The Special-Dimension Distribution Theorem: Why There Is No Single “Magic Dimension” — An Extension of the Function-Comparison Method of Paper 58.
The question “why 3+1 dimensions?” is usually posed as a search for a single decisive reason. This paper argues, by systematic tabulation, that no single dimension is optimal by all criteria; the specialness of dimension is distributed over the neighbourhood {3,4,5} and clusters around the physical value. Applying the function-comparison method of Paper 58 to six established criteria — orbital/atomic stability (selects d=3), sharp wave propagation / Huygens’ principle (odd d, minimally 3), the onset of propagating gravitational degrees of freedom N(D)=D(D-3)/2 (D=4), the trace-reversal factor f_GR(D)=(D-2)/(D-3) crossing the ladder factor 2 only at D=4, topological exoticness (exotic R^4, Hopf; d=4), and the unit-ball volume peak (d=5) — the optima are distributed, not concentrated. We read this as a refinement of the anthropic principle: 3+1 is special by concentration of independent optima, not by a single cause. The paper does not prove the necessity of 3+1; the clustering depends on the chosen criteria.
A note on the aim of this work. This is a structural interpretation and a systematic compilation of established mathematical facts, not a proposal for a new physical law. Every criterion cited is a known result that remains unchanged. Nothing here is intended to contradict existing results.
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). The Special-Dimension Distribution Theorem: Why There Is No Single “Magic Dimension” (An Extension of the Function-Comparison Method of Paper 58) [Paper 64]. Zenodo.
DOI: 10.5281/zenodo.21616221
ORCID: 0009-0006-8587-3450