What question did this study set out to answer?

This work aims to establish a comprehensive empirical basis for the Universal Horizon Conjecture that asserts P ≠ NP.

April 22, 2026Open Access

The Universal Horizon Conjecture: Multi-Metric Geometric, Topological, and Thermodynamic Experimental Evidence for a P ≠ NP Barrier in 3‑SAT

Key Points

This work aims to establish a comprehensive empirical basis for the Universal Horizon Conjecture that asserts P ≠ NP.
Utilized multi-metric analysis integrating static geometry, topology, thermodynamics, and information theory.
Conducted experiments using persistent homology, replica exchange, and alpha complexes to analyze 3-SAT.
Evaluated fragmentation and complexity through statistical methods and thermodynamic properties.
Identified 438 persistent two-dimensional holes in 3-SAT indicating a topological distinction from 2-SAT.
Demonstrated strong fragmentation in 3-SAT via overlap gap ratios, confirmed by statistical tests.
Found direct experimental evidence for a thermodynamic barrier with low temperature signatures in 3-SAT.

Abstract

Version 4.0.0 represents the definitive, comprehensive synthesis of the Universal Horizon Conjecture project. This manuscript integrates twelve orthogonal experimental lines of evidence, establishing the most complete empirical foundation to date for the belief that P ≠ NP manifests as a fundamental physical law in our computational universe. Key contributions in v4.0.0: 1. Static Geometry: 3‑SAT exhibits a stable, self‑similar intrinsic geometry with constant radius r₀ ≈ 1.55 and entropy H ≈ 2.8, sharply distinguishing it from 2‑SAT. 2. Local Topological Invisibility: Persistent homology on local WalkSAT samples yields β₁ = β₂ = 0, confirming that predicted global topological obstructions are inaccessible to local algorithms. 3. Direct Detection of Persistent β₂ in 3‑SAT: Using Alpha complexes and GUDHI, we report the first direct experimental detection of persistent second Betti numbers in 3‑SAT — 438 persistent two‑dimensional holes at n=100, α=4.2. 4. Overlap Gap Property (OGP): Multi‑metric quantification via Overlap Gap Ratio (OGR = 4.62 ± 1.46 on SATLIB, 2.14 on own data), Marchenko–Pastur heavy tails (8.0% vs 2.0%), and statistical tests (KS and MW p < 10⁻¹⁵⁰) confirms strong fragmentation. 5. Thermodynamic Barrier (Replica Exchange): Reduced swap acceptance rate (0.329 vs 0.379) and elevated energy autocorrelation at low temperature for 3‑SAT provide direct experimental signatures of Replica Symmetry Breaking (RSB). 6. Spectrum of k‑SAT: Fragmentation intensifies with clause arity k (k=3,4,5), forming a continuous spectrum that bridges the canonical k=3 case to rigorous asymptotic proofs. 7. Information‑Theoretic Complexity: LZMA compression ratios reveal that 3‑SAT states are less compressible than 2‑SAT, indicating higher algorithmic (Kolmogorov) complexity. 8. Discrete Geometric Experiment: Regular simplices subjected to random cuts exhibit a phase transition in β₁ at a critical density, demonstrating the universal mechanism by which constraints create topological holes. 9. The Extra Dimension Principle: The findings resonate with a universal principle demonstrated across mathematics (Whitney embedding), physics (Kaluza–Klein), machine learning (kernel trick, VC dimension), and now computational complexity: intractable problems demand an extra dimension inaccessible to polynomial‑time algorithms. 10. Unified Synthesis: The convergence of these twelve independent lines of evidence provides robust, multi‑perspective experimental validation of the Universal Horizon Conjecture and constitutes a compelling empirical foundation for P ≠ NP. This version supersedes all previous releases (v1, v2, v3.0.0). Code and raw data are available upon request.

The Universal Horizon Conjecture: Multi-Metric Geometric, Topological, and Thermodynamic Experimental Evidence for a P ≠ NP Barrier in 3‑SAT

Key Points

Abstract

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