P ≠ NP under the Davis Information-Geometric Axioms: Geometric Separation of Energy Landscapes

We prove P ≠ NP within the Davis Information-Geometric Framework, conditional on two physically-motivated axioms. The geometric structure of the energy landscape—exponentially many Ω(1)-separated metastable basins—is established rigorously via Kac-Rice theory. The axioms provide the interpretive bridge connecting this geometry to computational complexity.

The Two Axioms:
1. Davis Law: C = τ/K(A,B) — information flow across a barrier is inversely proportional to curvature
2. Tolerance Intensivity: τ = Θ(1) — the tolerance budget does not grow with problem size

Proven Results:
- Glassy Phase: Random 3-SAT has 2^Ω(n) metastable basins (Kac-Rice formula, five supporting lemmas complete)
- Barrier Structure: Distinct basins are separated by Ω(1) curvature barriers
- Solution Basin Sparsity: Derived from entropy deficit in the frozen regime
- Shattered Structure: No partition variable exists (derived from frozen phase properties)

The proof technique is non-relativizing, non-natural, and non-algebrizing, thereby avoiding the classical barriers to P vs NP proofs. Curvature scales as K ~ k(k-1)/2 for k-SAT, yielding a 3× separation between 2-SAT (in P) and 3-SAT (NP-complete).

Keywords: P vs NP, computational complexity, energy landscapes, Kac-Rice formula, curvature barriers, satisfiability, geometric complexity theory, Davis Field Equations

MSC 2020: 68Q15 (Complexity classes), 68Q17 (Computational difficulty of problems), 53B20 (Local Riemannian geometry), 90C27 (Combinatorial optimization)

Related:

Davis, B. R. (2025). The Field Equations of Semantic Coherence: A Geometric Theory of Meaning, Curvature, and Reasoning in Transformer Architectures. Zenodo. https://doi.org/10.5281/zenodo.17771796

Davis, B. R. (2025). The Davis Manifold: Geometry-First Detection with Compositional Error Budgets. Zenodo. https://doi.org/10.5281/zenodo.17642038

Davis, B. R. (2025). The Geometry of Sameness: An ε-Equivalence of Translation and Distance. Zenodo. https://doi.org/10.5281/zenodo.17642422

Geometry of Intelligence Series:

• Hidden Variable: Unlocking Patterns in a World Obsessed with Structure (2025)
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