{
  "abstract": "We study how the noncentral zeros of an elliptic-curve $L$-function enter the Birch--Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed $L$-function yields an exact Hadamard decomposition, and the zero statistics $J_E$ and $V_E$ satisfy the positive identity $J_E=V_E/8+Δ_E$. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average $82.4%$ of $Δ_{E,20}$, while independent edge values reveal a discrete $35/36$ zero-count split beyond height $20$. For Haar $SO(2N+1)$ we prove a fixed-dimensional hard-edge law $\\Pr{J_N>y}\\sim C_Ne^{-3y/2}$. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27962",
  "author_total": 1,
  "authors": [
    {
      "name": "Michel Planat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/10x6kd9p68m25.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27962",
  "fulltext_url": null,
  "id": "10x6kd9p68m25",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27962",
  "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",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214208978",
  "pdf_url": "https://arxiv.org/pdf/2609.27962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/10x6kd9p68m25.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Morse Topology, Zero Products, and Elliptic Prime Crossings",
  "title_text": "Morse Topology, Zero Products, and Elliptic Prime Crossings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/10x6kd9p68m25",
  "venue": "arXiv",
  "versions": []
}