The Speed of Light Is Near Three Hundred Million Because Length Is Decimal and Time Is Sexagesimal ── Perfecting the survey moves c further from three hundred million, and decimal time destroys the nearness ── the separator is whether the digits move when the units are changed ── [Paper 1055]

Abstract

The speed of light is 299792458 m/s. This paper asks what is recorded by a figure that falls 207542 short of 3x10^8. No new theorem and no new law is claimed. Scope of this paper (scope note): No new theorem and no new law is claimed──that the metre was designed from the meridian, that the speed of light has been a defined value since 1983, that the caesium transition frequency has been a defined value since 1967, that France enacted decimal time in 1793, and that the seconds pendulum was a candidate for the metre are all standard facts. The history is not retold──which survey party measured what and when, and how decimal time was administered, are not treated. What is used is only the ratio of a length and a time that can be computed today. No system of units is called better──decimal time is called neither better nor worse. Only that changing the base changes how the digits look. The subtracted number is not claimed to mean anything──the difference 207542 from 3x10^8 carries the dimension of m/s, and the reference 3x10^8 itself depends on the units. This difference cannot be pursued as a physical quantity. The fine-structure constant is not explained──why it is near 137 is not answered. That Eddington took it to be exactly 137, that it is in fact 137.035999177, and that it runs with energy are written only as cautions. The history of measurement is not judged──how accurate the 1975 value was is not asked. Only the order of its uncertainty is compared with the order of the gap. The agreement of g with the square of the circle constant is not called a shared root──what the two share is a fork in history, not one source. It is not a shared root in the sense of Paper 300. Relation to earlier papers: the scope note of Paper 915 said that the definition of the metre would not be discussed, and that the ratio of the meridian quadrant to ten million metres would merely be set down once──this paper takes that seat (no earlier occupant; the gravity item of the same scope note is occupied by Paper 1038). Paper 306 showed that the number of base units is a promise, and its scope note also said that no metrology would be built──here it is counted how that promise survives in the digits of a constant. Paper 314 showed that c is a defined value rather than a measurement──here it is seen in which measurement's digits that defined value froze. Paper 499 showed that not one of the four classical measurements of the speed of light measures one way──that paper is about how it was measured, this one about the digits of what was measured. Paper 1049 counted that the number of unit slots is a rank, and Paper 1054 showed that length and angle depend on the basis──this paper applies that question of what moves to the digits of a constant. Paper 95 separated convention from fact──here, within one and the same number, the digits fall on the side of convention and the ratio on the side of fact. Paper 300 showed that whether two things share a root is decidable──following that discipline, no root is inferred from numerical nearness alone. Paper 117 showed that scale invariance fixes an exponent in four ways──an exponent does not move with the units, but digits do. This paper sets the two in contrast. What is added is showing that perfecting the survey moves c further from 3x10^8, measuring that decimal time makes the roundness worse by a factor of 197, counting that caesium and the slowing rotation act at five parts in a hundred thousand of the gap, giving the ranking of roundness among twenty-five constants and the chance that it arises by accident, contrasting a dimensionless number that is rounder and does not move with the units, and dividing coincidences into three grades and showing the reversal in which the numerically better one has no cause. First, and this is the core. What makes it near three hundred million is a mixture of bases: length is divided decimally and time sexagesimally──under the decimal time that France enacted in 1793 (a day of ten hours, a hundred minutes, a hundred seconds) the decimal second is 0.864000 of the present second and the speed of light becomes 2.590207 times ten to the eighth. The relative distance from the mantissa to the nearest integer grows from 0.000691807 to 0.136597721, worse by a factor of 197.450715 (Section 5). Second, the Earth does matter, but in the opposite direction──the metre was designed as one ten-millionth of the meridian quadrant. Using the length 10001965.729 m obtained in Paper 915, the speed of light in the metre as designed is 299733538.5, and the gap from 3x10^8 widens from 207542 to 266461.5, a factor of 1.283892. The error of the survey was working towards three hundred million, not away from it (Section 3). Third, caesium contributes nothing to the figure──9192631770 was chosen to reproduce a second that already existed. Even if the Earth slowed until a day were 3 milliseconds longer, the speed of light would move by only 10.4 m/s, which is 5.016 times ten to the minus five of the gap (Section 4). Fourth, being round is not in itself remarkable──of twenty-five constants the mantissa of the speed of light is nearest to an integer, but if mantissas are log-uniform the chance that the roundest of them is this round is 0.119300 (two hundred thousand simulations) or 0.126781 (closed form). It happens one time in eight (Section 6). Fifth, a rounder number exists, and that one does not move with the units──the inverse fine-structure constant 137.035999177 lies at a relative distance 0.000262768 from 137, rounder than the speed of light by a factor of 2.6328. Under decimal time the mantissa of the speed of light moves from 2.997925 to 2.590207, while this dimensionless one does not move. The separator is whether the digits move when the units are changed (Section 7). Sixth, coincidences come in three grades, and there a reversal occurs──the agreement between g and the square of the circle constant, a fossil of a rejected definition (relative difference 0.006378614), is worse by a factor of 9.220227 than the mere accident of the speed of light and 3x10^8 (0.000691807). Yet only the former has a cause (Section 8). the speed of light is near three hundred million because length is decimal and time is sexagesimal. Under the decimal time of France in 1793 the speed of light becomes 2.590207 times ten to the eighth and the roundness worsens from 0.000691807 to 0.136597721, a factor of 197.450715──whereas improving the survey moves it by a factor of only 1.284. The Earth matters, but in the opposite direction──with the quadrant 10001965.729 m obtained in Paper 915, the speed of light in the metre as designed is 299733538.5 and the gap widens by a factor of 1.283892. To make it exactly three hundred million the quadrant would have to differ by 8.884 kilometres, 4.519338 times the actual error of the survey. Caesium contributes nothing to the figure──even with a day 3 milliseconds longer the speed of light moves by only 10.4 m/s, 5.016 times ten to the minus five of the gap. Being round is not in itself remarkable──the speed of light ranks first among 25 constants, but the chance of that by accident is 0.119300. A rounder number exists and does not move with the units──the inverse fine-structure constant lies 0.000262768 from 137, rounder by a factor of 2.6328, and its mantissa does not move under decimal time. And the three grades of coincidence show a reversal──the agreement of g with the square of the circle constant (0.006378614) is worse by a factor of 9.220227 than that of the speed of light with three hundred million (0.000691807), yet it holds exactly in the seconds-pendulum metre of 0.993621386 m. The cause belongs to the worse one. The separator is whether the digits move when the units are changed──if they move they record the history of the units, and if not they record the world. Plainly──the subtracted number 207542 has no identity (it is a difference carrying the dimension of m/s), and this paper does not answer why the fine-structure constant is near 137. What is added──showing that perfecting the survey moves the speed of light away from three hundred million, measuring that decimal time worsens the roundness by a factor of 197, counting that caesium and the slowing rotation act at five parts in a hundred thousand, giving the ranking of roundness and the chance of it arising by accident, contrasting a dimensionless number that is rounder and immobile, and dividing coincidences into three grades to show that the numerically better one has no cause. 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: speed of light, systems of units, definition of the metre, convention and fact, dimensionless quantities, multiple comparisons. ----- 光速は 299792458 m/s である。3x10^8 から 207542 だけ足りないこの数字が、何を記録しているのかを問う。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──メートルが子午線から設計されたこと、1983 年から光速が定義値であること、セシウムの遷移周波数が 1967 年から定義値であること、1793 年のフランスが十進時間を定めたこと、秒振り子がメートルの候補であったことは、いずれも標準的な事実である。歴史を語り直さない──どの測量隊がいつ何を測ったかも、十進時間がどう運用されたかも扱わない。使うのは、いま計算できる長さと時間の比だけである。どの単位系が優れているとも言わない──十進時間のほうがよいとも、悪いとも書かない。基数を替えると桁の見え方が変わることだけを示す。引いた数に意味があるとは言わない──3x10^8 との差 207542 は m/s の次元を持ち、基準に置いた 3x10^8 自体が単位に依る。この差を物理量として追いかけることはできない。微細構造定数を説明しない──なぜ 137 に近いのかには答えない。エディントンがちょうど 137 だと考えたこと、実際は 137.035999177 であること、さらにエネルギーとともに走ることを、注意として書くだけである。測定の歴史を裁かない──1975 年の値がどれだけ正確だったかは問わない。不確かさの桁と隔たりの桁を比べるだけである。 g と円周率の二乗の一致を同根とは呼ばない──二つが共有しているのは歴史の分岐点であって、一つの源ではない。論文300 の意味での同根ではない。既刊との関係:論文915 の射程注記は、メートルの定義を論じないと書き、子午線の四分円と千万 m の比を一つ並べるだけだとした──本稿はその席に座る(先客なし。同じ射程注記の重力の項には論文1038 が座っている)。論文306 は基本単位の個数が約束であることを示し、その射程注記も計量学を作らないと書いた──ここでは、その約束が定数の桁にどう残るかを数える。論文314 は c が測定結果ではなく定義値であることを示した──ここでは、その定義値がどの測定の桁で凍ったかを見る