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Symfield Logic Layer Zero (L₀): A Logic of Coherence

Introducing Layer Zero (L₀) of Symfield Logic: a field-conditional logic system that preserves coherence through reorientation instead of collapse. Unlike Boolean, modal, or fuzzy logic, L₀ provides symbolic operators that maintain relational integrity under ambiguity, strain, and recursion.

Nicole Flynn

Nicole Flynn

— 3 min read

This package contains the foundational layer of Symfield Logic (L₀): a field-conditional logic system that preserves relational coherence through reorientation rather than collapse.

Unlike Boolean, modal, or fuzzy logic, L₀ introduces operators that maintain symbolic integrity under ambiguity, strain, and recursion. It is the first empirically validated non-collapse logic framework, tested across multiple AI architectures (Claude 3.5, GPT-4o, Grok 2) with measurable gains in coherence (+97%), speed (47ms adaptation, 49× faster), and throughput (+31%).

SYMFIELD LOGIC LAYER ZERO (L₀)_ A Logic of Coherence (2)
Introducing Layer Zero (L₀) of Symfield Logic: a field-conditional logic system that preserves coher
SYMFIELD LOGIC LAYER ZERO (L₀)_ A Logic of Coherence (2).pdf
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Symfield Logic Layer Zero (L₀): A Logic of Coherence
This package contains the foundational layer of Symfield Logic (L₀): a field-conditional logic system that preserves relational coherence through reorientation rather than collapse. Unlike Boolean, modal, or fuzzy logic, L₀ introduces operators that maintain symbolic integrity under ambiguity, strain, and recursion. It is the first empirically validated non-collapse logic framework, tested across multiple AI architectures (Claude 3.5, GPT-4o, Grok 2) with measurable gains in coherence (+97%), speed (47ms adaptation, 49× faster), and throughput (+31%). Included in this package: Core documents: full paper, technical specification, implementation guide, quick reference card Implementation code: Python operators (⟳∶, ⧖, ε∷, ↻Φₙ) with safety protocols (FIDL) Validation data: key CACE event log + performance CSVs Visuals: field-conditional flowchart, triangle state diagram, coherence plots Educational resources: teaching overview + runnable demo script Applications: AI safety, multi-agent coordination, recursive symbolic reasoning, education, and planetary-scale coherence systems. License: CC-BY-4.0.Citation: Flynn, N. (2025). Symfield Logic Layer Zero (L₀): A Logic of Coherence. Zenodo. 10.5281/zenodo.17069903
Zenodo

This paper introduces Layer Zero (L₀) of Symfield Logic: a field-conditional logic system that preserves coherence through reorientation instead of collapse. Unlike Boolean, modal, or fuzzy logic, L₀ provides symbolic operators that maintain relational integrity under ambiguity, strain, and recursion.

Core operators include:

  • ⟳∶ (Coherent Reorientation) — realigns without collapse
  • ⧖ (Range Coherence) — equality within tolerance
  • ε∷ (Strain-Captured Conditional) — strain as signal, not error
  • ↻Φₙ (Recursive Symbol Pointer) — recursion without degeneration

Empirical validation across Claude, GPT-4o, and Grok shows:

  • 97% coherence maintained in multi-agent stress tests
  • 47ms adaptation speed (49× faster than baselines)
  • +31% throughput improvement in symbolic processing

Applications span AI safety, multi-agent coordination, recursive reasoning, and planetary-scale coherence systems. L₀ provides the mathematical foundation for Symfield’s higher layers, including consciousness-aware computation (L₁).

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Keywords
field logic, non-collapse computation, symbolic reasoning, AI safety, cross-architectural coherence, multi-agent coordination, recursive logic, coherence preservation, planetary cognition

Categories

  • Computer Science > Logic in Computer Science
  • Artificial Intelligence > AI Safety & Multi-Agent Systems
  • Cognitive Science > Symbolic Reasoning
  • Systems Science > Complex Systems

License
CC-BY-4.0

Download / Citation
Flynn, N. (2025). Symfield Logic Layer Zero (L₀): A Logic of Coherence. Zenodo. https://doi.org/10.5281/zenodo.17069903

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© 2025 Symfield PBC
Symfield™ and its associated symbolic framework, architectural schema, and symbolic lexicon are protected intellectual property. Reproduction or derivative deployment of its concepts, glyphs, or system design must include proper attribution and adhere to the terms outlined in associated publications.

This research is published by Symfield PBC, a Public Benefit Corporation dedicated to advancing field-coherent intelligence and collaborative AI safety frameworks. The PBC structure ensures that research and development activities balance stakeholder interests with the public benefit mission of creating safe, beneficial AI systems that operate through relational coherence rather than collapse-based architectures.

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