The P versus NP Problem
Field
Computational complexity theory
Clay problem statement (informal)
Does every problem whose solutions can be verified in polynomial time also admit a polynomial-time algorithm for finding a solution?
Equivalently, is P = NP?
Why the problem is difficult
SAT and related NP-complete problems appear to require searching through exponentially many possible assignments in the worst case. Although verifying a proposed solution can be fast, no general polynomial-time method is known for finding one.
Over many decades, researchers have developed powerful techniques in algorithms, logic, proof complexity, optimization, and heuristics, yet the central question remains unresolved.
One reason the problem is so resistant is that difficult instances often combine:
- many locally plausible choices
- long-range logical dependencies
- rugged search landscapes with traps and near-solutions
- combinatorial growth of possibilities as system size increases
This makes P versus NP a natural testing ground for new structural viewpoints on complexity.
Coherence Geometry approach (summary)
This project does not claim a proof of P = NP or a general solution to SAT.
Instead, it investigates whether a coherence-based framework can reveal hidden structure inside satisfiability problems that are usually viewed only through combinatorial search.
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.
Author’s Intuition
One way to visualize the SAT dynamics in this paper is as a flowing current. Even when a river appears turbulent, it can contain recurring vortices, eddies, and localized patterns hidden inside the motion. In the same spirit, satisfiable clause systems may contain coherent local structures that are not obvious in their raw combinatorial form.
The field-based dynamics attempts to reveal those structures. Most of the assignment rapidly settles into smooth large-scale flow, while unresolved contradictions concentrate into small localized defect regions with a known and repeating pattern. In the structured examples studied here, those defects behave like isolated eddies in the current. By identifying them and applying local repair moves, the larger flow becomes globally consistent.
More difficult SAT instances may contain more intricate turbulence, but the broader idea remains the same: apparent combinatorial disorder may hide exploitable geometric structure when viewed through the right dynamical lens.
Projection into standard complexity theory
Expressed in standard satisfiability language, the paper investigates a deterministic heuristic procedure for a specific structured 3SAT family:
- clauses define local influence scores on variables
- assignments evolve by greedy field-alignment updates
- most violated clauses disappear quickly during descent
- unresolved violations collapse into recurring localized patterns
- predefined bounded repair rules resolve those patterns
Under exhaustive evaluation over all 2^n initial assignments, the combined descent-and-repair procedure achieves complete solution for tested instances through system size n = 18.
Across tested sizes, the support of the residual defect region grows approximately linearly with system size, suggesting controlled geometric scaling rather than uncontrolled proliferation of unrelated cases.
The significance of this result is structural, not definitive. It does not establish polynomial-time solvability of SAT in general, but suggests that some satisfiability instances may contain low-dimensional internal organization that standard formulations do not make explicit.
Status
Exploratory study available
Technical Review
Readers are encouraged to evaluate this work according to its stated claims and mathematical arguments. Particular points of interest include:
- validity of the coherence-field SAT formulation
- reproducibility of the reported defect states
- correctness of the repair operators
- scaling behavior of the residual defect region
- possible extension beyond the studied family
Corrections, counterexamples, simplifications, and independent verification are welcome.
Available document
- Title: A Coherence Field Approach to SAT: Linear-Scale Defect Structure and Clause-Centered Repair
- Identifier: CGI-RSR-000002
- Author: Barry L. Petersen
- Length: 27 pages
- Version: v1.0
- Repository: Zenodo (DOI link)
- 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
The Zenodo-hosted PDF is the authoritative technical record. This page is descriptive only.
Computational Supplement
A research-state Jupyter notebook used during development is provided alongside the paper repository.
This notebook is shared intentionally in near-working form rather than as a polished software package. It contains exploratory experiments, debugging instrumentation, parameter sweeps, intermediate checks, and implementation remnants that were part of the discovery process.
For computational readers, these artifacts may be as informative as the final manuscript, since they show how the observed defect structures, repair rules, and exhaustive tests were actually developed and validated.
The notebook should be understood as a transparency and replication resource rather than a finished product. Users are encouraged to inspect, adapt, simplify, or extend it independently.
Relationship to other CG work
This result uses the same organizing principles that appear in coherence-governed dynamics and basin stabilization, but is expressed entirely within classical complexity theory.
It serves as a canonical example of projection: a coherence-based organizing idea expressed fully in silo language, without appeal to external frameworks.
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