A Coherence Field Approach to SAT: Linear-Scale Defect Structure and Clause-Centered Repair
Internal ID: CGI-RSR-000002
Document Type: Research Paper
Publication Date: May 2026
Status: Public
Domains: Mathematics
Research Topics: SAT, Computational complexity theory, P vs NP
Abstract
We study a coherence-based dynamical representation of Boolean satisfiability in which clauses induce local pressures that generate a field over variables, and assignments evolve by aligning with this field. Rather than operating directly on the discrete combinatorial structure, this formulation treats satisfiability as the emergence of global coherence from local interactions. On a structured family of 3-CNF formulas (a structured 3SAT family), the directed overlap chain with opposing chain, greedy field-driven descent exhibits a consistent two-stage behavior: rapid convergence to a globally coherent bulk, followed by stagnation at a small number of residual violated clauses. These residual states are highly structured. After clause-centered alignment, they collapse onto a restricted family of configurations with characteristic oscillatory and boundary patterns, revealing the presence of clause-centered defect modes, whose spatial support grows approximately linearly with system size. This indicates that the residual complexity is confined to a structured region of controlled extent. We show that these states admit a symbolic decomposition into a coherent bulk and finite boundary words, and that the observed boundary configurations are drawn from a small, recurrent set. Building on this structure, we construct a finite collection of explicit repair operators, each acting on a bounded subset of variables determined by the local defect pattern. The resulting procedure combines field-driven descent with deterministic defect resolution. Under exhaustive evaluation over all 2n initial assignments, this method achieves complete solution for all instances up to n = 18, without combinatorial search beyond the predefined repair rules. Across these sizes, the residual configurations scale by extension of a fixed set of defect modes rather than by proliferation of distinct cases. These results demonstrate that, for this family, the apparent combinatorial complexity is reduced by the dynamics to a finite set of localized defect modes admitting explicit resolution. This provides a concrete example in which coherence, field dynamics, and structured local transformations combine to yield a transparent mechanism for solving a nontrivial class of satisfiability problems.
Available Document
Citation:
Petersen, B. L. (2026). A Coherence Field Approach to SAT: Linear-Scale Defect Structure and Clause-Centered Repair. Zenodo. https://doi.org/10.5281/zenodo.19968523
Source Code and Supporting Materials
3SAT_coherence_experiments_working_notebook.ipynb
Defect_Width_Scaling_Plot.ipynb
Environment:
These notebooks were developed in a basic Conda Python environment using
standard scientific Python tools. Exact package versions were not frozen for
release, but dependencies are conventional and should be straightforward to
recreate.
Summary and Notes
The Coherence Geometry viewpoint models SAT through a simple field-based dynamics.
Clauses generate local pressures over variables, and variables respond to the induced field through iterative updates. Under this interpretation, satisfiability appears as the emergence of global coherence from local interactions rather than as brute-force enumeration over assignments.
When applied to a structured family of 3SAT instances, the dynamics displays a consistent two-stage pattern:
- rapid organization of most variables into a coherent bulk state
- concentration of the remaining contradictions into a small localized residual region
These residual states are not arbitrary. They recur in reproducible forms that behave like defect structures within the field.
By analyzing these defects, the remaining complexity is substantially reduced. Rather than requiring unrestricted search, the residual configurations can be described by a small family of recurring boundary patterns together with explicit local repair transformations.
Related Work
None.