The Fine-Structure Constant Does Not Move with the Units, but It Moves with Energy ── A factor of 4π sits in its formula and leaves no trace in the value, and whether a formula in π gives 137 is decided by how loose a tolerance is allowed ── the separator is whether the number moves when the units change or when the energy changes ── [Paper 1056]

Abstract

Paper 1055 divided numbers by whether they move when the units are changed──those that move record the history of the units, and those that do not record the world. This paper measures that the fine-structure constant falls into neither box. No new theorem and no new law is claimed. Scope of this paper (scope note): No new theorem and no new law is claimed──the definition of the fine-structure constant, that SI and Gaussian units place the rationalising factor of 4π differently, that a coupling runs with energy through vacuum polarisation, and the form of the one-loop leptonic contribution are all standard. Why it is near 137 is not answered──no attempt is made to derive the inverse from theory. The road that Paper 101 treated as Eddington's slip is not entered. It is not claimed that 4π yields 137──the measurement points the other way: the factors of 4π this corpus places do not reach the constant at all. Wyler's expression is not called a derivation──its closeness is granted, but as Section 4 shows, that closeness is a function of how many forms of expression are permitted. The hadronic contribution is not computed here──the value 0.02766 is taken from the literature. Only the one-loop leptonic part is computed here, and that too is an approximation. Variation over cosmic time is not treated──whether the constant changed with epoch is the subject of Paper 314; what is measured here is the running with energy scale alone. The two are different questions. No renormalisation group is built──the general theory of running is not entered; one number for one coupling is produced. Eddington is not judged──this paper too, on the same day, wrote Wyler's expression from memory and got it wrong. A caution is not aimed only at others. Relation to earlier papers: the scope note of Paper 1055 said that the fine-structure constant would not be explained, and that its running with energy would be written only as a caution──this paper takes that seat (no earlier occupant). The running set down there as a caution is measured here. Paper 314 showed that in any system of units the constant is 1/137.035999 and cannot be moved by redefining units──here that immobility is confirmed by actually changing the scales, and another axis along which it does move is added. Paper 101 treated as a caution that Eddington derived 136 and added to it once the measurement was known to be near 137──here that construction is actually computed, and an error of the same kind is recorded of the author. Paper 17 showed that 8π is an artefact of the units and fixed the corpus's unit budget──here it is counted that the same budget does not reach the constant. Paper 10 established that only dimensionless numbers can be fine-tuned──this paper shows that such a number can still move along another axis. Paper 15 computed the inverse independently──its digits are used here. Paper 44 showed that the power of 4π is the mass dimension──the factor here differs: it leaves neither dimension nor value. Paper 110 showed that Ricci flow is a renormalization group flow──that paper is the running of geometry, this one the running of a coupling. Paper 300 showed that whether two things share a root is decidable──following that discipline, no root is inferred from numerical nearness alone. What is added is measuring the constant along the two axes of units and of energy, naming the place inside this corpus where a factor of 4π enters a formula and leaves no trace, listing the distances from the corpus's factors of 4π to 137, counting how easily a formula in π hits 137, across tolerances and across sizes of the space of formulas, against controls, and showing that the separator of Paper 1055 needs a third box. First, and this is the core. The constant does not move with the units but moves with energy──scaling mass by 10^7, length by 10^-9 and time by 10^3 leaves the spread in its inverse at a relative 2.074 times ten to the minus sixteen (one spacing of double precision), whereas a one-loop vacuum polarisation gives 128.939785 at the mass of the Z boson, a shift of 8.096214 from 137.035999177, or 0.059081 in relative terms (Sections 2 and 6). Second, a factor of 4π sits in the formula and leaves no trace in the value──the SI form and the Gaussian form, which has no such factor, agree to a relative 0.000 times ten to the zero, not differing in a single digit of double precision. As a control, removing only that factor from the SI form shifts the value by exactly 12.566370614 (Section 2). Third, the factors of 4π this corpus places do not reach it──however the unit budget fixed in Paper 17 is moved, the inverse does not change, and of the numbers built from 4π that the corpus uses, the nearest is 124.025107, still 9.4945 per cent away. Dividing by powers of π does not approach an integer either, the smallest relative distance being 0.008239 (Section 3). Fourth, whether a formula in π gives 137 is decided by how loose a tolerance is allowed──among 10769 distinct expressions built from fractions and powers of π, none falls within the present precision (a relative 0.000262768), while 81 do once the tolerance is loosened to 0.05. For forty targets of the same magnitude the averages are 0.3750 and 77.2000, so 137 behaves exactly like the controls at every tolerance (Section 4). Fifth, a close formula exists, but its closeness is a function of how wide a space of formulas is permitted──Wyler's expression comes within a relative 6.077 times ten to the minus seven, yet the inverse is measured to ten to the minus ten, so it remains about four thousand times away. Widening the space from 2184 to 52872 expressions raises the hits from 0 to 1, and the control average then stands at 1.0750 (Sections 4 and 5). Sixth, the separator is whether the number moves when the units change or when the energy changes──the proton to electron mass ratio 1836.152673 moves under neither, the digits of the speed of light move under the units, and this constant moves under energy alone. A third box is needed (Section 7). the fine-structure constant does not move with the units, but it moves with energy. Scaling mass by 10^7, length by 10^-9 and time by 10^3 leaves the spread in its inverse at a relative 2.074 times ten to the minus sixteen (one spacing of double precision), while a one-loop vacuum polarisation gives 128.939785 at the mass of the Z boson, a shift of 8.096214, or 0.059081 relative. A factor of 4π sits in the formula and leaves no trace──the SI and Gaussian forms agree to a relative 0.000 times ten to the zero, not differing in a single digit of double precision, and removing that factor alone shifts the value by exactly 12.566370614. The factors of 4π the corpus places do not reach it either──the nearest, 4π^3, is 124.025107, still 9.4945 per cent away, and the smallest relative distance from dividing by powers of π is 0.008239. Whether a formula in π gives 137 is decided by the tolerance──of 10769 expressions none falls within the present precision 0.000262768 and 81 do at 0.05, against control averages of 0.3750 and 77.2000. Wyler's expression is close but is not a derivation──it comes within a relative 6.077 times ten to the minus seven, yet remains about four thousand times farther than the precision of measurement, and widening the space from 2184 to 52872 raises the hits from 0 to 1 while the control average becomes 1.0750. The separator is whether the number moves when the units change or when the energy changes──the proton to electron mass ratio 1836.152673 moves under neither, the digits of the speed of light move under the units, and this constant moves under energy alone. Plainly──why it is near 137 is not answered here either, the hadronic contribution 0.02766 is taken from the literature, and while writing this paper the author set down Wyler's expression from memory and got it wrong (87.239880, a relative 0.363380 away). What is added──measuring the constant along the two axes of units and of energy, naming the place in this corpus where a factor of 4π enters a formula and leaves no trace, listing the distances from the corpus's factors of 4π to 137, counting how easily a formula in π hits it across tolerances and sizes of space against controls, and showing that the separator of Paper 1055 needs a third box. 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. Keywords: fine-structure constant, dimensionless quantities, systems of units, vacuum polarisation, renormalization group, multiple comparisons. ----- 論文1055 は、数字を「単位を替えると動くか」で二つに分けた──動けば単位の履歴の記録、動かなければ世界の記録である。本稿は、微細構造定数がそのどちらの箱にも入らないことを測る。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──微細構造定数の定義、SI とガウス単位系で有理化の 4π の置き場が違うこと、結合定数が真空偏極でエネルギーとともに走ること、一ループのレプトンの寄与の形は、いずれも標準的である。なぜ 137 なのかには答えない──本稿は 1/α を理論から出そうとしない。論文101 がエディントンの滑落として扱った道には入らない。4π から 137 が出るとは言わない──測った結果はむしろ逆で、体系が置いた 4π は α に一切届かない。ワイラーの式を導出とは呼ばない──近さは認めるが、第4節が示すとおり、その近さは許した式の形の数の関数である。ハドロンの寄与は自前で計算していない──0.02766 は文献から借りた数である。自前で計算したのは一ループのレプトンの分だけで、それも近似である。宇宙時間での変化は扱わない──α が時代とともに変わったかは論文314 の主題であり、本稿が測るのはエネルギースケールでの走りだけである。二つは別の問いである。くりこみ群を作らない──走りの一般論には立ち入らず、一つの結合定数の一つの数を出すだけである。エディントンを裁かない──本稿も同じ日に、ワイラーの式を記憶で書いて外した。戒めは他人にだけ向けない。既刊との関係:論文1055 の射程注記は、微細構造定数を説明しないと書き、エネルギーとともに走ることを注意として書くだけだとした──本稿はその席に座る(先客なし)。注意として置かれた走りを、ここで測る。論文314 は、どんな単位系を取っても α は 1/137.035999 であり単位の再定義で動かせないと示した──本稿はその「動かない」を目盛りを実際に替えて確かめ、動く向きがほかに在ることを足す。論文101 はエディントンが 1/α を 136 と導き、測定が 137 と分かってから足したことを戒めとして扱った──ここではその構成を実際に計算し、同じ型の誤りを自分でも記録する。論文17 は 8π が単位の産物であることを示し、体系の単位の予算を確定した──ここでは同じ予算が α には届かないことを数える。論文10 は調整できるのが無次元量だけであることを確立した──本稿はその無次元量が、別の軸では動きうることを示す。論文15 は 1/α を自前で計算した──本稿はその桁を使う。論文44 は 4π の冪が質量次元であることを示した──α の 4π はそれと違い、次元も値も残さない。論文110 はリッチ・フローがくりこみ群フローであることを示した──