{
  "abstract": "Matrix $B=(b_{ij})$ with $b_{ij}=(2j+1)(2j+2i-1)$ was introduced in \\cite{Shi2024} as an additive sieve for odd primes. In this paper we introduce the minimal-anchor function $\\pmin(d)$, the least odd prime $p$ such that $p+d$ is prime (sequence A020483 of the OEIS at index $d/2$), whose finiteness for all even $d$ is exactly the weak Polignac (Maillet) conjecture, i.e.\\ the statement that every row of $B$ contains a semiprime. We prove that for each fixed $z$ the set $\\{d\\ \\text{even}:\\pmin(d)\\le z\\}$ has density zero, with the asymptotic $(π(z)-1)X/\\log X$; consequently no fixed finite set of anchor primes can cover a positive proportion of the rows. Density-one coverage of the rows nevertheless holds, by a classical theorem of Lavrik which we restate in the matrix-$B$ framework: almost every row contains the number of semiprimes predicted by the Hardy--Littlewood conjecture. We formulate quantitative conjectures on $\\pmin$ and support them with numerical data. Every-row coverage (= weak Polignac) is explicitly left open. We prove unconditional lower bounds for $\\pmin$: for every $ψ\\to0$, $\\pmin(d)>ψ(d)\\log d\\log\\log d$ for almost all even $d$, which is the conjectured typical order; and $\\max_{d\\le X}\\pmin(d)\\ge(\\frac12+o(1))\\log X\\log\\log X$. The same counting gives the corresponding lower bounds for the least prime in a Goldbach partition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.21529",
  "author_total": 1,
  "authors": [
    {
      "name": "Wujie Shi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4myt4dz32ey5d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.21529",
  "fulltext_url": null,
  "id": "4myt4dz32ey5d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.21529",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7162104673",
  "pdf_url": "https://arxiv.org/pdf/2605.21529",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4myt4dz32ey5d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2605.21529",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.21529"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares",
  "title_text": "A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/4myt4dz32ey5d",
  "venue": "arXiv",
  "versions": []
}