Abstract
Asked why the gaussian turns up everywhere, one usually answers with the central limit theorem alone. This paper writes down four unrelated reasons, each holding a different thing fixed. NO NEW MATHEMATICAL THEOREM IS CLAIMED.
First, finite variance is a condition and not a decoration. Averaging a thousand normal draws shrinks the spread from $1.345$ to $0.0427$; averaging a thousand Cauchy draws moves it from $2.003$ to $2.017$, which is nothing at all — the quartiles matched those of a single draw (Section 2).
Second, stability alone does not pick the gaussian. Both the normal and the Cauchy are stable, with relative gaps under $0.004$ — stability is a family, not a point (Section 3).
Third, maximum entropy depends on what is held fixed. Fix the variance and the gaussian wins (measured $1.418$ against a theoretical ceiling of $1.419$); fix the mean and the exponential wins; fix the range and the uniform does (Section 4).
Fourth, and this is the core. Rotational invariance together with independent coordinates forces the gaussian — turning the product density by forty-five degrees changed it by $5.6\times10^{-17}$ for the gaussian alone. No sum and no limit appears anywhere (Section 5).
Fifth, the first attempt to check a fourth characterisation failed. Measuring the correlation of the sample mean with the sample variance gave zero for the normal, the uniform and the Laplace alike. Every symmetric law does that — correlating with the square of the mean instead gives $+0.0006$ for the gaussian against $-0.300$ and $+0.299$ (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM IS CLAIMED — the central limit theorem, stable laws, maximum entropy, the Herschel–Maxwell argument, and the characterisation by independence of the sample mean and variance are all standard. No distribution is judged “correct”. On Paper 139: that paper treated the rate at which empirical measures converge. The question here is why the limit is gaussian, which is a different one. What is added is setting four reasons side by side by what each holds fixed, confirming with quartiles that averaging Cauchy variables does nothing, measuring the force of rotational invariance to machine precision, and extracting from a failed test of my own the fact that uncorrelated and independent are different.
1. Notation and position
Everything sampled was sampled for this paper, with sample sizes stated. Laws are centred at zero and spread is measured by quantiles, since the Cauchy has no variance.
Position: Paper 164 wrote that “probability” has four addresses. This paper stands beside it with one distribution reached by four doors — a case where it is the object rather than the word that has several entrances.
2. Finite variance is a condition, not a decoration
Verification 2.1: Over two hundred thousand trials, the interquartile range of the mean of $n$ draws.
| $n$ | normal: spread of the mean | Cauchy: the same |
|---|---|---|
| $1$ | $1.345$ | $2.003$ |
| $10$ | $0.425$ | $1.995$ |
| $100$ | $0.135$ | $2.003$ |
| $1000$ | $\mathbf{0.0427}$ | $\mathbf{2.017}$ |
(QED)
Verification 2.2: For the Cauchy, compare a single draw with the mean of a thousand.
| quantile | one draw | mean of a thousand |
|---|---|---|
| $25\%$ | $-1.003$ | $-1.000$ |
| $50\%$ | $-0.0006$ | $-0.003$ |
| $75\%$ | $1.004$ | $0.996$ |
(QED)
Observation 2.3 The normal mean shrinks like one over the square root of $n$ and the Cauchy mean does nothing — a thousand of them average to the same law as one. “Collect enough and the average settles” is true only when the variance is finite.
3. Stability alone does not pick the gaussian
Observation 3.1 A law is stable when a sum of independent copies is the same law up to shift and scale.
Verification 3.2: Interquartile ranges over a million trials.
| law | $k$ | scale | relative gap |
|---|---|---|---|
| normal | $2$ | $\sqrt{k}$ | $0.00075$ |
| normal | $100$ | $\sqrt{k}$ | $0.00060$ |
| Cauchy | $2$ | $k$ | $0.00213$ |
| Cauchy | $100$ | $k$ | $0.00244$ |
(QED)
Observation 3.3 Both are stable, with different exponents — the square root of $k$ for the normal and $k$ for the Cauchy. Stability is a family and not a point, and what picks the gaussian out of it is the extra condition of finite variance.
4. Maximum entropy: the answer is decided by what is held fixed
Verification 4.1: Differential entropy of laws with variance one, from histograms of two million points.
| law | variance | entropy |
|---|---|---|
| normal | $0.999$ | $\mathbf{1.4184}$ |
| Laplace | $1.001$ | $1.3468$ |
| uniform | $1.001$ | $1.2423$ |
| two-point | $1.000$ | $-3.929$ |
The theoretical ceiling at variance one is $1.4189$. (QED)
Observation 4.2 The gaussian wins — and only because the variance was what was fixed.
| what is held fixed | the winner |
|---|---|
| the variance | normal |
| the mean, on the positive half-line | exponential |
| the range | uniform |
Observation 4.3 “The gaussian is the least informative distribution” is a half-sentence. The full one is “least informative given that the variance is known”, and changing what is known changes the answer.
5. The core: symmetry alone forces the gaussian
Observation 5.1 The argument of Herschel and Maxwell. If the two coordinates of a point in the plane are independent and the density depends only on the distance from the origin, the law must be gaussian. No sum and no limit appears.
Verification 5.2: Turn the product density $f(x)f(y)$ by forty-five degrees and measure the largest change, over a grid of forty-one by forty-one.
| law | largest change under the turn |
|---|---|
| normal | $\mathbf{5.6\times10^{-17}}$ |
| Cauchy | $9.1\times10^{-3}$ |
| Laplace | $6.3\times10^{-2}$ |
| uniform | $8.3\times10^{-2}$ |
(QED)
Observation 5.3 Only the gaussian is unchanged, and it is unchanged to machine precision. What selects it here is symmetry — nothing was added up and no limit was taken. What is being demanded is quite unlike Section 2.
6. Uncorrelated is not independent
Observation 6.1 There is a fourth characterisation — the gaussian is the only law for which the sample mean and the sample variance are independent. The first attempt to check it failed.
Verification 6.2: Four hundred thousand samples of size eight, measuring the correlation.
| law | correlation of mean and variance |
|---|---|
| normal | $-0.003$ |
| uniform | $-0.0007$ |
| Laplace | $+0.003$ |
| exponential | $+0.694$ |
The uniform and the Laplace read as zero, exactly like the normal. (QED)
Observation 6.3 This does not refute the characterisation — the instrument was wrong. For a symmetric law the third central moment governing this correlation vanishes, so all of them read zero. Being uncorrelated and being independent are different statements.
Verification 6.4: So correlate the square of the mean with the variance, which independence would also force to zero.
| law | correlation of the squared mean with the variance |
|---|---|
| normal | $\mathbf{+0.0006}$ |
| uniform | $\mathbf{-0.300}$ |
| Laplace | $\mathbf{+0.299}$ |
| exponential | $+0.714$ |
(QED)
Verification 6.5: Look directly: the spread of the variance among samples whose mean was small against those whose mean was large.
| law | ratio of the two spreads |
|---|---|
| normal | $\mathbf{1.002}$ |
| uniform | $0.935$ |
| Laplace | $1.496$ |
| exponential | $7.482$ |
(QED)
Observation 6.6 Only the gaussian stays at zero. The uniform and the Laplace were uncorrelated and were not independent. And this characterisation touches neither sums, nor entropy, nor symmetry — it is a fourth and separate reason.
7. The census
Observation 7.1
| why the gaussian | what is held fixed | what happens without it |
|---|---|---|
| a limit of sums | finite variance | another stable law, such as Cauchy |
| stability alone | nothing | a whole family survives |
| maximum entropy | the variance | mean gives exponential, range gives uniform |
| rotation and independence | symmetry only | no sum and no limit needed |
| mean and variance independent | a property of samples | unrelated to the others |
Observation 7.2 These five are not a staircase. The limit of sums is a statement about a process, maximum entropy about knowledge, rotational invariance about symmetry, and independence about samples. One distribution with four quite different entrances.
Observation 7.3 So “assume a gaussian” does not name one assumption. Was finite variance assumed, or was it said that only the variance is known, or was symmetry demanded? Each fails in a different place.
8. Conclusion
- Averaging does nothing to a Cauchy (Ver. 2.1): $2.003$ to $2.017$.
- Stability is a family (Ver. 3.2): the Cauchy is stable too, scaling by $k$.
- Maximum entropy is decided by what is fixed (Ver. 4.1): measured $1.4184$ against a ceiling of $1.4189$.
- Symmetry alone forces the gaussian (Ver. 5.2): a change of $5.6\times10^{-17}$ under rotation.
- Uncorrelated is not independent (Ver. 6.2, 6.4): all correlations zero, yet $-0.300$ and $+0.299$ against the squared mean.
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Reference
Yamagishi, Y. (2026). The Gaussian Is Special for More Than One Reason [Paper 187]. YOI: https://www.yamagishiyuuki.com/yoi/187
DOI: 10.5281/zenodo.22105639
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 187. This paper is a preprint and has not been peer reviewed.