{
  "abstract": "We introduce \\textsc{LatentGamma}, a differentiable operator on the unit cube $[0,1]^M$ designed as a smooth surrogate of Tarski's immediate consequence operator $\\TP$ associated with a definite Horn program $P$ on $M$ atoms. The operator is built as a five-stage composition combining sigmoidal gating, softmax routing, and a residual update that enforces monotonicity by construction. We establish four theoretical results. First, the iterated sequence is coordinate-wise non-decreasing and bounded by $\\ind$, hence converges to a fixed point of \\textsc{LatentGamma}; the operator itself is lattice-monotone on $[0,1]^M$. Second, our \\emph{constructive expressivity theorem} shows that for every definite Horn program $P$ there exists a closed-form parameter assignment $θ^*(P)$ and an explicit time bound $T_{\\max}(P)$ such that the iterated sequence \\emph{exactly} reproduces $\\TPinf(F_0)$ for every initial fact set $F_0$ throughout the window $[D(P, F_0), T_{\\max}(P)]$, where $D(P, F_0) \\leq M$ is the derivation depth. The window $T_{\\max}$ is large in practice ($\\geq 10^5$ for sparse programs) and reflects the finite-time nature of computation by smooth sigmoidal gates. Third, the oracle is robust to Gaussian noise on its body and head logits, with explicit non-asymptotic bounds. Fourth, we prove a matching information-theoretic lower bound on the parameter count. We provide a complete numerical validation on programs ranging from $M = 33$ to $M = 504$ atoms: oracle accuracy reaches $1.0000$ on $2000$ test cases with zero false-positive and false-negative rates, and the empirical noise tolerance $σ_{\\max}$ scales precisely as the union bound $RM \\cdot Φ(-10/σ_η)$ predicts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06235",
  "author_total": 1,
  "authors": [
    {
      "name": "Aymen Mejri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0njt94n0jdhr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06235",
  "fulltext_url": null,
  "id": "0njt94n0jdhr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06235",
  "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.IT"
    ],
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Logic, Reasoning, and Knowledge"
    ]
  ],
  "openalex": "W7212081380",
  "pdf_url": "https://arxiv.org/pdf/2609.06235",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0njt94n0jdhr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06235",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06235"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Differentiable Horn Programs: A Constructive Expressivity Theorem for Latent Rule Operators",
  "title_text": "Differentiable Horn Programs: A Constructive Expressivity Theorem for Latent Rule Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/0njt94n0jdhr6",
  "venue": "arXiv",
  "versions": []
}