On the Integrality Gap Barrier for Spectral Embeddings of 3-SAT: A Falsifiable Framework at the Edge of the Sum-of-Squares Hierarchy

We study a spectral embedding of 3-SAT instances via a clause–variable incidence matrix M(Φ) and its associated Gram/Laplacian operators. We show that simple spectral properties (e.g., singularity of A = MM⊤ or low quadratic energy) do not characterize satisfiability. Explicit counterexamples demonstrate an integrality gap: continuous relaxations admit low- energy solutions that round to false positives. We connect this failure to the inability of quadratic forms to encode cubic clause interactions and relate the phenomenon to the Sum-of-Squares (SoS) hierarchy. We introduce the Symmetric Satisfiability Operator L(Φ) and analyze its spectral properties. We propose a falsifiable experimental program to map instance structure to the minimal SoS degree required for exact decision. Finally, we define the Cubic Deficit as a measure of the gap between quadratic relaxation and Boolean satisfiability, and conjecture that this deficit is the true computational barrier separating P from NP.