification never permanently exhausts novelty regeneration capacity. This mirrors risk budgeting in financial systems.

2. Strategic Inefficiency: Periodically introduce controlled inefficiencies (e.g., random exploration, objective perturbation, or temporary performance degradation) to prevent premature convergence to low-entropy manifolds. Biological systems employ similar strategies through mechanisms such as genetic recombination and somatic hypermutation.

3. Multi-Scale Entropy Monitoring: Implement entropy metrics at multiple scales-from micro (agent strategies) to macro (system-wide diversity)-to detect early warning signs of collapse before critical thresholds are crossed. This requires developing domain-appropriate entropy measures that capture effective adaptive dimensionality rather than superficial variability.

These principles shift the design paradigm from maximizing short-term performance to optimizing long-term adaptability trajectories.

This work establishes entropy collapse as a universal dynamical failure mode of intelligent systems. By distilling the phenomenon to three minimal assumptions-state diversity, feedback amplification, and bounded novelty regeneration-we reveal a shared geometry underlying diverse collapses across artificial, social, and biological domains.

Our contributions are fourfold: (1) theoretical-proving existence of collapse thresholds and irreversibility; (2) computational-demonstrating universality across update mechanisms; (3) interpretive-unifying seemingly disconnected domain-specific failures; and (4) prescriptive-proposing entropy-aware design principles for sustainable intelligence.

Crucially, entropy collapse reframes fundamental trade-offs: systems that optimize for short-term performance inevitably contract their adaptive dimensionality, becoming trapped on low-entropy manifolds. This explains the widespread failure of late-stage interventions and highlights the need for early entropy governance-monitoring and regulating entropy dynamics before critical thresholds are crossed.

Looking ahead, this framework provides a new lens for understanding the long-term evolution of complex adaptive systems. By recognizing entropy collapse as the hidden cost of intelligence, we can design systems that balance efficiency with adaptability, sustaining their capacity for innovation and resilience across extended horizons.