“Are the copper's two odd patterns odd at all once every width is counted?”
Experiments 18 and 27 found two patterns in K4 that no construction reproduced well: five of its six doubled letters fall in one column when the text is written in rows of 7, and pairs of letters 21 places apart repeat eleven times. Both were found by looking at many widths. This asks how often a scrambled copper shows a pattern that strong at some width, which is the fair comparison.
If you write the copper in rows of 7 letters, five of its six doubled lettersthe doublets Places where a carved letter is immediately followed by the same letter. line up in one column, and in rows of 21 the same letter pairs stack up more than usual. That looks like a clue to how it was made. But we found those widths by trying many widths, and if you try enough widths something always lines up. So we scramble the copper ten thousand times, try every width from 2 to 48 on each scramble, and see how often a scramblea grid Writing the message into rows of some width and reading it out in another order, column by column for instance, which shuffles the letters without changing them. looks at least this odd at its best width.
Words in the margin are defined on the glossary page.
1.1
The method at work
The carved letters go in at the top; the reading comes out at the bottom.
the column
The copper has six doubled letters. Written in rows of 7, five of them start in the same column.
the repeats
Written in rows of 21, a pair of letters one above the other turns up twice 11 times.
the test
Both were found by trying many widths, so the fair comparison is with shuffled coppers at their own best width. After that correction the column happens about 0.5 times in a hundred and the repeats about 1: both are real features of the copper.
2
Best candidate against the known letters
This experiment produced no candidate: the search ruled everything out before a full text was built. Its product is the bounded negative in section 7.
3
Walkthrough, by hand
Redo the method on a few letters with pencil and paper.
Find the doubled letters in the copper: BB at places 19-20, QQ at 26-27, SS at 33-34, SS at 43-44, ZZ at 47-48 and TT at 68-69.
Write the copper in rows of 7: the first letter of each doublet sits in column 19 mod 7 = 5, 26 mod 7 = 5, 33 mod 7 = 5, 43 mod 7 = 1, 47 mod 7 = 5, 68 mod 7 = 5.
Count: column 5 holds five of the six doublets.
Shuffle the 97 letters, find the doubled letters again, and count the fullest column at every width from 2 to 48.
Note how odd the fullest column is at its best width.
Do it ten thousand times and see where the copper falls.
the worksheet: the first steps by hand, on squared paper
4
Worked example, generated
The same steps run by the code on the real letters.
doublets (place of first letter, 1-based):
BB at 19-20: column (0-based index mod 7) 4
QQ at 26-27: column (0-based index mod 7) 4
SS at 33-34: column (0-based index mod 7) 4
SS at 43-44: column (0-based index mod 7) 0
ZZ at 47-48: column (0-based index mod 7) 4
TT at 68-69: column (0-based index mod 7) 4
5
Hypothesis, for a cryptanalyst
Family, key space, the constraint under test, and the clues it reads.
The width-7 doublet column (5 of 6 doublets at one position mod 7) and the width-21 vertical repeats (11 repeated pairs (c[i], c[i+21])) were selected after scanning widths, so their nominal p-values (0.01-0.1 in 0018) overstate them. For every width w in 2-48 compute both statistics on K4 and on 10,000 letter-preserving shuffles; turn each into a per-width tail probability from the shuffle distribution; take the smallest tail probability over widths as the scan statistic; and compare K4's scan statistic with the shuffles' (the look-elsewhere correction). Also condition the doublet statistic on shuffles with exactly six doublets, since K4 has six. If K4's corrected p-values exceed 0.05, family 7's two unexplained features are not evidence for any construction. Bounds: widths 2-48, 10,000 controls.
6
Prediction
Written before the run; the result decides it.
After the correction neither pattern is unusual: K4's scan p-value is above 0.05 for both, so the two features need no explanation. Zero candidates.
7
Method
For widths 2-48 compute, on K4, the largest number of doublets sharing a column and the number of repeated vertical pairs.
Do the same on 10,000 shuffles.
Per width, the tail probability of each value among the shuffles.
Per text, the smallest tail probability over widths.
K4's rank among the shuffles is the corrected p-value; repeat with six-doublet shuffles only.
8
Result and bounds
The harness's record: candidates, gates, loss, and the negative result this experiment leaves behind.
10000 shuffles, widths 2-48. Doublet column: nominal p at width 7 0.0005, corrected over widths 0.0049 (0.0178 among the 729 shuffles with 6 doublets). Vertical repeats: nominal p at width 21 0.0004, corrected 0.0099.
What was searched, as the harness recorded it.
family
look-elsewhere test of the width fingerprints
widths
2-48
controls
10,000
k4 doublets
6
k4 col7
5
k4 rep21
11
nominal p col7
0.001
nominal p rep21
0
k4 best width col
7
k4 best width rep
21
corrected p doublet column
0.005
corrected p doublet column six doublet shuffles
0.018
six doublet shuffles
729
corrected p vertical repeats
0.01
How this experiment sits among the others is on the lineage page; its family's open hypotheses are on the families page.
Rendered from the record on 2026-10-18 12:26 UTC; updated after every experiment.