Abstract
That a weather forecast cannot reach beyond a certain horizon is due neither to missing equations nor to slow computers. This paper asks what sets the limit——the answer is one number, the error doubling time $\tau_d$. No new mathematical theorem and no new law is claimed.
First, the price per day is a fixed factor. With $\tau_d=1.5$ days, each extra day costs $\mathbf{1.5874}$ times the initial accuracy (Section 2).
Second, this is the core of the paper. Going from $14$ to $21$ days costs $\mathbf{25.40}$ times; to $30$ days, $\mathbf{1625.5}$ times; to $60$ days, $\mathbf{1.70\times10^{9}}$ times (Section 2).
Third, read backwards, the return is logarithmic. Observing $\mathbf{10}$ times more precisely gains only $\mathbf{4.98}$ days (Section 3).
Fourth, even $10^{9}$ times gains only $44.85$ days (Section 3).
Fifth, the familiar “about two weeks” comes from here. If the initial error is $10^{-3}$ of saturation, the forecastable span is $\mathbf{14.95}$ days (Section 4).
Sixth, the separator is $\tau_d$ itself. At $\tau_d=1.0$ day the same extension costs $\mathbf{65536}$ times; at $2.5$ days only $\mathbf{84.4}$——everything rides on one number (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——exponential error growth, the doubling time, the limit of predictability, and the value $\tau_d\approx1.5$ days are all standard. We do not build meteorology——all we use is one exponential and its inverse. We do not discuss chaos——the Lorenz equations, attractors, and bifurcations are not treated at all. We do not discuss numerical weather prediction——grid resolution, parameterisation, and data assimilation are not treated. We do not say the error grows exactly exponentially——$e^{\lambda t}$ holds only while the error is small, and growth stops near saturation. The computations here are confined to the linear-growth regime. We assert no value for $\tau_d$——$1.5$ days is a representative value widely used in the literature, and it moves from about $1$ to $2.5$ days with season, region, and variable. Section 5 shows the size of that dependence itself. We do not say there is a single exponent——the real atmosphere has different growth rates at different scales, with smaller eddies growing faster. A single $\tau_d$ is a crude approximation. We do not deny that forecasts improve——forecasts have in fact grown longer. What this paper says is only that the growth is logarithmic, not that improvement is pointless. Relation to earlier papers: Paper 253 showed that time is one-dimensional because prediction demands it, not because a law says so——this paper turns how far that demand can be met into a number. Paper 195 separated “stable” into six words——that paper is a classification of stability; this one is a time scale of predictability, the same hyperbolicity as material with a different question. Paper 190 measured “rare” on a logarithmic scale——the return here is likewise logarithmic. Paper 266 showed that the premise of the sampling theorem is never met——“knowing the initial state exactly” here is likewise a premise never met, the same figure. What is added is writing the price per day as the fixed factor $1.5874$, computing the accuracy needed for $14\to21\to30\to60$ days as $25.40$ / $1625.5$ / $1.70\times10^{9}$, writing backwards that $10$ times the observation gains only $4.98$ days, and sweeping $\tau_d$ from $1.0$ to $2.5$ to show the answer moving from $65536$ to $84.4$.
1. Notation, and One Number
Setting: A small error $\varepsilon_0$ in the initial state grows as
$$\varepsilon(t)=\varepsilon_0\,e^{\lambda t}=\varepsilon_0\,2^{t/\tau_d}$$Here $\tau_d$ is the time for the error to double, taken as about $1.5$ days for the atmosphere.
Check 1.1:
| Quantity | Value |
|---|---|
| Doubling time $\tau_d$ | $1.5$ days |
| $\lambda=\ln2/\tau_d$ | $\mathbf{0.462098}$ per day |
Question: To extend a forecast, what must be improved, and by how much?
Observation 1.2 The span $T$ over which a forecast holds runs until the error reaches a useless size $\varepsilon_{\max}$. Solving $\varepsilon_0 2^{T/\tau_d}=\varepsilon_{\max}$,
$$T=\tau_d\log_2\!\left(\frac{\varepsilon_{\max}}{\varepsilon_0}\right)$$——the initial accuracy enters only inside a logarithm.
2. The Core: $1.5874$ Times per Day
Check 2.1: The factor needed to extend $T$ by one day is $2^{1/\tau_d}$.
| Quantity | Value |
|---|---|
| Accuracy factor per day, $2^{1/1.5}$ | $\mathbf{1.5874}$ |
Observation 2.2 $1.5874$ per day, and the factor is the same wherever the day is added. The day at $14$ and the day at $60$ cost the same $1.5874$——but they compound.
Check 2.3: We compound from a base of $14$ days.
| Extension | Doublings $(T-14)/\tau_d$ | Accuracy factor required |
|---|---|---|
| $14\to16$ days | $1.3333$ | $2.5198$ |
| $14\to18$ days | $2.6667$ | $6.3496$ |
| $14\to21$ days | $4.6667$ | $\mathbf{25.398}$ |
| $14\to24$ days | $6.6667$ | $101.59$ |
| $14\to28$ days | $9.3333$ | $645.08$ |
| $14\to30$ days | $10.6667$ | $\mathbf{1625.5}$ |
| $14\to45$ days | $20.6667$ | $1.6645\times10^{6}$ |
| $14\to60$ days | $30.6667$ | $\mathbf{1.7045\times10^{9}}$ |
| $14\to90$ days | $50.6667$ | $1.7873\times10^{15}$ |
Observation 2.4 This is the core of the paper. One more week costs $\mathbf{25.4}$ times, two weeks $\mathbf{645}$, six weeks $\mathbf{1.7}$ billion——the extension adds while the price multiplies.
Observation 2.5 $14\to16$ days costs a mere $2.52$. The same “two more days” from $88$ to $90$ requires a starting accuracy already $10^{15}$ times better——the factor is constant; the starting point is not.
3. Read Backwards: $10$ Times the Observation, $4.98$ Days
Setting: In practice the question runs the other way——make the observing network $N$ times more precise, and how many days does the forecast gain? The answer is $\tau_d\log_2 N$.
Check 3.1:
| Accuracy factor $N$ | Days gained, $\tau_d\log_2N$ |
|---|---|
| $2$ | $\mathbf{1.5000}$ |
| $10$ | $\mathbf{4.9829}$ |
| $100$ | $9.9658$ |
| $1000$ | $14.9487$ |
| $10^{6}$ | $29.8974$ |
| $10^{9}$ | $\mathbf{44.8460}$ |
Observation 3.2 Observing $10$ times more precisely gains only $\mathbf{4.98}$ days. Halving the error buys exactly $\tau_d=1.5$ days——which is the definition itself.
Observation 3.3 Even $10^{9}$ times gains $\mathbf{44.85}$ days. Put an instrument in every cubic metre on Earth and you still do not reach two months——because the return is logarithmic.
Observation 3.4 Paper 190 wrote that “rare” is measured on a logarithmic scale. Here the relation between resource and result is itself logarithmic——the input multiplies while the output only adds.
Observation 3.5 A point to write honestly——forecasts have in fact grown longer. What this paper describes is the shape of that growth, not that improvement is pointless——five days of improvement is the fruit of the enormous labour of a $\mathbf{10}$-fold gain in accuracy.
4. “About Two Weeks” Comes from Here
Setting: The forecastable span is set by how small the initial error $\varepsilon_0$ is against the saturation error $\varepsilon_{\max}$.
Check 4.1:
| $\varepsilon_0/\varepsilon_{\max}$ | Forecastable span $\tau_d\log_2(\varepsilon_{\max}/\varepsilon_0)$ |
|---|---|
| $10^{-2}$ | $9.9658$ days |
| $\mathbf{10^{-3}}$ | $\mathbf{14.9487}$ days |
| $10^{-4}$ | $19.9316$ days |
| $10^{-6}$ | $29.8974$ days |
Observation 4.2 If the initial error is $\mathbf{10^{-3}}$ of saturation, the forecastable span is $\mathbf{14.95}$ days. The familiar “the limit of forecasting is about two weeks” comes from this single line——that two-week figure is fixed by just two things: $\tau_d$ and the initial accuracy.
Observation 4.3 What acts here is not the complexity of the equations. Solve the Navier--Stokes equations perfectly and you still exhaust the forecast in $14.95$ days at $10^{-3}$ initial accuracy——the limit lies on the input side, not the solver side.
Observation 4.4 Paper 266 showed that the premise of the sampling theorem is never met. “Knowing the initial state exactly” is likewise a premise never met——the theorem correct and the premise out of reach, the same figure.
5. Separator: Everything Rides on $\tau_d$
Observation 5.1 Separator——what must be moved for the answer to change? Only $\tau_d$.
Check 5.2: We sweep $\tau_d$ and look at the answer to the same question.
| $\tau_d$ [days] | Accuracy factor for $14\to30$ days | Days gained by $10\times$ observation |
|---|---|---|
| $1.0$ | $\mathbf{6.5536\times10^{4}}$ | $3.3219$ |
| $1.5$ | $1.6255\times10^{3}$ | $4.9829$ |
| $2.0$ | $2.5600\times10^{2}$ | $6.6439$ |
| $2.5$ | $\mathbf{8.4449\times10^{1}}$ | $8.3048$ |
Observation 5.3 Moving $\tau_d$ from $1.0$ to $2.5$ takes the required accuracy from $\mathbf{65536}$ to $\mathbf{84.4}$, a factor of $776$. The whole enterprise of forecasting rides on this one number——so measuring $\tau_d$ matters as much as extending the forecast.
Observation 5.4 A point to write honestly——the real atmosphere has no single $\tau_d$. Growth rates differ by scale, with smaller eddies growing faster——the single $\tau_d$ here is a crude approximation, and what can be shown is the structure, not the value.
Observation 5.5 Paper 195 separated “stable” into six words. That $\tau_d$ is finite at all is due to hyperbolic instability, but this paper does not enter that classification——that paper separates by what is perturbed; this one only turns the speed of growth into a number.
Observation 5.6 Paper 253 showed that time is one-dimensional because of the demand that prediction be possible, not because a law says so. This paper is its continuation——that demand is met only out to $14.95$ days.
6. Roll Call and Closing
- $\lambda=0.462098$ per day (Check 1.1): From $\tau_d=1.5$ days.
- $1.5874$ per day (Check 2.1): The same factor wherever added.
- $14\to21$ days costs $25.398$ (Check 2.3): The price of one week.
- $14\to30$ days costs $1625.5$ (Check 2.3): Two and a half weeks.
- $14\to60$ days costs $1.70\times10^{9}$ (Check 2.3): $1.7$ billion for six weeks.
- $10\times$ observation gains $4.9829$ days (Check 3.1): The return is logarithmic.
- $10^{9}\times$ gains $44.846$ days (Check 3.1): Not even two months.
- $10^{-3}$ gives $14.9487$ days (Check 4.1): The source of “about two weeks.”
- $\tau_d=1.0$ costs $65536$ (Check 5.2): $2.5$ costs $84.4$.
- A factor of $776$ (Observation 5.3): All of it on one number.
Closing: What sets the limit of forecasting is neither the equations nor the computers, but one number, the error doubling time $\tau_d$. At $\tau_d=1.5$ days, each extra day costs $\mathbf{1.5874}$ times the initial accuracy——the factor is the same wherever the day is added, but the extension adds while the price multiplies, so one week costs $\mathbf{25.4}$, two weeks $\mathbf{645}$, six weeks $\mathbf{1.7}$ billion. Read backwards, observing $\mathbf{10}$ times more precisely gains only $\mathbf{4.98}$ days, and even $10^{9}$ times gains $\mathbf{44.85}$. The familiar “about two weeks” comes from this one line——$\mathbf{14.95}$ days at an initial error of $10^{-3}$ of saturation. One thing separates them——$\tau_d$ itself. At $1.0$ day the same extension costs $\mathbf{65536}$; at $2.5$ days, $\mathbf{84.4}$. A factor of $776$ arises from a single number. So the work of extending forecasts and the work of measuring $\tau_d$ carry the same weight.
References
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- Lorenz, E. N. (1969). The Predictability of a Flow Which Possesses Many Scales of Motion. Tellus, 21(3).
- Palmer, T. N. and Hagedorn, R. (eds.) (2006). Predictability of Weather and Climate. Cambridge: Cambridge University Press.
- Zhang, F. et al. (2019). What Is the Predictability Limit of Midlatitude Weather? Journal of the Atmospheric Sciences, 76(4).
- Bauer, P., Thorpe, A. and Brunet, G. (2015). The Quiet Revolution of Numerical Weather Prediction. Nature, 525(7567).
- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). “Stable” Is Six Separate Words, and What Separates Them Is What You Perturb [Paper 195]. Zenodo. 10.5281/zenodo.22107170
- Yamagishi, Y. (2026). Time Is One-Dimensional Because Prediction Demands It, Not Because a Law Says So [Paper 253]. Zenodo. 10.5281/zenodo.22161926
- Yamagishi, Y. (2026). The Sampling Theorem Is Correct and Its Premise Is Never Met [Paper 266]. Zenodo. 10.5281/zenodo.22169765
Reference
Yamagishi, Y. (2026). One Number Sets the Limit of Forecasting [Paper 289]. YOI: https://www.yamagishiyuuki.com/yoi/289
DOI: 10.5281/zenodo.22171367
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 289. This paper is a preprint and has not been peer reviewed.