What Freezes 137 Is Not a Number but a Single Mass ── The screening stops because no charge is lighter than the electron, and the fall is quadratic rather than logarithmic ── the separator is whether the screening is made by one particle or by a pair ── [Paper 1057]

Abstract

Paper 1056 measured that the fine-structure constant runs with energy and set aside why it stops at low energy. This paper measures the shape of that stopping. 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 form of the one-loop vacuum polarisation, that heavy fields decouple at low energy, Uehling’s correction to the Coulomb law, Debye and Hückel screening, and the 2019 revision of the units are all standard. Why it is 137 is not answered──what is answered here is only why it freezes. The two are different questions, and answering the first leaves the second standing. For 4π and the knot of charge, no measurable content was found──Section 8 says so. Only three things could be measured, and none of them corresponds to a knot. Nothing beyond one loop──higher corrections and effects on bound states are not treated. The approximation is stated. Debye screening and vacuum polarisation are not called a shared root──what they share is only that screening needs a length; the exponents come from different places. This is not a shared root in the sense of Paper 300. The revision of the units is not called an error──the permeability ceased to be 4π times ten to the minus seven because the definition changed, not because the old value was wrong. Neither hadrons nor bound states are included──only the three leptons are treated. Relation to earlier papers: the scope note of Paper 1056 said that no renormalisation group would be built, that the general theory of running would not be entered and that one number for one coupling would be produced──this paper takes that seat (no earlier occupant). From where 1056 stopped, it goes on to the mechanism by which the running stops. Paper 1055 showed that numbers divide by whether they move when the units change──here it is shown that a number on the immobile side also has a floor, and that a mass makes it. Paper 1049 counted that each independent constant fixed removes one unit slot──this paper confirms that the subtraction never touches a dimensionless quantity. Paper 17 showed that 8π is an artefact of the units and Paper 44 that the power of 4π is the mass dimension──here it is counted that one of those factors acquired a date. Paper 804 treated how far oil drops establish the quantisation of charge──that the elementary charge became a defined value in 2019 is taken here as given. Paper 22 treated a plasma equation using the Debye length──here that screening and the screening of the vacuum are compared by their far-field exponents. Paper 300 showed that whether two things share a root is decidable──the two kinds of screening are separated by that discipline. Paper 101 treated numerology as a caution and Paper 314 showed that the constant cannot be moved by redefining units──both are the ground this paper stands on. What is added is measuring that the fall which stops the screening is quadratic, showing by moving the mass that the floor is made by a mass and not by a number, counting that almost all of that floor is made by one species, measuring that the far-field exponents of the two screenings differ by a factor of two and placing the separator at one particle versus a pair, and counting the amount by which one factor of 4π ceased to be exact. First, and this is the core. The fall is quadratic, not logarithmic──computing the one-loop vacuum polarisation below the electron mass, as the momentum falls from one tenth to one thousandth of that mass the screening falls from 1.546890 times ten to the minus six to 1.548546 times ten to the minus ten, a log-log slope of 1.999767585. A logarithm keeps moving; a square stops (Section 2). Second, what sets the floor is not a number but the mass of the lightest charged particle──at the same momentum, making the mass a thousand times smaller raises the screening from 1.548530 times ten to the minus eight to 2.319313 times ten to the minus three, a factor of 149775.1686, and nothing freezes (Section 3). Third, almost all of that floor is made by the electron alone──at a momentum equal to the electron mass the electron accounts for 0.999974105 of the total, the muon for 0.000025804 and the tau for 0.000000091 (Section 4). Fourth, the two kinds of screening differ by a factor of two in the exponent──the far-field exponent of the vacuum polarisation approaches 2.029698057 while the Yukawa control approaches 1.020273, a ratio of 1.989367110. Debye screening rearranges charges one at a time, while the vacuum can only make them in pairs (Section 5). Fifth, in 2019 one factor of 4π ceased to be exact──against the permeability 1.256637061435917 times ten to the minus six, exact by definition, the present measured value is 1.256637062120000 times ten to the minus six, a relative difference of 5.444 times ten to the minus ten, which is 3.629 times the uncertainty (Section 6). Sixth, fixing three constants exactly still does not reach the constant──the speed of light, the reduced Planck constant and the elementary charge are independent and remove three slots, yet the dimension vector of the constant is zero and does not move. And why it freezes and why it is 137 are different questions; only the former is answered here (Sections 7 and 8). what freezes 137 is not a number but a single mass. Computing the one-loop vacuum polarisation below the electron mass, the screening falls from 1.546890 times ten to the minus six to 1.548546 times ten to the minus ten with a log-log slope of 1.999767585──a logarithm keeps moving, a square stops. What sets the floor is a mass──at the same momentum, a mass one thousand times smaller raises the screening by a factor of 149775.1686 and nothing freezes. Almost all of that floor is made by one species──the electron accounts for 0.999974105, against 0.000025804 for the muon and 0.000000091 for the tau. The two kinds of screening differ by a factor of two in the exponent──the vacuum polarisation approaches 2.029698057 and the Yukawa form 1.020273, a ratio of 1.989367110. The separator is whether the screening is made by one particle or by a pair. In 2019 one factor of 4π ceased to be exact──the permeability moved from 1.256637061435917 times ten to the minus six to 1.256637062120000 times ten to the minus six, a relative difference of 5.444 times ten to the minus ten, or 3.629 times the uncertainty. Fixing three constants still does not reach it──three slots go, but the dimension vector is zero; instead, measuring the permittivity of the vacuum has become the same measurement (a relative difference of 3.011 times ten to the minus twelve). Plainly──why it is 137 is not answered here either. Answering why it freezes leaves that question standing. And no measurable quantity in this paper corresponds to a knot of 4π and charge. What is added──measuring that the fall which stops the screening is quadratic, showing by moving the mass that the floor is made by a mass and not by a number, counting that almost all of it is made by one species, measuring that the far-field exponents of the two screenings differ by a factor of two and placing the separator at one particle versus a pair, and counting the amount by which one factor of 4π ceased to be exact. 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: vacuum polarisation, decoupling, screening, fine-structure constant, Compton wavelength, revision of the units. ----- 論文1056 は、微細構造定数がエネルギーで走ることを測り、なぜ低いところで止まるのかは外した。本稿はその「止まる」の形を測る。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──一ループの真空偏極の形、重い場が低いエネルギーで効かなくなること、ウーリングによるクーロン則の補正、デバイとヒュッケルの遮蔽、2019 年の単位の改定は、いずれも標準的である。なぜ 137 なのかには答えない──本稿が答えるのは「なぜ凍るか」だけである。二つは別の問いであり、前者に答えても後者は残る。4π と電荷の結び目については、測れる中身が見つからなかった──第8節にそう書く。測れたのは三つだけで、「結び目」に対応する量は本稿には無い。一ループより先へは行かない──高次の補正も、束縛状態への効果も扱わない。近似であることを明記する。デバイ遮蔽と真空偏極を同根とは呼ばない──共有しているのは「遮蔽に長さが要る」ことだけで、指数の出どころが違う。論文300 の意味での同根ではない。単位の改定を誤りとは呼ばない──透磁率が 4π×10⁻⁷ でなくなったのは定義が変わったからであり、以前の値が間違っていたのではない。ハドロンも束縛状態も入れない──扱うのはレプトン三種だけである。既刊との関係:論文1056 の射程注記は、くりこみ群を作らないと書き、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だとした──本稿はその席に座る(先客なし)。1056 が止めたところから、走りが止まる仕組みへ進む。論文1055 は、数字が単位を替えて動くかどうかで分かれることを示した──ここでは、動かない側の数にも床があり、その床を作るのが質量だと示す。論文1049 は、独立な定数を一つ固定するたびに単位の枠が一つ減ることを数えた──本稿はその引き算が無次元量には一度も触れないことを確かめる。論文17 は 8π が単位の産物であることを示し、論文44 は 4π の冪が質量次元であることを示した──ここではその 4π の一つに日付が付いたことを数える。論文804 は油滴が電荷の量子化をどこまで示すかを扱った──その電気素量が 2019 年から定義値になったことを、本稿は前提として受け取る。論文22 はデバイ長を使うプラズマの式を扱った──ここではその遮蔽と真空の遮蔽を、遠方の指数で比べる。論文300 は同根か別根かは判定できると示した──その規律に従って二つの遮蔽を分ける。論文101 はエディントンの数合わせを戒めとして扱い、論文314 は α が単位の再定義で動かせないと示した──どちらも本稿の土台である。加えたのは、遮蔽が止まる落ち方が二乗だと測ったこと、床を作るのが数ではなく質量だと質量を動かして示したこと、その床のほとんどを電子一種が作っていると数えたこと、デバイと真空の遠方の指数が二倍違うと測り、分離子を一粒か対かに置いたこと、一つの 4π が厳密をやめた量を数えたことである。 第一に、これが本稿の芯である。落ち方は対数ではなく二乗である──一ループの真空偏極を電子質量より下まで計算すると、運動量を電子質量の十分の一から千分の一まで下げるあいだ、遮蔽の大きさは 1.546890×10⁻⁶ から 1.548546×10⁻¹⁰ へ落ち、両対数の傾きは 1.999767585 である。対数なら動き続け、二乗なら止まる(第2節)。 第二に、床を決めているのは数ではなく、いちばん軽い荷電粒子の質量である──同じ運動量で質量だけを 1000 分の 1 にすると、遮蔽は 1.548530×10⁻⁸ から 2.319313×10⁻³ へ、149775.1686 倍に増えて凍らない(第3節)。 第三に、その床を作っているのはほぼ電子だけである──電子質量の運動量で、電子の寄与が全体の 0.999974105 を占め、ミューオンは 0.000025804、タウは 0.000000091 しかない(第4節)。 第四に、遮蔽の切れ方は、デバイと真空で指数が二倍違う──真空偏極の遠方の指数は 2.029698057 に寄り、対照の湯川型は 1.020273 に寄る。比は 1.989367110 である。デバイは電荷が一粒ずつ並び替わるので一つぶん、真空は対でしか作れないので二つぶんになる(第5節)。 第五に、2019 年に、一つの 4π が「厳密」をやめた──定義により厳密だった透磁率 1.256637061435917×10⁻⁶ に対し、いまの測定値は 1.256637062120000×10⁻⁶ で、相対差 5.444×10⁻¹⁰ は不確かさの 3.629 倍である(第6節)。 第六に、定数を三つ厳密にしても、α には届かない──光速と換算プランク定数と電気素量は独立で枠を三つ消すが、α の次元ベクトルはゼロなので動かない。そして「なぜ凍るか」と「なぜ 137 か」は別の問いであり、本稿が答えるのは前者だけである(第7節・第8節)。 137 を凍らせているのは数ではなく、一つの質量である。一ループの真空偏極を電子質量より下まで計算すると、遮蔽は 1.546890×10⁻⁶ から 1.548546×10⁻¹⁰ へ落ち、両対数の傾きは 1.999767585 である──対数なら動き続け、二乗なら止まる。床を決めているのは質量である──同じ運動量で質量だけを 10