Abstract
“It is random” carries two separate meanings: that it passes tests and that it cannot be written briefly. This paper exhibits in numbers a sequence that passes all of the first and breaks the second spectacularly. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, it passes every test. RANDU has a mean of $0.501293$, a runs statistic of $-0.543$, a serial correlation of $-0.000029$ and a chi-square on bytes of $256.9$ against an expected $255$ — indistinguishable from a modern generator (Section 2).
Second, compression does not detect it either. Under zlib, RANDU comes to $0.9993$ of its size against $1.0004$ for a modern generator, all but the same (Section 3).
Third, taken two at a time almost nothing shows. Searching small relations between consecutive pairs gives $25353$ distinct values for RANDU against $59878$ for the comparison — a faint trace, and not enough to raise an alarm (Section 4).
Fourth, this is the core. Taken three at a time, $x_{i+2}-6x_{i+1}+9x_i$ takes exactly $15$ values, and they are the whole numbers from $-5$ to $9$, with a departure from a whole number of $\mathbf{0}$ — sixty thousand points lying on fifteen parallel planes (Section 5).
Fifth, the information actually held is $\mathbf{70}$ bytes. The procedure that makes $60002$ bytes is $66$ bytes plus a seed of $4$, a ratio of $0.00117$ — a thousandth of the $0.9993$ that the best compressor reached (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — Marsaglia's “random numbers fall mainly in the planes”, Kolmogorov complexity and its uncomputability are all standard. No generator is designed — only two old ones and one modern one appear. No survey of tests is offered — four tests and one compressor are used, and that some test would catch it is of course true (a test aimed at lattice structure finds it at once). Kolmogorov complexity is not computed — only an upper bound can be. Cryptographic security is not treated. Relation to Paper 220: there, all three pairs were independent and the three together were not — RANDU is likewise clean in pairs and broken in triples. One shape appearing in two places, probability and the generation of numbers. And 220 wrote that a search can rule out small relations and never settle the question — Section 7 here is the general form of that asymmetry. What is added is measuring that four tests and a compressor fail to separate three generators, measuring that pairs show almost no trace, showing that the triples take exactly fifteen whole-number values, and giving the length of the short procedure against what compression achieved.
1. Notation and Position
There are at least four senses of “random”: passing tests (statistical), not being writable briefly (Kolmogorov), not being predictable, and having come from such a mechanism (a statement about a process). This paper compares the first two.
The generators used are RANDU, $x_{i+1}=65539\,x_i \bmod 2^{31}$; a comparison, $x_{i+1}=(1103515245\,x_i+12345)\bmod 2^{31}$; and a modern generator.
Position: In Paper 220 all three pairs were independent and the three together were dependent — here too it is clean in pairs and broken in triples. In Paper 221 passing the test of being diagonalisable guaranteed nothing — the same holds here.
2. It Passes Every Test
Check 2.1: generate $60002$ values and apply four tests.
| Generator | Mean | Runs, $z$ | Serial correlation | Chi-square on bytes |
|---|---|---|---|---|
| RANDU | $0.501293$ | $-0.543$ | $-0.000029$ | $256.9$ |
| Comparison | $0.500159$ | $-0.143$ | $-0.000639$ | $256.0$ |
| A modern generator | $0.500654$ | ── | $0.000053$ | ── |
(The expectation of a chi-square on $255$ degrees of freedom is $255$.)
Observation 2.2 Not one column separates the three. RANDU is the most notorious generator there is, and it passes all four.
3. Compression Does Not Detect It
Check 3.1: write the values as bytes and apply zlib.
| Generator | Raw size | After zlib | Ratio |
|---|---|---|---|
| RANDU | $60002$ | $59961$ | $\mathbf{0.9993}$ |
| Comparison | $60002$ | $60028$ | $1.0004$ |
| A modern generator | $60002$ | $60028$ | $1.0004$ |
Observation 3.2 None of them shrinks. Compression is a test like any other and misses in the same way — “incompressible, therefore random” says no more than that this compressor sees no structure.
4. Taken Two at a Time, Almost Nothing Shows
Check 4.1: search exhaustively for relations with small coefficients between consecutive pairs.
| Generator | Fewest distinct values | Coefficient |
|---|---|---|
| RANDU | $25353$ | $c_1=-3$ |
| Comparison | $59878$ | $c_1=1$ |
Observation 4.2 RANDU is the lower of the two ($25353$ against $59878$), and that is a faint trace and not a collapse. Both have tens of thousands of values, and so long as one looks two at a time no alarm sounds.
5. The Core: Taken Three at a Time, Fifteen Planes
RANDU satisfies $x_{i+2}=6x_{i+1}-9x_i \pmod{2^{31}}$ exactly. Hence, once normalised to lie between $0$ and $1$, the combination $x_{i+2}-6x_{i+1}+9x_i$ can only be a whole number.
Check 5.1:
| Measured | Value |
|---|---|
| Number of triples | $60000$ |
| Distinct values of $x_{i+2}-6x_{i+1}+9x_i$ | $\mathbf{15}$ |
| Which values | $-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9$ |
| Largest departure from a whole number | $\mathbf{0}$ |
| The same search on the comparison | $59874$ values |
Observation 5.2 All sixty thousand points lie on fifteen parallel planes. Placed in three-dimensional space, the points do not fill it — and not one of the tests of Sections 2 to 4 noticed.
Observation 5.3 This is the shape of Paper 220. There, all three pairs were independent and the three together dependent; here, clean in pairs and broken in triples — one way of slipping through appearing in two different places.
6. The Information Actually Held Is Seventy Bytes
Check 6.1:
| Measured | Value |
|---|---|
| Bytes produced | $60002$ |
| The procedure that makes them | $66$ bytes (plus a seed of $4$) |
| Ratio | $\mathbf{0.00117}$ |
| The ratio zlib achieved | $0.9993$ |
Observation 6.2 The shortest description is a thousandth of the length the best general compressor reached. A short description certainly exists — and no test found it.
7. The Asymmetry That Cannot Be Crossed
Observation 7.1 exhibiting a short procedure gives an upper bound on the complexity. A lower bound is another matter: Kolmogorov complexity is uncomputable (Chaitin).
| What can be shown | How |
|---|---|
| That a short description exists | exhibit one (done in Section 6) |
| That a short description does not | impossible in general |
Observation 7.2 So a test can say “this can be written briefly” and can never say “this cannot”. Passing carries only the meaning “no structure has been found yet” — the general form of what Paper 220 wrote, that a search rules out small relations and never settles the question.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the four senses of “random” is random with respect to whom.
| Phrase | With respect to what |
|---|---|
| Passes tests | that list of tests |
| Cannot be compressed | that compressor |
| Cannot be written briefly | every procedure (uncomputable) |
| Came from such a mechanism | a statement about a process, not about the sequence |
Observation 8.2 (the fence) The fence around “this sequence is random” is saying with respect to what. Outside it the sentence is not even false but empty — the same shape as “independence was checked” in 220 and “the blackbody peaks here” in 222.
Observation 8.3 And the most widely used tests are the ones that see least. Mean, runs, correlation, chi-square and compression all look at properties of one or two dimensions — and the defect of RANDU first appears in three. In 220 the correlation sat among the fourteen of twenty that held: the easier a condition is to check, the harder it is to break.
9. The Census and the Close
What this paper measured:
- It passes every test (Check 2.1): a runs statistic of $-0.543$.
- Compression does not detect it (Check 3.1): $0.9993$ against $1.0004$.
- Pairs show only a trace (Check 4.1): $25353$ against $59878$.
- Triples give fifteen planes (Check 5.1): the whole numbers $-5$ to $9$, departure $0$.
- The information is seventy bytes (Check 6.1): a ratio of $0.00117$.
- Upper bounds can be shown and lower bounds cannot (Observation 7.1).
References
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Reference
Yamagishi, Y. (2026). It Passes Every Test, and the Points Lie on Fifteen Planes [Paper 223]. YOI: https://www.yamagishiyuuki.com/yoi/223
DOI: 10.5281/zenodo.22120728
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 223. This paper is a preprint and has not been peer reviewed.