Coherence Under Transformation

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1. Systems as Graphs Think of any system as a network of nodes (ideas, agents, or subsystems) connected by edges that represent relationships or influences. Each graph can be “deformed” in ways that preserve the type and structure of nodes, and we can measure coherence, which captures how much of the system’s structure survives under these deformations. 2. Principle of Evolution Systems evolve along paths that minimize internal tension while preserving coherence. Intuitively, tension is how much a node or connection “wants” to reorganize due to inconsistencies. Systems naturally find trajectories where tension is balanced and coherence is maintained. 3. Curvature and Structure In this framework: Areas with high tension act like “mass” in physics—they curve the system’s structure. Coherence is like “energy” that resists distortion. The evolution of the system is determined by the interplay between tension and coherence, shaping the overall network geometry. This mirrors Einstein’s insight in physics: structure emerges from interactions between curvature and energy. Here, logic and reasoning emerge from interactions between tension and coherence. 4. Key Properties Uniqueness: For any system following these rules, there’s a unique evolution path consistent with coherence and tension minimization. Substrate-neutral: The framework works for any system that can be represented as a network—brains, AIs, or abstract idea graphs. Empirical support: Patterns in AI reasoning networks (like LLM hidden states) are consistent with the framework’s predictions about coherence and curvature. 5. Conceptual Takeaways Systems that preserve coherence under all legal transformations and minimize tension naturally evolve along a unique, substrate-neutral structure. Coherence and curvature are two sides of the same coin: structure emerges as the system optimizes itself. Even without computing every detail, this gives a unified lens for reasoning, logic, and emergent structure. ________________________________ Recent studies provide partial validation of the framework’s principles in real-world reasoning systems: Logical Flow & Geometry – Zhou et al. (2025) show that LLM hidden states trace smooth trajectories in embedding space that reflect logical structure rather than surface content (https://arxiv.org/abs/2510.09782?utm_source=chatgpt.com). Persistent Homology – Fay et al. (2025) demonstrate that topological invariants in LLM activations persist under perturbations, supporting the idea of multiscale structural features (https://arxiv.org/abs/2505.20435?utm_source=chatgpt.com). Memory as Topological Cycles – Xin Li (2025) models memory and inference as stable loops in activation space, aligning with the framework’s “protected 1-cycles” concept (https://arxiv.org/abs/2508.11646?utm_source=chatgpt.com). Together, these studies show that logical coherence, tension, and persistent topological structure are observable in real systems, echoing the principles of the Einstein–Persistence framework.

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Coherence Under Transformation

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