hbar Is Not on the Mathematical Side of the Inequality ── The Uncertainty Principle Is Not One Theorem: A Distinct-Root Census over Variance, Decay, Support, Entropy and Primality (the Yamagishi Addresses of Uncertainty) ── [Paper 129]
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
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
In 1927, Werner Heisenberg published the inequality that bears his name: Δx · Δp ≥ ℏ/2. The product of position and momentum uncertainties has a floor. No matter how the state is prepared or how the measurement is performed, the uncertainty in position multiplied by the uncertainty in momentum cannot drop below half of Planck’s reduced constant. The floor is sharp — Gaussian wavepackets saturate it, and no state goes below. The inequality is one of the most directly empirical structures in physics. It governs the diffraction of single particles through slits, the resolution limits of microscopes, the zero‑point energy of harmonic oscillators, the stability of atomic ground states, and the sensitivity of interferometric measurements. Any technology pushing against the limits of precision measurement runs into ℏ/2 sooner or later. The question is where the bound comes from, and why the constant is exactly ℏ/2. In standard QM, the inequality follows from the canonical commutation relation [x̂, p̂] = iℏ plus Cauchy–Schwarz applied to variance operators. The argument is short, mechanical, and gives the right answer — but leaves deeper questions unanswered. Why is position–momentum a conjugate pair? Why does the commutation relation hold? Why is the constant in the commutator exactly ℏ? Each points back to structural commitments standard QM postulates: the operators, the commutation relation, the Hilbert space they act on. This walkthrough shows how the Event Density framework derives the structural commitments themselves. The Hilbert space comes from U2 (forced inner product on the participation‑measure space). The momentum operator p̂ = −iℏ∇ comes from Stone’s theorem applied to spatial translation symmetry (U5). The canonical commutation relation [x̂ᵢ, p̂ⱼ] = iℏ δᵢⱼ comes automatically from that operator form. The Fourier conjugacy of position and momentum representations comes from the same Stone identification. Like Bell–Tsirelson, Heisenberg is a place where the framework’s contribution is upstream. The bound itself is mathematical physics — a theorem about Fourier transforms on L² spaces. What the framework adds is forcing the structures the standard argument requires. With Hilbert space, momentum operator, and Fourier conjugacy all derived rather than postulated, the standard Hardy–Folland–Sitaram Fourier‑uncertainty argument applies, and Δx · Δp ≥ ℏ/2 emerges as a derived consequence of substrate ontology, all the way down.
In 1927, Werner Heisenberg published the inequality that bears his name: Δx · Δp ≥ ℏ/2. The product of position and momentum uncertainties has a floor. No matter how the state is prepared or how the measurement is performed, the uncertainty in position multiplied by the uncertainty in momentum cannot drop below half of Planck’s reduced constant. The floor is sharp — Gaussian wavepackets saturate it, and no state goes below. The inequality is one of the most directly empirical structures in physics. It governs the diffraction of single particles through slits, the resolution limits of microscopes, the zero‑point energy of harmonic oscillators, the stability of atomic ground states, and the ultimate sensitivity of interferometric measurements. Any technology pushing against the limits of precision measurement runs into ℏ/2 sooner or later. The question is where the bound comes from, and why the constant is exactly ℏ/2. In standard QM, the inequality follows from the canonical commutation relation [x̂, p̂] = iℏ together with Cauchy–Schwarz applied to variance operators. The argument is short, mechanical, and gives the right answer — but leaves deeper questions unanswered. Why is position–momentum a conjugate pair? Why does the canonical commutation relation hold? Why is the constant in the commutator exactly ℏ? Each points back to structural commitments standard QM postulates: the operators, the commutation relation, the Hilbert space they act on. This walkthrough shows how the Event Density framework derives the structural commitments themselves. The Hilbert space comes from U2 (forced inner product on the participation‑measure space). The momentum operator p̂ = −iℏ∇ comes from Stone’s theorem applied to spatial translation symmetry (U5). The canonical commutation relation [x̂ᵢ, p̂ⱼ] = iℏ δᵢⱼ comes automatically from that operator form. The Fourier conjugacy of position and momentum representations comes from the same Stone identification. Like Bell–Tsirelson, Heisenberg is a place where the framework’s contribution is upstream. The bound itself is mathematical physics — a theorem about Fourier transforms on L² spaces. What the framework adds is forcing the structures the standard argument requires. The walkthrough does not introduce a new derivation hinge in the way Born or Schrödinger did; it shows that an established result of standard QM survives and remains derivable when the underlying structure is itself derived from substrate primitives. With Hilbert space, momentum operator, and Fourier conjugacy all derived rather than postulated, the standard Hardy–Folland–Sitaram argument applies, and Δx · Δp ≥ ℏ/2 emerges as a derived consequence of substrate ontology, all the way down.
There are states about which the whole is completely known and the parts are not known at all. Paper 88 wrote that under the single word entropy there are two different quantities, the coarse-grained one and the horizon-area one. This paper adds a third address: the von Neumann entropy. NO NEW PHYSICAL LAW IS CLAIMED. The von Neumann entropy, subadditivity, the Araki-Lieb triangle inequality and the negativity of the conditional entropy are all standard. Nothing in Paper 88 is retracted: the separation of the two entropies stands, and a third is simply added. First, it is not Shannon's entropy quantised. Shannon's entropy requires a probability distribution, and a quantum state does not come with one, since a different distribution appears for each basis one measures in. Collecting the Shannon quantity over twenty thousand measurement bases for random single-qubit mixed states, the value measured in the eigenbasis equalled the von Neumann entropy to machine precision, with a difference of zero in all four cases, while random bases only approached it from above. The von Neumann entropy is therefore the smallest Shannon quantity obtainable by measurement: in any other basis, more disorder is reported than the state actually holds. The two start from different data - Shannon is handed a distribution and returns a number, while von Neumann is handed a state and chooses the distribution himself. Second, unitary evolution does not change it at all. Over four thousand random four-dimensional density matrices and random unitaries, the largest change was 2.33 times ten to the minus fifteen. It is exactly conserved for a closed system. So the von Neumann entropy cannot by itself be the arrow of time: the arrow requires something non-unitary, a partial trace, a measurement or a coarse-graining (Papers 77, 78 and 88). Third, and this is the core. The whole can carry less than a part. For the singlet state the entropy of the pair is zero, being a pure state, while either half alone has entropy 0.6931471806, that is the natural logarithm of two. Classically this cannot happen: drawing two hundred thousand random two-by-two joint distributions and measuring the entropy of the whole minus the larger of the two parts, the smallest value found was 0.007226 and it was never once negative. Classically, knowing the whole includes knowing the parts; quantum mechanically it does not, and there are states about which the pair is completely known and either half alone is not known at all. This is not a statement that quantum systems are more disordered: an entropy of zero for the pair says there is no disorder whatever. It says only that the information has been placed in the correlation rather than in the parts. Fourth, the conditional entropy therefore turns negative. For the singlet it is minus 0.6931471806, that is minus the logarithm of two, while over the same two hundred thousand classical draws the smallest value was 0.009555. The classical conditional entropy is bounded below by zero and the quantum one reaches minus the logarithm of two - the two do not even share a sign convention, being written with the same symbol and called by the same name while having different ranges. Negative entropy is not a figure of speech but has an operational meaning, readable as a resource for future use in quantum state merging; this paper does not enter that interpretation, since what can be written is that the value is negative, and its address. Fifth, what stands in its place. Over twenty thousand random two-qubit states of random rank, subadditivity and the Araki-Lieb triangle inequality were violated zero times, and the singlet lies exactly on the boundary of the triangle inequality, the difference between the parts being zero and the whole being zero as well. The classical inequality is not weakened but replaced. The census. The coarse-grained entropy lives in the observer's resolution and is moved by changing the coarse-graining (Papers 77, 78, 88). The horizon-area entropy lives in the boundary area and dimension and is moved by changing the area (Papers 73, 88). The von Neumann entropy lives in the spectrum of the density matrix and is moved only by a non-unitary operation. Three quantities under one word: the first lives in the observer, the second in the boundary, and the third in the state itself. Of the three, only the third is unchanged by the passage of time. Entanglement entropy is known to obey an area law in field theory, so there is a rhyme between the third and the second. This paper does not write that they share a root: one is fixed by the choice of subsystem, the other by geometry. The line Paper 88 drew, that the only relation is a bound, is not crossed here either. IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new physical law (there is none) / any retraction of Paper 88 (there is none) / an interpretation of negative entropy (not entered). What is added: measuring that the eigenbasis Shannon quantity equals the von Neumann entropy, checking unitary invariance numerically, confirming on two hundred thousand samples that the classical inequality never fails and setting it against the quantum minus logarithm of two, and writing for each of the three entropies what must be moved for it to move. What remains: there are states about which the whole is completely known and the parts are not known at all. In classical language that sentence contradicts itself. It does not contradict itself because the information has been placed not in the parts but between them. 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. ----- 全体について完全に知っていながら、部分については何も知らない——そういう状態がある。 論文88 は「エントロピー」という一語のもとに二つの別物——粗視化のものと地平線の面積のもの——があると書いた。本稿は第三の住所を加える:フォン・ノイマンのエントロピーである。新しい物理法則は主張しない。フォン・ノイマンのエントロピー、劣加法性、荒木=リーブの三角不等式、条件付きエントロピーの負値は、いずれも標準的な内容である。⚠️ また論文88 の内容を撤回しない——二つのエントロピーの分離はそのままで、三つ目を足すだけである。 第一に、これはシャノンを「量子化」したものではない。シャノンのエントロピーには確率分布が要る。ところが量子状態には、分布が一つ付いてはいない——どの基底で測るかによって違う分布が出るからである。無作為な一量子ビットの混合状態について二万通りの基底で測ってみると、固有基底で測ったときのシャノン量がフォン・ノイマンのエントロピーに機械精度で一致し(四例とも差ゼロ)、無作為な基底は上からしか近づかなかった。すなわちフォン・ノイマンのエントロピーは「測ったときのシャノン量の最小値」である。どの基底で測っても、状態が実際に持っている以上の乱雑さが報告される。⭐ 出発点が違うのである——シャノンは分布を渡されて数を返し、フォン・ノイマンは状態を渡されて分布そのものを自分で選ぶ。 第二に、ユニタリ発展では一切変わらない。四次元の無作為な密度行列と無作為なユニタリを四千組取って測ると、最大変化は 2.33 かける十のマイナス十五乗であった。閉じた量子系では厳密に保存される。ゆえにフォン・ノイマンのエントロピーは、それ自体では時間の矢になりえない——矢が要るのは非ユニタリな操作、すなわち部分トレース・測定・粗視化のほうである(論文77・78・88)。 第三に、これが本稿の芯である。全体のほうが部分より小さいことがある。一重項状態では、二つ合わせた系のエントロピーがゼロ(純粋状態)なのに、片方だけを見るとその値は 0.6931471806、すなわち ln 2 になる。古典では起こりえない:二かける二の同時分布を二十万個無作為に引いて、全体のエントロピーから部分の大きいほうを引いた値を測ると、最小でも 0.007226 で、一度も負にならなかった。古典では、全体を知ることは部分を知ることを含む。量子ではそうならない——二つを合わせた系については完全に知っていて、片方だけを見ると何も知らない、という状態が存在する。⚠️ ただしこれは「量子のほうが乱雑だ」という話ではない。全体のエントロピーがゼロとは、乱雑さが皆無だということである。情報が部分にではなく相関の側に置かれている、というだけである。 第四に、したがって条件付きエントロピーが負になる。一重項では、全体から部分を引いて マイナス 0.6931471806、すなわち マイナス ln 2 である。同じ二十万個の古典分布では最小でも 0.009555 だった。古典の条件付きエントロピーは下からゼロで抑えられ、量子のそれは マイナス ln 2 に届く——二つは符号の約束すら共有していない。同じ記号で書かれ、同じ名前で呼ばれているが、値域が違う。⚠️ なお「負のエントロピー」は言葉の綾ではなく操作的な意味を持つ(量子状態併合における「将来使える資源」として読める)が、本稿はその解釈には立ち入らない。書けるのは、値が負になるという事実と、その住所までである。 第五に、代わりに何が残るか。無作為な二量子ビット状態を二万個(階数も無作為に)取って調べると、劣加法性も荒木=リーブの三角不等式も破れがゼロだった。しかも一重項は三角不等式の境界にちょうど乗っている(部分どうしの差がゼロ、全体もゼロ)。古典の不等式は弱められたのではなく、別の不等式に置き換わっているのである。 点呼をしておく。粗視化のエントロピーは観測者の解像度に住み、動かすには粗視化を変える(論文77・78・88)。地平線の面積エントロピーは境界の面積と次元に住み、動かすには面積を変える(論文73・88)。フォン・ノイマンのエントロピーは密度行列のスペクトルに住み、動かすには非ユニタリな操作が要る。一語の下に三つの量があり、一つ目は観測者に、二つ目は境界に、そして三つ目は状態そのものに住んでいる。三つのうち、時間の経過で変わらないのは三つ目だけである。 ⚠️ 絡み合いエントロピーが場の理論で面積則を示すことは知られており、三つ目と二つ目のあいだに韻がある。だが本稿はそれを同根とは書かない——片方は部分系の選び方で決まり、他方は幾何が決める。論文88 が「結ぶのは上限だけ」と書いた線を、ここでも越えない。 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)/論文88 の撤回(一つもない)/負のエントロピーの解釈(立ち入らない)。加えたもの:固有基底のシャノン量が S(ρ) に一致することを測ったこと、ユニタリ不変性を数で確かめたこと、古典側の不等式が破れないことを二十万個の標本で確認して量子側の マイナス ln 2 と対置したこと、三つのエントロピーに「動かすと動くもの」を書いたこと。残るもの:全体について完全に知っていながら、部分については何も知らない——そういう状態がある。古典の言葉では、この文は矛盾している。矛盾していないのは、情報が部分にではなく、部分と部分のあいだに置かれているからである。 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。
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 1927 Werner Heisenberg derived the uncertainty principle delta_x * delta_p >= hbar/2 from the commutation relation [x, p] = ihbar and the Cauchy-Schwarz inequality. He motivated it through the gamma-ray microscope thought experiment: measuring position precisely requires high-energy photons that impart momentum kicks. But the principle holds even for ideal measurements that cause no disturbance — it is a property of quantum states, not of measurement apparatus. The true origin is deeper: the commutation relation [x, p] = ihbar itself. What is its physical origin? Why ihbar and not zero? For 100 years this has had no mechanical answer. This paper establishes that within the Photonic Universe Hypothesis, the Heisenberg uncertainty principle is a geometric property of the E8 Phase I condensate. Position x and momentum p are canonically conjugate variables in the E8 Lie-Poisson bracket. The commutator [x, p] = ihbar is the quantisation of the classical Poisson bracket {x, p} = 1, with hbar = E_Pt_P being the minimum E8 lattice action (Theorem 176). The uncertainty product delta_x * delta_p >= hbar/2 is derived three independent ways: (1) from the E8 Lie-Poisson bracket and canonical quantisation; (2) from the Fourier duality of E8 bow waves (Theorem 187): sharp position requires a localised bow wave packet and sharp momentum requires a plane wave — these are mutually exclusive; (3) from the U(1) phase-number uncertainty of the E8 phase mode. The factor hbar/2 (not hbar) comes from the QHO ground state energy hbaromega/2, arising from the canonical commutation relation [a, a+] = 1 (Theorem 174). The energy-time uncertainty delta_E * delta_t >= hbar/2 follows from the E8 phase-mode dispersion relation E = hbar*omega. The uncertainty principle is not about measurement disturbance. It is about E8 lattice geometry. Four falsifiable predictions follow.
The uncertainty principle (linked to p,x or E,t) plays an important role in quantum mechanics. For example, an electron with an exact momentum p is described by a wavefunction exp(ipx). This function exists at all x points and so it is suggested in the literature that there is complete uncertainty as to the electron’s position. (More specifically P(x)=exp(-ipx)exp(ipx) normalization constant=constant and so all x points have the same probability.) Thus if p is completely certain, it is said that position x is completely unknown. We note, however, that exp(ipx) consists of cos(px) and sin(px) which are repeated probability distributions (with zero endpoints). Thus even though they exist in all of space, due to repetition which in turn is due to spatial invariance, there is a specific length number, namely the wavelength. Thus we argue that in classical physics there is a set of points (x,p), but in quantum mechanics ( p , x with its dx interval), with the latter being equal to hbar/p. Then the only uncertainty is within the dx interval i.e. the wavelength and spatial density is linked to momentum i.e. there is dynamical probability. In particular, there is a frequency hbar/E and a wavelength hbar/p which define an internal clock and ruler and the two are independent. Thus within a wavelength one has a probabilistic picture (in order to have endpoints defining the dx interval) and no sense of time, yet on average time exists because of hbar/E and the particle moves in space with a constant v. This begs the question: How do time and space x arise in such a scenario? We speculate the following. Assume there exists a property called energy which defines an object. This object, however, may break into two different objects. As a result, one introduces a parameter called time to distinguish between the two systems (E) and (E1,E2). For a certain time range one has a single object, while for a later time interval, one has two objects. Each of the two objects in the second system, however, needs to be distinguished as well and so another parameter, namely space x is used for this. If the two objects existed at the same x and t, how would one know there were two unless another parameter were introduced? We use t and x as the distinguishing parameters and furthermore give these a measure i.e. dt and dx. We then argue that from these ideas one may show the notion of speed and direction (because there are two objects) as well as conservation of momentum arises. Given the notion of an interval, we argue it must be described by a probability distribution and we link this to constant/p and exp(ipx). We finally note that since spatial density is linked to dynamical probability, one must consider motion associated with a density which may lead even to collective motion.