Certified Stopping for Fixed-Codebook Quantization

When pairwise refinement stops in fixed-codebook quantization, how much joint improvement remains possible? We develop a certified stopping framework that separates conditional search, certificate optimization, and relaxation strength, connecting each source of uncertainty to the computation that can reduce it. A budgeted pair oracle maintains a feasible incumbent and a valid conditional lower bound after any number of finite branch scores. A globally valid curvature split extends these bounds to the coupled quadratic. Optimizing signed block curvature and the anchor recovers a block-moment relaxation when $H\succ 0$ and every block codebook has full affine span. Its certified optimization interval decomposes exactly into restricted-master and finite-separation terms. On eight exactly enumerable six-coordinate restrictions of captured GPT-2 reconstruction objectives, one pair-fixed point remains above the finite optimum, while four objectives retain strictly positive block-moment relaxation gaps after certificate optimization. Across all eight stored rational objectives, the optimized certificate is tighter than continuous-box and fixed-curvature controls. The hierarchy distinguishes when to score more candidates, improve the conic state, or change the relaxation.