Scale-Invariant Resonance (SIR): A Multiscale Information-Theoretic Framework for Pattern Emergence

Scale-Invariant Resonance (SIR): A Multiscale Information-Theoretic Framework for Pattern Emergence Kirill Vasilchenko Holon Institute of Technology, Holon, Israel kiri[email protected] Vasilchenko, K. (2025). Scale-Invariant Resonance: A Unified Criterion for Pattern Emergence in Physical and Cognitive Systems. Zenodo. https://doi.org/10.5281/zenodo.17526709 Abstract: The problem of distinguishing genuine emergent structures from stochastic fluctuations remains a central challenge in the study of complex systems, spanning statistical physics, computational neuroscience, and information theory. Classical frameworks of self-organization typically define order through thermodynamic stability or the minimization of variational free energy at a specific level of description. However, these approaches often implicitly assume a fixed scale of observation. In this work, we propose a multi-scale framework termed ScaleInvariant Resonance (SIR). We posit that "true" self-organizing forms—unlike random noise—are characterized by their ability to maintain structural integrity and low algorithmic complexity across varying resolutions of observation. This aligns with the intuition that meaningful patterns are robust to coarse-graining, whereas noise is intrinsically "scale-fragile". Keywords: multiscale entropy; Lempel-Ziv complexity; self-organization; renormalization; EEG analysis; causal emergence. 1. Introduction The distinction between genuine emergent structures and stochastic fluctuations remains a fundamental challenge in the study of complex systems, spanning statistical physics, computational neuroscience, and information theory [1, 2]. Classical frameworks of selforganization typically define order through thermodynamic stability or the minimization of variational free energy at a specific level of description [3]. However, these approaches often implicitly assume a fixed scale of observation, ignoring the multi-scale nature of complex phenomena. In this work, we propose a multi-scale framework termed Scale-Invariant Resonance (SIR). We posit that "true" self-organizing forms—unlike random noise—are characterized by their ability to maintain structural integrity and low algorithmic complexity across varying resolutions of observation. This aligns with the intuition that meaningful patterns are robust to coarse-graining, whereas noise is intrinsically "scale-fragile." 1.1. Theoretical Context and Related Work The concept of scale-dependence in complexity measures has been explored extensively in the journal Entropy and the broader literature. Multiscale Entropy (MSE). Costa et al. [4, 5] demonstrated that biological signals carry information across multiple temporal scales, and that disease states (e.g., heart failure) often involve a loss of complexity. Our framework extends this by linking multiscale stability directly to the concept of resonance in dynamical systems, specifically differentiating between "ordered" (compressible) and "disordered" (incompressible) states under renormalization. Unlike MSE, which measures statistical unpredictability, SIR combines thermodynamic order parameters with algorithmic compressibility to identify structures that are both physically stable and informationally efficient. Causal Emergence. Hoel et al. [6] introduced the theory of causal emergence, arguing that macroscale descriptions can possess greater causal power (effective information) than their micro-scale constituents. While Hoel’s approach relies on mutual information and transition probabilities, our SIR framework combines thermodynamic order parameters (e.g., synchronization, magnetization) with algorithmic complexity to identify states that are both energetically stable and informationally efficient. Renormalization and Criticality. In statistical physics, the Renormalization Group (RG) flow is used to extract relevant macroscopic variables by iteratively integrating out short-range degrees of freedom [7]. Near critical points, systems exhibit scale invariance (e.g., in the 2D Ising model). We adopt a simplified block-averaging renormalization operator to test whether a system's state retains its informational identity ("self-cleans") or degenerates into noise ("scale fragility"). Algorithmic Complexity vs. Shannon Entropy. Traditional Shannon entropy is maximized for random noise, making it a poor metric for structural order [8]. Instead, we employ Lempel-Ziv (LZ) complexity [9], which estimates the number of unique patterns in a sequence. LZ complexity has proven to be a robust proxy for Kolmogorov complexity in neuroscience, successfully distinguishing wakefulness from anesthesia or disorders of consciousness [10, 11, 13]. Unlike previous studies that often use heuristic compression (e.g., zlib), we utilize a rigorous, normalized LZ-76 algorithm to ensure our metric is invariant to the sequence length reduction caused by coarse-graining. 1.2. Paper Contribution and Structure We operationalize the SIR framework by applying a coarse-graining operator to three distinct dynamical systems and one empirical dataset: Physical Layer: The Kuramoto model of coupled oscillators and the 2D Ising model of ferromagnetism. We show that synchronization and magnetic order appear as scale-invariant energetic minima. Cognitive Layer: The Hopfield network serves as a proxy for associative memory. We demonstrate that retrieval states (attractors) remain algorithmically simple across scales, while spurious states (noise) act as high-entropy fluctuations. Empirical Validation: Using human EEG data [12], we show that the brain's Alpha rhythm creates a statistically significant "complexity gap" (p<0.001) compared to desynchronized activity, validating SIR as a biomarker of neural organization. This study provides a quantitative methodology for filtering meaningful structure from noise, bridging energetic cost and algorithmic compressibility in self-organizing systems. While thermodynamic stability and algorithmic simplicity are often studied separately, this work unifies them into a single observable – scale-invariant resonance – providing a model-agnostic tool to detect emergence without tuning parameters. 2. Methodology In this work, we propose the Scale-Invariant Resonance (SIR) framework to quantitatively distinguish emergent structures from stochastic noise. Our approach integrates thermodynamic order parameters with an information-theoretic metric of algorithmic complexity, calculated across a range of spatiotemporal scales (renormalization procedure). 2.1. The Coarse-Graining Operator We employ a real-space coarse-graining procedure inspired by Kadanoff's block-spin transformation [15]. While this does not constitute a full Renormalization Group flow in the fieldtheoretic sense, it effectively captures the macroscopic scaling properties relevant for pattern detection in finite systems. To analyze system properties at various levels of resolution, we apply a renormalization operator Mσ. For a discrete time series or spatial lattice X={x1,x2,…,xN}, the coarse-grained version X(σ) at scale σ is obtained by averaging blocks of size σ: 𝒙𝒌 (𝛔)=𝟏 𝛔∑ 𝒙𝒋 𝒌𝛔 𝒋=(𝒌−𝟏)𝛔+𝟏 , 𝒌=𝟏,…,⌊𝑵 𝝈⌋ This block-averaging approach represents a phenomenological real-space renormalization, focusing on the preservation of informational structure rather than the exact invariance of the Hamiltonian. For the subsequent complexity analysis (LZ), the resulting continuous signal is binarized relative to the median of the distribution m=median(X(σ)): 𝒔𝒌 (𝛔)=if 𝒙𝒌 (𝛔)>𝒎 then 𝟏 else 𝟎 Binarization via median thresholding is a standard procedure for Lempel-Ziv complexity analysis in neuroscience [10, 13]. We utilize median binarization to maximize the entropy of the resulting symbolic sequence, ensuring that any reduction in complexity is due to temporal structure rather than amplitude bias. 2.2. Normalized Lempel-Ziv Complexity (LZ-76) Unlike heuristic compression methods (e.g., DEFLATE/zlib), which may introduce systematic bias, we utilize the rigorous Lempel-Ziv (LZ-76) algorithm to estimate algorithmic complexity. The complexity CLZ(S) is defined as the number of unique substrings (patterns) required to generate the sequence S. 𝑪norm=𝑪LZ(𝑺) 𝐥𝐨𝐠𝟐𝑵 𝑵 Where N is the length of the sequence at the given scale. Theoretically, for random noise Cnorm →1, whereas for regular or resonant structures Cnorm→0. 2.3. Modeling Physical Systems To validate the SIR hypothesis, we employ three model systems covering distinct classes of dynamics. A. Kuramoto Model (Synchronization) We simulate a network of N=256 phase oscillators with all-to-all coupling. The phase evolution θi is governed by the equation: 𝒅𝛉𝒊 𝒅𝒕=𝛚𝒊+𝑲 𝑵∑ 𝑵 𝒋=𝟏 𝐬𝐢𝐧(𝛉𝒋−𝛉𝒊) where ωi∼N(0,1) are the natural frequencies, and K is the coupling strength. We investigate the transition from incoherence (K=0) to global synchronization (K>Kc≈1.6). The macroscopic order parameter R(σ) is calculated at each scale. B. 2D Ising Model (Criticality) To test the universality of the approach (generalization to spatial lattices), we use the 2D Ising model on an L×L lattice (L=24) with the Hamiltonian: 𝑯=−𝑱∑𝛔𝒊𝛔𝒋 ⟨𝒊,𝒋⟩ The dynamics are simulated using the Metropolis algorithm. We compare three regimes: ordered (ferromagnetic, T<Tc), critical (T≈2.27), and disordered (noise-like, T≫Tc). C. Hopfield Network (Cognitive Layer) To investigate the scale-invariance of associative memory structures, we simulated a standard Hopfield network consisting of N=1024 binary neurons si∈{−1,+1}. We stored P=3 random orthogonal patterns {ξμ} using the Hebbian learning rule: 𝑾𝒊𝒋=𝟏 𝑵∑𝛏𝒊𝛍 𝑷 𝛍=𝟏 𝛏𝒋𝛍, 𝑾𝒊𝒊=𝟎 The network dynamics evolve via asynchronous Glauber updates at zero temperature, minimizing the energy function: 𝒔𝒊(𝒕+𝟏)=sgn(∑𝑾𝒊𝒋𝒔𝒋(𝒕) 𝑵 𝒋=𝟏 ) We compared two initialization regimes: • Resonant State (Memory Retrieval): The system is initialized with a stored pattern ξ1 corrupted by 25% random bit-flip noise. This state represents the robust retrieval of information (an attractor). • Desynchronized state (Spurious): The system is initialized with a completely random state uncorrelated with any stored pattern. Similar to the physical models, we apply the renormalization operator and calculate the normalized LZ complexity Cnorm(σ) across scales to distinguish the attractor dynamics from transient noise. 2.4. Empirical Validation: Neurophysiological Data To verify the biological relevance of the criterion, we utilized real EEG data from the UCI Machine Learning Repository: EEG Eye State Data Set. • Data: Continuous EEG recording (14 channels, sampling rate fs=128 Hz), duration 117 seconds.sp • States: The subject alternates between two states: "Eyes Open" (desynchronization, noise-like activity) and "Eyes Closed" (Alpha rhythm dominance, resonant activity). While the 'Eyes Open' state contains functional beta/gamma activity, from a renormalization perspective it exhibits lower structural compressibility compared to the hypersynchronous Alpha state. We treat it as a high-entropy reference baseline rather than pure stochastic noise. • Preprocessing: To remove artifacts and low-frequency drift, we applied a 4th-order Butterworth bandpass filter in the 1−40 Hz range. • Analysis: The signal was segmented into 2-second epochs. For each epoch and each scale σ∈[1,2,…,32], the normalized complexity Cnorm was calculated. 2.5. Statistical Analysis To confirm the significance of differences between resonant and desynchronized states, we used Welch's t-test for independent samples. In all experiments (model and empirical), we report mean values and standard deviation (Mean±SD) over an ensemble of realizations (Monte Carlo method, Ntrials=20) or recording epochs. Statistical significance was established at the level of p<0.001. 3. Results To validate the Scale-Invariant Resonance (SIR) framework, we performed a multiscale analysis of algorithmic complexity and order parameters across three domains: continuous physical synchronization (Kuramoto model), discrete spatial lattices (2D Ising model), associative memory (Hopfield network), and empirical neurophysiological dynamics (Human EEG). 3.1. Physical Layer: Synchronization and Criticality A. Kuramoto Model. We first analyzed the emergence of synchronization in a network of N=256 phase oscillators. As shown in Figure 1, in the non-resonant regime (coupling K=0.0, noisedriven), the system exhibits low thermodynamic order (R≈0) and high normalized Lempel-Ziv complexity (Cnorm→1) across all scales. In contrast, in the resonant regime (K=4.0), the system maintains a high order parameter (R≈1.0) and consistently low algorithmic complexity (Cnorm <0.1) throughout the coarse-graining process. This confirms that synchronization is a scaleinvariant feature, whereas incoherent noise is scale-dependent. Figure 1. Thermodynamic stability and algorithmic complexity in the Kuramoto model (N=256). (Left) The order parameter R(σ) as a function of the coarse-graining scale σ. In the resonant state (K=4.0, purple), macroscopic order persists across scales. (Right) Normalized Lempel-Ziv complexity Cnorm(σ). The resonant state maintains low information content (compressibility), distinguishable from the high-entropy noise state (K=0.0, blue). Error bars indicate ±1 SD over 5 Monte Carlo trials. B. 2D Ising Model. To test the generalization of the framework to spatial lattices, we simulated a 24×24 Ising model (Figure 2). At high temperatures (T=5.0, green), the system is in a disordered paramagnetic phase, characterized by near-zero magnetization and maximal algorithmic complexity. At low temperatures (T=1.5, blue), the system settles into ferromagnetic order (Resonance), characterized by stable magnetization ∣M∣≈1 and low complexity. Crucially, even near the critical point (T=2.3, orange), the SIR metric distinguishes ordered domains from thermal noise. Figure 2. Generalization to spatial systems: 2D Ising Model. (Left) Magnetization ∣ M ∣ versus coarse-graining block size σ. (Right) Normalized LZ complexity. The resonant ferromagnetic state (T=1.5, blue) is separated from the thermal noise state (T=5.0, green) by a significant complexity gap across all observed spatial scales. 3.2. Cognitive Layer: Associative Memory We applied the multiscale renormalization operator to a Hopfield network (N=1024 neurons) to differentiate between robust memory retrieval (Resonance) and spurious states (Noise). Figure 3 demonstrates that while a noise-driven state may exhibit incidental correlations at the microscopic scale (σ=1), it lacks structural integrity at higher levels of abstraction. The resonant state (retrieval of a stored pattern) maintains low normalized complexity across scales. Figure 3. Information structure in the Hopfield network. Comparison of normalized LZ complexity for a Resonant state (pattern retrieval, blue) and a Noise state (random attractor, orange). The resonant state remains algorithmically simple under renormalization, whereas the noise state exhibits high entropy. Error bars: ±1 SD (Ntrials=20). 3.3. Empirical Validation: Human EEG Finally, we tested the biological relevance of SIR using real human EEG data from the UCI repository (14 channels, fs=128 Hz). We compared epochs of "Eyes Closed" (characterized by Alpha rhythm) against "Eyes Open" (desynchronized activity). Previous studies [13] have shown lower LZ in anesthesia/sleep. Our contribution extends this by characterizing the scaling behavior of this complexity. We show that resonant states are not just 'simpler' at one scale, but maintain a stable informational topology across the renormalization flow, unlike noise which diverges. As illustrated in Figure 4, the resonant Alpha state exhibits consistently lower normalized algorithmic complexity than the noise-like "Eyes Open" state. Crucially, this separation is observed across all temporal scales σ∈[1,32]. We performed Welch’s t-test at each scale, confirming that the difference in complexity is statistically significant (p<0.001). This empirical evidence supports the hypothesis that functional neural patterns are encoded as scale-robust, compressible structures. Figure 4. Empirical validation on human EEG data. Multiscale normalized Lempel-Ziv complexity for "Eyes Open" (Noise-like, orange) and "Eyes Closed" (Resonant Alpha, blue) states. The resonant state creates a "complexity gap," maintaining lower information content across temporal scales. Asterisks ( ∗∗∗ ) indicate statistical significance (p<0.001, Welch’s t-test). 3.4. Qualitative Evidence: Structure Preservation To provide an intuitive visualization of the "self-cleaning" property of resonance, we reconstructed the 2D lattice states of the Hopfield network at scales σ=1,4,16 (Figure 5). The resonant state, initially corrupted by 25% noise, reveals its underlying structure as the scale increases. In contrast, the noise state degenerates into unstructured fluctuations, confirming that noise is intrinsically "scale-fragile."