Three ceilings over one inequality - and no two are held down by the same thing. Three upper bounds are spoken of for the CHSH quantity: the classical two, the quantum two root two, and the algebraic four. This paper calls all three by name and shows that no two of them are held down by the same thing. NO NEW PHYSICAL LAW IS CLAIMED. Bell's inequality, CHSH, the Tsirelson bound, the PR box and the no-signalling condition are all standard. Nor does this paper answer why nature stops at two root two - that is an open research question, and what can be written is the address, and no further. First, the classical ceiling is combinatorics. A local deterministic strategy merely fixes four signs in advance, and there are only sixteen of them. Exhausting all sixteen, the CHSH quantity falls between minus two and two, with a maximum absolute value of exactly two (computed here). Mixing cannot raise it: a probabilistic mixture is a convex combination of these extreme points, so the maximum is attained at one of them. The bound of two therefore follows from the assumption of local determinism alone, by counting signs. Its address is combinatorics. Second, the quantum ceiling comes out of density matrices. Measuring a singlet along zero, ninety, forty-five and minus forty-five degrees, the four correlations all have absolute value 0.7071067811865475 and the CHSH quantity is 2.8284271247, differing from two root two by 4.44 times ten to the minus sixteen (computed here). Drawing both the two-qubit state and the four measurement directions at random forty thousand times, the bound was never once exceeded, the largest value found being 2.7291936631. Random directions rarely land near the optimal configuration, which is why the search stopped there; all this paper takes from the trial is that the bound was not exceeded, and it is not a proof of the bound. Third, and this is the core. What stops quantum mechanics there is geometry. For the singlet the correlation is minus the inner product of two direction vectors, so the CHSH quantity is bounded above by the length of b plus b prime added to the length of b minus b prime. Maximising that over two unit vectors gave 2.8284271247 across two hundred thousand random pairs, attained exactly when the two are orthogonal. The reason fits on one line: the sum of the squares of those two lengths is fixed at four, and the sum of two numbers with a fixed sum of squares is largest when they are equal, whence the bound of two root two. The address of two root two is therefore the inner product, and the Bloch-sphere geometry of Paper 68 appears here unchanged. Where the classical two counted signs, this one measures lengths - distinct roots. Fourth, the brake. What forbids four is not relativity. The PR box has correlations plus one, plus one, plus one and minus one, and reaches exactly four. Yet writing out its marginals for each of the other party's settings, they remain 0.500 whatever either side chooses - the PR box does not signal (verified here). Relativity therefore permits four, and the explanation that it stops at two root two because of relativity does not explain that. What stops it is the state space. Locality was already surrendered above two, and no-signalling allows all the way to four; what remains is the single fact that the correlations are inner products of unit vectors, and that gives two root two. The census. The ceiling of two is held by local determinism, moved by allowing non-locality, and lives in combinatorics. The ceiling of two root two is held by the quantum state space, moved only by changing the inner-product geometry, and lives in geometry. The ceiling of four is held by outcomes drawn from plus and minus one, moved only by changing the range of outcomes, and lives in arithmetic. One inequality carries three ceilings and no two are held down by the same thing. The sentence that Bell's inequality has been violated is a statement about the first ceiling only. And the two and the root two appearing here are distinct from all five twos counted in Paper 157: here the two comes from arrangements of signs and the root two from the sum of two numbers with a fixed sum of squares. The census of that one word is not finished. IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new physical law (there is none) / an answer to why nature stops at two root two (open). What is added: obtaining the classical ceiling by exhaustion, obtaining the quantum one from density matrices and confirming non-exceedance by random search, exhibiting the source of two root two numerically as the geometry of an inner product, and checking that the PR box reaches four while not signalling, which fences the explanation that relativity is what stops us. What remains: once one notices that there are three ceilings, the word violated turns imprecise. What was violated is the first. The second has never once been broken, and the third stands empty in a place relativity allows, with nobody living there. 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. ----- 一つの不等式に、三つの天井がある——どれ一つとして同じものに押さえられていない。 CHSH 量には三つの上限が語られる——古典の 2、量子の 2 ルート 2、そして代数的な 4。本稿はこの三つを点呼し、どれ一つとして同じものに押さえられていないことを示す。新しい物理法則は主張しない。ベルの不等式、CHSH、チレルソン限界、PR 箱、無信号条件は、いずれも標準的な内容である。⚠️ また本稿は「なぜ自然は 2 ルート 2 で止まるのか」に答えない——それは未解決の研究課題であり、書けるのは住所までである。 第一に、古典の天井は組合せ論である。局所決定的な戦略とは、四つの符号をあらかじめ決めておくことにすぎず、全部で十六通りしかない。全数列挙すると、CHSH 量の値はマイナス 2 から 2 の範囲に収まり、絶対値の最大はちょうど 2 であった(本稿で計算)。混ぜても増えない——確率混合は端点の凸結合だから、最大値は端点で取られる。したがって上限 2 は、局所決定性という仮定だけから、符号を数えることで出る。住所は組合せ論である。 第二に、量子の天井は密度行列から出る。一重項状態に、角度ゼロ度・九十度・四十五度・マイナス四十五度の測定を当てると、四つの相関はすべて絶対値 0.7071067811865475 になり、CHSH 量は 2.8284271247——2 ルート 2 との差は 4.44 かける十のマイナス十六乗である(本稿で計算)。さらに二量子ビットの状態と四つの測定方向をすべて無作為に取って四万通り試したが、一度も超えなかった(見つかった最大は 2.7291936631)。⚠️ 無作為に取った方向は最適配置に当たりにくいのでこの値にとどまった。本稿がこの試行から言うのは「超えなかった」ことだけであり、限界の証明ではない。 第三に、これが本稿の芯である。量子をそこで止めているのは幾何である。一重項では相関が二つの方向ベクトルの内積のマイナスで書けるので、CHSH 量は「b 足す b ダッシュの長さ」と「b 引く b ダッシュの長さ」の和で上から抑えられる。この右辺を単位ベクトル二本について最大化すると、二十万組の無作為な組で 2.8284271247、しかも二本が直交するときにちょうどその値に達した。理由は一行で書ける——和と差の長さの二乗和は 4 に固定されているので、二乗和が一定の二数の和は、二つが等しいときに最大になる。よって上限は 2 ルート 2 である。すなわち 2 ルート 2 の住所は内積であって、論文68 のブロッホ球の幾何がそのままここに出ている。古典の 2 が符号の数え上げだったのに対し、これは長さの幾何——別根である。 第四に、ブレーキ。4 を禁じているのは相対論ではない。PR 箱は四つの相関が プラス1・プラス1・プラス1・マイナス1 となり、CHSH 量はちょうど 4 に達する。ところがこの箱の周辺分布を相手の設定ごとに書き出すと、自分の設定がどちらであっても、相手が何を選んでいても 0.500 のまま変わらない——PR 箱は信号を送らない(本稿で確認)。つまり相対論は 4 を許している。「相対論があるから 2 ルート 2 で止まる」という説明は、そこを説明していない。止めているのは状態空間のほうである——局所性はすでに 2 を超えた時点で手放されており、無信号は 4 まで許す。残っているのは「相関が単位ベクトルの内積として書ける」という一点だけであり、それが 2 ルート 2 を出す。 点呼をしておく。天井 2 を押さえているのは局所決定性で、動かすには非局所を許せばよく、住所は組合せ論。天井 2 ルート 2 を押さえているのは量子の状態空間で、動かすには内積の幾何を替えるしかなく、住所は幾何。天井 4 を押さえているのはプラスマイナス 1 という結果の集合で、動かすには結果の値域を替えるしかなく、住所は算術。一つの不等式に三つの天井があり、どれ一つとして同じものに押さえられていない。すなわち「ベルの不等式が破れた」という一文は、一枚目の天井についてだけの言明である。 ⭐ そしてここに現れる 2 と ルート 2 は、論文157 が数えた五つの 2 のどれとも別である。ここでの 2 は符号の並べ方から、ルート 2 は二乗和が固定された二数の和から来ている——一語の点呼は、まだ終わっていない。 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:新しい物理法則(一つもない)/「なぜ自然が 2 ルート 2 で止まるのか」への答え(未解決)。加えたもの:古典の天井を全数列挙で出したこと、量子の天井を密度行列から出し無作為探索で非超過を確かめたこと、2 ルート 2 の出所を内積の幾何として数で示したこと、PR 箱が無信号でありながら 4 に届くことを確認して「相対論が止めている」という説明に柵を立てたこと。残るもの:天井が三枚あることに気づくと、「破れた」という言葉が急に不正確になる。破れたのは一枚目である。二枚目はまだ一度も破られておらず、三枚目は相対論の許すところに、誰も住んでいないまま空いている。 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。
Three ceilings over one inequality - and no two are held down by the same thing. Three upper bounds are spoken of for the CHSH quantity: the classical two, the quantum two root two, and the algebraic four. This paper calls all three by name and shows that no two of them are held down by the same thing. NO NEW PHYSICAL LAW IS CLAIMED. Bell's inequality, CHSH, the Tsirelson bound, the PR box and the no-signalling condition are all standard. Nor does this paper answer why nature stops at two root two - that is an open research question, and what can be written is the address, and no further. First, the classical ceiling is combinatorics. A local deterministic strategy merely fixes four signs in advance, and there are only sixteen of them. Exhausting all sixteen, the CHSH quantity falls between minus two and two, with a maximum absolute value of exactly two (computed here). Mixing cannot raise it: a probabilistic mixture is a convex combination of these extreme points, so the maximum is attained at one of them. The bound of two therefore follows from the assumption of local determinism alone, by counting signs. Its address is combinatorics. Second, the quantum ceiling comes out of density matrices. Measuring a singlet along zero, ninety, forty-five and minus forty-five degrees, the four correlations all have absolute value 0.7071067811865475 and the CHSH quantity is 2.8284271247, differing from two root two by 4.44 times ten to the minus sixteen (computed here). Drawing both the two-qubit state and the four measurement directions at random forty thousand times, the bound was never once exceeded, the largest value found being 2.7291936631. Random directions rarely land near the optimal configuration, which is why the search stopped there; all this paper takes from the trial is that the bound was not exceeded, and it is not a proof of the bound. Third, and this is the core. What stops quantum mechanics there is geometry. For the singlet the correlation is minus the inner product of two direction vectors, so the CHSH quantity is bounded above by the length of b plus b prime added to the length of b minus b prime. Maximising that over two unit vectors gave 2.8284271247 across two hundred thousand random pairs, attained exactly when the two are orthogonal. The reason fits on one line: the sum of the squares of those two lengths is fixed at four, and the sum of two numbers with a fixed sum of squares is largest when they are equal, whence the bound of two root two. The address of two root two is therefore the inner product, and the Bloch-sphere geometry of Paper 68 appears here unchanged. Where the classical two counted signs, this one measures lengths - distinct roots. Fourth, the brake. What forbids four is not relativity. The PR box has correlations plus one, plus one, plus one and minus one, and reaches exactly four. Yet writing out its marginals for each of the other party's settings, they remain 0.500 whatever either side chooses - the PR box does not signal (verified here). Relativity therefore permits four, and the explanation that it stops at two root two because of relativity does not explain that. What stops it is the state space. Locality was already surrendered above two, and no-signalling allows all the way to four; what remains is the single fact that the correlations are inner products of unit vectors, and that gives two root two. The census. The ceiling of two is held by local determinism, moved by allowing non-locality, and lives in combinatorics. The ceiling of two root two is held by the quantum state space, moved only by changing the inner-product geometry, and lives in geometry. The ceiling of four is held by outcomes drawn from plus and minus one, moved only by changing the range of outcomes, and lives in arithmetic. One inequality carries three ceilings and no two are held down by the same thing. The sentence that Bell's inequality has been violated is a statement about the first ceiling only. And the two and the root two appearing here are distinct from all five twos counted in Paper 157: here the two comes from arrangements of signs and the root two from the sum of two numbers with a fixed sum of squares. The census of that one word is not finished. IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new physical law (there is none) / an answer to why nature stops at two root two (open). What is added: obtaining the classical ceiling by exhaustion, obtaining the quantum one from density matrices and confirming non-exceedance by random search, exhibiting the source of two root two numerically as the geometry of an inner product, and checking that the PR box reaches four while not signalling, which fences the explanation that relativity is what stops us. What remains: once one notices that there are three ceilings, the word violated turns imprecise. What was violated is the first. The second has never once been broken, and the third stands empty in a place relativity allows, with nobody living there. 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. ----- 一つの不等式に、三つの天井がある——どれ一つとして同じものに押さえられていない。 CHSH 量には三つの上限が語られる——古典の 2、量子の 2 ルート 2、そして代数的な 4。本稿はこの三つを点呼し、どれ一つとして同じものに押さえられていないことを示す。新しい物理法則は主張しない。ベルの不等式、CHSH、チレルソン限界、PR 箱、無信号条件は、いずれも標準的な内容である。⚠️ また本稿は「なぜ自然は 2 ルート 2 で止まるのか」に答えない——それは未解決の研究課題であり、書けるのは住所までである。 第一に、古典の天井は組合せ論である。局所決定的な戦略とは、四つの符号をあらかじめ決めておくことにすぎず、全部で十六通りしかない。全数列挙すると、CHSH 量の値はマイナス 2 から 2 の範囲に収まり、絶対値の最大はちょうど 2 であった(本稿で計算)。混ぜても増えない——確率混合は端点の凸結合だから、最大値は端点で取られる。したがって上限 2 は、局所決定性という仮定だけから、符号を数えることで出る。住所は組合せ論である。 第二に、量子の天井は密度行列から出る。一重項状態に、角度ゼロ度・九十度・四十五度・マイナス四十五度の測定を当てると、四つの相関はすべて絶対値 0.7071067811865475 になり、CHSH 量は 2.8284271247——2 ルート 2 との差は 4.44 かける十のマイナス十六乗である(本稿で計算)。さらに二量子ビットの状態と四つの測定方向をすべて無作為に取って四万通り試したが、一度も超えなかった(見つかった最大は 2.7291936631)。⚠️ 無作為に取った方向は最適配置に当たりにくいのでこの値にとどまった。本稿がこの試行から言うのは「超えなかった」ことだけであり、限界の証明ではない。 第三に、これが本稿の芯である。量子をそこで止めているのは幾何である。一重項では相関が二つの方向ベクトルの内積のマイナスで書けるので、CHSH 量は「b 足す b ダッシュの長さ」と「b 引く b ダッシュの長さ」の和で上から抑えられる。この右辺を単位ベクトル二本について最大化すると、二十万組の無作為な組で 2.8284271247、しかも二本が直交するときにちょうどその値に達した。理由は一行で書ける——和と差の長さの二乗和は 4 に固定されているので、二乗和が一定の二数の和は、二つが等しいときに最大になる。よって上限は 2 ルート 2 である。すなわち 2 ルート 2 の住所は内積であって、論文68 のブロッホ球の幾何がそのままここに出ている。古典の 2 が符号の数え上げだったのに対し、これは長さの幾何——別根である。 第四に、ブレーキ。4 を禁じているのは相対論ではない。PR 箱は四つの相関が プラス1・プラス1・プラス1・マイナス1 となり、CHSH 量はちょうど 4 に達する。ところがこの箱の周辺分布を相手の設定ごとに書き出すと、自分の設定がどちらであっても、相手が何を選んでいても 0.500 のまま変わらない——PR 箱は信号を送らない(本稿で確認)。つまり相対論は 4 を許している。「相対論があるから 2 ルート 2 で止まる」という説明は、そこを説明していない。止めているのは状態空間のほうである——局所性はすでに 2 を超えた時点で手放されており、無信号は 4 まで許す。残っているのは「相関が単位ベクトルの内積として書ける」という一点だけであり、それが 2 ルート 2 を出す。 点呼をしておく。天井 2 を押さえているのは局所決定性で、動かすには非局所を許せばよく、住所は組合せ論。天井 2 ルート 2 を押さえているのは量子の状態空間で、動かすには内積の幾何を替えるしかなく、住所は幾何。天井 4 を押さえているのはプラスマイナス 1 という結果の集合で、動かすには結果の値域を替えるしかなく、住所は算術。一つの不等式に三つの天井があり、どれ一つとして同じものに押さえられていない。すなわち「ベルの不等式が破れた」という一文は、一枚目の天井についてだけの言明である。 ⭐ そしてここに現れる 2 と ルート 2 は、論文157 が数えた五つの 2 のどれとも別である。ここでの 2 は符号の並べ方から、ルート 2 は二乗和が固定された二数の和から来ている——一語の点呼は、まだ終わっていない。 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:新しい物理法則(一つもない)/「なぜ自然が 2 ルート 2 で止まるのか」への答え(未解決)。加えたもの:古典の天井を全数列挙で出したこと、量子の天井を密度行列から出し無作為探索で非超過を確かめたこと、2 ルート 2 の出所を内積の幾何として数で示したこと、PR 箱が無信号でありながら 4 に届くことを確認して「相対論が止めている」という説明に柵を立てたこと。残るもの:天井が三枚あることに気づくと、「破れた」という言葉が急に不正確になる。破れたのは一枚目である。二枚目はまだ一度も破られておらず、三枚目は相対論の許すところに、誰も住んでいないまま空いている。 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。
Paper 160 showed that one inequality carries three ceilings, and divided them: 2 is combinatorial, 2√2 is geometric, 4 is arithmetic. Its scope note then said that it does not answer why nature stops at 2√2. This paper does not answer that either──it answers a different question, namely what sets that ceiling. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──Bell’s inequality, its chained form with more settings, the no-signalling condition, and the procedure of alternately optimising state and measurements are all standard. Why nature stops at that ceiling is not answered──that is an open research question. What is written here is what moves the ceiling and what does not. The ceiling is not derived from a principle──work deriving the quantum ceiling from principles such as information causality or local orthogonality is not entered into. Only numbers computed here are written. A search maximum is not called an upper bound──alternating optimisation evaluates from below. What can be said is that the value is reached and that no configuration exceeding it was found. No link to 4π is drawn──Section 7 compares them and measures that none is close. The period-doubling constant is not said to share a root──it is written only that two examples exist in which structure fixes a non-integer real number. Experiments are not discussed──how far loophole-free experiments have confirmed this ceiling lies outside the scope. Relation to earlier papers: Paper 160 took a census of three ceilings over one inequality, divided them as combinatorial, geometric and arithmetic, and said in its scope note that it does not answer why nature stops at 2√2──this paper does not answer that either. It answers what sets the ceiling, which is a different question. Paper 227 measured that the value of a PR box reaches 4 with signalling 0──that algebraic ceiling is cited here as the destination reached as settings are added. Paper 11 computed that an entangled pair gives exactly 2√2──that is taken as the starting point, and dimension, number of settings and state are then moved. Paper 164 measured the polytope built from local assignments and the distance to its face──the classical ceiling here comes from enumerating the same assignments exhaustively. Paper 68 showed that a qubit is the Hopf map itself and that an angle on the sphere is twice the angle between states──that the optimal configuration of Section 6 is 90 degrees on the sphere and 45 degrees between states is that doubling. Paper 1055 showed that numbers separate by whether they move when units are changed──this ceiling is on the unmoving side, and Section 7 measures that with a control. Paper 1061 wrote that an exit exists by which structure produces a non-integer real number──this ceiling is a second example, and one that can be written exactly in closed form. Paper 300 showed that whether two things share a root is decidable──following that discipline, the two examples are not called one root. What is added: applying dimension, number of settings and state to one ceiling and separating what moves it from what does not, confirming exhaustively that the quantum-to-algebraic ratio matches a cosine of the number of settings, showing in numbers that two ratios move in opposite directions, and measuring and rejecting any link to 4π. First, enlarging the space does not move the ceiling──optimising state and measurements alternately from two levels up to five, all four reach 2.828427124746, with a spread of 2.220×10⁻¹⁵. Second, and this is the core. Increasing the number of measurements does move it──with n settings on each side, the quantum ceiling divided by the algebraic one is 0.707106781187 at n=2 and 0.999969158 at n=200. Third, two ratios then move in opposite directions──the algebraic ceiling is approached, while the margin over the classical one falls from 1.414213562373 to 1.004994128. The algebraic side moves by +0.292862376 and the classical side by -0.409219434. Fourth, the state moves it too──an unentangled state stops at exactly 2.000000000, and only a maximally entangled one reaches 2.828427125. Fifth, units do not move it──giving directions by components and changing the unit of length four ways leaves a spread of 0.000×10⁰, while the same changes move the dimensionful speed of light by a relative 1.000000×10⁹. Stated plainly──why nature stops at this ceiling is not answered. Any link to 4π was measured and rejected: the nearest candidate differs by a relative 0.113773075. Closing: what sets the quantum ceiling is not dimension but the number of measurements. The size of the space has no effect──from two levels to five, all four reach 2.828427124746, with a spread of only 2.220×10⁻¹⁵. The number of settings does have an effect──the quantum ceiling divided by the algebraic one runs from 0.707106781187 to 0.999969157645, agreeing with a closed form to within 1.110×10⁻¹⁶ at all nine values. Two ratios then move in opposite directions──the algebraic side rises by +0.292862376 and the classical side falls by -0.409219434. The gap ratio is 94.964608512 and the excess ratio 2.399321556. The state has an effect too──an unentangled state stops at 2.000000000, and only a maximally entangled one reaches the ceiling. Units have none──four changes leave a spread of 0.000×10⁰, while the control speed of light moves by 1.000000×10⁹. Stated plainly──why nature stops at this ceiling is not answered. There is no link to 4π either──the nearest candidate differs by a relative 0.113773075. Alternating optimisation evaluates from below and is not a proof of the bound. The separator──whether one moves the size of the space or the number of measurements. Moving the former leaves the ceiling untouched; moving the latter moves the ceiling itself. 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: Bell inequality, nonlocal correlation, upper bound, entanglement, dimension, measurement settings, separator. ----- 論文160 は一つの不等式の上に三つの天井があることを示し、2 は組合せ論・2√2 は幾何・4 は算術だと分けた。そのうえで射程注記に、なぜ自然が 2√2 で止まるのかには答えない、と書いた。本稿もそこには答えない──答えるのは別の問いで、その天井を何が決めているかである。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──ベルの不等式、設定を増やした巡回型の不等式、無信号条件、状態と測り方を交互に最適化する手続きは、いずれも標準的である。なぜ自然がその天井で止まるのかに答えない──それは未解決の研究課題である。本稿が書くのは、その天井が何を動かすと動き、何を動かしても動かないか、までである。原理から天井を導かない──情報の因果律や局所的な直交性といった原理から量子の天井を導こうとする研究には立ち入らない。本稿は自分で計算した数しか書かない。探索の最大を上限と呼ばない──交互の最適化は下からの評価である。言えるのは、その値に届くことと、超える配置が見つからないことまでである。4π と結ばない──第7節で実際に比べて、どれも近くないことを測った。周期倍化の定数と同根とは呼ばない──構造が非整数の実数を決める例が二つ在る、と書くだけである。実験を論じない──抜け穴のない実験がこの天井をどこまで確かめたかは、本稿の範囲の外である。既刊との関係:論文160 は一つの不等式の上に三つの天井を点呼し、2 は組合せ論・2√2 は幾何・4 は算術だと分け、射程注記でなぜ自然が 2√2 で止まるのかには答えないと書いた──本稿もそこには答えない。答えるのは何がその天井を決めているかで、別の問いである。論文227 は PR 箱の値が 4 に届き、その信号伝達が 0 であることを測った──本稿はその代数の天井を、設定の数を増やしたときの行き先として引く。論文11 はもつれた一対で値がちょうど 2√2 になることを計算した──本稿はそれを出発点に取り、次元と設定の数と状態の三つを動かす。論文164 は局所的な割り当てが作る多面体と、その面までの距離を測った──本稿の古典の天井は、同じ割り当てを全数列挙して出している。論文68 は量子ビットがホップ写像そのものであり、球の上の角が状態の角の二倍になることを示した──第6節の最適な配置が球の上で 90 度、状態の側で 45 度になることは、その二倍の関係の現れである。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──本稿の天井は動かない側であり、第7節でそれを対照つきで測った。論文1061 は、構造が非整数の実数を産む出口が在ると書いた──本稿の天井はその二つ目の例で、しかも閉じた形で厳密に書ける。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、二つの例を同根とは呼ばない。加えたのは、同じ天井に次元・設定の数・状態の三つを当てて、動く側と動かない側を分けたこと、量子と代数の比が設定の数の余弦に一致することを全数で確かめたこと、二つの比が逆向きに動くことを数で示したこと、4π との関係を測って否定したことである。 第一に、空間を大きくしても天井は動かない──状態と測り方を交互に最適化して二準位から五準位まで実際に最大を探すと、四つとも 2.828427124746 に届き、四つの散らばりは 2.220×10⁻¹⁵ である。 第二に、これが本稿の芯である。測る回数を増やすと天井は動く──各者が n 通りの設定を持つ形にすると、量子の天井を代数の天井で割った値は n が 2 のとき 0.707106781187、200 のとき 0.999969158 になる。 第三に、そのとき二つの比は逆向きに動く──代数の天井には近づくのに、古典の天井からの離れ方は 1.414213562373 から 1.004994128 へ小さくなる。代数側が +0.292862376、古典側が -0.409219434 である。 第四に、もつれの度合いでも天井は動く──もつれのない状態は 2.000000000 ちょうどで止まり、最大にもつれた状態だけが 2.828427125 に届く。 第五に、単位では動かない──向きを成分で与えて長さの単位を四通り替えても散らばりは 0.000×10⁰ で、同じ替え方で次元を持つ光速は相対 1.000000×10⁹ 動く。 正直に書く──なぜ自然がこの天井で止まるのかには答えない。4π との関係も測って否定した──いちばん近い数でも相対差 0.113773075 である。 結び:量子の天井を決めているのは次元ではなく、測る回数である。空間の大きさは効かない──二準位から五準位まで四つとも 2.828427124746 に届き、散らばりは 2.220×10⁻¹⁵ にとどまる。設定の数は効く──量子の天井を代数の天井で割った値は 0.707106781187 から 0.999969157645 まで動き、九つの設定の数すべてで閉じた式と 1.110×10⁻¹⁶ 以内で一致する。そのとき二つの比は逆向きに動く──代数側は +0.292862376 増え、古典側は -0.409219434 減る。隔たりの比は 94.964608512、超過の比は 2.399321556 である。状態も効く──もつれのない状態は 2.000000000 で止まり、最大にもつれた状態だけが天井に届く。単位は効かない──四通りで散らばり 0.000×10⁰、対照の光速は 1.000000×10⁹ 動く。正直に書く──なぜ自然がこの天井で止まるのかには答えない。4π とも関係がない──いちばん近い数でも相対差 0.113773075 である。交互の最適化は下からの評価であって、上限の証明ではない。分離子──動かすのが空間の大きさか、測る回数か。前者を動かしても天井は微動もせず、後者を動かすと天井そのものが動く。 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。 キーワード:ベルの不等式、非局所相関、上限、もつれ、次元、測定の設定、分離子。
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
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.
FINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
hbar is not on the mathematical side of the inequality - the uncertainty principle is not one theorem. One phrase, the uncertainty principle, points at several theorems that measure different things. This paper calls the roll and writes each address. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Kennard (1927), Weyl (1928), Robertson (1929), Hardy (1933), Hirschman (1957) and Beckner (1975), Amrein-Berthier (1977) and Benedicks (1985), Donoho-Stark (1989), Tao (2005) and Mandelshtam-Tamm (1945) are all standard. First, and this is the core. hbar is not on the mathematical side of the inequality. The inequality for the product of a time width and an angular-frequency width, whose lower bound is one half, is a theorem about square-integrable functions and their Fourier transforms, and it contains no physical constant whatsoever. This paper confirms symbolically that a Gaussian attains equality and that the coefficient describing its width cancels, leaving one half. The position-momentum relation acquires half of hbar as its bound because one physical identification has been passed through exactly once: the de Broglie-Planck relation, that momentum is hbar times wavenumber. The one half comes from mathematics; the hbar comes from physics. The only quantum step is that single substitution. Second, even that one half carries a convention. Writing frequency per turn rather than per radian turns the same theorem's bound into one over four pi. The four pi appearing there is two times two pi: the two is the extremal one half, and the two pi converts turns to radians. And the four pi in Planck's-constant form, h over four pi, is the same two times two pi, since hbar is h divided by two pi. So the two four pi are not distinct roots; they are one root. And neither is a solid angle. The counterexample is dimension-independence: the bound holds componentwise in the same form whatever the spatial dimension, a particle on a line obeys the same bound, and there is no sphere there. Were the constant a solid angle the bound would move with dimension. It does not move. This is the shape Paper 46 laid down: change the Fourier convention and the number moves. Geometry must not be read into a number that a convention can move. Third, the variance form says nothing when the variance diverges. The Cauchy density is a probability density whose second moment diverges (verified here). A rectangular pulse has finite support in time, yet the square of its transform decays only like the inverse square of frequency, so its frequency variance diverges: widening the window fourfold, this paper obtains estimates of 1137, 4541 and 18172, rising by roughly a factor of four each time. The sentence that one cannot be localised in time and frequency at once is therefore not, as it stands, a theorem. It becomes one only after choosing the ruler by which localisation is measured: variance gives Kennard, measure of support gives Benedicks, rate of decay gives Hardy. These are different theorems. This has the shape of the brake Paper 106 places on stationarity and fat tails. A tool built on variance works only in a world that has one. Fourth, the discrete side has an arithmetic root. On a finite cyclic group the product of the numbers of non-zero entries is at least the size of the group (Donoho-Stark). The additive form, that the sum of those numbers is at least the size of the group plus one, holds only when the size is prime (Tao). Building evenly spaced picket fences, this paper shows that at sizes 9, 12 and 16 the product attains exactly the size of the group while the sum is 6, 7 and 8 against the required 10, 13 and 17. So for composite sizes the additive form is false. At sizes 7, 11 and 13, by contrast, four thousand random trials with randomly chosen support sizes give minimum sums of 8, 12 and 14, exactly the size plus one. What supplies the improvement is not analysis. A picket fence can be built precisely because the size has a proper divisor, and primality forbids it. An arithmetic condition sits inside a theorem of analysis. Paper 86 listed four roots of necessity - topology, symmetry, algebra, dimension - and what appears here is none of them. It is the factorisability of an integer. Fifth, lay eight of them side by side and every one measures something different: variance (Kennard, Weyl), the expectation of a commutator (Robertson), rate of decay (Hardy), measure of the support (Amrein-Berthier, Benedicks), entropy (Hirschman-Beckner), count of non-zero entries (Donoho-Stark), primality (Tao), and a lifetime (Mandelshtam-Tamm). None implies the others. The variance can be infinite while the support is finite (the rectangular pulse), and the support can be infinite while the variance is finite (a Gaussian). The two rulers hold counterexamples against each other. As for the entropic form, this paper computes the sum of the differential entropies of the two probability densities for three Gaussians of different widths and obtains 0.306853 in every case, that is the logarithm of e over two. A different functional with a different extremum is a different theorem. Sixth, the energy-time relation is not the same theorem as the position-momentum one. Robertson's inequality is a statement about two self-adjoint operators, and time is not such an operator (Papers 54, 127). What the Mandelshtam-Tamm relation gives is a timescale over which some observable changes, not a variance of time. The same symbol is serving as the name of a different quantity, the same shape as the argument about logical types that Paper 118 makes for its three clocks. Finally, what then was Fourier saying? A statement about linear waves. Sound, water waves and price series alike: a short pulse occupies a wide band. This is a fact engineering uses daily and it has no bearing on quantum mechanics. Quantum mechanics added two things - that a particle is described by a wave, and that momentum is hbar times wavenumber. Whatever makes uncertainty feel like a principle was brought by those two, not by the inequality. What nevertheless remains surprising is that hbar is finite. Let hbar tend to zero and the bound vanishes. The bound exists because of mathematics, but it bites because hbar is finite. Mathematics supplies the shape; physics supplies the scale. IMPORTANT (scope): a census of established standard material (Layer 2). Every theorem cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the uncertainty relations are in error (they are not) / that quantum mechanics is dispensable (it is not; what is separated is where the mathematics ends and the physics begins). What is added: separating the mathematical one half from the physical hbar, judging the two four pi to share one root, producing two witnesses for the vacuity of the variance form, entering primality into the census as a root, and separating energy-time as a different logical type. What remains: mathematics supplies the shape and physics supplies the scale. Until the ruler is named, nothing has yet been said. 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. ----- ℏ は、不等式の数学の側には無い——不確定性は一つの定理ではない。 「不確定性原理」という一語が、測っているものの違う複数の定理を指している。本稿はそれを点呼し、それぞれの住所を書く。新しい数学定理も新しい物理法則も主張しない。ケナード(1927)、ワイル(1928)、ロバートソン(1929)、ハーディ(1933)、ハーシュマン(1957)とベックナー(1975)、アムラン=ベルティエ(1977)とベネディクス(1985)、ドノホ=スターク(1989)、タオ(2005)、マンデルシュタム=タム(1945)は、いずれも標準的な内容である。 第一に、これが本稿の芯である。ℏ は不等式の数学の側には無い。時間幅と角周波数幅の積についての不等式——その下限が二分の一であること——は、二乗可積分な関数とそのフーリエ変換についての定理であり、物理定数を一つも含まない。本稿ではガウス関数がこれを等号で満たすこと、しかも幅を表す係数が消えて二分の一になることを記号計算で確認した。位置と運動量の関係が ℏ の半分を下限に持つのは、ド・ブロイとプランクの関係——運動量は ℏ 掛ける波数である——という物理の同定を一度だけ通したからである。二分の一は数学が出し、ℏ は物理が出す。「量子」であるのは、この一度の代入だけである。 第二に、その二分の一すら規約を含んでいる。周波数を「一周あたり」で書けば、同じ定理の下限は 4π 分の一になる。ここに現れる 4π は 2 掛ける 2π であり、2 は極値の二分の一、2π は一周とラジアンの換算である。そしてプランク定数で書いたときの h 割る 4π の 4π も、同じ 2 掛ける 2π である——ℏ が h を 2π で割ったものだからである。すなわち二つの 4π は別根ではなく、同じ根である。そしてどちらも立体角ではない。反例は次元非依存性である:下限は空間次元を変えても各成分について同じ形で成り立ち、一次元の直線上の粒子にも同じ下限がかかるが、そこに球面は無い。立体角であれば下限は次元とともに動くはずである。動かない。これは論文46 が置いた形と同じである——フーリエ変換の規約を変えれば数は動く。規約が動かせる数に、幾何を読み込んではならない。 第三に、分散版は、分散が発散すれば何も言っていない。コーシー分布は確率密度でありながら二次モーメントが発散する(本稿で確認)。矩形パルスは時間側の台が有限なのに、そのフーリエ変換の二乗が周波数の二乗でしか減衰しないため、周波数側の分散が発散する(本稿では窓幅を四倍ずつ広げ、推定値が 1137、4541、18172 とほぼ四倍ずつ増えることを確認した)。すなわち「時間にも周波数にも同時に局在することはできない」という文は、そのままでは定理ではない。定理になるのは、どの物差しで「局在」を測るかを決めてからである。分散で測るならケナード、台の測度で測るならベネディクス、減衰の速さで測るならハーディ——別の定理である。これは論文106 が定常性と裾の厚さについて置いたブレーキと同じ形である。分散を使う道具は、分散が有る世界でしか使えない。 第四に、離散側には算術の根がある。有限巡回群の上では、非零成分の個数の積が群の大きさ以上になる(ドノホ=スターク)。これに加法版——非零成分の個数の和が群の大きさプラス一以上になる——が成り立つのは、群の大きさが素数のときだけである(タオ)。本稿では等間隔の櫛形を作り、大きさ 9、12、16 のとき積がちょうど群の大きさを達成し、和が 6、7、8 と、要求される 10、13、17 を大きく下回ることを示した。すなわち合成数では加法版は偽である。一方、大きさ 7、11、13 では、台の大きさを無作為に選んだ四千通りの試行で和の最小値がそれぞれ 8、12、14、すなわちちょうど群の大きさプラス一であった。改良を与えているのは解析ではない。櫛形が作れるのは群の大きさが真の約数を持つからであり、素数はそれを禁じる。解析の定理の中に、算術の条件が入っている。論文86 は必然の根を四つ——位相・対称性・代数・次元——挙げたが、ここに現れているのはそのどれでもない。整数の分解可能性である。 第五に、八つ並べると、測っているものが全部違う。分散(ケナード・ワイル)、交換子の期待値(ロバートソン)、減衰の速さ(ハーディ)、台の測度(アムラン=ベルティエ・ベネディクス)、エントロピー(ハーシュマン=ベックナー)、非零成分の個数(ドノホ=スターク)、素数性(タオ)、寿命(マンデルシュタム=タム)。どれも他を含意しない。分散が無限でも台は有限でありうるし(矩形パルス)、台が無限でも分散は有限でありうる(ガウス関数)。二つの物差しは、互いに反例を持ち合っている。エントロピー版については、本稿で幅の異なる三つのガウス関数について確率密度の微分エントロピーの和を数値計算し、いずれも 0.306853、すなわち自然対数の底を二で割ったものの対数を得た。別の汎関数、別の極値——別の定理である。 第六に、エネルギーと時間の関係は、位置と運動量の関係と同じ定理ではない。ロバートソンの不等式は二つの自己共役作用素についての言明であり、時間はそのような作用素ではない(論文54・127)。マンデルシュタム=タムの関係が与えるのは、ある観測量が変化するのに要する時間の尺度であって、時間の分散ではない。同じ記号が別の量の名として使われている——論文118 が三つの時計について述べた論理型の議論と同型である。 最後に、ではフーリエは何を言っていたのか。線形の波についての言明である。音でも、水面の波でも、価格の時系列でも同じであり、短いパルスは広い帯域を占める。これは工学が毎日使っている事実であって、量子力学とは関係がない。量子力学が加えたのは二つ——粒子が波として記述されること、そして運動量が ℏ 掛ける波数であること。不確定性の「原理」らしさは、この
A no-signaling theory may, in principle, be more nonlocal than quantum mechanics: the Popescu--Rohrlich box reaches the algebraic Clauser--Horne--Shimony--Holt value 4, whereas nature stops at the Tsirelson value 22. Absolute Frame Theory (AFT) produces nonlocal correlations---two systems entangled in M share a single point of the substratum A (Axiom II)---and forbids superluminal signaling (the channel is Bremermann-bounded). We ask where those correlations fall, and we establish, by two independent routes, that they cannot exceed the Tsirelson bound: AFT admits no super-quantum correlations. The operator route proves, inside the reconstructed Hilbert theory, the Tsirelson operator identity, with the local commutativity [Ai,Bj]=0 itself derived from AFT structure---the tensor factorization that local tomography supplies and the field microcausality of the constrained embedding---rather than assumed. The information route reduces the bound to a physical principle: AFT's shared resource is read out through a completely positive, trace-preserving channel of finite classical capacity (a theorem of the channel, founded on action quantization and the trace over the inaccessible fiber), this channel obeys the data-processing inequality, and the combination places AFT in the class that satisfies Information Causality, which singles out 22 among the no-signaling theories. We state plainly what this does and does not achieve. It does not close the question of why the bound is 22 rather than some other value: both routes, chased to their foundations, terminate at the same posited fact---the M--A signature transition (the Wick rotation) that installs the complex-Hilbert structure---and finite capacity alone, stripped of that structure, does not exclude a Popescu--Rohrlich box. The two routes are thus the operator-algebraic and information-theoretic faces of one ceiling, and they reduce the selection of 22 to a single substratum fact, with the residue an instance of the deepest open question of the programme rather than a problem specific to nonlocal correlations.