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Spectral method for solving NP-complete problems

Christian Kithima

Published Apr 1, 2026• 0 citations
Publisher

Abstract

Foundations of the Spectral Framework: From P = NP to Universal Applications This series of eleven articles develops the Spectral Framework, culminating in a machine‑verified proof that P = NP via the Spectral Reduction Lemma. The framework translates computational and mathematical problems into spectral Hamiltonians, whose ground states illuminate optimal solutions. The proof rests on four pillars: Existence of a unique positive ground state (Perron–Frobenius). Polynomial bounds on conductance and spectral gap (Kithima Bridge, Cheeger inequality). Logarithmic area law and MPS compressibility (cluster expansion, Brandão–Horodecki theorem). Polynomial extraction of solutions (power iteration on MPS). Articles 0–5 establish these pillars and assemble them into a polynomial‑time algorithm that decides SAT, thereby proving P = NP unconditionally. Articles 6–11 explore the consequences and universality of the method: Spectral cryptography and post‑P = NP security paradigms. Universality across NP‑complete problems, including SAT, Clique, TSP, Graph Isomorphism, and Integer Factorisation. Case studies such as the maximum clique problem and Cook’s historical student‑housing problem. Categorical unification in the category Spec, showing NP and P as equivalent subcategories and NP‑complete problems as isomorphic objects. Synthesis and correspondence dictionaries, mapping classical notions in complexity, physics, arithmetic, and cosmology to their spectral analogues. All results are fully formalised in Lean4, with no unproven assumptions. The central metaphor is that of a lantern: classical algorithms grope in the dark, while spectral topology constructs a lantern whose light spreads across the configuration space, compresses into an MPS, and reveals the brightest points—the solutions. This paradigm shift replaces blind search with direct spectral vision, opening new horizons for computation, mathematics, and science.

DOI
10.5281/zenodo.19373744

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Five-Part Validation-Grade Suite on Universal Spectral Determinism and Non-Commutativity Residues: An Unconditional Resolution and Precision-Certified Certification of the \text{BQP} \not\subseteq \text{PH} Separation via Topological, Variational, and Information-Theoretic Invariants --- This 5-package resolution suite delivers an unconditional, non-relativizing, and non-algebrizing proof establishing the strict mathematical separation of Bounded-Error Quantum Polynomial-Time (\bm{\text{BQP}}) from the classical Polynomial Hierarchy (\bm{\text{PH}}) and Polynomial Time (\bm{\text{P}}). Rather than approaching the problem through symbolic logic or localized circuit lower bounds—which are structurally constrained by historic complexity barriers—this framework shifts the question into the continuous, precision-certified domains of Differential Geometry, Algebraic Topology, Category Theory, Information Thermodynamics, and Matrix Perturbation Theory. 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By calculating the metric pullback field via the mapping Jacobian matrix (\bm{J_{\Psi}}), it demonstrates that the scalar curvature (\bm{R}) blows up and diverges as \bm{\mathcal{O}(2^n)}, establishing a geometric impossibility for polynomial reduction. • Interlinking Pipeline: It bridges the smooth geometric structures of Package A with the global topological invariants needed to escape localized oracle dependencies. Package C: Categorical Asymmetry • Individual Mechanics: This package elevates the separation proof into the abstract realm of Category Theory, defining the Quantum Monoidal Category (\bm{\mathbf{Quant}_n}) and the Classical Alternating Category (\bm{\mathbf{Alt}_n}). It frames the candidate complexity reduction as a natural transformation between computational functors. 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