The first edition of this paper presented the one-line chain p=4πv=e/r=c√n_opt and claimed that four domains of physics were thereby "fully synchronised". The second edition found that the three equals signs have different logical standings and withdrew the "grand unification" claim. The 3rd edition turns that finding into a census the series can use.
First, the equals sign has distinct roots too. "=" is not one kind of assertion. This paper takes a census of six roots: (a) definition (p≡mv; true by stipulation, cannot be false), (b) consequence of substitution (with m=4π, p=4πv; automatic once the convention is adopted), (c) identity (sin²θ+cos²θ=1; true for all values), (d) conditional (mv²/r=e²/4πε₀r²; holds only on solutions), (e) empirical law (v_phase=c/n_opt; could have been false, and has been tested), (f) dimensionally void (the sides differ in dimension, so it is not even a candidate for truth). There is an ordering of content among the six: (a) and (b) say nothing about the world, (c) is said by mathematics, (d) and (e) are said by the world, and (f) says nothing at all. Chaining quantities with "=" therefore makes them look as though they had the same standing — the symbol hides the difference in standing. This is the same shape as the 3rd edition of Paper 15, which wrote that "a name can conceal a distinction": there it was a name, here it is a symbol.
Second, this paper's chain set three different kinds of relation side by side under one symbol. p=4πv is (b) — it follows once m=4π is adopted, and is not a discovery. 4πv=e/r is (d) — the centripetal balance on a circular orbit, not valid for arbitrary v and r. e/r=c√n_opt is (f) — the dimensions disagree and √n_opt has no source. Note especially the character of (d): the 3rd edition of Paper 15 narrowed its range further still, to a single radius, from which it can neither be differentiated with respect to r nor extended to other radii. The middle link of the chain is joined only at a point.
Third, the dimensional mismatch the second edition recorded as unresolved does resolve — but the way it resolves is the problem. In the system of units identified in the 3rd edition of Paper 15 (ε₀=r so [ε₀]=L; m=4π so M=1; hence C=L²T⁻¹, charge as area per unit time), p=mv is (1)(L/T)=L/T, 4πv is L/T, e/r is C/L=(L²T⁻¹)/L=L/T, and c√n_opt is L/T: all four agree at L/T. But the condition that buys this agreement is ε₀=r, which the 3rd edition of Paper 15 showed is not a convention — a convention must be a number, and something assigning a different value at every radius is not a choice of units. So "the dimensions agree" is not evidence for the chain; it is a consequence of the same single illegitimate move. Two results of one operation must not be used as evidence for each other. As an aside, writing "in natural units" does not fix the equation either: in Heaviside–Lorentz units α=e²/4π so e=√(4πα)=0.3028, while in Gaussian units α=e² so e=√α=0.0854 (computed here).
Fourth, the third equals sign is not "future work". The standard optical relation in which a square root of the refractive index arises is Maxwell's relation n_opt=√(ε_r μ_r), so that √n_opt=(ε_r μ_r)^(1/4) — the fourth root of ε_r μ_r (computed here: 1.2247 for ε_r=2.25, i.e. n=1.5, and 1.4142 for ε_r=4, i.e. n=2). This quantity occurs in no standard relation of optics, whereas the phase velocity in a medium is v_phase=c/n_opt, with no square root and with n_opt in the denominator. Since no source for the square root can be identified, this is not an equation whose derivation is incomplete but an equation with no root; leaving it as future work would imply that the gap will one day be filled, and the 3rd edition removes that implication. A fourth root also appears in the 3rd edition of Paper 03 (the Kolmogorov length), but it is a distinct root: there it is forced by the determinant |det|=4 of the dimension matrix, here it is an ungrounded square root of a square root. The two 1/4s must not be joined.
Fifth, what remains is that one substitution appears in several places. As verified here, putting m=4π into E_k=½mv² gives E_k=2πv² (Paper 18); into P=⅓nm⟨v²⟩ gives P=(4/3)πnv² (Paper 19); into (3/2)kT=½m⟨v²⟩ gives kT=(4/3)πv² (Paper 20). All three are one substitution into a standard formula, and no new relation appears; this is not a "transfer of machinery" in the sense of Paper 100. Moreover the 3rd edition of Paper 35 sorted the two 4π's inside 4πG: the 4π of Poisson's equation is forced by Gauss's flux and cannot be chosen, whereas the 4π of G=4π is a choice of units and can be. The 4π of p=4πv is the latter. Being a chosen number, its appearance in several domains is no cause for surprise — the same number was put in four places, so it appears in four places. A unification built on a choice of units unifies the notation. It does not unify the physics.
Finally, this paper places the series' third check. Following the 3rd edition of Paper 11 ("a quantity entering a normalised probability only as an overall factor cannot be observed, by definition") and the 3rd edition of Paper 15 ("a convention must be a number"), this paper adds: write out the root of each equals sign. When several quantities are chained with "=", state for each equality whether it is (a) a definition, (b) a consequence of substitution, (c) an identity, (d) a conditional, (e) an empirical law, or (f) dimensionally void. Without that, the symbol hides the difference in standing. All three checks were extracted from the author's own failures: in Paper 11 a quantity that cancels was called a mechanism; in Paper 15 a quantity that varies was called a constant; here different kinds of relation were set side by side under one symbol. Being wary of numbers is not enough — "=" too makes different things look the same. This paper claims no unification of physical domains.
(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 Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo.
DOI: 10.5281/zenodo.21600471
ORCID: 0009-0006-8587-3450