{
  "abstract": "For $r\\ge1$, let $K=2r+14$ and $d=\\dim S_K$. We prove strict positivity for determinants of Mellin period functionals, yielding a full-spark theorem for periods in a fundamental half-range and, in particular, the mixed odd--even period independence conjecture of Xue. As a consequence, the consecutive first Rankin--Cohen brackets $[E_{2r+10-2j},E_{2j+2}]_1$, $1\\le j\\le d$, form a basis of $S_K$, and we determine the signs of all admissible ordered determinants of first Eisenstein brackets. This gives a uniform exact all-weight formula for the divisor--tau convolution $C_r(n)=\\sum_{m=1}^{n-1}σ_{2r+1}(m)τ(n-m)$. Reducing the same coordinate identity modulo primes, we obtain canonical prime-wise reductions for infinitely many primes in every weight and characterize sparse Ramanujan-type specializations through vanishing Cramer coordinates. Finally, for homogeneous $f,g\\in\\mathbf Q[E_4,E_6]$, we prove $[f,g]_1/Δ=-3456\\det((f_{E_4},f_{E_6}),(g_{E_4},g_{E_6}))$, reducing the Cramer system to a one-variable coordinate problem in $T=E_6^2/E_4^3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10052",
  "author_total": 1,
  "authors": [
    {
      "name": "Kelvin Lam"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chg44qrjfkgpr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10052",
  "fulltext_url": null,
  "id": "chg44qrjfkgpr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10052",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7221044047",
  "pdf_url": "https://arxiv.org/pdf/2610.10052",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chg44qrjfkgpr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10052",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10052"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Consecutive Rankin-Cohen Bases, Full-Spark Periods, and Divisor-Tau Congruences",
  "title_text": "Consecutive Rankin-Cohen Bases, Full-Spark Periods, and Divisor-Tau Congruences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/chg44qrjfkgpr",
  "venue": "arXiv",
  "versions": []
}