The Euler-Characteristic Origin of the 4π Patch: Reading m = 4π as 2π·χ(S²) via Poincaré–Hopf.
The constant 4π — the total solid angle of the 2-sphere and the limit mass m=4π of Paper 01 — anchors the Yamagishi system. This paper observes that 4π is topological, not merely geometric: by Gauss–Bonnet ∫K dA = 2π·χ, and by the Poincaré–Hopf theorem the Euler characteristic χ(S²)=2 is fixed by topology alone (every vector field on the sphere must vanish — the “hairy ball theorem”). Hence 4π = ∫K dA = 2π·χ(S²) = 2π×2, so m = 4π = 2π·χ(S²): the “2” in 4π=2π·2 is the Euler characteristic of the sphere. The cornerstone value is therefore forced by topology, not arbitrary. The anchor exists only at even sphere dimensions (χ(Sⁿ)=1+(−1)ⁿ), echoing the even/odd split of Paper 57, and generalizes via Chern–Gauss–Bonnet to the (4π)ⁿn! ladder of Paper 48. Deliberately, this reaches Hopf through the Euler-characteristic door, not through the (distinct) mass–instanton 16π² bridge.
A note on the aim of this work. This is a structural interpretation (a topological re-reading) of an established constant, not a proposal for a new physical law. The representation m=4π remains a choice; only the value of the solid angle is shown to be topological. 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 Euler-Characteristic Origin of the $4\pi$ Patch: Reading $m = 4\pi$ as $2\pi\,\chi(S^2)$ via Poincaré–Hopf (A Topological Anchor for the Yamagishi Solid-Angle Constant) [Paper 67]. Zenodo.
DOI: 10.5281/zenodo.21642170
ORCID: 0009-0006-8587-3450