Time Is One-Dimensional Because Prediction Demands It, Not Because a Law Says So ── The Same Procedure, With Only the Signature Changed, Gives Amplifications of 0.012 and 3.5 × 10^32 ── [Paper 253]

Abstract

Why is time one-dimensional? The answer offered here is that no law decrees it; the requirement that one be able to predict allows nothing else. The equations of a universe with two times can be written, and their solutions exist. What breaks is prediction. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): no new mathematical theorem and no new law is claimed. Hadamard's three conditions for well-posedness, the ill-posedness of the Cauchy problem for Laplace's equation, the ill-posedness of the backward heat equation, the Cauchy problem for ultrahyperbolic equations, and the classification of second order operators by signature are all standard. It is not proved that spacetime must be 3+1 ── the discipline is that of Paper 69, and what is done here is to add one entry to its ledger. No anthropic argument is made ── the phrase because there are observers is never used. No measured value is cited ── every number is computed from a definition. No theory of partial differential equations is built ── only exact solutions and a finite Fourier representation are used. Prediction is not defined ── what is treated is the single point of continuous dependence on the data. It is not claimed that ultrahyperbolic equations have no solutions ── solutions exist; what fails is uniqueness and continuous dependence. The arrow of time is not solved ── what Paper 98 recorded as open is left untouched. Quantum theory is not treated. The relation to earlier papers. Paper 69 wrote why 3+1 as an overdetermination and assembled five independent roots selecting four dimensions; not one of them selects which of the four is time, and this paper fills that empty place. Paper 159 showed that ill-posed is not one word and placed the separator in the decay of the singular values; this paper applies that same separator to the number of time dimensions. Paper 154 showed that a limit without its order is not a quantity; that concerns the order of limits and this the signature. Papers 118, 119 and 122 showed that the one word time covers four logical types; what is asked here is not the type but the number. Paper 251 separated special in two dimensions; this paper stands beside it, on the side of special in one. First, being solvable and being predictable are different demands. Hadamard wrote what it is for a problem to be well-posed as three conditions: that a solution exists, that it is unique, and that it depends continuously on the data. The three are independent; the first two are about whether it can be solved, and what corresponds to prediction is the third alone. Why the third is prediction: initial data is measured and then entered, and measurement always carries error. If the error changes the answer, then holding a formula for the solution one still cannot state tomorrow's value. Second, with no time dimension the initial value problem explodes. Take Laplace's equation and, treating the vertical coordinate as the time, solve it as an initial value problem (Hadamard's example). With data of size one over n, the amplification at unit height is 1.101 times ten to the third at n=10, 1.213 times ten to the seventh at n=20, 2.942 times ten to the fifteenth at n=40, and 3.463 times ten to the thirty-second at n=80. The data tends to zero and the solution diverges. Neither existence nor uniqueness has failed ── there is a solution and it is unique. What has failed is the third condition alone. Third, with one time dimension the same procedure stays bounded. Change the equation to the wave equation; one sign has been changed and nothing else. With the same data the amplification is 0.0544 at n=10, 0.0457 at n=20, 0.0186 at n=40 and 0.0124 at n=80. At the same n=80 that is 3.463 times ten to the thirty-second against 0.0124 ── thirty-four orders of magnitude. What changed is one sign in the equation, and not the data, not the method, not the precision. What makes the difference is the signature. Fourth, this is the core. One and the same heat equation exchanges well-posedness for ill-posedness when only its direction is reversed. Mode n is multiplied by the exponential of minus n squared t. At n=80 that is 1.604 times ten to the minus twenty-eighth forwards against 6.235 times ten to the twenty-seventh backwards. Running it on a grid of 512 points, the maximum going forwards stays below one at 0.9759, 0.8944 and 0.8165, while backwards it grows to 2.073 times ten to the eleventh, 2.417 times ten to the hundred and twenty-third, and 1.304 times ten to the two hundred and sixty-fourth, overflowing double precision before reaching t=0.05. To measure what this means for prediction, relative noise of ten to the minus tenth ── standing for observational error ── is added to the data and that component alone is sent both ways: forwards it decays to 9.398, 5.403 and 3.835 times ten to the minus eleventh, while backwards it has grown to 4.135 times ten to the seventeenth already at t=0.001. The signal is buried; one holds the same formula for the solution and cannot state a value. Here is the core: there is no asymmetry on the side of the law. The heat equation is one equation, and reversing time does not turn it into another. The asymmetry is on the side of well-posedness. This does not explain the arrow of time ── as Paper 98 recorded honestly, why there is a low entropy past is unsolved. What can be said here is one step short of that: the phenomenon of being able to predict one way and not the other does not itself require an asymmetric law. Fifth, with two time dimensions what happens next is not determined. Giving time two dimensions, a plane wave gives the dispersion relation that the sum of the squares of the two frequencies equals the square of the wave number. With one time the same procedure returns two values, plus and minus the wave number; with two it returns a whole circle in the frequency plane, a continuum. So the data does not determine what happens next. What has failed this time is uniqueness ── in the third section it was continuous dependence. One word, ill-posed, is naming two different failures (Paper 159). Sixth, one measure separates them: the signature of the principal symbol. Counting the signs of the eigenvalues, (4,0) is elliptic and ill-posed, (3,1) is hyperbolic and well-posed, and (2,2) and (1,3) are ultrahyperbolic and ill-posed. Only one time dimension is well-posed. Since (1,3) is (3,1) with the overall sign reversed and means the same physics, what is to be counted is the size of the smaller sign class. One thing follows: it is not that time is special. The sign class with only one member is what we call time. The direction looks reversed because the definition comes first and time second. Seventh, one independent entry is added to the census of Paper 69. The roots it assembled ── conformal invariance of the Maxwell action, graviton degrees of freedom, exotic four-space, Bertrand and Ehrenfest stability, the maximum of the ball volume ── all select how many, and not one of them selects which of them is time. That is the empty place this paper fills. And this root cannot be derived from the other five: conformal invariance, graviton degrees of freedom and exotic four-space are all statements made after a signature has been assumed, and well-posedness stands on the side of that assumption. They are distinct roots (Paper 58). Even so, 3+1 is not proved here. What can be said reaches no further than that prediction is not an available activity unless there is exactly one time dimension, and adds nothing about why space has three. Closing. Time is one-dimensional, and not because time is special. The equations of a two-time universe can be written and their solutions exist. What breaks is prediction ── add a dimension and the answer stops being unique; remove one and measurement error buries it. There is exactly one signature in which the word tomorrow means anything. And the fifth section showed that even inside that one, one direction permits prediction and the other does not ── without once invoking an asymmetric law. 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. ----- 時間はなぜ一本なのか。本稿の答は、そう決めた法則があるからではなく、「予言できる」という要求がそれしか許さないから、である。時間が二本ある宇宙の方程式は書ける。解も存在する。壊れるのは予言のほうである。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない。アダマールの適切性の三条件、ラプラス方程式のコーシー問題の不適切性、後ろ向き熱方程式の不適切性、超双曲型方程式のコーシー問題、二階作用素の符号数による分類は、いずれも標準的である。時空が 3+1 でなければならないとは証明しない──論文69 と同じ規律であり、本稿がするのはその台帳に一本足すことだけである。人間原理を立てない──「観測者がいるから」とは一度も言わない。測定値を引かない──本稿の数はすべて定義から計算したものである。偏微分方程式の理論を作らない──使うのは厳密解と有限次元のフーリエ表示だけである。「予言」を定義しない──扱うのは初期条件への連続依存という一点であって、認識論には立ち入らない。超双曲型方程式に解が無いとは言わない──解は存在する。壊れるのは一意性と連続依存である。時間の矢を解かない──論文98 が未解決と書いたことに手をつけない。量子論を扱わない。 既刊との関係。論文69 は「なぜ 3+1 次元か」を過剰決定として書き、D=4 を選ぶ五つの独立な根を並べた。だがそのどれも、四つのうちどれが時間かを選んでいない。本稿はその空いた位置に一本足す。論文159 は「不適切」が一語でないことを示し、分離子を特異値の落ち方に置いた。本稿は同じ分離子を時間の本数に当てる。論文154 は順序を書かない極限は量ではないと示した。あちらは極限の順序、こちらは符号数であり、別の軸である。論文118・119・122 は「時間」という一語が四つの論理型を覆うことを示した。本稿が問うのは型ではなく本数である。論文251 は「二次元だけ特別」を分けた。本稿はその隣、一次元だけ特別の側に立つ。 第一に、「解ける」と「予言できる」は別の要求である。アダマールは、問題が適切であることを三つの条件で書いた──解が存在すること、解が一つに決まること、初期条件に連続に依存すること。三つは独立であり、前二つは解けるかどうかの話で、予言に対応するのは三つ目だけである。なぜ三つ目が予言なのか。初期条件は測って入れるものであり、測定には必ず誤差がある。誤差が答を変えてしまうなら、解の式を持っていても、明日の値を言うことができない。 第二に、時間が 0 本だと初期値問題が爆発する。ラプラス方程式を取り、縦の座標を時間だと思って初期値問題として解く(アダマールの例)。初期値の大きさが 1/n のとき、y=1 での増幅率は n=10 で 1.101×10^3、n=20 で 1.213×10^7、n=40 で 2.942×10^15、n=80 で 3.463×10^32 であった。初期値は 0 に向かっているのに、解は発散する。しかも存在も一意性も壊れていない──解はあり、一つに決まる。壊れているのは三つ目だけである。 第三に、時間が 1 本だと同じ手順が有界に収まる。方程式を波動方程式に替える。符号を一つ変えただけである。同じ初期値を入れると、増幅率は n=10 で 0.0544、n=20 で 0.0457、n=40 で 0.0186、n=80 で 0.0124 になった。同じ n=80 で 3.463×10^32 と 0.0124 ──34 桁の差である。替えたのは方程式の一つの符号であって、初期条件でも解き方でも精度でもな