A Structural Study of the 4π Patch: of the four claims about the Yamagishi patch, two have been overtaken by their own author. Second edition.
In its first edition this paper organised the structure of the Yamagishi patch (G=4π, hbar=1/4π) into four claims: (A) the closure of the Planck quantities is not peculiar to 4π; (B) the synchronisation equation e=4π v r makes charge consistent as a geometric quantity; (C) the 16π² falls into two distinct roots; (D) four independent requirements all select d=3. What the second edition does is set those four against the papers of this corpus written afterwards. The outcome is asymmetric: (A) and (C) were generalised and sharpened by the later work, (B) was narrowed by it, and (D) survives untouched. NO NEW PHYSICAL LAW IS CLAIMED.
First, (A) was a special case of a theorem in Paper 116. The first edition said that geometric quantities carry a to the zero and material ones a to the minus one; Paper 116 proved the general statement epsilon = mu, that the power of the patch constant is the mass dimension itself, half-integers included. The first edition’s dichotomy is merely mu = 0 versus mu = 1. Indeed the Planck length and time have mu = 0 and carry a to the zero; the Planck mass, energy, force and hbar have mu = 1 and carry a to the minus one; and G has mu = minus one and carries a to the plus one. The first edition found the dividing line, and Paper 116 gave that line a name; and once named it turned out to be not a dichotomy but a continuous scale, since mu takes minus one and one half as well as zero and one. That generalisation decisively changes the next point. The first edition’s main conclusion is unaffected: closure follows from the reciprocal patch hbar = 1/G and is not peculiar to 4π, so the closure of the Planck quantities must not be used as evidence for the specialness of 4π.
Second, the problem that (B) set out to solve was not a problem. The first edition called the half-integer power arising from the Gaussian Planck charge something that falls outside Theorem 1. It does not. Gaussian charge has dimension M to the one half, L to the three halves, T to the minus one, that is mass dimension one half, and by epsilon = mu the power of a must be minus one half. That the Planck charge carries it is not a defect of the patch but the half-integer mass dimension of charge. The first edition’s theorem had merely been stated for mu in zero and one, and in Paper 116’s form mu = one half is inside it.
And e = 4π v r is built from lengths and times alone, so it has mu = 0 by construction; its independence of a is a consequence of the definition, not a discovery. The half-integer power has not been removed. It appears to have been removed only because a quantity with half-integer mass dimension was replaced by one without. It is Gaussian charge that carries mu = one half, not 4π v r. This is the same shape as Paper 45’s verdict that S = 1/(2 alpha) is a restatement of a definition.
Further, the source of e = 4π v r, Paper 15, has been substantially narrowed in its third edition: epsilon-zero = r is a function of the configuration and not a choice of units; read as a modification of the force law it gives an inverse cube, for which no stable circular orbit exists; the extension to a general principal quantum number was withdrawn; and what remains is that the relation is a restatement holding at one radius, and that in the unit system it implies, charge has the dimension of area per second. The first edition used it as a general representation of charge, a reading wider than what Paper 15 now leaves standing, and the second edition returns it to that range. The corollary that under m = 4π charge takes the form of the angular momentum m v r is correct as a numerical restatement at one radius, but does not stand as the general claim that charge is a geometric quantity: the mass dimension of charge is one half and does not change with the representation. What changed was the quantity, not the dimension of the quantity.
Third, Family II in (C) was not a single root. The first edition placed the loop factor, Gauss-Bonnet, the second Chern number and instanton, and the Hopf invariant together as normalisations of characteristic classes and measures in four dimensions; that merger is too strong. As the third edition of Paper 46 showed, the loop factor’s 16π² decomposes as 2 times (2π) to the fourth divided by Omega-three, where Omega-three is 2π²: the 2 is the Jacobian of a change of variable, the (2π) to the fourth is the convention of where the 2π of the Fourier transform is placed, and Omega-three is the surface area of the unit three-sphere, that is the geometry. And changing the convention changes the number: with d-four-k over (2π) squared it becomes 4, and with plain d-four-k it becomes one over π squared (verified here). No physical amplitude changes, but the number does. Gauss-Bonnet, Chern and Hopf, by contrast, are characteristic-class normalisations and do not move with the Fourier convention. The loop factor therefore has a different standing from the other three, and Family II splits into at least two.
The first edition’s table also gave the source of 4π in the Gauss-Bonnet row as the Pfaffian normalisation; but by the third edition of Paper 48 the Chern-Gauss-Bonnet normalisation is (2π) to the n times two-to-the-n n factorial, one block, and peeling the two-to-the-n off it to manufacture (4π) to the n was cutting in the wrong place. For n equal to one, two and three the block is 4π, 32π² and 384π³, containing no (4π) to the n (verified here). The distinctness of Family I, the algebraic square of the mass patch, from Family II holds for a stronger reason: by the third edition of Paper 46 the left side is convention times geometry times Jacobian while the right side is convention times convention, the patch constant being any positive a (Paper 116). The compositions are entirely different.
Fourth, (D) is untouched. The generalisation to n = d minus one, the selection of d = 3 by four requirements (orbital stability via Bertrand’s theorem, atomic stability, wave propagation via Huygens’ principle, and topological normalisation), and the limitation that this is a selection by consistency and not a derivation, all stand. Paper 114 later placed n = d minus one on the side that does not select the dimension, in agreement with what the first edition had said of itself. The second edition does note that the fourth requirement contains a Fourier convention and cannot be counted with the same force as the other three. The negative result from the independence of the Casimirs is likewise untouched: the two occurrences of 16π² inside the atom, the Bohr-Yamagishi one and the genuine Hopf invariant of electron spin, have distinct roots, and identifying them would confuse mass with spin. Since a genuine Hopf 16π² does exist in electron spin, the natural entrance for any future Hopf-based formulation is spin and the Berry phase, not the mass representation.
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). All external physics is standard (Bertrand 1873, Ehrenfest 1917, Hadamard 1923, Hopf 1931, BPST 1975, Wigner 1939). NO NEW THEOREM AND NO PREDICTION. What the second edition adds is an internal reconciliation, and every correction comes from later papers of this corpus (15, 46, 48, 116). What is not claimed: any new physical law (there is none) / closure as evidence for the specialness of 4π / the general claim that charge is a geometric quantity (withdrawn) / a single root uniting the loop factor with the characteristic-class normalisations / a decomposition of Gauss-Bonnet into (4π) to the n (the normalisation is (2π) to the n times two-to-the-n n factorial) / a derivation that space must be three-dimensional (it is a selection by consistency) / the existence of a single magic dimension (change the criteria and the special dimension changes). What remains: turning one’s own discipline on oneself is how one finds out what is left.
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 Structural Study of the $4\pi$ Patch [Paper 56]. Zenodo.
DOI: 10.5281/zenodo.21597500
ORCID: 0009-0006-8587-3450