Substituting the Yamagishi gravitational patch G_geom = 4π = Vol(S²), obtained from the n = 2 uniqueness theorem, into Newtonian gravity and Poisson’s equation gives ∇²Φ = 16π²ρ; taking M = m = 4π as well gives F = 64π³/r² and U = −64π³/r. These substitutions are correct and the two are consistent via F = −dU/dr.
[What the 3rd edition adds] — an identification of the two 4π’s contained in the product 4πG.
Root 1 — the 4π of Poisson’s equation. It comes from ∇²(1/r) = −4π δ³(r), that is, the total solid angle Vol(S²) forced by Gauss’s flux theorem. It cannot be chosen: once three-dimensional space and the inverse-square law are given, it is fixed. Measured by Paper 114, it is also dimension-blind — in d dimensions the corresponding factor becomes Vol(S^(d−1)), so it is the d = 3 member of a family with one member per dimension.
Root 2 — the 4π substituted for G. This is a choice of units, a convention of exactly the same standing as G = 1 (geometrised units) or G = 6.674×10⁻¹¹ (SI). Physics does not decide which is correct (Paper 95).
Hence 16π² = (4π)(4π) is the product of a “4π of fact” and a “4π of convention”, and the two must not be merged (Papers 86, 100). Reading them as “the same 4π appearing twice” would grant a convention the standing of a fact — the failure mode against which Paper 101 warned.
[Four addresses of 16π²] — this paper’s product; Vol(S³ at R = 2) (Paper 08); the instanton-number normalisation (Papers 45, 48); and the one-loop geometric measure 1/16π² (Paper 110). In particular Vol(S³ at R = 2) = Vol(T² at a = 2) = 16π² is an instance in which two manifolds of different dimension — three and two — carry the same number (157.9137…). There is no cleaner specimen of “a signature, not a root”.
[Product versus recursion] — the chain 4π → 16π² → 64π³ is the flat-torus sequence Vol(Tⁿ at a = 2) = (4π)ⁿ, a direct product. Sphere volumes instead obey Vol(Sⁿ) = (2π/(n−1)) Vol(S^(n−2)), whose step is two and which splits by parity into an even and an odd staircase (Paper 115). Moreover, in the Hopf bundle S³ is not S² × S¹ (Papers 67, 109): the sphere side contains a counterexample to the product picture. The generating rules differ, so the two must not be read as one series.
[What a choice of units cannot do] — the dimensionless gravitational coupling α_G = G m_p²/(ħc) ≈ 5.906×10⁻³⁹ is dimensionless and therefore unchanged by any choice of units (Duff–Okun–Veneziano 2002). The numerical value of G can be set to 1, to 4π, or to 6.674×10⁻¹¹ at will; α_G does not move at all. The content of the question “why is gravity so weak?” resides in α_G, not in the numerical value of G, so the substitution of this paper does not touch that question.
[The dimension of G is the physics] — in natural units G carries mass dimension −2 (G = 1/M_Pl²), and a coupling of negative mass dimension means divergences worsen at higher orders; pure gravity was shown to diverge at two loops (Goroff–Sagnotti 1986). Setting G = 4π tidies the appearance of the equations but hides that mass dimension from view — and what is hidden is where the difficulty of quantum gravity lives.
[On 8πG] — the coefficient 8π of the Einstein equations is fixed by the Newtonian limit, so under G = 4π one may write 8πG = 32π². The factor 2 here comes from that limit; the 2 in Paper 48’s 32π² = (4π)²·2! comes from the antisymmetrisation of the Pfaffian. Same value, different root; they must not be merged.
Scope: a structural interpretation of established physical law (Layer 2). No new physical law and no prediction is claimed. The measured value of G is not derived, the weakness of gravity is not explained, quantum gravity is not solved, and gravity is not called a “geometric pressure” (no quantity with the dimensions of force per area is defined here; F is a force). The main result and the consistency F = −dU/dr are unchanged from the second edition. The Universe OS is a non-scientific Layer-3 lens.
(scope note) Positioning of this paper: This paper offers a structural interpretation of established physical law. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are substitutions into known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
Note on units (important): The substitutions m=4π, G=4π and the like employed here constitute a choice of natural units. G and ħ are dimensionful quantities whose numerical values depend entirely on the unit system. Setting G=4π is possible in exactly the sense that G=1 (geometrised units) or G=6.674×10⁻¹¹ (SI) is possible; physics does not determine which is “correct.”
What this paper exhibits is therefore how the equations look once this unit system is adopted, not a new fact about nature. Moreover the relations obtained hold in natural units, and conversion factors are required to carry them into SI. Whatever value this paper has lies in making geometric structure visible — rendering the provenance of π and the role of solid angle easier to read off the equations.
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 Gravitational Constant Unpacking Theory [Paper 35]. Zenodo.
DOI: 10.5281/zenodo.21430118
ORCID: 0009-0006-8587-3450