This paper applied two "patches", m=4π and ε₀=r, to the balance equation of the Bohr model and obtained the form 16π²v²r²=e², that is (4πv)r=e. The 3rd edition shows that these two patches are not operations of the same kind, and fixes the range in which the equation holds.
First, a convention is a number, not a function of the configuration. m=4π fixes the unit of mass once; thereafter, for every electron and at every radius, m is the same 4π — a legitimate choice of units, of the same standing as G=4π and ħ=1/(4π) in Papers 35, 38 and 39. But ε₀=r assigns a different value at every radius: the permittivity of the vacuum would differ by a factor of ten between an electron at r=10⁻¹¹ m and one at r=10⁻¹⁰ m. A system of units cannot depend on where the electron is. Hence ε₀=r is not a choice of units. Among the three roots set out in the 3rd edition of Paper 38 — forced / choice of units / choice of configuration — m=4π belongs to the second and ε₀=r to the third. The first and second editions grouped the two under the single word "Yamagishi patch", grouping what must not be grouped. Including charge, the base dimensions are M, L, T, Q, so at most four values may be freely assigned; this series has already spent three (m=4π, G=4π, ħ=1/(4π)). ε₀ could occupy the fourth slot — but only so long as it is a number.
Second, read as a change in the force law, ε₀=r brings on the very collapse it was meant to prevent. Substituting ε₀=r into the Coulomb force e²/(4πε₀r²) gives e²/(4πr³) — an inverse cube, not an inverse square. For an attractive F=−K/r³ the effective potential is U_eff=(L²/m−K)/(2r²), which has no minimum because the centrifugal term and the force term both go as r⁻². Only three cases exist: for L²/m>K it is repulsive and no bound circular orbit exists (escape); for L²/m=K it vanishes identically and every radius is neutrally in equilibrium; for L²/m<K it tends to −∞ as r→0 and the particle falls to the centre. An inverse-cube force therefore admits no stable circular orbit (the classic "fall to the centre" of Landau and Lifshitz). Bertrand's theorem (1873) permits closed orbits only for the inverse square and for the harmonic oscillator; the inverse cube lies outside it. The 3rd edition of Paper 28 cited Bertrand's theorem as a signature of what is special about the Kepler problem; here the same theorem works on the side of prohibition. Read as a change in the force law, the patch produces the very collapse the first edition set out to prevent — already at the classical level, before quantum mechanics is reached.
Third, the extension to general principal quantum number cannot be derived. Substituting m=4π gives 4πv²=e²/(4πε₀r), and then ε₀=r gives 16π²v²r²=e² — in which the principal quantum number n appears nowhere on the right. The balance equation does not know about n. Meanwhile the quantisation condition mvr=nħ with m=4π gives 4πv_n r_n=nħ, while the balance gives 4πv_n r_n=e; for both to hold one needs e=nħ for every n. As verified here, nħ is 1.05×10⁻³⁴, 2.11×10⁻³⁴ and 3.16×10⁻³⁴ (J·s) for n=1,2,3, whereas e=1.60×10⁻¹⁹ (C) — neither the values nor the units agree. Theorem 3.1 of the second edition (16π²v_n²r_n²=e²/n²) therefore cannot be derived, and is withdrawn; the n in e²/n² was written by analogy with the quantisation condition, not obtained from the balance. The claim that "the discrete spectral structure of the Balmer series and the like is perfectly explained" is withdrawn with it — the theorem it presupposed cannot be derived, and even had the n-dependence been right, it would only have rewritten mvr=nħ, leaving the origin of the discreteness as Bohr's postulate. A change of notation, not an added explanation.
Fourth, something does survive. Read as numerical substitutions at that radius, m=4π and ε₀=r put the Bohr balance condition into the form (4πv)r=e. This is a restatement valid at one radius; it can neither be differentiated with respect to r nor extended to other radii. The equation is also dimensionally consistent within the system of units it implicitly defines: from [ε₀]=C²T²M⁻¹L⁻³, setting [ε₀]=L gives C²=ML⁴T⁻², so C=M^(1/2)L²T⁻¹, and with m=4π (dimensionless) M=1, so C=L²T⁻¹ — charge becomes an area per unit time. Thus (4πv)r=e is consistent as (L/T)(L)=L²/T, but in that system of units charge is an area per unit time. This is a worked instance of Paper 95 (Poincaré's conventionalism): one is free to adopt a convention, but not free to choose what the convention drags along with it.
Fifth, why atoms do not collapse is already explained by quantum mechanics. The probability density of a stationary state ψ_n(r)e^(−iE_n t/ħ) is |ψ_n(r,t)|²=|ψ_n(r)|², independent of time; the charge density therefore does not vary in time and no oscillating dipole moment exists. Since radiation arises from time-varying currents and charge distributions, a stationary state cannot radiate in principle (Schrödinger 1926). The electron is not "cancelling its radiation geometrically while orbiting": in a stationary state it is not orbiting at all. Bohr's circular orbit was a provisional hypothesis of 1913, superseded by the Schrödinger equation, and this paper does not replace that explanation. Consistency with the mode criterion of Paper 14 was also confirmed: the field at the Bohr radius is 5.142×10¹¹ V/m, the Schwinger limit is 1.323×10¹⁸ V/m, and the ratio is 3.886×10⁻⁷, so the atomic scale lies in the ordinary regime.
Finally, this paper sets out the series' second check. Following the 3rd edition of Paper 11, which established that "a quantity entering a normalised probability only as an overall factor cannot be observed, by definition", this paper places beside it: a convention must be a number. When one says that a quantity has been fixed by convention, check whether its value depends on the configuration (position, time, state); if it does, that is not a convention but a change of the theory. Both checks were extracted from the author's own failures — in Paper 11 a quantity that cancels was called a mechanism; here a quantity that varies was called a constant. Section 5 of the second edition, "forming the backbone infrastructure network of the universe's OS" and "connects perfectly", is withdrawn as well: the former is Layer-3 (non-scientific) vocabulary and is removed from the body under the discipline of Paper 82, while the latter treats the same number as evidence of a common origin, contrary to the discipline of this series. The classical collapse time 1.556×10⁻¹¹ s, the Bohr radius and velocity, the fine structure constant (1/α=137.035999) and the ratio of the atomic field to the Schwinger limit were all computed here (no external tables consulted). This paper claims no explanation of atomic stability, none of spectra, and no new physical law.
(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). A Convention Is a Number, Not a Function of the Configuration [Paper 15]. Zenodo.
DOI: 10.5281/zenodo.21600352
ORCID: 0009-0006-8587-3450