This paper introduced ε₀=r (the sync patch) and ε₀=c (the overclock patch), and the second edition organised them as a mode-switching specification driven by the state of the system, supplying quantitative criteria for the switch. The 3rd edition evaluates the switch and its two ends separately.
First, the two patches are not the same kind of thing. ε₀=r is a function of the configuration (of position), giving a different value at every radius; ε₀=c is a single number, the same everywhere. In the three roots sorted by the 3rd edition of Paper 38 (forced / choice of units / choice of configuration), ε₀=c belongs to the second and ε₀=r to the third. ε₀=c could be a convention if the slot for the unit of charge is used, but ε₀=r could not — as the 3rd edition of Paper 15 established, a convention must be a number and must not be a function of the configuration. The second edition ranged the two as "two modes of one specification": it ranged together what must not be ranged together. The same type of error recurs across the series — the one-word grouping "Yamagishi patch" in the 3rd edition of Paper 15, the single list of c, ħ, G, α, Λ in the 3rd edition of Paper 10, the single chain of equals signs in the 3rd edition of Paper 16. In each case the format concealed a difference of category.
Second, the mode switch is itself configuration-dependence at a coarser grain. Prescribing ε₀=r in the near zone and ε₀=c in the far zone is nothing other than saying that the value of ε₀ changes with position (r). Where ε₀=r changes continuously, the mode switch jumps at a boundary — the difference is only one of grain, and the judgement of the 3rd edition of Paper 15 still applies when the grain is made coarser. The mode switch therefore does not remove the defect of ε₀=r; it merely discretises the defect and hides it. A necessary qualification: the idea that the response of the vacuum depends on the state has genuine physical warrant — as the second edition rightly noted, the running coupling, the Euler–Heisenberg effective Lagrangian and the Schwinger limit are all established facts. The problem is not the idea but writing it on the side of ε₀, a constant that fixes the units. A state-dependent response is written as a non-linear term in an effective Lagrangian, not as a switch of the unit system.
Third, the two require different systems of units. What is equated to [ε₀]=C²T²M⁻¹L⁻³ fixes the dimension of charge itself: with ε₀=r (so that [ε₀] carries L) one gets C=M^(1/2)L²T⁻¹ (area per unit time), while with ε₀=c (so that [ε₀] carries LT⁻¹) one gets C=M^(1/2)L²T^(−3/2). Though both are "a substitution into ε₀", charge becomes a different thing: the two patches are not merely incompatible but require different systems of units. Furthermore, as the 3rd editions of Papers 38 and 39 established, at most four values may be freely assigned against the base dimensions M, L, T, Q, and this series has already spent three on m=4π, G=4π, ħ=1/(4π). Only one slot therefore remains for ε₀, and only "a single number" can go into it — ε₀=c could be a candidate, ε₀=r could not, and using both at once is putting two things into one slot. This is the same type of oversight the 3rd edition of Paper 17 identified for ħ=1 versus ħ=1/(4π). As an aside, the second edition's "overclock permeability theorem" μ₀=1/c³ is correct as algebra, obtained by putting ε₀=c into μ₀=1/(ε₀c²); but numerically 1/c³=3.711×10⁻²⁶, utterly unlike the SI value μ₀=1.2566×10⁻⁶ — the manifestation of changing units, not a claim about permeability (root (b), a consequence of substitution, in the census of the 3rd edition of Paper 16).
Fourth, this paper wrote down an inverse-cube force with its own hand and never checked the consequence. Section 3 of the second edition derives, under ε₀=r, the Coulomb force F_E(r)=Q₁Q₂/(4πr³) — an inverse cube, not an inverse square. As the 3rd edition of Paper 15 showed, the effective potential of an attractive F=−K/r³ 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 — escape, fall to the centre, or neutral equilibrium at every radius — and no stable circular orbit (Landau and Lifshitz's "fall to the centre"; Bertrand's theorem permits closed orbits only for the inverse square and Hooke's law). This paper had written F∝r⁻³ with its own hand and had not looked at what an inverse cube means. Paper 15 applied this patch to the atom in order to explain why the electron does not fall in; but read literally, the patch drops the electron — and already at the classical level, before quantum mechanics is reached. As the 3rd edition of Paper 10 wrote, stopping at "I have not shown this yet" is not safe; here it was not even "not shown": the equation was written and not read.
Fifth, the criteria nevertheless survive. Criterion I, the near/far zone boundary, is given by r~λ/(2π)=c/ω for wavelength λ and angular frequency ω — a standard distinction of electromagnetism (computed here: r=0.159 m for λ=1 m, r=1.59×10⁻⁴ m for λ=1 mm) defined without reference to any patch. Criterion II, the Schwinger limit, is E_c=m_e²c³/(eħ)=1.3233×10¹⁸ V/m, above which pair creation from the vacuum becomes appreciable and the linear-response picture breaks down (Schwinger 1951); computed here, the field at the Bohr radius is E=5.142×10¹¹ V/m, giving E/E_c=3.886×10⁻⁷. Both criteria hold irrespective of what is substituted for ε₀, and both are facts internal to established electromagnetism rather than discoveries of this paper — but setting them in one table and attaching numbers to them was this paper's doing, and that remains useful. In particular the number E/E_c=3.886×10⁻⁷ shows concretely that the atomic scale stands seven orders of magnitude away from the non-linear response of the vacuum, and it is a dimensionless ratio, independent of the system of units (3rd edition of Paper 38). This paper built a good switch and wired it to a bad pair of settings; the 3rd edition keeps the switch and removes the ends.
Finally, the second edition's "module block theory — declaration of indivisibility" is removed from the body. The second edition presented it as a theorem, holding that 2π, 4π, 8π and 16π² must not be arbitrarily divided and that stripping the topological meaning from these factors irreversibly destroys the topological structure of the manifold. But the later work of this series consists precisely in dividing them: Paper 48 separated the three factors of 2 in Gauss–Bonnet, Paper 86 took a census of four roots of necessity, Paper 115 showed that 2π and 4π belong to different staircases, the 3rd edition of Paper 35 sorted the two 4π's inside 4πG into a fact and a convention, the 3rd edition of Paper 17 gave each of three twos its own address, and the 3rd edition of Paper 34 showed that the 5 and the 1/5 of G/(5c⁵) are distinct roots. "Do not divide" and "one must divide" collide head-on within one series, and the series chose the latter. The declaration stood in a theorem environment without a proof, and "irreversible destruction of the topological structure" is undefined: it is a claim, not a theorem, and is moved to Layer 3, the non-scientific layer (the discipline of Paper 82 and of the 3rd edition of Paper 10). The core of the intuition, however — that these factors carry geometric meaning and that something is lost by treating them as bare numbers — was inherited by the series in another form. The form inherited is not "do not divide" but "write down which root it came from" (Papers 86, 100). The way to protect the meaning was the opposite of what was written. All numbers were computed here (no external tables consulted). This paper does not claim that the permittivity of the vacuum equals a distance or the speed of light.
(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. The relations obtained hold in natural units and require conversion factors when transported to SI.
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 Two Patches Are Not the Same Kind of Thing [Paper 14]. Zenodo.
DOI: 10.5281/zenodo.21600268
ORCID: 0009-0006-8587-3450