CIv7-ECA: Structural Break Detection via Symbolic Substrate Compression and Fault Geometry

Hypothesis Structural breaks in univariate time series can be robustly detected by encoding the input as symbolic sequences (e.g., via permutation entropy, delta-sign encoding), evolving these sequences through Elementary Cellular Automata (ECA), and analysing the resulting 2D symbolic evolution as an algorithmic and topological substrate. This substrate exposes latent causal dynamics, semantic irregularities, and algorithmic fault lines. These transitions manifest as discontinuities in:

• Algorithmic compressibility (e.g., via BDM or CTM)
• Joint compression failure (predictive collapse à la Sutskever)
• Torsion and persistent topological invariants (Walch)
• Motif circuit reconfiguration and symbolic attention rewiring (Circuit Tracer analogy)
• Local motif entropy shifts and permutation entropy gradients
• Collapse of symbolic attractors and bifurcation points in 2D dynamics
• Failure of logical boundary preservation (torsion and curvature loss per Hodge)
• Reality signal collapse (based on perceptual vs. imagined structure loss)
• Prediction breakdown at the edge of chaos (Zhang et al.)
• Simulability gaps in world models built from symbolic evolution (Ha & Schmidhuber)
• Emergence of intelligent representation capacity from non-intelligent but complex symbolic sources (Zhang et al.)
• Reversible inference layers from local symbolic MCMC transitions (Vivier-Ardisson et al.)
• Self-replicating symbolic agent emergence in open-ended improvement loops (Darwin Gödel Machine)
• LLM-guided symbolic alpha search and functional meta-discovery (AlphaEvolve, DeepMind)
• Symbolic rule factorisation and causal primes behind complexity emergence (Riedel & Zenil)
• Compression-meaning divergence between human and LLM semantics in cluster formation and fidelity trade-offs (Shani et al.)
• Human-guided symbolic segmentation and conceptual scaffolding for efficient reasoning emergence (OpenThoughts)
• Negative complexity zones from local motif degeneracy and symmetry folding (Grosse et al.)
• Symbolic fault geometry as hybrid MDL-complexity ridge between manifold sectors (Grünwald & Roos)
• Dynamic curriculum-guided symbolic evolution using SFT-RL hybrid training (SASR)
• Curriculum-aware adaptive strategy switching for symbolic exploration vs. memorisation (Chen et al., SASR)
• Multi-agent R\&D orchestration for factor-model co-evolution in symbolic financial substrates (RD-Agent(Q))
• Self-supervised symbolic encodings across structural graph domains with transferable motif-awareness (GFSE, Chen et al.)
• Causal fidelity loss from symbolic information bottlenecks in compression vs. meaning trade-offs (Shani et al.)
• Step-wise adaptation of symbolic reasoning dynamics using gradient-informed switching (SASR)
• Bandit-driven symbolic substrate scheduling for hypothesis refinement (RD-Agent(Q))
• Universal motif encoders for symbolic graph-to-sequence compression (GFSE)
• Latent alignment instabilities and symbolic leakage paths across representation manifolds (Jha et al., vec2vec)
• Algebraic and topological substrate bias as a default generalisation prior in LLMs (Hodge et al.)
• Emergent symbolic coherence and crystalline reasoning structures across attention-MLP duality (Hodge, Langlands view)
• Forecast error divergence as a symbolic signal of model inadequacy and time series regime shift (Grundy et al.)
• Symbolic rule primality and causal decomposition as the algebraic scaffolding underlying composite complexity emergence (Riedel & Zenil)

The symbolic substrate is not merely representational—it is causally expressive, revealing failure surfaces where data-driven structure collapses. The substrate thus functions as a model-agnostic detection layer for algorithmic, topological, and semantic regime shifts.

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Rationale

• Sutskever’s theory of unsupervised learning demonstrates that compression is prediction; joint compression failure between segments of symbolic time series reflects true structural discontinuity.
• Walch’s torsion analysis confirms that fragile, high-dimensional topological signatures collapse under even minor perturbations—making torsion an excellent early-warning detector.
• Sakabe et al. show that BDM captures algorithmic changes in symbolic representations better than Shannon entropy, making it ideal for symbolic phase-shift detection in ECA substrates.
• Anthropic’s Circuit Tracer demonstrates that influence graphs can localise semantic drift in LLMs; we extend this metaphor to symbolic motifs in ECA evolution, where rewiring of symbolic circuits signals change.
• Grosse et al. provide a geometric generalisation of Occam’s Razor showing how singular models can have negative complexity; in symbolic substrates, degeneracy or symmetry in motif transitions may represent low-complexity zones that are broken by regime change.
• Grünwald & Roos (2019) offer a modern MDL theory emphasizing model selection and predictive robustness via universal distributions. Their use of NML and luckiness-weighted codes supports our treatment of symbolic substrates as sequential predictors whose compressibility divergence reflects structural breakpoints.
• BrightStar Labs (EMs) use CA-like substrates for generalisation via initial state optimization. Our use is inverse: symbolic inputs are evolved under fixed rules, making the substrate a diagnostic canvas rather than a policy machine.
• OpenThoughts (2024) emphasises the critical role of structured symbolic input for training high-fidelity reasoning systems; our substrate acts as a pre-LLM semantic preprocessor.
• Dijkstra et al. (2025) on the neural basis of reality monitoring shows that additive imagery-perception confusion maps directly onto symbolic substrate failures, where local motif reinforcement mimics perceptual signal misclassification.
• Zhang et al. (2024) explore intelligence at the edge of chaos, showing that exposure to structured yet unpredictable systems (like Class IV ECAs) enables models to develop generalisable reasoning capabilities. ECA-evolved substrates inhabit this complexity sweet spot—structured enough to encode causality, complex enough to force predictive reasoning.
• Ha & Schmidhuber’s World Models highlight the importance of a compressed latent space capable of simulating the environment. ECA substrates can similarly act as symbolic generative scaffolds, allowing detection of divergences between real and simulated dynamics.
• Vivier-Ardisson et al. (2025) demonstrate that symbolic substrates based on local MCMC layers can serve as reversible reasoning scaffolds. The symbolic dynamics of ECAs can be understood as heuristic proposal distributions whose temperature-driven collapse or bifurcation traces the onset of algorithmic phase transitions.
• Darwin Gödel Machine (2024) introduces a framework for autonomous, open-ended self-improvement through empirical feedback loops. The symbolic agent maintains a lineage of increasingly capable variants via archive-driven selection and self-modification, mirroring symbolic motif evolution within the ECA substrate.
• AlphaEvolve (2025) demonstrates the potential of LLM-guided, feedback-grounded evolutionary code improvement. Within symbolic substrates, Alpha-style LLM agents can discover compressible, high-performance motif configurations that exceed static design patterns.
• Riedel & Zenil (2025) introduce the concept of ECA rule primality, showing that all Class 4 (complex) ECA rules are composites of simpler Class 1 or 2 primes. Their causal decomposition framework demonstrates that complexity in symbolic dynamics emerges from non-commutative compositions of simpler primitives—providing an algebraic foundation for fault geometry and symbolic fault line inference. These prime rules, especially diagonal shifters like 15 and 170, function as symbolic basis vectors whose combinatorics drive phase transitions in the ECA evolution.
• Shani et al. (2024) reveal how LLMs favour statistically efficient clustering at the expense of human-aligned typicality and semantic richness. This compression-meaning trade-off becomes visible as a form of structural distortion in symbolic substrates, where meaning-preserving clusters are sacrificed for global compactness.
• Chen et al. (2024) present a hybrid SFT-RL framework (SASR) that guides training via gradient-informed transitions between learning modes. This idea parallels symbolic evolution driven by motif divergence minimisation, where symbolic substrates adaptively switch between memorisation (fine-tuning) and generalisation (policy exploration).
• RD-Agent(Q) (2024) demonstrates that a multi-agent R\&D framework for financial factor and model co-evolution outperforms monolithic baselines via closed-loop synthesis, validation, and scheduling. This architecture offers a template for symbolic substrate orchestration where hypothesis generation, code realisation, and fault pattern evaluation are delegated to symbolic agents whose interactions mirror substrate evolution.
• GFSE (Chen et al., 2024) establishes a universal graph encoder that captures domain-agnostic structural patterns across graphs. When integrated with symbolic substrates, GFSE offers structure-preserving positional encodings, transferable across motif classes, enabling motif-aware compressibility analysis and topological fault detection in hybrid graph-symbolic systems.
• Jha et al. (2024) demonstrate that embedding spaces across different models share a universal geometric alignment, even without paired data. This implies that symbolic motifs, once encoded in a universal latent manifold, may be projected across architectures while retaining semantic fidelity. The vec2vec framework highlights potential leakage pathways and causal geometry transfer that reveal otherwise hidden symbolic structure breaks.
• Hodge et al. (2024) argue that transformer representations encode powerful algebraic and topological priors, not because they learn language per se, but because they exploit latent symmetry, closure, and group-theoretic regularities. ECA substrates foreground these algebraic invariants, making faults in the symmetry group interpretable as causal discontinuities.
• Hodge et al. (2024) also describe transformers as geometric machines of meaning: Attention layers perform Picard-style geometric operations while MLPs encode Néron-Severi algebraic invariants, stabilised through Hodge numbers and harmonic lattices, and unified by a Langlands-type correspondence. The ECA substrate provides an empirical lens to test and visualise this algebraic-topological alignment.

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Supporting Literature

• Zenil et al. – Algorithmic Information Dynamics
• BrightStar Labs (2025) – Emergent Models and symbolic computation substrates
• Sakabe et al. – Attribution Drift from symbolic perturbations
• Maria Walch (2024) – Torsion as topological signal in high-dimensional symbolic dynamics
• Ilya Sutskever (2024) – Unsupervised Learning as Compression (joint vs. separate encoding)
• Anthropic (2025) – Circuit Tracer and latent attribution path rewiring
• Grosse et al. – Occam’s Razor in Geometric-Algorithmic Deep Models
• Peter Grünwald & Teemu Roos – MDL Revisited
• OpenThoughts (2024) – Recipes for compositional reasoning over symbolic sequences
• Dijkstra et al. (2025) – Neural Basis for Reality Signal Formation and Binary Judgement
• Zhang et al. – Intelligence at the Edge of Chaos
• Ha & Schmidhuber – World Models and Evolutionary Symbol Machines
• Zhang et al. – DGM and Self-Modifying Substrate Architectures
• Vivier-Ardisson et al. – Physics of Learning & MCMC Layers
• DeepMind (2025) – AlphaEvolve: LLM-supervised evolutionary algorithm discovery
• Riedel & Zenil (2025) – Rule Primality, Turing Composition, and Causal Decomposition in ECA
• Shani et al. (2024) – From Tokens to Thoughts: Compression-Meaning Divergence in LLM vs Human Concepts
• Chen et al. (2024) – SASR: Adaptive Integration of SFT and RL
• Li et al. (2024) – RD-Agent(Q): Multi-Agent System for Symbolic Financial Factor-Model Co-Design
• Chen et al. (2024) – GFSE: Universal Graph Structural Encoder for Cross-Domain Symbolic Motif Integration
• Jha et al. (2024) – Universal Geometry of Embeddings and Platonic Alignment
• Hodge et al. (2024) – Algebraic and Topological Inductive Bias in Transformer Representations
• Grundy et al. (2025) – Forecast Error-Based Online Changepoint Detection

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CIv7-ECA: Symbolic Substrate Compression and Fault Geometry in Algorithmic Systems

Hypothesis

Complex systems can be understood and diagnosed by encoding temporal or spatial sequences into symbolic representations (e.g., delta-signs, permutation sequences), evolving these through Elementary Cellular Automata (ECA), and analysing the resulting 2D symbolic evolution as a causally expressive substrate. This symbolic substrate makes observable the emergence, collapse, and mutation of structures—revealing failure modes or discontinuities as algorithmic, geometric, and semantic phase transitions.

These transitions are not limited to any one domain. They may manifest in:

• Time series (e.g., structural breaks)
• Graph dynamics (e.g., motif degeneracy)
• LLM-guided code evolution
• Self-reflective symbolic agents
• Perception-to-imagination overlays in human-AI cognition

ECA-based symbolic substrates, as deterministic rule-based expanders of semantic signal, make fault geometry and algorithmic misalignment visible and measurable. We posit that symbolic motif evolution via fixed ECA rules forms an interpretable diagnostic plane over which discontinuities appear as phase shifts, compressibility divergences, and topological inflections.

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Distinguished Application: Structural Break Detection

Structural breaks in univariate time series can be robustly detected by:

• Symbolically encoding the series (e.g., delta-signs, permutation entropy)
• Evolving the sequences through Class IV ECAs
• Analysing the resulting symbolic 2D evolution for latent fault geometry

Discontinuities manifest as:

• Sudden changes in algorithmic compressibility (via BDM or CTM)
• Local semantic motif rewiring
• Fault lines traced via torsion, entropy gradients, or symbolic attractor collapse

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Discontinuities as Phase Shifts in Symbolic Substrate

Key transition signatures include:

• Algorithmic compressibility shift (BDM/CTM/entropy gradient)
• Joint compression failure between adjacent symbolic segments (Sutskever)
• Collapse of motif circuits and symbolic attractor basins
• Torsional bifurcation (Walch): topological instability under semantic perturbation
• Rewiring of symbolic attention flows (in analogy to Circuit Tracer)
• Local motif entropy spikes or symmetry folds (Grosse et al.)
• Bifurcations and edge-of-chaos attractor instabilities (Zhang et al.)
• Perception-imagination confusion signals (Dijkstra et al.)
• Simulability divergence in symbolic world models (Ha & Schmidhuber)
• Latent steering irreliability analogs in symbolic motif shifts (Braun et al.)
• MCMC layer temperature collapse analogs in symbolic phase shifts (Vivier-Ardisson et al.)
• Negative complexity emergence in degenerate motif zones
• Universal geometry leakage and motif projection drift (Jha et al.)
• Substrate collapse under symbolic agent self-modification (Darwin Gödel Machine)
• Motif classifier failure under category compression vs. semantic fidelity (Shani et al.)
• Failure of symbolic curriculum stability under hybrid SFT-RL learning (SASR)

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Rationale and Theoretical Foundations

• Compression is Prediction (Sutskever): Symbolic substrates act as joint compressors; failure to compress adjacent regions signals regime change.
• Algorithmic Complexity (Sakabe, Zenil): Symbolic evolution under ECA rules yields robust estimates of local information content, superior to Shannon entropy.
• Torsion (Walch): A sensitive indicator of latent topological stress; local symbolic torsion marks instability zones.
• Circuit Drift Analogy (Anthropic): Semantic attention drift in LLMs has symbolic analogs in motif reconfiguration over time.
• Negative Complexity (Grosse): Low-complexity symbolic zones can suddenly destabilise under symmetry breakage.
• Reality Signal (Dijkstra): Symbolic motifs sometimes mimic imagined, not real, patterns—leading to symbolic hallucinations.
• Edge of Chaos (Zhang): Class IV ECAs provide maximally informative symbolic dynamics—structured enough to encode causality, chaotic enough to surface meaningful divergence.
• World Models (Ha & Schmidhuber): Symbolic ECA evolution provides an alternate substrate for generative modeling and mismatch detection.
• vec2vec Leakage (Jha): Symbolic motifs exhibit cross-architecture alignment potential—vulnerable to projection drift and causality loss.
• Langlands Duality (Hodge): Transformers encode geometric-algebraic duality; symbolic substrates externalise this as motif attractor duality.
• Steering Vector Failure (Braun et al.): Symbolic motifs resist coherent steering when their local evolution manifolds lose separability or directional alignment—analogous to latent vector drift in LLMs.

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