{
  "abstract": "Let $5\\leq p_1<\\cdots<p_r$ be primes, put $P=p_1\\cdots p_r$, and let $g_P(2n)$ count the unordered decompositions $2n=h+k$ for which both summands are coprime to $6P$. We derive an explicit finite inclusion--exclusion formula for $g_P(2n)$ by parametrizing the admissible residue classes and combining the local restrictions through the Chinese remainder theorem. The associated complete-modulus correlation is a specialization of Nagell's totient function and determines both the finite local density and the affine increment $g_P(2(n+6P))-g_P(2n)=κ_P(n)$. A parameter-density decomposition gives a residue-sensitive positivity criterion and the uniform sufficient condition $n\\geq 3(3^r-1)\\prod_{i=1}^{r}\\frac{p_i}{p_i-2}+2\\Longrightarrow g_P(2n)>0$. Maximizing this bound over squarefree moduli whose prime factors do not exceed a prime $p$ gives an explicit threshold $\\mathcal N(p)$ satisfying $\\log\\mathcal N(p)\\sim(\\log3)p/\\log p$. We also describe the complete prime cutoff at $\\sqrt{2n}$ and its exact additive-prime interpretation. We also evaluate $g_P(2tP)$ explicitly using the real character of period $3$. The positivity thresholds are elementary and fully explicit; lower-bound sieves provide smaller asymptotic scales, and the present thresholds do not establish positivity at the complete cutoff for all $n\\geq13$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22305",
  "author_total": 1,
  "authors": [
    {
      "name": "Andres M. Salazar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7jtv36z6ame66.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22305",
  "fulltext_url": null,
  "id": "7jtv36z6ame66",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22305",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220663630",
  "pdf_url": "https://arxiv.org/pdf/2609.22305",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7jtv36z6ame66.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.22305",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22305"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Restricted Coprime Representations Modulo 6P: Exact CRT Formulas and Positivity Thresholds",
  "title_text": "Restricted Coprime Representations Modulo 6P: Exact CRT Formulas and Positivity Thresholds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7jtv36z6ame66",
  "venue": "arXiv",
  "versions": []
}