Continuous Symmetry Emerges from Weak Assumptions: Hidehiko Yamabe and Hilbert’s Fifth Problem (The Deep Cousin of Ignatowski).
This corpus keeps meeting the theme that rich structure emerges, inevitably, from very weak assumptions — Ignatowski (Paper 83: relativity principle + homogeneity/isotropy + group structure alone give only two relativities and the Lorentz group), Galois-Lie-Noether (Paper 85), Gleason (Paper 71). This paper finds the deepest mathematical statement of that theme in the work of the Japanese mathematician Hidehiko Yamabe (1923-1960). Hilbert’s fifth problem — is a continuous group automatically differentiable (a Lie group)? — was solved affirmatively by Gleason, Montgomery-Zippin, and Yamabe. By Yamabe’s theorem in particular, a topological group that is locally Euclidean (a manifold) is automatically a Lie group, without assuming differentiability: the smooth structure of a continuous symmetry is FORCED by weak topological assumptions. This is the general form, in pure mathematics, of Ignatowski’s (83) derivation of the Lorentz group from a few kinematic axioms — one lineage of “weak assumptions to symmetry,” on a different stage (kinematics vs the topology of groups; distinct roots, Papers 58/48). Yamabe’s second gift, the Yamabe problem (every conformal class contains a metric of constant scalar curvature; initiated by Yamabe, completed by Trudinger-Aubin-Schoen), is a second bridge to the corpus’s curvature papers (67 Gauss-Bonnet/4pi, 07 Ricci flow/Perelman, 60 velocity-space curvature). IMPORTANT (scope): a Layer-2 structural interpretation of borrowed standard results; no new mathematics and no new prediction; it does not solve or extend Hilbert’s fifth or the Yamabe problem. No conflation: Gleason-Yamabe (Hilbert 5) vs Gleason’s theorem (71, quantum) are distinct theorems by the same person; the group result vs the Yamabe problem are the same person’s different works; Ignatowski’s kinematics vs Yamabe’s topology is a shared idea, not identical theorems. The Universe OS is a Layer-3 non-science lens.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Continuous Symmetry Emerges from Weak Assumptions [Paper 90]. Zenodo.
DOI: 10.5281/zenodo.21758361
ORCID: 0009-0006-8587-3450