Experimental Evidence of a Topological Obstruction Separating P and NP: Connections to Geometric Complexity Theory, the Padding Barrier, and Algebraic Multiplicity Gaps

We present comprehensive experimental evidence that the separation between P and NP manifests as a double obstruction: a topological obstruction (higher Betti numbers β2 > 0 in NP-complete solution spaces) and an algebraic obstruction (a positive multiplicity gap q −t > 0 in Geometric Complexity Theory). Building cubical complexes from SAT solutions and their Hamming neighbours, we consistently find β2 = 0 for 2-SAT (P) and β2 > 0 for 3-SAT and Vertex Cover (NP), with a clear scaling law and direct H1 → H2 correlations. Independently, we verify the algebraic obstruction on two micro-models: the perfect matchings of K3,3 (β1 = 1, q − t = 5) and the cube graph (β2 = 4, q − t = 40). In both cases, the presence of topological voids coincides with a strictly positive multiplicity gap. We formulate the Bridge Conjecture linking the two phenomena and outline a complete road-map for a formal proof of P ̸= NP, conditional on a cohomological injectivity lemma yet to be proved. All experimental data are fully reproducible and provide concrete targets for future theoretical work.