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.
Abstract by Takuya Nemoto, given by the authors in the submission issue, because the DOI metadata has none, shown as plain text without other changes, under CC BY 4.0.
This is a preprint. It has not been peer-reviewed. Listing on Sophaxis means it passed moderation, not that it is correct. Sophaxis does not guarantee its quality. The authors bear full responsibility for its content.
Unmodified copies of the files the authors deposited at Zenodo, kept so that search engines can index the full text. They are shared under CC BY 4.0, by Takuya Nemoto. Cite the DOI.
Use of language models and other generative models
This manuscript was prepared with drafting, mathematical-formalization, and computational-verification assistance from Claude (Anthropic, Claude Sonnet 5). Every numerical claim in Section 3 was computed by executing the operators exactly as defined in the text; the verification script (series3_verification.py) is distributed together with this PDF. The author reviewed all content and is responsible for its claims.
Notes and corrections
2026-11-02 The author's errata for this series (https://github.com/cpsbvbng26-dotcom/trinity-infinity/blob/main/ERRATA.md) record that the verification script that this paper says is distributed with the PDF does not exist (E3).
2026-11-02 The same errata record that the statement on language models covers only the revision. The mathematical content of the first draft, published in October 2025, came from a language model (ChatGPT, OpenAI), and the paper does not say so (E4).
Cite
APA
Nemoto, T. (2026). The Trinity-Infinity Framework, Series III: Beyond Three — A Generalization and Retrospective [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22058964
Chicago (author-date)
Nemoto, Takuya. 2026. “The Trinity-Infinity Framework, Series III: Beyond Three — A Generalization and Retrospective.” Preprint, Zenodo. https://doi.org/10.5281/zenodo.22058964.
BibTeX
@misc{nemoto2026sophaxis20260008,
author = {Takuya Nemoto},
title = {The Trinity-Infinity Framework, Series III: Beyond Three — A Generalization and Retrospective},
year = {2026},
howpublished = {Zenodo},
doi = {10.5281/zenodo.22058964},
note = {Preprint, not peer-reviewed. Listed on Sophaxis as 2026.0008}
}