What question did this study set out to answer?

This study aims to provide a structural resolution to the computational hardness distinction between VP and VNP using geometric insights.

June 15, 2026Open Access

The Geometric Substrate of Computational Hardness: A Structural Resolution of the VP vs. VNP Phase Transition via 6-Dimensional Riemannian Foliation

Key Points

This study aims to provide a structural resolution to the computational hardness distinction between VP and VNP using geometric insights.
Developed a 6-dimensional Riemannian manifold framework for constraint satisfaction problems.
Applied Anderson Operator Framework (AOF 25.05) for proving polynomial vs. factorial complexity.
Utilized simulation data vectors with parameters up to N=1024 for validation.
Established a continuous phase transition boundary between polynomial tractability (VP) and factorial complexity (VNP).
Identified a critical threshold (k_c ≈ 4.267) leading to singularity characterized by a vanishing minimal eigenvalue (λ_min → 0).
Demonstrated the capability for independent laboratory replication of the phase transition via a comprehensive validation suite.

Abstract

The Geometric Substrate of Computational Hardness: A Structural Resolution of the VP vs. VNP Phase Transition via 6-Dimensional Riemannian Foliation --- This repository presents the complete 18-package resolution suite establishing the Universal Phase-Transition Law for Computation. Moving beyond traditional asymptotic step-counting and discrete Turing-machine abstractions, this work formalizes the configuration space of combinatorial constraint satisfaction problems as a smooth, 6-dimensional compact orientable Hantzsche-Wendt Riemannian manifold (M₇ₖ^6D). Through the mathematical architecture of the Anderson Operator Framework (AOF 25. 05), we prove that the boundary separating polynomial tractability (VP) from factorial complexity (VNP) is a continuous, physically observable geometric phase transition governed by the independent constraint density parameter (k). Under sub-critical loading (k < kc), the manifold maintains a Ricci-flat state (R₈₉ 0), guaranteeing polynomial-time geodesic pathfinding. At the critical threshold (kc 4. 267), a localized metric collapse ("neck-pinch" singularity) occurs, characterized by a vanishing minimal eigenvalue (_ 0) of the self-adjoint Anderson Operator and a power-law divergence of the Jacobian condition number (). The suite provides a comprehensive, unforgeable verification architecture consisting of 5 Standard Academic Core (SAC) technical reports and 12 Agnostic Replication Kit (ARK) packages. These modules provide the theoretical proofs, explicit domain maps (including algebraic complexity, tensor decomposition, and arithmetic sieves), native compiler API binaries (MDEV23 and SGAV23), standardized simulation data vectors (N 128), automated troubleshooting rails, and cryptographic final validation seals. This integrated framework enables independent, high-fidelity laboratory replication of the phase-transition crossing prior to formal institutional peer-to-peer review. Manifest of Deposited ModulesIndependent verifiers and institutional review panels can audit the suite via five structured operational categories: 1. Foundational Theorems & Definitions: Theorem - Universal Phase-Transition Law for Computation. pdf (The core tripartite analytical proof) SAC-01 Technical Report. pdf (Continuous substrate mappings and Perelman-style singularity tracking) SAC-03 Appendix A. pdf (Theoretical supplementary proofs, homology derivations, and Whitney embedding validations) SAC-04 Executive Summary. pdf (High-level institutional abstract) 2. Translational & Application Infrastructures: SAC-05 Lexicon Bridge. pdf (Term-for-term mapping from standard complexity nomenclature to AOF geometric primitives) ARK Package 01 Summary & Educational Academic Instructional. pdf (Pedagogical guide to differential evolution equations) ARK Package 02 Application Atlas. pdf (Concrete domain parameters for Tensor Decomposition, K-SAT, and prime-base sieves) 3. Automated Execution Engines & APIs: ARK Package 07 API Documentation. pdf (Low-level C/Python operator bindings for MDEV23 and SGAV23 environments) ARK Package 10 Required Tool Registry & Reference List. pdf (Hardened environmental software dependencies) ARK Package 12 Common Toolchain, Environment & Infrastructure Engineering Specifications. pdf (Quad-precision 128-bit verification configurations) 4. Empirical Benchmarks & Datasets: SAC-02 Technical Report - Simulation & Empirical Data. pdf (Quantitative validation logs showing finite-size scaling limits up to N=1024) ARK Package 11 Simulated Inputs & Target Coordinate Profiles. pdf (Machine-parseable test vectors for sub-critical, critical, and super-critical runs) 5. Validator Diagnostic Rails & Sealing Protocols: ARK Package 03 Failure Mode and Effects Analysis (FMEA). pdf (Systemic risk matrices and mitigation operators) ARK Package 05 Troubleshooting Manual – Stall & Recovery. pdf (Automated logic gates for horizon stalls and metric trapping) ARK Package 06 Emergency Logic Core. pdf (Hardware-abstracted topological clamping overrides) ARK Package 08 Reviewer Packet. pdf (Explicit objective validation gates tracking the spectral gap and Ricci spike) ARK Package 09 One-Page Reviewer Packet. pdf (Rapid validation scorecards and execution of the final automated AOF-25. 05 cryptographic seal) ---

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