There Are Two Kinds of Exactness: One Is Set by the Units, the Other Is Unmoved by Continuous Deformation ── The quantum Hall resistance is the product of the two, and the fine-structure constant can be neither, so it runs ── the separator is whether a quantity has a dimension, is an integer, or is neither ── [Paper 1058]

Abstract

Paper 1056 and Paper 1057 measured that the fine-structure constant does not move with the units but moves with energy, and why it stops at low energy. Why, then, is there a quantity built from the same charge and the same Planck constant that does not move at all? This paper answers that. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──the quantisation of the Hall resistance, the quantisation of flux, the integrality of the Chern number and the 2019 revision of the units are all standard. It is not claimed that topology froze the value──what froze the value of the von Klitzing constant was the decision of 2019, not topology. What topology freezes is the integer. No index theorem is derived──no proof that the Chern number is an integer is given. Here it is only observed numerically that a plain integral approaches one. No mechanism of the quantum Hall effect is derived──why it quantises, and how edge states or disorder act, are not the subject. The fractional case is not treated──only the integer case is examined. Experimental accuracy is not discussed──how closely real measurements agree is not the subject; only what enters the formula is asked. Why it is 137 is not answered──following the caution of Paper 101 and Paper 1056, no explanation of the value is attempted. Relation to earlier papers: the scope note of Paper 514 says that no index theorem is derived, that the integrality of the invariant belongs to the territory of the paper numbered 70, and that it only borrows──this paper takes that seat (no earlier occupant). The integer side that was only borrowed is computed here, and shown to be exact in a different sense from the unit side. Paper 514 showed that no property of the sample enters the quantised resistance, and listed the values divided by integers──this paper does not produce that list anew but rereads it as a product of two exactnesses. Paper 270 showed that the two in the flux quantum is a count of carriers and that the 2019 revision made its value exact──here what the same revision gave up is added. Paper 10 established that only dimensionless numbers can be fine-tuned──this paper uses the converse: only a quantity with a dimension can be fixed by definition. Paper 1057 measured that in 2019 one factor of 4π ceased to be exact──here that is identified as the price of the trade. Paper 1056 showed that the fine-structure constant does not move with the units but moves with energy──this paper adds, on the side that does not move, the reason it cannot be moved. Paper 1055 divided numbers by whether they move when the units change──here the side that does not move is shown to split further into two. Paper 1049 showed that fixing an independent constant removes exactly one unit slot──the 2019 revision is an instance of that subtraction. Paper 300 showed that whether two things share a root is decidable──following that discipline, the two kinds of exactness are not called one root. What is added: separating the word exact into two different things, writing the 2019 revision as a trade in numbers, confirming by actually moving the parameter that topological exactness is exactness under continuous deformation, showing by mesh convergence that the departure from an integer is numerical error, and writing the separator that the fine-structure constant can be exact in neither sense. First, and this is the core. The word exact is doing two different jobs──one is that a definition of the units has fixed the value, the other is that continuous deformation does not move it. The first is available only to quantities with a dimension, the second happens only to integers (Sections 2 and 5). Second, a quantity with a dimension can be made exact by definition──the von Klitzing constant is 25812.807459 ohms, its inverse 3.874045865×10⁻⁵ siemens, the flux quantum 2.067833848×10⁻¹⁵ weber, all from defined values alone. Moving the fine-structure constant by a relative 1×10⁻⁶ changes it by 0.000×10⁰ (Section 2). Third, the 2019 revision was a trade──the permeability of vacuum is recovered from the fine-structure constant to a relative 5.718×10⁻¹², and moving that constant by a relative 1×10⁻⁶ moves it by 1.000×10⁻⁶. The departure from 4π×10⁻⁷ is 5.444×10⁻¹⁰ with the CODATA 2018 recommended value and 1.320×10⁻¹⁰ (0.825 times the uncertainty) with CODATA 2022, moving, even in sign, each time the recommended values are revised──that is the price of the trade (Section 3). Fourth, a dimensionless quantity cannot be made exact by definition──across five choices of scale the spread of the inverse fine-structure constant is 0.000×10⁰, while the control, the von Klitzing constant, has a spread of ratios of 1.000×10¹⁴ (Section 4). Fifth, topological exactness is not the fixing of a value──computing the Chern number of a two-band lattice model by plain integration gives -0.999891028 and the like, and the departure from an integer shrinks as 4.358×10⁻⁴, 1.090×10⁻⁴, 2.724×10⁻⁵ for meshes of 200, 400 and 800. Refining by a factor of four shrinks it by 15.996, so it is numerical error. Meanwhile moving the parameter continuously does not move the integer (Section 5). Sixth, the fine-structure constant can be exact in neither sense──being dimensionless it cannot be fixed by definition, and not being an integer it is not protected by topology. Its departure from the nearest integer, 0.035999178, is 1.6419×10⁶ times its own measurement uncertainty and does not shrink with finer computation (Section 7). Closing: there are two kinds of exactness, one set by the units and one unmoved by continuous deformation. A quantity with a dimension can be made exact by definition──the von Klitzing constant is 25812.807459 ohms and the flux quantum 2.067833848×10⁻¹⁵ weber, both from defined values alone. The 2019 revision was a trade──the permeability of vacuum is recovered from the fine-structure constant to a relative 5.718×10⁻¹², moving by 1.000×10⁻⁶ when that constant moves by a relative 1×10⁻⁶, and the departure from 4π×10⁻⁷, which moved with the revision of the recommended values from 5.444×10⁻¹⁰ (CODATA 2018) to 1.320×10⁻¹⁰ (CODATA 2022, 0.825 times the uncertainty), is the price. A dimensionless quantity cannot be made exact by definition──across five scales the spread of the inverse fine-structure constant is 0.000×10⁰ against a control spread of 1.000×10¹⁴. Topological exactness is not the fixing of a value──the Chern number comes out as -0.999891028 and the like, but the departure is numerical error shrinking from 4.358×10⁻⁴ to 2.724×10⁻⁵ by a factor of 15.996, while continuous motion of the parameter leaves the integer unmoved (spread 1.882×10⁻³). The measured quantum Hall resistance is the product of the two──the list divided by integers that Paper 514 produced is reread as a product of exactness by definition and exactness by topology. The fine-structure constant can be neither──dimensionless, so not definable, and with a departure from an integer of 0.035999178 that is 1.6419×10⁶ times its own uncertainty, so not protected. Stated plainly──the proposal that arrived had the order reversed, but it was the proposal that made visible that two exactnesses share one formula. The separator──whether the quantity has a dimension, is an integer, or is neither. With a dimension it can be fixed by definition, as an integer by topology, and a quantity that is neither can take no road at all. 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, Chern number, quantum Hall effect, definition of the units, topological invariant, dimensionless quantity, permeability of vacuum. ----- 論文1056 と論文1057 は、微細構造定数が単位では動かずエネルギーで動くこと、そして低いエネルギーで止まる理由を測った。では、同じ電荷とプランク定数からできていながら、まったく動かない量があるのはなぜか。本稿はその問いに答える。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──量子ホール効果の量子化、磁束の量子化、チャーン数が整数であること、2019 年の単位の改定は、いずれも標準的である。トポロジーが値を凍らせたとは書かない──フォン・クリッツィング定数の値を凍らせたのは 2019 年の決定であって、位相ではない。位相が凍らせているのは整数のほうである。指数定理を導かない──チャーン数が整数になることの証明はしない。ここでは素の積分が整数に寄ることを数で見るだけである。量子ホールの機構を導かない──なぜ量子化するのか、端状態や不純物がどう効くかは主題ではない。分数の場合を扱わない──整数の場合だけを見る。実験の精度を論じない──実際の測定でどこまで一致するかは主題ではなく、式に何が入っているかだけを問う。なぜ 137 なのかには答えない──論文101 と論文1056 の戒めに従い、値の説明には入らない。既刊との関係:論文514 の射程注記は、指数定理を導かないと書き、整数が位相不変量であることは番号 70 の編が扱う領域であり本稿は借りるにとどめるとした──本稿はその席に座る(先客なし)。借りるにとどめられた整数の側を実際に計算し、単位の側の厳密とは別種であることを示す。論文514 は、量子化された抵抗の式に試料の量が一つも入らないことを示し、整数で割った列も出した──本稿はその列を新しく出さず、二つの厳密の積として読み直す。論文270 は、磁束量子の 2 が担い手の個数であり、2019 年の改定でその値が厳密になったことを示した──ここでは、同じ改定が何を手放したかを足す。論文10 は、調整を語れるのが無次元量だけであることを確立した──本稿はその裏を使う。定義で固定できるのは次元を持つ量だけである。論文1057 は、2019 年に一つの 4π が厳密をやめたことを測った──ここではそれが取引の代金であったことを書く。論文1056 は、微細構造定数が単位では動かずエネルギーで動くことを示した──本稿は「単位では動かない」の側に、動かせない理由を足す。論文1055 は、数字が単位を替えて動くかで二つに分けた──ここでは「動かない」側がさらに二つに割れることを示す。論文1049 は、独立な定数を固定するたびに単位の枠が一つ減ることを示した──2019 年の改定はその引き算の実例である。論文300 は、同根か別根かは判定できると示した──その規律に従い、二つの厳密を同根とは呼ばない。加えたのは、「厳密」が二つの別のことを指していると分けたこと、2019 年の改定を取引として数で書いたこと、位相の側の厳密が連続変形に対するものであることを実際に動かして確かめたこと、整数からの隔たりが数値誤差であることを刻みの収束で示したこと、微細構造定数がどちらの意味でも厳密になれないと分離子を書いたことである。 第一に、これが本稿の芯である。「厳密」という語が二つの別のことを指している──片方は単位の定義が値を決めたことであり、片方は連続変形で動かないことである。前者は次元を持つ量にしかできず、後者は整数にしか起きない(第2節・第5節)。 第二に、次元を持つ量は、定義で厳密にできる──フォン・クリッツィング定数は 25812.807459 オーム、その逆数は 3.874045865×10⁻⁵ ジーメンス、磁束量子は 2.067833848×10⁻¹⁵ ウェーバで、どれも定義値だけから出る。微細構造定数を相対 1×10⁻⁶ 動かしても、変化は 0.000×10⁰ である(第2節)。 第三に、2019 年の改定は取引だった──真空の透磁率は微細構造定数から相対 5.718×10⁻¹² で再現し、微細構造定数を相対 1×10⁻⁶ 動かすと 1.000×10⁻⁶ だけ動く。4π×10⁻⁷ からのずれは、CODATA 2018 の推奨値で 5.444×10⁻¹⁰、CODATA 2022 で 1.320×10⁻¹⁰(不確かさの 0.825 倍)と、推奨値が改められるたびに向きまで変えて動く──それがその代金である(第3節)。 第四に、無次元量は、定義では厳密にできない──目盛りを五通り替えても微細構造定数の逆数の散らばりは 0.000×10⁰ で、対照のフォン・クリッツィング定数の数値は比の散らばりが 1.000×10¹⁴ になる(第4節)。 第五に、位相の「厳密」は、値が定まっていることではない──二本バンドの格子模型のチャーン数を素の積分で