The Dimension Census: Why 3+1 Dimensions, as an Over-Determination by Independent Roots.
Paper 64 showed that the “special” dimension is not unique but distributed over {3,4,5}. This paper sharpens that at D=4. We assemble a census: a systematic accounting of dimension-selecting criteria, each with its mathematical root. Space (d=3) is selected by orbital/atomic stability (Bertrand; Ehrenfest) and sharp wave propagation (Huygens); capacity (d=5) by the unit-ball volume peak (Paper 59); and spacetime (D=4) by FIVE criteria — the conformal (Weyl) invariance of the free Maxwell action (only at D=4), the onset of a propagating graviton N=D(D−3)/2, the exotic smooth structures of R⁴ (Donaldson) and the Hopf/instanton content, the unique crossing of the trace-reversal factor with the ladder factor (Paper 58), and the even-dimensional topological anchor χ(S^2n)=2 (Paper 67). The central result certifies, by the distinct-roots method of Paper 58, that these five roots are pairwise independent (conformal weight, little-group representation theory, differential topology, the GR trace, and the Euler characteristic have distinct origins, none reducible to another). Hence D=4 is not special for a single reason, nor loosely distributed, but OVER-DETERMINED by independent roots — a descriptive strengthening of the anthropic reading, not a proof of necessity. It adds two criteria (Maxwell conformal; the Paper-67 anchor) beyond Paper 64 and claims no new physics.
A note on the aim of this work. This is a structural interpretation and systematic compilation of established facts, not a proposal for a new physical law and not a proof that D=4 is necessary. 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 Dimension Census: Why $3{+}1$ Dimensions, as an Over-Determination by Independent Roots (Certifying, by the Method of Paper 58, that the Selections of $D=4$ Are Distinct) [Paper 69]. Zenodo.
DOI: 10.5281/zenodo.21650011
ORCID: 0009-0006-8587-3450