This paper is a comparison table listing how representative formulas of mechanics and astrophysics look once the Yamagishi patches (m = 4π, G = 4π) are applied.
[What the 3rd edition adds] — a column of roots. Earlier editions had only two columns, “standard form” and “after the patch”, but each row in fact mixes three kinds of number of different standing.
(i) Forced. Numbers settled by geometry or analysis, with no choice available: the 4π of Poisson’s equation (Gauss’s flux, Vol(S²)); the 4π² of Kepler’s third law (the square of the angular 2π); the ½ of the kinetic energy (the result of an integration); the ⅓ of the gas pressure (isotropy in three dimensions); the 3/2 of the equipartition relation (three degrees of freedom).
(ii) Choice of units. A convention fixing the numerical value of a dimensionful quantity: G = 4π, m = 4π.
(iii) Choice of configuration. A specification of which system is being described, distinct from a choice of units: M = m, i.e. taking both point masses to be the unit mass. Setting m = 4π fixes one unit of mass; it does not make every mass equal to 4π. Accordingly F = 64π³/r² is not the general form of Newtonian gravity but the expression for two point masses that both carry the unit mass.
A consequence: in the Kepler row, π disappears entirely because the forced 4π² of the numerator is divided by the chosen 16π² of the denominator, giving ¼. This is not π “being eliminated”, and no meaning should be read into the disappearance.
[Is this unit system admissible?] With base dimensions (M, L, T), the dimension exponents of G, m and c are (−1, 3, −2), (1, 0, 0) and (0, 1, −1); the determinant of that matrix is 1, so the three are dimensionally independent and their numerical values may be assigned simultaneously. The unit system of this table is therefore not inconsistent. At the same time there are only three base dimensions, so at most three dimensionful quantities may be assigned: imposing a value on a fourth is no longer a choice of units but a claim about nature. Cross that boundary and a convention turns into a prediction.
[What the substitution cannot move] The two columns look different; what is the same are the dimensionless ratios. The ratio of escape to orbital velocity is √2 in both columns; kT/E_k is 2/3 in both; and the equation of state P = nkT holds in both, since ⅓·4π·nv² = n·(4/3)πv² exactly. The eye-catching numbers 4π, 16π² and 64π³ are units and change appearance at will; what does not move are √2, 2/3 and the relation P = nkT. What Paper 35 (3rd edition) said of α_G ≈ 5.9×10⁻³⁹ holds for the table as a whole: dimensionless quantities are invariant under a choice of units, and physics lives nowhere else.
[A sharper diagnosis of g = 16π²/r²] The second edition withdrew this expression (Paper 36) because GM varies by 7.4 orders of magnitude between the Moon, the Earth and the Sun. The 3rd edition supplies the structural reason. g is not an independent constant but the derived quantity g = GM/r², so once G and M are fixed its value is already settled — the operation “set g equal to 16π²/r²” does not exist as a choice of units. Indeed G = 4π and M = 4π make g = 16π²/r² follow automatically; the problem is that this holds only for a body whose mass is exactly the unit mass. The substance of the error was to use a choice of configuration as a universal law, and the 7.4 orders of magnitude are the symptom by which it surfaced, not the cause. Relatedly, the 2π of the pendulum period is a forced number: numerical integration gives T/√(L/g) = 6.283185, agreeing with 2π to eight digits, so inserting 16π² into g to cancel it covers a forced structure with a choice.
[Why (4π)ⁿ agrees at 16π² and diverges at 64π³] The chain is the flat-torus sequence Vol(Tⁿ at a = 2) = (4π)ⁿ, a direct product — powers of one number. Sphere volumes instead obey Vol(Sⁿ) = (2π/(n−1)) Vol(S^(n−2)), whose step is two, whose coefficient changes at each stage, and which splits by parity into two staircases (Paper 115). At step two the values agree (Vol(S³ at R = 2) = 157.914 = Vol(T² at a = 2)); at step three they part (Vol(S⁴ at R = 2) = 421.103 against 64π³ = 1984.402). Since the generating rules are fundamentally different, the divergence is normal and it is the agreement that requires explanation: it is a signature, not evidence of a common origin. Moreover, in the Hopf bundle S³ is not S² × S¹ (Papers 67, 109), so the sphere side contains a counterexample to the product picture, and the reading “the two sequences are one thing in two guises” is not available. Whenever (4π)ⁿ is discussed, radius 2 must be stated explicitly: at radius 1 one has Vol(T³) = 8π³ = 248.05.
[Audit of sources] The papers referenced by this table are 12, 16, 18, 19, 20, 28 and 35, and none of the eight discarded papers of this series (23, 24, 25, 27, 31, 33, 36, 37) appears among them. The check left pending at the second edition — whether discarded papers had been excluded — is hereby complete.
Scope: a list of rewritings that follow a choice of units (Layer 2). This is a list, not a proof; constants are not eliminated, their numerical values merely change with the choice of units; the g-dependent formulas remain excluded, though they may return if g is supplied correctly; the (4π)ⁿ sequence is not merged with the sphere sequence; no new physical law and no prediction is claimed. 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 Master Table of Classical-Physics Geometric Unpacking [Paper 38]. Zenodo.
DOI: 10.5281/zenodo.21430234
ORCID: 0009-0006-8587-3450