I’ve been working on a structural result for QUBO and higher-order pseudo-Boolean optimization that started from a question about escaping certified local minima, but ended up giving something substantially stronger. For an incumbent solution (x), write the exact change caused by flipping a set (S) in its local Möbius expansion. Build a graph/hypergraph containing only negative interactions—terms that can actually create cooperative improvement. The main theorem is that for any move (S), [\Delta_x(S)\ge \sum_C \Delta_x(S\cap C),] where (C) ranges over the connected components of the negative-interaction structure. This immediately implies: If any globally improving move exists, one negative component already contains an improving submove. If every negative component has been optimized and none improves the incumbent, the incumbent is globally optimal over the entire Boolean cube. The independently optimized component values give a rigorous global lower bound and an explicit certificate on the remaining optimality gap. For QUBO, if the largest negative component has size (c), global certification can be performed by enumerating only those components rather than arbitrary subsets of (n) variables. Repeatedly optimizing the best negative component gives globally convergent exact descent; the proof gives geometric contraction of the true objective gap. Dense positive interactions do not destroy the decomposition: the ordinary QUBO interaction graph may even be complete while the useful negative-interaction structure remains highly fragmented. There is also a hardness boundary: global certification remains coNP-hard when the negative-interaction graph collapses into one small-diameter connected component, so the structural parameter is doing genuine computational work rather than disguising a generic polynomial-time algorithm for QUBO. I’ve written a self-contained paper with the proofs, QUBO specialization, higher-order pseudo-Boolean generalization, complexity results, separation examples, and an independent exact-integer verifier. I’d particularly appreciate scrutiny of: whether this local-to-global negative-component theorem is already known under another terminology; whether the global-gap certificate connects to an established pseudo-Boolean/roof-duality result I have missed; whether there are counterexamples to any of the stated generalizations; which benchmark families would be the strongest empirical test of the resulting algorithm. I’m deliberately treating novelty/priority as unresolved until people familiar with the optimization literature have checked it.