The Adler–Bardeen theorem (1969) states that the coefficient of the chiral anomaly is fixed at one loop and receives no corrections from higher orders. This paper examines the geometric meaning of the 16π² appearing in its denominator.
Principal withdrawal in the 2nd edition: The first edition claimed that higher-order corrections vanish because 16π² carries a "Read-Only" attribute. This is withdrawn. The same 1/(16π²) is a universal loop factor appearing in essentially every one-loop quantity in four-dimensional field theory, and most of those — the QCD beta function among them — do receive higher-order corrections. If 16π² conferred protection, the QCD beta function would be one-loop exact; it is not. The value 16π² is therefore not the cause of non-renormalisation (Section 4). The identification 16π² = c²n is also withdrawn, being dimensionally ill-formed.
What the 2nd edition retains: 16π² nonetheless has a different and more accurate role. Since ∫ F F/(16π²) yields an integer (a winding number, the Atiyah–Singer index), 16π² is the normalisation constant that makes the index integral. What guarantees non-renormalisation is that integrality rather than the value of 16π² — though the two are not unrelated. The paper retains its claim in this limited form (Section 5).
(scope note) Scope of this paper: This paper offers a structural interpretation of existing physical law. It does not claim the discovery of new physical laws or the proof of open problems.
Note on units: Substitutions such as m=4π are a choice of natural units. G and ħ are dimensionful; their numerical values depend entirely on the unit system. Setting G=4π is legitimate in exactly the same sense as setting G=1 (geometric units) or G=6.674×10⁻¹¹ (SI); physics does not single out one of these as "correct." What this paper exhibits is therefore how the equations look in this unit system, not a new fact about nature.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. 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 Yamagishi–Adler–Bardeen Note [Paper 29]. Zenodo.
DOI: 10.5281/zenodo.21601581
ORCID: 0009-0006-8587-3450