{
  "abstract": "We show that replication carries congruence symmetry down an explicit level tower. If a normalized replicable function $f$, holomorphic on the upper half-plane, is invariant under $Γ_0(N)$, then its $n$th replicate is invariant under $Γ_0(N/(N,n))$. Thus every replicate whose index is divisible by $N$ is the normalized modular invariant $J=j-744$. This gives a direct and classification-free proof of $J$-finality for congruence-invariant replicable functions. The argument uses only the replication identities and an elementary generation theorem for congruence subgroups; it requires neither complete replicability nor arithmetic hypotheses on the Fourier coefficients. We then apply the descent law to completely replicable functions of finite replication order. Their replication towers have a canonical terminal replicate, and the only possible terminal functions are $J$, $q^{-1}$, and $q^{-1}+q$. Moreover, a finite-order completely replicable function is $J$-final precisely when its terminal replicate is $J$, and any symmetry beyond translations forces this alternative.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08243",
  "author_total": 2,
  "authors": [
    {
      "name": "Eric Culf"
    },
    {
      "name": "Abdellah Sebbar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dek196k60cqx7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08243",
  "fulltext_url": null,
  "id": "dek196k60cqx7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08243",
  "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.GR"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7212040867",
  "pdf_url": "https://arxiv.org/pdf/2609.08243",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dek196k60cqx7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08243",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08243"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Replication Descent and $J$-Finality of Replicable Functions",
  "title_text": "Replication Descent and J-Finality of Replicable Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/dek196k60cqx7",
  "venue": "arXiv",
  "versions": []
}