The Home of the Constants: 4π and 32π² as Index-Theorem Normalizations — Placing the Yamagishi Patch in the Atiyah–Singer Family, Without Merging Its Branches.
Paper 67 anchored the cornerstone as 4π = 2π·χ(S²) (the 2D Gauss–Bonnet case); Paper 48 unpacked the higher normalization (4π)ⁿ·n! of Chern–Gauss–Bonnet. This paper names their common home: the Atiyah–Singer index theorem (analytic index = topological integral of a characteristic class). Three points. (i) Placement: 4π and (4π)ⁿ·n! are the Euler-class branch normalizations; Poincaré–Hopf (Hopf 1926) is the prototype of the word “index”. (ii) Two rooms, one house: the instanton 16π² of Paper 45 (the second Chern class, k = (1/16π²)∫tr(F∧F)) also lives in the theorem, but in the Chern-character branch of the gauge-coupled Dirac operator — a DIFFERENT branch. So the index theorem both unifies (a common home) and certifies the distinctness Paper 48 found between the Euler 32π² and the instanton 16π². (iii) Consequence: the patch constants are named members of a classical family, topological in all even dimensions. A common home is not a common origin — the index theorem is the family house; the constants are siblings, not the same person. This is a placement, not a derivation; the spin/Dirac (Â-genus) branch, linking to the qubit of Paper 68, is a direction, not a claim. No new mathematics or physics; the branches are not merged (distinct roots, Papers 48/58).
A note on the aim of this work. This is a structural interpretation (a placement of established constants in an established theorem), not new mathematics or a new physical law. The index theorem is assumed and cited. 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 Home of the Constants: $4\pi$ and $32\pi^2$ as Index-Theorem Normalizations (Placing the Yamagishi Patch in the Atiyah–Singer Family, Without Merging Its Branches) [Paper 70]. Zenodo.
DOI: 10.5281/zenodo.21650347
ORCID: 0009-0006-8587-3450