[
  {
    "id": "sophaxis-2026.0008",
    "type": "article",
    "genre": "Preprint",
    "title": "The Trinity-Infinity Framework, Series III: Beyond Three — A Generalization and Retrospective",
    "author": [
      {
        "family": "Nemoto",
        "given": "Takuya"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          22
        ]
      ]
    },
    "publisher": "Zenodo",
    "DOI": "10.5281/zenodo.22058964",
    "URL": "https://doi.org/10.5281/zenodo.22058964",
    "abstract": "The first two series of the Trinity-Infinity framework prove that a recursion combining a cyclic permutation of three elements with integration toward a fixed anchor converges geometrically to a unique fixed point. We show this result never depended on there being three elements: the identical argument holds for any number of elements n ≥ 2 (Theorem 1), which we verify computationally for n=2 through 8. We then identify precisely the one respect in which n=3 is structurally distinguished — it is the smallest n for which the cyclic permutation is not its own inverse — and are explicit that this is a narrow structural fact, not a justification for the broader cultural or philosophical resonance of triads invoked in Series I's introduction. We close with a retrospective distinguishing what this series has actually proven and verified from what the original preprints claimed but did not establish.",
    "note": "Preprint, not peer-reviewed. Listed on Sophaxis as 2026.0008"
  }
]