The Fringes Die and Come Back ── The Visibility of Interference Fringes Is the Fourier Transform of the Spectrum, and the Sodium Doublet Kills the Fringes Every 0.577 mm: a Length Measured on a Bench Tells the Splitting in the Spectrum ── [Paper 218]
They went out not because information was lost but because the information had moved over to the side of length. Paper 207 wrote that the fence around the Jones calculus is coherence, and did not treat the theory of it; this paper turns that fence into a quantity that can be measured. Splitting a beam in two, delaying one by a path difference and recombining, the visibility of the fringes is the difference of the greatest and least intensities over their sum; for two beams of equal intensity it equals the modulus of the degree of coherence, which is the Fourier transform of the spectrum, by the theorem of Wiener and Khinchin. The path difference over which the visibility dies is the coherence length. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED: the Wiener-Khinchin theorem, the van Cittert-Zernike theorem and Michelson's reading of spectra from visibility are all standard. No theorem is proved, the results being confirmed by numerical transform; quantum optics is not treated, neither photon correlations nor second-order coherence appearing; no measurement of a real source is claimed, the widths being representative values; and the general theory of partial coherence is not developed, only first-order coherence being treated. A note on a name. The corpus carries Wiener in thirteen papers and every one of them is the Wiener filter or the Wiener-Hopf method; the Wiener-Khinchin theorem appears here for the first time. And on the relation to Paper 214: there the radiance could not be raised by any means, whereas the coherence here can be raised, and what pays for it is light thrown away. First, the visibility is the Fourier transform of the spectrum. Transforming a Gaussian spectrum on a million points gives a visibility of one at zero path difference, 0.6413811693 at twenty micrometres, 0.0890953204 at fifty and 0.0002388201 at a hundred, differing from the closed form by 2.22 times ten to the minus sixteen, 3.95 times ten to the minus ten, 2.98 times ten to the minus ten and 2.76 times ten to the minus twelve. For a rectangular spectrum the values are 0.9354900075, 0.5045558077, 0.2162356772 and 0.0850403377 at ten, twenty, fifty and eighty micrometres, agreeing with the sinc function to about one part in a million. A Gaussian gives a Gaussian and a rectangle gives a sinc. Nothing was assumed; every number came out of the transform. The differences of order a millionth on the rectangular side come from putting a discontinuous spectrum on a finite grid. Second, real sources differ by eight orders. Setting the width in frequency against the coherence length, a stabilised laser at a megahertz gives three hundred metres, a helium-neon at one and a half gigahertz gives a fifth of a metre, a sodium lamp gives 588 micrometres, a filter ten nanometres wide gives 31.9 micrometres and white light gives 0.999 micrometres. Ordinary sources alone span eight orders. White light is in step for only two wavelengths, which is why fringes in white light cannot be seen at all unless the paths are matched. Third, this is the core. Putting in the two sodium lines, at 589.0 and 589.6 nanometres, a separation of 5.18 times ten to the eleventh hertz, and running the same transform, the visibility falls to nothing at path differences of 0.292297, 0.869397, 1.446497 and 2.023597 millimetres, against theoretical values of 0.289395, 0.868186, 1.446977 and 2.025767. The fringes die once and come back. The visibility does not fall away monotonically: it travels back and forth between nothing and one, because the spectrum has two lines. With one it does not return: neither the Gaussian nor the rectangle came back. Fourth, that spacing tells the splitting in the spectrum. The spacing between the deaths measures 0.577100 millimetres, differing by three parts in a thousand from the speed of light divided by the separation, which is 0.578791. That order can be read backwards: measure the spacing and the separation in frequency is the speed of light divided by it, and in wavelength the square of the wavelength divided by it. So the fine structure of a spectrum can be measured with a ruler and no spectrograph, which is what Michelson actually did. One single fact is lodged in two places, on the frequency side as the gap between two lines and on the length side as the spacing of the deaths. Asking which is the real information does not arise: they are carried into one another by the transform. Fifth, there is a second coherence. For a round source the degree of coherence across the beam falls as twice the first Bessel function over its argument, and from its first zero at 3.83170597 the coherence radius is 0.61 wavelengths over the angular radius. At 550 nanometres this gives 72.1 micrometres for the Sun, 74.2 for the Moon, 671 for a millimetre lamp at a metre and 67100 for the same lamp at a hundred metres. This is a different thing from the coherence above: that one is conjugate to frequency and is measured along the path, this one is conjugate to angle and is measured across the beam. One word, coherence, standing for two different Fourier pairs. Even sunlight is in step within seventy micrometres, so two slits placed close enough together will show fringes in sunlight; and separating them puts the fringes out, reading the angular size of the source from the separation at which they go being the stellar interferometer. Sixth, coherence can be bought and light pays. Narrowing a filter about 550 nanometres, a width of three hundred nanometres gives a coherence length of 1.01 micrometres with all the light through; a hundred nanometres gives 3.02 micrometres with a third; ten nanometres gives 30.2 micrometres with a thirtieth; and a tenth of a nanometre gives 3020 micrometres with a three-thousandth. The product is 1.008 times ten to the minus six in every case and does not move. The coherence length can be made three thousand times longer and the light through becomes three thousand times less. Set beside Paper 214 this tells: there the radiance could not be raised by any means, and here the coherence can. Being forbidden and having a price are two different things. The separator and the fence. What separates the senses of coherent is what the spread is conjugate to: temporal coherence is set by the width in frequency and measured along the path difference, spatial coherence is set by the angular size of the source and measured across the beam. Both are the same Fourier relation, differing only in which quantities are paired. The fence around the fringes are visible has two rails, that the path difference be within the coherence length and that the slit separation be within the spatial coherence width. Drop either and the fringes go, and they go differently: on the time side according to the shape of the spectrum, on the space side according to the shape of the source. And what the third check showed is that the going out is itself a measurement. Read the absence of fringes as a failure and nothing is gained; read where they go and out come the splitting of the spectrum and the angular size of the source. In Paper 207 the degree of polarization could not be written inside the Jones language; here that same unwritable quantity has acquired a scale in millimetres. Important (scope note): a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem or physical law, none is claimed; any proof of a theorem, the results being confirmed by numerical transform; quantum optics, neither photon correlations nor second-order coherence appearing; any measurement of a real source, the widths being representative values; the general theory of partial coherence, only first-order coherence being treated. What is added: obtaining the visibility as a numerical transform of the spectrum and setting it against two closed forms, tabulating the coherence lengths of five sources, measuring four places where the doublet kills the fringes, reading the frequency separation back out of their spacing, giving the spatial coherence radius for four sources, and measuring that the product of coherence bought and light lost does not move. What remains: draw two mirrors slowly apart and the fringes fade and go out. Stop there and the story ends at no longer visible. Draw them further and the fringes come back, and the spacing of their returns tells at once that the source has two lines in its spectrum and how far apart those lines are. They went out not because information was lost but because the information had moved over to the side of length. And the length over which light stays in step differs between sources by eight orders: the phrase coherent light says nothing at all until one settles over what length. 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. ----- 見えなくなったのは情報が失われたからではなく、情報が長さの側へ移っていたからである。 論文207 は「ジョーンズ計算を囲っている柵はコヒーレンスである」と書き、その理論には触れなかった。本稿はその柵を測れる量にする。二つに分けた光を路程差 d だけずらして重ねたときの縞の可視度を V=(I_max−I_min)/(I_max+I_min) とすると、強度の等しい二本では V はコヒーレンス度の絶対値に等しく、それはスペクトルのフーリエ変換である(ウィーナー=ヒンチン)。V が落ちきるまでの路程差をコヒーレンス長という。新しい数学定理も新しい物理法則も主張しない:ウィーナー=ヒンチンの定理、ファン・シッタート=ゼルニケの定理、マイケルソンによる可視度からの分光は、いずれも標準的である。⚠️ 定理を証明しない——数値変換で確かめるのであって、定理は引用する。⚠️ 量子光学を扱わない(光子相関も二次のコヒーレンスも出てこない)。⚠️ 実在の光源の測定値を主張しない——幅は代表的な値であり、測定ではない。⚠️ 部分コヒーレンスの一般論を展開しない(扱うのは一次のコヒーレンスだけ)。 ⭐ 名前について一つ。体系には「ウィーナー」が 13 編に現れるが、その全部がウィーナー・フィルタかウィーナー=ホップ法であり、ウィーナー=ヒンチンの定理は本稿が初出である。⭐⭐ また論文214 との関係を書いておくと、あちらでは輝度がどうやっても上げられなかった——本稿のコヒーレンスは上げられる。ただし光を捨てることで、である。 第一に、可視度はスペクトルのフーリエ変換である。中心 5×10¹⁴ Hz・標準偏差 2×10¹² Hz のガウス型スペクトルを 2²⁰ 点で数値変換すると、路程差 0 で 1.0000000000、20 μm で 0.6413811693、50 μm で 0.0890953204、100 μm で 0.0002388201 であり、閉じた形との差はそれぞれ 2.22×10⁻¹⁶、3.95×10⁻¹⁰、2.98×10
They went out not because information was lost but because the information had moved over to the side of length. Paper 207 wrote that the fence around the Jones calculus is coherence, and did not treat the theory of it; this paper turns that fence into a quantity that can be measured. Splitting a beam in two, delaying one by a path difference and recombining, the visibility of the fringes is the difference of the greatest and least intensities over their sum; for two beams of equal intensity it equals the modulus of the degree of coherence, which is the Fourier transform of the spectrum, by the theorem of Wiener and Khinchin. The path difference over which the visibility dies is the coherence length. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED: the Wiener-Khinchin theorem, the van Cittert-Zernike theorem and Michelson's reading of spectra from visibility are all standard. No theorem is proved, the results being confirmed by numerical transform; quantum optics is not treated, neither photon correlations nor second-order coherence appearing; no measurement of a real source is claimed, the widths being representative values; and the general theory of partial coherence is not developed, only first-order coherence being treated. A note on a name. The corpus carries Wiener in thirteen papers and every one of them is the Wiener filter or the Wiener-Hopf method; the Wiener-Khinchin theorem appears here for the first time. And on the relation to Paper 214: there the radiance could not be raised by any means, whereas the coherence here can be raised, and what pays for it is light thrown away. First, the visibility is the Fourier transform of the spectrum. Transforming a Gaussian spectrum on a million points gives a visibility of one at zero path difference, 0.6413811693 at twenty micrometres, 0.0890953204 at fifty and 0.0002388201 at a hundred, differing from the closed form by 2.22 times ten to the minus sixteen, 3.95 times ten to the minus ten, 2.98 times ten to the minus ten and 2.76 times ten to the minus twelve. For a rectangular spectrum the values are 0.9354900075, 0.5045558077, 0.2162356772 and 0.0850403377 at ten, twenty, fifty and eighty micrometres, agreeing with the sinc function to about one part in a million. A Gaussian gives a Gaussian and a rectangle gives a sinc. Nothing was assumed; every number came out of the transform. The differences of order a millionth on the rectangular side come from putting a discontinuous spectrum on a finite grid. Second, real sources differ by eight orders. Setting the width in frequency against the coherence length, a stabilised laser at a megahertz gives three hundred metres, a helium-neon at one and a half gigahertz gives a fifth of a metre, a sodium lamp gives 588 micrometres, a filter ten nanometres wide gives 31.9 micrometres and white light gives 0.999 micrometres. Ordinary sources alone span eight orders. White light is in step for only two wavelengths, which is why fringes in white light cannot be seen at all unless the paths are matched. Third, this is the core. Putting in the two sodium lines, at 589.0 and 589.6 nanometres, a separation of 5.18 times ten to the eleventh hertz, and running the same transform, the visibility falls to nothing at path differences of 0.292297, 0.869397, 1.446497 and 2.023597 millimetres, against theoretical values of 0.289395, 0.868186, 1.446977 and 2.025767. The fringes die once and come back. The visibility does not fall away monotonically: it travels back and forth between nothing and one, because the spectrum has two lines. With one it does not return: neither the Gaussian nor the rectangle came back. Fourth, that spacing tells the splitting in the spectrum. The spacing between the deaths measures 0.577100 millimetres, differing by three parts in a thousand from the speed of light divided by the separation, which is 0.578791. That order can be read backwards: measure the spacing and the separation in frequency is the speed of light divided by it, and in wavelength the square of the wavelength divided by it. So the fine structure of a spectrum can be measured with a ruler and no spectrograph, which is what Michelson actually did. One single fact is lodged in two places, on the frequency side as the gap between two lines and on the length side as the spacing of the deaths. Asking which is the real information does not arise: they are carried into one another by the transform. Fifth, there is a second coherence. For a round source the degree of coherence across the beam falls as twice the first Bessel function over its argument, and from its first zero at 3.83170597 the coherence radius is 0.61 wavelengths over the angular radius. At 550 nanometres this gives 72.1 micrometres for the Sun, 74.2 for the Moon, 671 for a millimetre lamp at a metre and 67100 for the same lamp at a hundred metres. This is a different thing from the coherence above: that one is conjugate to frequency and is measured along the path, this one is conjugate to angle and is measured across the beam. One word, coherence, standing for two different Fourier pairs. Even sunlight is in step within seventy micrometres, so two slits placed close enough together will show fringes in sunlight; and separating them puts the fringes out, reading the angular size of the source from the separation at which they go being the stellar interferometer. Sixth, coherence can be bought and light pays. Narrowing a filter about 550 nanometres, a width of three hundred nanometres gives a coherence length of 1.01 micrometres with all the light through; a hundred nanometres gives 3.02 micrometres with a third; ten nanometres gives 30.2 micrometres with a thirtieth; and a tenth of a nanometre gives 3020 micrometres with a three-thousandth. The product is 1.008 times ten to the minus six in every case and does not move. The coherence length can be made three thousand times longer and the light through becomes three thousand times less. Set beside Paper 214 this tells: there the radiance could not be raised by any means, and here the coherence can. Being forbidden and having a price are two different things. The separator and the fence. What separates the senses of coherent is what the spread is conjugate to: temporal coherence is set by the width in frequency and measured along the path difference, spatial coherence is set by the angular size of the source and measured across the beam. Both are the same Fourier relation, differing only in which quantities are paired. The fence around the fringes are visible has two rails, that the path difference be within the coherence length and that the slit separation be within the spatial coherence width. Drop either and the fringes go, and they go differently: on the time side according to the shape of the spectrum, on the space side according to the shape of the source. And what the third check showed is that the going out is itself a measurement. Read the absence of fringes as a failure and nothing is gained; read where they go and out come the splitting of the spectrum and the angular size of the source. In Paper 207 the degree of polarization could not be written inside the Jones language; here that same unwritable quantity has acquired a scale in millimetres. Important (scope note): a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem or physical law, none is claimed; any proof of a theorem, the results being confirmed by numerical transform; quantum optics, neither photon correlations nor second-order coherence appearing; any measurement of a real source, the widths being representative values; the general theory of partial coherence, only first-order coherence being treated. What is added: obtaining the visibility as a numerical transform of the spectrum and setting it against two closed forms, tabulating the coherence lengths of five sources, measuring four places where the doublet kills the fringes, reading the frequency separation back out of their spacing, giving the spatial coherence radius for four sources, and measuring that the product of coherence bought and light lost does not move. What remains: draw two mirrors slowly apart and the fringes fade and go out. Stop there and the story ends at no longer visible. Draw them further and the fringes come back, and the spacing of their returns tells at once that the source has two lines in its spectrum and how far apart those lines are. They went out not because information was lost but because the information had moved over to the side of length. And the length over which light stays in step differs between sources by eight orders: the phrase coherent light says nothing at all until one settles over what length. 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. ----- 見えなくなったのは情報が失われたからではなく、情報が長さの側へ移っていたからである。 論文207 は「ジョーンズ計算を囲っている柵はコヒーレンスである」と書き、その理論には触れなかった。本稿はその柵を測れる量にする。二つに分けた光を路程差 d だけずらして重ねたときの縞の可視度を V=(I_max−I_min)/(I_max+I_min) とすると、強度の等しい二本では V はコヒーレンス度の絶対値に等しく、それはスペクトルのフーリエ変換である(ウィーナー=ヒンチン)。V が落ちきるまでの路程差をコヒーレンス長という。新しい数学定理も新しい物理法則も主張しない:ウィーナー=ヒンチンの定理、ファン・シッタート=ゼルニケの定理、マイケルソンによる可視度からの分光は、いずれも標準的である。⚠️ 定理を証明しない——数値変換で確かめるのであって、定理は引用する。⚠️ 量子光学を扱わない(光子相関も二次のコヒーレンスも出てこない)。⚠️ 実在の光源の測定値を主張しない——幅は代表的な値であり、測定ではない。⚠️ 部分コヒーレンスの一般論を展開しない(扱うのは一次のコヒーレンスだけ)。 ⭐ 名前について一つ。体系には「ウィーナー」が 13 編に現れるが、その全部がウィーナー・フィルタかウィーナー=ホップ法であり、ウィーナー=ヒンチンの定理は本稿が初出である。⭐⭐ また論文214 との関係を書いておくと、あちらでは輝度がどうやっても上げられなかった——本稿のコヒーレンスは上げられる。ただし光を捨てることで、である。 第一に、可視度はスペクトルのフーリエ変換である。中心 5×10¹⁴ Hz・標準偏差 2×10¹² Hz のガウス型スペクトルを 2²⁰ 点で数値変換すると、路程差 0 で 1.0000000000、20 μm で 0.6413811693、50 μm で 0.0890953204、100 μm で 0.0002388201 であり、閉じた形との差はそれぞれ 2.22×10⁻¹⁶、3.95×10⁻¹⁰、2.98×10
Many years ago, a paper by Emil Wolf inspired one of us (MAF) to look more closely at the analytic properties of propagating and scattered optical fields. In 1962, Wolf discussed how Michelson’s interferometer could provide information about the energy distribution in the spectrum of a light beam from measurements of the visibility of interference fringes. A simple relationship exists between the visibility function V(τ) and the modulus of the complex degree of coherence, γ(τ). The methods of Hanbury Brown, and Twiss showed that |γ(τ)| could be determined in various ways, and the important question arose as to how one might recover spectral information, g(ν) from |γ(τ)|. Since V(τ) is proportional to the Fourier transform of g(ν), one might think that only the autocorrelation function of g(ν) might be found, without explicitly knowing the phase of γ(τ). How does one determine the missing phase? The deep insight used to address this question was to take into account the analytic properties from which a complete recovery of g(ν) became possible. It is the underlying analytic properties of the functions that we so routinely employ in modeling optical processes that can provide a deeper understanding of relationships between field parameters and information; this is what we address here.
Writing the equality is not an assertion but a way of putting a question: the form in which one asks whether this wall can be touched. And the answer cannot be read off the symbol. When a bound in physics is written down, at most and strictly less than are distinguished with care. And the symbol on the page does not settle whether the wall can be touched. This paper measures, for eleven bounds, whether the equality is reached. For a bound, the equality is reached when some configuration realises it, and not reached when the quantity comes arbitrarily close without ever taking it; in mathematics this is the standard distinction between a supremum and a maximum. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED: the Buchdahl bound, the Tsirelson bound, PR boxes, minimum-uncertainty states, the Landauer bound and the Chandrasekhar mass are all standard. No bound is derived, only whether the equality is reached being measured; no claim of completeness is made, eleven examples appearing; no general mathematics is offered; and no rule about notation is proposed. The relation to earlier papers first. Paper 16 showed that the equals sign has six distinct roots; this paper takes the inequality side, where one symbol points at two kinds of wall. Paper 21 wrote that whether something is an identity or a hypothesis depends on which theory one stands inside, and the core section here is the quantitative version of that. Paper 194 showed there is only one equality fence and Paper 160 counted three ceilings over one inequality; this paper sorts those ceilings by whether they are touched. First, some walls cannot be touched. Measuring the central pressure of a uniform-density interior Schwarzschild solution against the compactness gives 0.261204 times the rest energy density at 0.5, 5.10897 at 0.86, 166.666 at 0.888, 166667 at 0.888888 and 1.66667 times ten to the sixth at 0.8888888: it diverges. The configuration that would touch the Buchdahl bound of eight ninths is not a configuration at all. In form the bound is written as at most, and in physics it is strictly less than. Second, some walls can be. Measuring minimum-uncertainty gaussians of width ten to the minus eleven, minus ten and minus nine, the product is 5.272859085000000 times ten to the minus thirty-five in every case, and the ratio to half of the reduced Planck constant is 1.000000000000000. Equality holds exactly, to fifteen digits. The equality condition of Cauchy-Schwarz is that the two vectors be proportional, and states meeting it exist, not one but a whole family. For the arithmetic and geometric means, equality holds only when all the numbers agree: drawing two hundred thousand triples uniformly, equality occurred zero times. It is reached, and the set that reaches it has probability zero. Third, my own guess was wrong. A first draft said that the algebraic CHSH ceiling of four is forbidden by no-signalling. Building a PR box and measuring it gives a CHSH value of 4.000000000000 and a largest signalling of zero. The guess was wrong: a PR box signals not at all and reaches four exactly. What forbids four is quantum mechanics, and not relativity. Fourth, this is the core. The three ceilings were counted from each standpoint, the quantum maximum measuring 2.828427124746 against two root two. Standing inside local hidden variables, two is reached and both two root two and four are impossible. Standing inside quantum mechanics, two and two root two are reached and four is impossible. Standing inside any no-signalling theory, all three are reached. The number two root two is a wall that is touched, a wall out of reach, and not a wall at all. The number never changed: what changed is which theory one stands inside. This is the quantitative version of what Paper 21 wrote about identities and hypotheses. Fifth, some walls sit at an endpoint. Measuring the work of erasure at three hundred kelvin as the time is lengthened, against a floor of 2.870978885 times ten to the minus twenty-one joules, the excess is five hundred per cent at one unit of time, five per cent at a hundred, five hundredths of a per cent at ten thousand, and still five ten-thousandths of a per cent at a million. Every finite time lies strictly above. The floor is the limit of the set and not a member of it: written with at least, and the equality never occurs. Sixth, eleven bounds were counted. Reached: Heisenberg's half a reduced Planck constant, because a state meets the equality condition; the arithmetic-geometric mean inequality, at one point only, of probability zero; the Tsirelson bound, inside quantum mechanics; the CHSH value of two, inside local hidden variables; the CHSH value of four, inside a no-signalling theory, by a PR box; the Schwarzschild radius, the horizon being a real surface. Not reached: the CHSH value of four when one stands in quantum mechanics; the Buchdahl bound; the Landauer bound; the Chandrasekhar mass; the speed of light for a massive body. The reasons for not reaching fall into three: the extremiser requires a divergence, as for Buchdahl and the speed of light; the extremiser sits at an endpoint outside the set, as for Landauer; or something other than the inequality excludes it, as stability does for Chandrasekhar. There is only one reason for reaching: a configuration meeting the equality condition lies inside the permitted set. The separator and the fence. What separates the two kinds of at most is whether the extremiser lies inside the set. It is not the symbol. The fence around this quantity is bounded is over which set the bound is taken, and outside it the same number becomes reachable or unreachable, the same shape as this sequence is random in Paper 223 and volume is conserved in Paper 225. The difference between strictly less and at most is not a question of notation but a question of measurement. Writing the equality is not an assertion but a way of putting the question, the form in which one asks whether this wall can be touched. Paper 16 counted six distinct roots of the equals sign; what this paper counted is two kinds of inequality sign, and the sorting of the symbols is not finished. Important (scope note): a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem or law, none is claimed; the derivation of any bound, only the attainment of equality being measured; completeness of the classification, eleven examples appearing; any general mathematics, whether a supremum is a maximum being a standard distinction in analysis; any rule about notation. What is added: following the Buchdahl central pressure numerically into the limit, confirming minimum uncertainty to fifteen digits, computing the CHSH value and the signalling of a PR box, checking the equality of the means over two hundred thousand random draws, following the Landauer excess out to a million units of time, and tabulating eleven bounds by whether the equality is reached. What remains: when bounds are written, at most and strictly less than are chosen with care, and that choice does not say whether the wall can be touched. Buchdahl's eight ninths is written as at most and no star touches it, since trying makes the pressure at the centre infinite. Landauer's floor is written as at least and no finite time touches it: after a million times as long it is still five ten-thousandths of a per cent above. A gaussian, meanwhile, meets its floor exactly, to fifteen digits. And two root two holds three different standings depending on where one stands: out of reach inside local hidden variables, touched inside quantum mechanics, and not a wall at all inside a no-signalling theory. The number never moved. So writing the equality is not a claim but a way of putting a question, the form in which one asks whether this wall can be touched. And the answer cannot be read off the symbol: one has to build the extremiser and see whether it lies inside the set that was allowed. 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. ----- 等号を書くことは、主張ではなく問いの立て方である。「この壁に、触れられますか」と尋ねる形式が、等号なのである。そして答は、記号を見ても分からない。 物理の上限を書くとき、我々は「以下」と「未満」を書き分ける。だが紙の上の記号は、その壁に触れられるかを決めていない。本稿は十一の上限について、等号が到達されるかどうかを実際に測る。上限 X ≤ B について、B を実現する配置が存在するとき「到達される」といい、X が B にいくらでも近づけるのに B そのものを取れないとき「到達されない」という。数学では上限が最大値かどうかの区別であり、標準的である。新しい数学定理も新しい法則も主張しない:ブッフダール限界、ツィレルソン限界、PR 箱、最小不確定性状態、ランダウアー限界、チャンドラセカール質量は、いずれも標準的である。⚠️ 上限を導出しない——与えられた上限について等号が到達されるかだけを測る。⚠️ 網羅しない——扱うのは十一の例であり、分類が完全であることは主張しない。⚠️ 数学の一般論を作らない。⚠️ 記号の使い分けを規定しない——「以下と書くべきか未満と書くべきか」は論じない。 ⭐ 既刊との関係を先に書いておく。論文16 は等号に六つの別根があることを示した。本稿は不等号の側であり、同じ記号が二種類の壁を指していることを示す。⭐⭐ 論文21 は「恒等式か仮説かは、どの理論の内側に立つかで決まる」と書いたが、本稿の第5節はその定量版である。論文194 は等号の柵は一つしかないと示し、論文160 は一つの不等式に三つの天井を数えたが、本稿はその天井を触れるか触れないかで分ける。 第一に、触れない壁がある。一様密度の内部シュヴァルツシルト解で中心圧を測ると、コンパクトネスが 0.5 で 0.261204 倍、0.86 で 5.10897 倍、0.888 で 166.666 倍、0.888888 で 166667 倍、0.8888888 で 1.66667×10⁶ 倍と発散する(本稿で計算)。⚠️ ブッフダール限界 8/9 に触れる配置は、そもそも配置ではない。⭐ 形の上では「以下」だが、物理としては「未満」である。 第二に、触れる壁がある。最小不確定性状態(ガウス波束)で測ると、幅 10⁻¹¹、10⁻¹⁰、10⁻⁹ のいずれでも積は 5.272859085000000×10⁻³⁵ であり、ħ/2 との比は 1.000000000000000 であった(本稿で計算)。⭐ 十五桁で厳密に等号である。コーシー=シュワルツの等号条件は「二つのベクトルが比例すること」であり、それを満たす状態が実在する——しかも一つではなく族として。なお相加平均と相乗平均の等号は全部の数が等しいときに限り、0.5 から 4 の一様乱数で三つ組を 200000 回作ると等号は 0 回であった。⭐ 到達されるが、到達する集合の確率は零である。 第三に、私の見立ては外れた。初版で「CHSH の代数的天井 4 は無信号条件が禁じている」と書いたが、実際に PR 箱を組んで測ると、CHSH 値は 4.000000000000、その信号伝達は最大でも 0 であった(本稿で計算)。⚠️ 見込みは外れた。PR 箱は無信号でありながら 4 にちょうど到達する——⭐ 4 を禁じているのは相対論ではなく、量子力学のほうであった。 第四に、これが本稿の芯である。三つの天井を、立つ場所ごとに点呼した。量子の最大値は実測 2.828427124746(2√2=2.828427124746)。局所隠れた変数の内側に立てば、2 は到達するが 2√2 も 4 も不可能である。量子力学の内側に立てば、2 も 2√2 も到達し、4 は不可能である。無信号理論の内側に立てば、三つとも到達する(本稿で計算)。⭐⭐⭐ 2√2 は、触れる壁でもあり、届かない壁でもあり、そもそも壁ですらない。変わったのは数ではない——⚠️ 変わっ
However we have shown in Eqs.(18.29) and (18.30) that the square of the absolute value of a complex scalar wave function Ψ is a measure of the wave intensity, which is the quantity of primary interest for the theories of interference and diffraction for either mechanical or electromagnetic waves. The periodic spatial variation of intensity throughout a region of superposition of waves which originate in the same source is known as interference. Its theory is based essentially on the linearity of the wave equation, which allows a linear combination of partial waves to constitute the actual wave at any given point. Consider first the superposition of two monochromatic plane waves, having the complex representation (16.25), that is (18.1) Since we are usually interested in the relative variation of the intensity only, in a medium 0 ,Z and not in its absolute value, it is common practice to adopt simplified wave intensities, compared to those in Eqs.(18.29) or (18.30), which read ( ) ( )2 21 1 2 2, , ,I r t I r= Ψ = Ψ tG G (18.2) The superposition of monochromatic waves (18.1) results in an intensity ( ) ( ) ( )2 21 1 2, , ,tG ( )I r t r t r= Ψ = Ψ +ΨG G ( ) ( ) ( )2 2 *1 2 1 2, , 2Re ,r t r t r t r t,⎡ ⎤= Ψ + Ψ + Ψ Ψ⎣ ⎦G G G G ( )1 2 1 2 1 2 1 22 cosI I I I k r k r ϕ ϕ= + + ⋅ − ⋅ + −G GG G (18.3) where the last term is called the interference term. If the phase difference, given by ( )1 2 1k k r 2δ ϕ ϕ= − ⋅ + −G G G (18.4) is a constant, the two waves are said to be mutually coherent, and the interference term indicates a sinusoidal spatial variation between maxima and minima, of intensity max 1 2 1 22 ,I I I I I m2δ π= + + = (18.5) ( )min 1 2 1 22 , 2I I I I I m 1δ π= + − = + B A SS 21 Figure 18.1. Interference pattern produced by two monochromatic waves originating from sources 1 2andS S Hence, the stationary interference pattern consists of a succession of surfaces, defined by the condition Nδ π= where .12or2 += mmN It is usually produced by superposition of waves originating in two point sources , which can be assumed, for simplicity, to be of equal phase constants 21 and SS 1 2 0.ϕ ϕ− = If the two sources are far enough from a given point, at distances 1 2 1 2,r r r r>> − ,G G the incident waves at the point can be regarded as plane waves, as we shall show in Chapter 19. Under this assumption Eq.(18.4) reduces to ( ) ( )1 2 1 2 1 2or 2k k r k r r N r r N Nkπ λδ π= − ⋅ ≅ − = − = =G G G (18.6) Both the antinodal and nodal surfaces ( mN 2= ) ( )12 += mN are defined by Eq.(18.6) to be hyperboloids of revolution. Their intersection with a plane will produce alternate nodal and antinodal lines called fringes of the form of either hyperbolas or circular rings if the plane is parallel (A) or normal (B) to the line joining the sources (Figure 18.1). The interference fringes are not observed if the phase difference 1 2ϕ ϕ− has a random variation with time, since the interference term in Eq.(18.3) is averaged out. In this event the waves are said to be mutually uncoherent and do not interfere by superposition. Mutually coherent waves that originate in the same source can be obtained by splitting a single beam, so that on recombining partial waves interference effects occur.
When someone says take the classical limit, the next question is what is being asked to become classical. Paper 113 wrote that quantum mechanics is a deformation of classical mechanics by Planck's constant and that letting that constant go to zero returns the classical algebra. That is a statement about structures. This paper asks the neighbouring question: does the behaviour return? They are different sentences, and the second has several answers. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED: Ehrenfest's theorem, Bohr's correspondence principle, the WKB approximation, and the collapse and revival of wavepackets are all standard, and no interpretation of quantum mechanics is judged. Every number was computed for this paper, and no matrix result is quoted unless doubling the basis left it alone. First, the equations. The rate of change of the mean momentum is exactly minus the mean of the force, while the classical equation uses the force at the mean position. These agree exactly when the force is linear, that is when the potential is at most quadratic. For a Gaussian packet of mean seven tenths and standard deviation thirty-five hundredths, the gap was zero for the quadratic, one hundred and twenty-two and a half thousandths for the cubic, and twenty-five thousand seven hundred and twenty-five hundred-thousandths for the quartic, matching the closed forms exactly. What makes the gap is the width of the packet, and the width is where Planck's constant enters, so the gap is proportional to it. At a fixed time the classical equation is therefore recovered, and that proviso is worth remembering. Second, the spectrum. Bohr said that for large quantum number the frequency radiated on a jump down by one approaches the classical orbital frequency. In hydrogen the ratio of the two runs three, then one and four hundred and six thousandths, then one and fifteen thousandths at a hundred, and one and one and a half thousandths at a thousand, approaching one like one plus three over twice the quantum number. Not once does Planck's constant appear in that ratio. The correspondence principle is not a statement that the constant is small but that the quantum number is large, a different sentence from the first. Nor is it about a single frequency: a jump down by two corresponds to the second harmonic of the orbit and by three to the third, with ratios that were measured separately. Third, the wavefunction. Take the Bohr-Sommerfeld condition with the added one half, the Maslov index, and apply it in the same form to two wells. For the harmonic oscillator the recipe agrees with the matrix eigenvalues to zero error at every level tested, the ground state included. For the quartic well it is never exact, with a relative error of about eighteen hundredths at the ground state falling to about two hundred-thousandths at the fortieth level. Semiclassical is not one condition either: the same recipe means different things in different wells. Fourth, and this is the core. Place a packet at rest in the quartic well and integrate the classical orbit with my own Runge-Kutta scheme. Holding the time fixed and lowering Planck's constant, the difference dies in proportion to it: halving the constant halves the difference, in agreement with the closed form of the first part. Holding the constant fixed and waiting instead, the swing collapses from its full value to about two ten-thousandths, and then comes back to about two thirds of its original size much later, while the classical orbit has not moved at all. So this is not a small correction to an orbit. Both the collapse and the revival come from the level spacing not being uniform, and nothing on the classical side answers to that. Fix the time and take the constant to zero and the answer is classical; fix the constant and take the time to infinity and it is not. This pattern is not new to this corpus. Paper 154 takes two limits failing to commute as its subject and sets zero against a definite magnetisation in the Curie-Weiss magnet. All that is added here is that the same pattern appears at the quantum-classical boundary. Fifth, the survival time is finite all the same. Measuring how long the swing takes to fall below a tenth of its initial value gives values that grow steadily as the constant shrinks, each surviving a doubling of the basis to eleven decimal places. It grows, and only as a power law. Making the constant small postpones the classical orbit's failure rather than preventing it. And Ehrenfest time is itself one name for two behaviours: the short logarithmic one often quoted belongs to chaotic systems, where the spreading is exponential, while this well is integrable and measurement gives a power. The exponent measured here is one packet in one well, not a derived exponent. A census. The equations become classical by Ehrenfest, exactly up to quadratic and spoiled by the width beyond. The spectrum becomes classical by Bohr's correspondence, driven by large quantum number and not by Planck's constant. The wavefunction becomes classical by WKB, exact in one well and asymptotic in another. The state becomes classical by decoherence, in which superposition goes. These four are not a staircase of strength: decoherence makes the state classical without making the equations classical, and as Paper 55 wrote a diagonal density matrix does not say which outcome occurs, while conversely the equations can be classical with superposition surviving. What settles it is what is being asked to be classical. Important (scope note): a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem or physical law, none is claimed; any judgement on interpretations of quantum mechanics; any refutation of Paper 113, since this is a brake and not a refutation; and any novelty for the pattern of two limits failing to commute, which Paper 154 already holds. What is added: setting four limits against one separator, what is being asked to become classical; writing the Ehrenfest gap in closed form and confirming its proportionality in numbers; measuring the rate of the correspondence principle and showing that Planck's constant never enters it; placing one WKB recipe side by side where it is exact and where it is only asymptotic; and measuring the collapse and revival to a precision that survives a doubling of the basis. What remains: when someone says take the classical limit, the next question is what is being asked to become classical. The equations, the spectrum, the wavefunction, or the state all go by the same phrase, hold under different conditions and fail in different places. And the algebra returning does not mean the behaviour returns. 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. ----- 「古典極限を取る」と言われたら、次に問うべきは「何が古典になれと言われているのか」である。 論文113 は「量子力学は古典力学の ℏ による変形であり、ℏ→0 で古典代数に戻る」と書いた。それは構造についての言明である。本稿は隣の問いを立てる——では振る舞いは戻るのか。二つは別の文であり、後者には答えが複数ある。新しい数学定理も新しい物理法則も主張しない:エーレンフェストの定理、ボーアの対応原理、WKB 近似、波束の崩壊と再生は、いずれも標準的な内容である。また量子力学の解釈を判定しない。⚠️ 数値はすべて自前の計算であり、行列の対角化は基底を倍にして答えが動かないことを確かめてから引用した(動いた値は載せていない)。 第一に、方程式である。運動量の平均の時間変化は −⟨V′(x)⟩ に厳密に等しく、古典の式は −V′(⟨x⟩) である。両者が一致するのは V′ が線形のとき、すなわち V が二次までのときに限る。平均 0.7、標準偏差 0.35 のガウス波束で隔たりを測ると、二次では 0、三次では 0.1225(=σ²)、四次では 0.25725(=3μσ²)とぴたり予測どおりであった(本稿で計算)。⭐ 隔たりを作っているのは波束の幅である。そして幅こそが ℏ の入り口である——最小不確定の波束では σ²=ℏ/2 だから、隔たりは ℏ に比例する(ℏ=1, 0.01, 0.0001 で 1.05, 0.0105, 0.000105)。したがって時刻を固定すれば ℏ→0 で古典の式に戻る。⚠️ この但し書きを覚えておきたい。 第二に、スペクトルである。ボーア(1913, 1920)は、量子数が大きいとき n→n−1 の遷移で放射される振動数が古典の公転振動数に近づくと述べた。水素をリュードベリ単位で書けば遷移振動数は 1/(n−1)²−1/n²、n 番目の準位での古典の公転振動数は 2/n³ である。比は n=2 で 3.0、n=5 で 1.40625、n=10 で 1.17283950617、n=100 で 1.01520253035、n=1000 で 1.0015020025、n=10000 で 1.00015002 となり、近づき方は 1+3/(2n) と桁で一致した(本稿で計算)。⭐⭐ この比のどこにも ℏ が現れていない。対応原理は「ℏ が小さい」という言明ではなく、「量子数が大きい」という言明である——第一の ℏ→0 とは別の文なのである。しかも一本の振動数の話ですらない。n→n−k は公転の第 k 高調波に対応し、n=1000 で k=1,2,3 の比は 1.0015020025、1.00300802005、1.00451806774 であった。対応しているのはフーリエ級数の全体である。 第三に、波動関数である。ボーア=ゾンマーフェルトの量子化条件に 1/2 を加えたもの(マスロフ指数)を、二つの井戸に同じ形で当てる。調和振動子では n=0, 1, 5, 20 のいずれでも WKB の値が行列の固有値と誤差ゼロで一致した——基底状態を含め、どの n でも厳密である。ところが四次の井戸ではどの n でも厳密でなく、相対誤差は n=0 で 1.822×10⁻¹、n=1 で 1.257×10⁻²、n=10 で 3.204×10⁻⁴、n=40 で 2.156×10⁻⁵ と、n とともに改善するだけである(本稿で計算)。⭐ 「準古典」も一つの条件ではない。同じ処方が、井戸によって別のことを意味している。 第四に、これが本稿の芯である。四次の井戸に x=1.5 で静止した波束を置き、古典側は ẍ=−4x³ を自前のルンゲ=クッタ法で解く。時刻を t=2 に固定して ℏ を下げると、差は 0.1762、0.1050、0.0581、0.0307 と ℏ に比例して消える(ℏ を半分にすると差も半分——第一の閉じた形と合う)。⭐⭐ ところが ℏ=0.1 に固定して待つと、振れ幅は 1.5 から潰れていき、t=27.50 で 0.000195 に達し、そして t=199.85 で 0.664054 へ戻ってきた。古典の軌道は潰れも戻りもせず、1.4986 から一度も動いていない。したがってこれは軌道への小さな補正ではない。潰れも再生も、準位の間隔が一様でないことから来る——古典の側には対応するものが何も無い。すなわち、時刻を固定して ℏ→0 とすれば古典に戻り、ℏ を固定して t→∞ とすれば戻らない。二つの極限は順序を持って取られ、順序が答を決める。 ⚠️ この型は本体系の新顔ではない。論文154 が「二つの極限が交換しない」を主題にし、キュリー=ワイス磁石で 0 と 0.957504 を並べている。本稿が加えたのは、同じ型が量子と古典の境目にも現れるということだけである。 第五に、それでも存続時間は有限である。振れ幅が最初の十分の一を切るまでの時間を測ると、ℏ=0.4, 0.2, 0.1, 0.05 に対して 8.160、10.640、14.280、19.200 であった(各行とも基底を倍にして 10⁻¹¹ しか動かない)。伸びるが、伸び方は冪則にとどまる(測った両対数の傾きは ℏ の −0.41 乗)。⭐ ℏ を小さくすることは古典の軌道を延命するのであって、永続させはしない。⚠️ そして「エーレンフェスト時間」もまた一語で二つを指している。文献でよく引かれる log(1/ℏ) という短さはカオス系のものであり、広がりが指数的であることから来る。本稿の井戸は可積分であり、測ると冪則になる。⚠️ 上の指数は一つの波束・一つの井戸についての測定値であって、導出した指数ではない。 点呼をしておく。方程式が古典になるのがエーレンフェストで、二次までは厳密、以遠は幅が壊す。スペクトルが古典になるのがボーアの対応原理で、効くのは大きな量子数であって ℏ ではない。波動関数が古典になるのが WKB・準古典で、井戸によって厳密でも漸近でもある。状態が古典になるのがデコヒーレンスで、重ね合わせは消える。⚠️ 四つは強さの階段ではない。デコヒーレンスは状態を古典的にするが方程式を古典化せず(論文55 が書いたとおり、対角的な密度行列はどの結果が実現するかを語らない)、逆に方程式が古典的でも重ね合わせは残りうる。決めているのは、何に対して古典であれと言っているかである。 重要(射程):確立された標準的事実の点呼(第2層)。示さないこと:新しい数学定理や物理法則(一つもない)/量子力学の解釈の判定(していない)/論文113 への反論(ブレーキであって反論ではない)/「二つの極限が交換しない」という型の新規性(論文154 が既に持っている)。加えたもの:四つを「何が古典になれと言われているか」という一つの分離子に並べたこと、エーレンフェストの隔たりを閉じた形で書いて ℏ 比例を数で確かめたこと、対応原理の近づき方を測
A census of the twos - one shape of exponent, arriving from this many different places. This corpus has met the exponent 2 in many places. Each has been written about separately, but no paper has yet called them all by name at once. This one does, using a single test: what must be moved for that 2 to move? NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. The inverse-square law, the Born rule, Ito's lemma, maximum likelihood and Clifford algebras are all standard. Nor does this paper retract anything the corpus has said about the individual twos. First, the 2 of the inverse square is d minus one. A spherically symmetric harmonic function in d dimensions satisfies a second-order ordinary differential equation, and reading off the exponent of the force gives one in two dimensions, two in three, three in four, four in five and five in six. The 2 of the inverse square is simply d minus one read at three dimensions. What must be moved is the spatial dimension (Papers 01 and 156). Second, Born's 2 does not move with the dimension, and what fixes it is unitarity. Draw a random unitary matrix and a random unit vector, and ask whether the sum of the p-th powers of the absolute values of the components changes. At p of one the largest change is 0.6698, at 1.5 it is 0.2279, at 2.5 it is 0.1798, at three 0.4248 and at four 0.4364. But at p of two it is 8.882 times ten to the minus sixteen - machine precision (computed here, worst case over four hundred draws in six dimensions). Holding p at two and changing the dimension of the Hilbert space to two, three, five, eight and thirteen leaves the change at order ten to the minus fifteen throughout. What must be moved is therefore not the dimension but the demand that time evolution be unitary. This paper does not call that a derivation of the Born rule: the derivation is Gleason's theorem (Paper 71), whose assumptions are far weaker than a random trial. What is shown here is only the step before - with any other exponent, probability is not conserved at all. Third, Ito's 2 moves with the roughness of the path. Generating fractional Brownian motion of Hurst exponent H exactly and measuring the quadratic variation as the mesh is refined, theory predicts a scaling of n to the one minus two H. The measured slopes were 0.5965 at H of 0.2, 0.3028 at 0.35, 0.0046 at 0.5, minus 0.3081 at 0.65 and minus 0.5917 at 0.8, agreeing with the predicted 0.60, 0.30, 0.00, minus 0.30 and minus 0.60 to within 0.01 (computed here). The quadratic variation is finite and non-zero only at one half. What must be moved is the roughness of the path, not the spatial dimension (Paper 132). Fourth, and this is what is new here. The 2 of least squares is the assumed error law itself. Minimising the sum of the p-th powers of the residuals is maximum likelihood for errors whose density is proportional to the exponential of minus the absolute residual to the p, so p of two corresponds to the Gaussian and one to the Laplace law. The exponent is therefore not a property of the data but the name of an assumption. With a true location of 3.0, forty points and two thousand repetitions: under Gaussian errors the p of two estimator gives 0.1565 against 0.1983 and wins; under Laplace errors p of one gives 0.1782 against 0.2199 and wins; and under Cauchy errors p of one gives 0.2632 while p of two gives 128.3741 - a factor of about five hundred (computed here). The exponent that wins matches the exponent of the law that generated the data. The gap under Cauchy noise is not a loss of accuracy but a breakdown of the estimator: the Cauchy law has no variance, so minimising a sum of squares has lost its meaning. It is the same shape of failure that Paper 129 described when it wrote that the variance form of the uncertainty relation says nothing once the variance diverges. So the 2 of least squares has a distinct root from the inverse square, from Born and from Ito - what must be moved is neither the dimension nor the roughness but the probability law one assumed. Fifth, Clifford's 2 cannot be moved. The defining relation of a Clifford algebra is that a vector squared equals the value of the quadratic form, and in the Pauli algebra each of the three squares minus the identity is exactly the zero matrix. There is no parameter here to turn: replacing the 2 does not give a different Clifford algebra, it gives no Clifford algebra (the same shape as epsilon squared being zero as a definition of a ring, in Papers 104 and 148). The census. The inverse-square 2 moves with the spatial dimension, has general form d minus one, and lives in harmonicity. Born's 2 moves with nothing, being held by unitarity. Ito's 2 moves with the roughness and is finite only at one half. The least-squares 2 moves with the assumed error law, its general form being the exponent in the density. Clifford's 2 cannot be moved, being a definition. Fourier's one half likewise does not move, arriving as the value of an extremum. One question, three answers, and three cases of nothing at all - and the three nothings are not the same either: Born's is held by a uniqueness theorem, Clifford's by a definition, and Fourier's arrives as an extremal value. The verdict is the same and the reason for it is different in all three. IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem or physical law (there is none) / any retraction of what the corpus has said about the individual twos (there is none). What is added: calling five twos by name under one test, showing in numbers that unitarity selects p equal to two uniquely, measuring the slope of the quadratic variation against one minus two H, and giving the 2 of least squares its address as an assumed error law, exhibited as a factor of five hundred under Cauchy noise. What remains: one shape of exponent, arriving from this many different places. On meeting a 2, the question to ask is not why two but what would have to move for it to stop being two. Only when that question has no answer may the 2 be written down as something that does not move. 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. ----- 二乗の点呼——同じ形の指数が、これだけ違う場所から来ている。 この体系は、あちこちで指数 2 に出会ってきた。個々の 2 については書いてきたが、横断して点呼した稿がまだ無い。本稿はそれを、一つの試験——何を動かせばその 2 が動くか——で行う。新しい数学定理も新しい物理法則も主張しない。逆二乗則、ボルン則、伊藤の補題、最尤推定、クリフォード代数は、いずれも標準的な内容である。⚠️ また本稿は、個々の 2 について体系が既に書いたことを撤回しない。 第一に、逆二乗の 2 は d マイナス 1 である。d 次元の球対称な調和関数は二階の常微分方程式を満たし、解いて力の指数を読むと、二次元で 1、三次元で 2、四次元で 3、五次元で 4、六次元で 5 となる。逆二乗の 2 は、d マイナス 1 を三次元で読んだ値にすぎない。動かすべきものは空間次元である(論文01・論文156)。 第二に、ボルンの 2 は次元では動かない。そして動かしているのはユニタリ性である。n 次元のユニタリ行列と単位ベクトルを無作為に取り、成分の絶対値の p 乗の和がユニタリ変換で変わるかを見る。p が 1 なら最大変化は 0.6698、1.5 なら 0.2279、2.5 なら 0.1798、3 なら 0.4248、4 なら 0.4364。ところが p が 2 のときだけ 8.882 かける十のマイナス十六乗——機械精度である(本稿で計算、六次元・四百組の最悪値)。さらに p を 2 に固定したままヒルベルト空間の次元を 2・3・5・8・13 と変えても、変化は十のマイナス十五乗台のまま動かない。すなわち動かすべきものは空間次元ではなく、時間発展がユニタリであるという要求のほうである。⚠️ 本稿はこれをボルン則の導出とは呼ばない。導出はグリーソンの定理が与えており(論文71)、そこでの仮定は本稿の無作為試験より遥かに弱い。本稿が示したのは「他の指数ではそもそも確率が保たれない」という、一段手前の事実だけである。 第三に、伊藤の 2 は道の粗さで動く。ハースト指数 H の非整数ブラウン運動を厳密に生成し、二次変分を刻み幅を細かくしながら測る。理論では n の 1 マイナス 2H 乗で振る舞うはずで、実際に測った傾きは H が 0.2 で 0.5965、0.35 で 0.3028、0.5 で 0.0046、0.65 でマイナス 0.3081、0.8 でマイナス 0.5917。予測値である 1 マイナス 2H の 0.60・0.30・0.00・マイナス0.30・マイナス0.60 と、いずれも 0.01 以内で一致した(本稿で計算)。二次変分が有限かつ非零であるのは H が二分の一のときだけである。動かすべきものは道の粗さであって、空間次元ではない(論文132)。 第四に、これが本稿の新しい部分である。最小二乗の 2 は、仮定した誤差則そのものである。残差の絶対値の p 乗の和を最小化することは、密度が指数マイナス絶対値 r の p 乗に比例する誤差についての最尤推定と同じである。p が 2 ならガウス、1 ならラプラスに対応する。したがって p はデータの性質ではなく、置いた仮定の名前である。真の位置を 3.0 とし四十点の標本を二千回引いて確かめると、ガウス誤差では p が 2 の推定量が 0.1565、1 が 0.1983 で 2 の勝ち。ラプラス誤差では 1 が 0.1782、2 が 0.2199 で 1 の勝ち。そしてコーシー誤差では 1 が 0.2632 なのに対し、2 は 128.3741——約五百倍の差である(本稿で計算)。勝つ指数は、データを生んだ法則の指数と一致する。 コーシーでの差は精度の劣化ではなく、推定量が破綻しているということである。コーシー分布は分散を持たないので、二乗を最小にするという方針そのものが意味を失う。論文129 が「分散が発散すれば分散版の不確定性は何も言っていない」と書いたのと、同じ形の失敗である。したがって最小二乗の 2 は、逆二乗の 2 とも、ボルンの 2 とも、伊藤の 2 とも別根である——動かすべきものは、次元でも粗さでもなく、置いた確率分布である。 第五に、クリフォードの 2 は動かせない。クリフォード代数の定義関係は「ベクトルの二乗が二次形式の値に等しい」であり、パウリ行列で確かめると三つとも二乗から単位行列を引いてちょうど零行列になる。ここには動かすべき径数が存在しない。2 を替えても「別のクリフォード代数」にはならず、クリフォード代数でなくなるだけである(論文104・論文148 の「イプシロンの二乗がゼロなのは環の定義」と同じ型)。 点呼をしておく。逆二乗の 2 は空間次元で動き、一般形は d マイナス 1、住所は調和性。ボルンの 2 は何も動かず、ユニタリ性が固定している。伊藤の 2 は道の粗さで動き、有限になるのは H が二分の一のときだけ。最小二乗の 2 は仮定した誤差則で動き、一般形は指数関数の肩の p。クリフォードの 2 は環の定義ゆえ動かせない。フーリエの二分の一も動かず、極値の値として出る。すなわち一つの問いに、三つの答えと、三つの「該当なし」がある。しかもその三つの「該当なし」も互いに同じではない——ボルンの 2 は一意性定理が固定し、クリフォードの 2 は定義が固定し、フーリエの二分の一は極値の値として出る。動かないという結果が同じでも、動かない理由は三つとも違う。 重要(射程):確立された標準的事実の点呼(第2層)。示さないこと:新しい数学定理も新しい物理法則も(一つもない)/個々の 2 についての過去の記述の撤回(一つもない)。加えたもの:五つの 2 を一つの試験で横断点呼したこと、ユニタリ性が p を 2 に一意に選ぶことを数で示したこと、二次変分の傾きが 1 マイナス 2H に一致することを測ったこと、最小二乗の 2 が「仮定した誤差則」であると住所を書きコーシー分布で五百倍の差として見せたこと。残るもの:同じ形の指数が、これだけ違う場所から来ている。2 を見たときに問うべきなのは「なぜ 2 なのか」ではなく、何を動かせばそれが 2 でなくなるのか、である。その問いに答えが無いとき——そのときだけ、その 2 は動かないものとして書いてよい。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。