This paper introduces the Trinity-Infinity (III∞) framework, a conceptual model where three finite elements (Trinity) interact recursively to generate infinite dynamics (Infinity). The framework is applied to game theory, mathematical logic, and mechanical engineering, demonstrating how finite structures under recursion yield unbounded systemic behaviors.
Abstract by Takuya Nemoto, from the DOI metadata, 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 It has not been peer-reviewed.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 and closed-form claim in Section 2 (Remark 1) was computed by executing the operators exactly as defined in the text; the verification script (series1_verification.py) is distributed together with this PDF so the two results can be reproduced independently. 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: A Conceptual Model with Illustrative Applications in Game Theory, Mathematical Logic, and Mechanical Engineering [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.22058624
Chicago (author-date)
Nemoto, Takuya. 2026. “The Trinity-Infinity Framework: A Conceptual Model with Illustrative Applications in Game Theory, Mathematical Logic, and Mechanical Engineering.” Preprint, Zenodo. https://doi.org/10.5281/zenodo.22058624.
BibTeX
@misc{nemoto2026sophaxis20260006,
author = {Takuya Nemoto},
title = {The Trinity-Infinity Framework: A Conceptual Model with Illustrative Applications in Game Theory, Mathematical Logic, and Mechanical Engineering},
year = {2026},
howpublished = {Zenodo},
doi = {10.5281/zenodo.22058624},
note = {Preprint, not peer-reviewed. Listed on Sophaxis as 2026.0006}
}