Paper
The Trinity-Infinity Framework, Series III
Beyond Three — A Generalization and Retrospective
ORCID0009-0000-1406-0547
VersionAugust 2026
Shows that the convergence theorem does not need three elements — it holds for any n ≥ 2 — and looks back at what the series established and what it withdrew.
Where it lives
The same text exists in several places. Cite the Zenodo DOI.
Zenodo 10.5281/zenodo.22058964Canonical. It carries the DOI and fixes the version
PDF on GitHubThe distributed file itself — a copy, not the canonical record
The record
Defects found, and what could not be fixed. Nothing has been deleted.
ErrataDefects found, and what could not be fixed. Nothing has been deleted.
Route of the rediscoveryNine stages from the initial idea to identifying it as a rediscovery of standard results
Verification scriptsChecks the theorems, and every number printed on the page, by machine
Correction register（TI-…）Every error found after publication, recorded one by one
External evaluationsEvaluations received, and how they were checked
Use of language models
What language models were used for. A language model is not an author.
Three does no work. The one respect in which n = 3 is distinguished is that it is the smallest n for which the cyclic permutation is not its own inverse, and the paper explicitly denies that this justifies any cultural connotation of triads.
This repository's disclosure of language-model useWhat language models were used for. A language model is not an author.
Not peer-reviewedA DOI does not imply peer review
Citation
Cite in a form that makes the version clear.
Nemoto, T. (2026). The Trinity-Infinity Framework, Series III: Beyond Three — A Generalization and Retrospective. Zenodo. https://doi.org/10.5281/zenodo.22058964Cite in a form that makes the version clear.