Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

Whether a deep residual architecture trains stably is usually determined by training it, which is expensive and answers the question only for the architecture, depth, and floating-point format tested. In practice, stability is secured by heuristics for where to place normalization and which type to use, supported by experiments but lacking a common principle. We show that stability can instead be certified before training, directly from the architectural primitives of a residual block, as an explicit function of depth and floating-point format. The certificate has two elements: (a) a power-law growth bound, |v(x)|≤ c|x|q+b, whose exponent q is computed from the block's primitives by an arithmetic of exponents (forward); and (b) gradient bounds from a Lipschitz condition on the block over reachable states (backward). The certificate determines when the state can reach the largest finite floating-point value M: for q≤1, overflow requires at least order log M layers, whereas for q>1 it can occur within order logqlog M layers; the threshold q=1 is sharp. Every normalization and bounded activation sets q=0, and the arithmetic identifies minimal modifications that bring a block from q>1 to q=1 without normalization. Empirically, in forecasting, only q>1 blocks blow up, as predicted by the forward certificate. In GPT-2 on OpenWebText, q=1 blocks without normalization also diverge: the forward certificate holds, while the backward one fails in the attention block with the largest query-key coefficient. Relaxing normalization from q=0 to q=1 improves out-of-distribution generalization in operator learning and accuracy in time-series forecasting. As deep models become more complex and costly to train, our results provide a way to assess the trade-off between stability and representational flexibility directly from block primitives, before training.