Historical Genetic Logic as a Dynamical Coherence Judge for Large Language Models

Abstract

Historical Genetic Logic as a Dynamical Coherence Judge for Large Language Models A Rigorous Formalization of Xenopoulos’ Dialectical Operators and Experimental Validation on LLM Self‑Contradiction Katerina XenopoulouIndependent Researcher, Kefalonia, GreeceORCID: 0009-0004-9057-7432Correspondence: [email protected] Theoretical Foundation: Epameinondas Xenopoulos †Epistemology of Logic: Logic–Dialectic or Theory of Knowledge (2nd ed., 2024)ORCID: 0009-0000-1736-8555† In memoriam (1920–1994) DOI: 10.5281/zenodo.19263676 https://zenodo.org/uploads/19263676 ABSTRACT Internal self‑contradiction remains a critical failure mode in Large Language Models (LLMs), limiting their reliability in high‑stakes reasoning. While current mitigation strategies like Chain‑of‑Thought (CoT) prompting improve performance, they lack formal guarantees of logical stability. This paper introduces a novel framework for diagnosing and regulating LLM coherence by formalizing Historical Genetic Logic as a Nonlinear Dynamical System. We demonstrate that the reasoning process in autoregressive models can be modeled as a trajectory in a recursive metric space D=⋃n=0∞DnD=⋃n=0∞Dn with Dn+1=[0,1]2×Pfin(Dn)Dn+1=[0,1]2×Pfin(Dn). Our core theoretical contribution, the Xenopoulos Spectral Invariance Theorem (Theorem 7.1), proves that CoT prompting leaves the Lyapunov spectrum invariant, merely extending unstable trajectories without suppressing the underlying chaotic divergence. To address this, we propose the Xenopoulos Layer, a spectral feedback controller that dynamically intervenes in the Jacobian operator Fγ=F−γIFγ=F−γI. By enforcing a negative Lyapunov exponent λ1(γ)<0λ1(γ)<0, the controller provides formal guarantees of stability and coherence. The 34th Principle establishes that any sufficiently expressive autoregressive system with nonlinear reinforcement and memory feedback necessarily admits regions of positive Lyapunov growth—implying that absolute coherence is structurally unattainable, and spectral regulation is therefore essential. Experimental validation across GPT‑4, Claude, Gemini, and DeepSeek architectures shows an 80–100% reduction in logical contradictions compared to state‑of‑the‑art self‑correction methods. Scaling analysis on the Epistemology of Logic corpus (7,816 sentences) demonstrates zero XEPTQLRI instability and τ9τ9 meta‑transcendence, proving that Historical Genetic Logic provides the optimal structural foundation for coherent AI reasoning. The results suggest that transitioning from representation‑level prompting to operator‑level spectral control is essential for the next generation of safe and aligned Artificial Intelligence. Keywords: LLM Coherence, Nonlinear Dynamics, Lyapunov Exponents, Historical Genetic Logic, AI Safety, Spectral Control, Xenopoulos Layer, 34th Principle 1.1 The Problem of Dynamic Reasoning Classical logic was designed to formalize valid inference under the assumption of static propositions and reversible operations. In such systems, truth values are fixed, negation is involutive, and inference rules operate independently of historical accumulation. These assumptions ensure formal clarity but exclude a fundamental property of real reasoning processes: historical evolution. Modern reasoning systems—biological or artificial—do not operate in static propositional spaces. They accumulate memory, amplify internal tensions through nonlinear feedback, and remain subject to stochastic perturbations. Consequently, their behavior may exhibit sensitivity to initial conditions, bounded divergence, and regime transitions—phenomena typically studied in nonlinear dynamical systems rather than in formal logic. The central theoretical difficulty is therefore the following: How can reasoning be modeled as a mathematically rigorous dynamical process that incorporates memory growth, nonlinear reinforcement, and measurable stability properties without reducing it to static Boolean inference? 1.2 The Case of Large Language Models Autoregressive language models generate text by recursively predicting the next token based on previous context. This process can be viewed as a trajectory in a high‑dimensional space, where each step depends on the accumulated history. While such models achieve remarkable performance, they remain prone to internal contradictions, hallucinations, and logical inconsistencies—particularly in long‑form reasoning tasks. Current mitigation strategies, such as Chain‑of‑Thought (CoT) prompting, improve performance by encouraging intermediate reasoning steps but do not provide formal guarantees of logical stability. This gap motivates a dynamical systems approach to reasoning coherence. 1.3 The Theoretical Gap Existing approaches fall into three broad categories: Classical Logic Extensions: Extend Boolean systems but retain reversibility and static semantics. Probabilistic / Bayesian Models: Model uncertainty but not dynamical instability. Optimization‑Based Views: Focus on training dynamics, not reasoning trajectory dynamics. None of these frameworks provide a mathematical language for measuring, predicting, or controlling the emergence of self‑contradiction as a dynamical phenomenon. 1.4 Historical Genetic Logic as a Dynamical System Epameinondas Xenopoulos (1920–1994) developed Historical Genetic Logic as an alternative to static formal logic. His central thesis was that contradiction is not an error to be eliminated but a creative force that drives development. In his framework: Identity is genetic: A→A′A→A′, not A=AA=A Negation is dialectical: ¬D(A)¬D(A) preserves AA while generating its evolution Contradiction is tension: the product of a proposition and its dialectical negation Historicity is memory: the present state incorporates the past These philosophical principles were formalized in a system of 33 principles, 10 axioms, and 5 theorems (Xenopoulos, 2024; Xenopoulou, 2026). The present work builds upon this foundational framework, applying its dynamical core—specifically the memory‑structured recurrence and the instability functional—to model and regulate coherence in Large Language Models. Table 1 summarizes the structural correspondence between the philosophical principles and their mathematical counterparts as used in this work. Table 1: Structural Correspondence: Philosophy to Mathematics Philosophical Principle Mathematical Counterpart Historicity Ht={xτ:τ<t}Ht={xτ:τ<t} Memory‑structured evolution xt+1=F(xt,xt−1,…,xt−m+1)xt+1=F(xt,xt−1,…,xt−m+1) Dialectical intensity at=θt−Atat=θt−At Historical mean μt=1m∑i=1mat−iμt=m1∑i=1mat−i Nonlinear amplification Tt=κat2(1+βtanh⁡(μt))Tt=κat2(1+βtanh(μt)) For the complete mathematical formulation of the foundational system, we refer the reader to the cited works. 1.5 Main Contributions A. Foundational Framework (from Xenopoulos, 2024; Xenopoulou, 2026) A complete metric historical state space for reasoning systems. A non‑Boolean algebra (XLDA) with non‑involutive negation. An irreversible non‑reductive closure principle (INRC). A memory‑structured nonlinear recurrence with positive Lyapunov exponent. A compact partially hyperbolic attractor (XDA). An extended dialectical metric (XDM). A measurable instability functional (XEPTQLRI). B. Contributions of This Work (LLM Application)8. Proof of bounded divergence and analytic ceiling for the recurrence.9. A spectral feedback controller modifying the Jacobian spectrum, applied to LLM trajectories.10. A formal comparison showing that Chain‑of‑Thought does not alter Lyapunov structure.11. A phase transition theory of cognitive regimes in autoregressive models.12. An executable empirical validation protocol for LLM coherence. 1.6 Structure of the Paper Section 2 introduces the formal dialectical state space. Section 3 derives the memory‑structured nonlinear dynamics. Section 4 maps LLM outputs to dynamical trajectories. Section 5 presents the experimental validation framework and summary results. Section 6 develops spectral gap analysis and control. Section 7 compares the framework with Chain‑of‑Thought prompting. Section 8 establishes cognitive phase transition results. Section 9 provides comparative scaling analysis. Section 10 discusses practical logic and developmental interpretation. Section 11 formalizes structural guarantees. Section 12 provides comparative analysis. Section 13 discusses implications and limitations. Section 14 concludes. Section 15 lists references. SECTION 2: FORMAL DIALECTICAL STATE SPACE 2.1 Recursive Construction of the Historical Space Classical logical systems are defined over static propositional domains. In contrast, we define a historically expanding state space. Let D0=[0,1]2×{∅}D0=[0,1]2×{∅} For each n≥0n≥0, define recursively Dn+1=[0,1]2×Pfin(Dn)Dn+1=[0,1]2×Pfin(Dn) where Pfin(Dn)Pfin(Dn) denotes the set of all finite subsets of DnDn. Define the full dialectical space D=⋃n=0∞DnD=n=0⋃∞Dn Interpretation. Each state consists of two bounded components in [0,1]2[0,1]2 and a finite historical memory drawn from lower levels. Thus every element of DD is finitely generated but potentially unbounded in historical depth. 2.2 Dialectical State Definition 2.1 (Dialectical State). A dialectical state is a triple x=(θ,A,H)∈Dnx=(θ,A,H)∈Dn such that: θ,A∈[0,1],H⊂Dn−1,H is finite.θ,A∈[0,1],H⊂Dn−1,H is finite. We interpret θθ as primary assertion component, AA as opposing component, and HH as historical memory. No semantic interpretation is required for formal development. 2.3 Metric Structure We define a recursive metric. Base Level. For x,y∈D0x,y∈D0: d(x,y)=∣θx−θy∣+∣Ax−Ay∣d(x,y)=∣θx−θy∣+∣Ax−Ay∣ Recursive Level. For x,y∈Dn+1x,y∈Dn+1: d(x,y)=∣θx−θy∣+∣Ax−Ay∣+dH(Hx,Hy)d(x,y)=∣θx−θy∣+∣Ax−Ay∣+dH(Hx,Hy) where dHdH is the Hausdorff metric induced by dd: dH(Hx,Hy)=max⁡{sup⁡hx∈Hxinf⁡hy∈Hyd(hx,hy), sup⁡hy∈Hyinf⁡hx∈Hxd(