{
  "abstract": "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, $|v(x)|\\le c|x|^q+b$, 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 $q\\le1$, overflow requires at least order $\\log M$ layers, whereas for $q>1$ it can occur within order $\\log_q\\log M$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.14576",
  "author_total": 4,
  "authors": [
    {
      "name": "Hyemin Gu"
    },
    {
      "name": "Michael Tyrrell"
    },
    {
      "name": "Tuhin Sahai"
    },
    {
      "name": "Markos A. Katsoulakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mf7vvtvskmbc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.14576",
  "fulltext_url": null,
  "id": "0mf7vvtvskmbc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.14576",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Neural Networks and Applications"
    ]
  ],
  "openalex": "W7169542570",
  "pdf_url": "https://arxiv.org/pdf/2607.14576",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mf7vvtvskmbc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:58Z",
      "native_id": "2607.14576",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.14576"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Certifying Residual Architectures from Their Primitives: A Sharp Stability Threshold",
  "title_text": "Certifying Residual Architectures from Their Primitives: A Sharp Stability Threshold",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/0mf7vvtvskmbc",
  "venue": "arXiv",
  "versions": []
}