{
  "abstract": "We derive an exact gradient-step representation of the RoPE-softmax forward pass. For every deterministic RoPE-softmax attention head with arbitrary affine projection weights, we construct a query-dependent effective matrix $ΔM_i$ satisfying $y_i = μ_i + u_i^\\top ΔM_i$, where $μ_i$ is the uniform mean of the attended values and $u_i$ is the augmented query input. The construction applies the classical exponential divided difference $ρ= ϕ_1$ to retain the softmax exactly. Its positive coefficients give a unit gradient-step representation on a query-conditioned quadratic objective. The same function connects the RoPE generator to exact positional finite differences. We derive a tokenwise formula for the error of reusing one query's matrix and prove that a nonconstant finite-cache head cannot admit a globally exact affine query readout. Reconstruction checks and frozen-reuse calibration on one pretrained Qwen2.5-0.5B layer verify the representation and quantify the correction required when one query's matrix is reused.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06685",
  "author_total": 3,
  "authors": [
    {
      "name": "Julie Huang"
    },
    {
      "name": "Maggie Chlon"
    },
    {
      "name": "Leon Chlon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/06c1kr5kejy42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06685",
  "fulltext_url": null,
  "id": "06c1kr5kejy42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06685",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "1702|Topic Modeling"
    ]
  ],
  "openalex": "W7211995705",
  "pdf_url": "https://arxiv.org/pdf/2609.06685",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/06c1kr5kejy42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:07Z",
      "native_id": "2609.06685",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06685"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "RoPE attention is an exact forward-pass gradient step with softmax intact",
  "title_text": "RoPE attention is an exact forward-pass gradient step with softmax intact",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/06c1kr5kejy42",
  "venue": "arXiv",
  "versions": []
}