This paper examines the geometric constant 32π² common to the instanton topological charge density q(x) of quantum chromodynamics and the Pontryagin density q_grav(x) of general relativity. We give the factorisation 32π² = 2 × (4π)² under the Yamagishi mass patch m=4π and ask why the same constant occurs in both.
Important confirmation in the 2nd edition: this coincidence is genuine. Where 16π² or 32π² recur across this corpus, the recurrences are usually numerical coincidences of distinct origin that must not be merged (error pattern C; see Papers 22, 29, 30, 45, 48). The coincidence treated here is an exception, and merging is justified. The QCD instanton number and the gravitational Pontryagin index are both normalisations of Chern–Pontryagin characteristic classes to integer values, so their shared 32π² descends from one and the same mathematical structure (Section 5). It is the only legitimate unification identified in this review.
Withdrawal in the 2nd edition: On the other hand the first edition's substitution 32π² → 2c²n (a "VRAM rendering budget") is withdrawn. The paper's own synchronisation relation yields 2n, not 2c²n; the c² is a spurious factor left behind when 1/v² was dropped. In addition 32π² is dimensionless while 2c²n is not (Section 4).
(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 Unified Budget Canvas Theory [Paper 32]. Zenodo.
DOI: 10.5281/zenodo.21601849
ORCID: 0009-0006-8587-3450