The Cross-Alignment Matrix
Divide 1 by 7 and you get a repeating pattern. Divide 2 by 7 and you get another. How similar are they? The alignment measures that: compare the two patterns digit by digit, count the matches, and you get a number between 0 and 1. Do this for every pair of fractions with denominator 7 and you know everything about how the field holds together.
But if you average all those pairwise comparisons into a single number, you lose the map. You know how much agreement there is, but not where it lives. Which fractions agree with which? Are the agreements clustered into groups, or scattered evenly?
To keep the map, you keep every comparison. You arrange them in a grid. That grid is the cross-alignment matrix.
Building the matrix
Take $p = 7$ in base 10. Six fractions:
1/7 => 0.|142857|
2/7 => 0.|285714|
3/7 => 0.|428571|
4/7 => 0.|571428|
5/7 => 0.|714285|
6/7 => 0.|857142|
Compare 1/7 and 2/7. Position by position: 1 vs 2, 4 vs 8, 2 vs 5, 8 vs 7, 5 vs 1, 7 vs 4. Zero matches out of six. Now compare 1/7 and 3/7: 1 vs 4, 4 vs 2, 2 vs 8, 8 vs 5, 5 vs 7, 7 vs 1. Zero again. Every pair, zero matches.
Do this for all pairs and arrange the results in a grid. Row $k$, column $j$ records the fraction of positions where $k/7$ and $j/7$ share the same digit:
1/7 2/7 3/7 4/7 5/7 6/7
1/7 [ 1 0 0 0 0 0 ]
2/7 [ 0 1 0 0 0 0 ]
3/7 [ 0 0 1 0 0 0 ]
4/7 [ 0 0 0 1 0 0 ]
5/7 [ 0 0 0 0 1 0 ]
6/7 [ 0 0 0 0 0 1 ]
Ones on the diagonal (every fraction matches itself). Zeros everywhere else. When I say two fractions are strangers, this is what I mean: if you lined up their repetends and compared them digit by digit, you would never find a match. At $p = 7$, every fraction is a stranger to every other.
Mathematicians call this the identity matrix: the matrix that says every element recognizes only itself and no one else. It is the mathematical way of saying there is nothing to see here. Everyone is alone.
When the strangers meet
At $p = 13$, the fractions are no longer alone:
1/13 => 0.|076923|
2/13 => 0.|153846|
3/13 => 0.|230769|
4/13 => 0.|307692|
5/13 => 0.|384615|
6/13 => 0.|461538|
7/13 => 0.|538461|
8/13 => 0.|615384|
9/13 => 0.|692307|
10/13 => 0.|769230|
11/13 => 0.|846153|
12/13 => 0.|923076|
Twelve fractions, ten possible digits. Some pairs must share digits somewhere. Look at the fourth digit of each repetend: 9, 8, 7, 6, 6, 6, 4, 3, 3, 2, 1, 0. The digit 6 appears three times. The digit 3 appears twice. Those shared digits produce nonzero entries in the matrix. The grid is no longer empty off the diagonal.
And the nonzero entries are not scattered randomly. The first six fractions (1/13 through 6/13) are rotations of 076923. The last six (7/13 through 12/13) are rotations of 538461. Two groups, two parallel orbits through the remainders. Fractions within the same orbit share more digits because their repetends are rotations of the same string. The matrix develops two bright blocks on the diagonal, with dimmer entries between them.
The field of 13 has internal structure that the field of 7 does not. The matrix makes that structure visible.
Blocks
At composites, the structure gets even cleaner. Consider the field of 12:
1/12 => 0.08|3| (mixed)
2/12 => 0.1|6| (mixed)
3/12 => 0.25 (open)
4/12 => 0.|3| (closed)
5/12 => 0.41|6| (mixed)
6/12 => 0.5 (open)
7/12 => 0.58|3| (mixed)
8/12 => 0.|6| (closed)
9/12 => 0.75 (open)
10/12 => 0.8|3| (mixed)
11/12 => 0.91|6| (mixed)
Three kinds of fractions. The terminating ones (3/12, 6/12, 9/12) have no repeating block. When you compare them to each other in the repeating part, there are no digits to disagree on. They agree by absence.
The fractions that repeat 3 (1/12, 4/12, 7/12, 10/12) all share the same repeating digit. Perfect agreement within the group. The fractions that repeat 6 (2/12, 5/12, 8/12, 11/12) are the same. Perfect agreement within their group.
But a fraction repeating 3 and a fraction repeating 6 share nothing. They are strangers.
The matrix of 12 is three bright squares on the diagonal, separated by darkness. One square for the terminators. One for the 3s. One for the 6s. Within each square, full agreement. Between squares, nothing. The prime 3 determines that there are three blocks. The smooth factor 4 determines that each block holds three or four fractions. The structure comes from the prime. The resolution comes from the smooth factor.
The chord
Here is where the matrix begins to speak in a language beyond blocks and zeros.
Think of a musical chord. A chord is a combination of pure tones. Each tone has a frequency and a volume. If every tone is equally loud, you hear something like white noise: no structure, no pattern. If one tone dominates, you hear a note.
A matrix has something analogous. Its eigenvalues are like the volumes of the individual tones. They tell you which patterns are strong and which are weak. The identity matrix at $p = 7$ has every eigenvalue equal to 1: every tone equally loud, no pattern standing out. That is total independence, heard as a perfectly balanced chord.
When collisions appear, some tones get louder. The collisions concentrate energy at specific frequencies. The chord develops character: peaks where the repetend echoes itself, valleys where everything moves and nothing matches.
The eigenvalue spectrum is the harmonic portrait of the fractional field. The paper proves that for primes where 10 is a primitive root, this portrait can be computed exactly by the Fourier transform. The coherence geometry of the field is a frequency spectrum, and the digit-partitioning property is a flat one.
What the matrix gives you
If you average the entries of the cross-alignment matrix in different ways, you recover the scalars: $\alpha$ (the average of one row), $\sigma$ (the average of the off-diagonal), $F$ (the difference). Those numbers are useful. But they are summaries of the matrix, not the matrix itself. They tell you how much coherence there is. The matrix tells you how that coherence is organized.
At $p = 7$, the matrix is the identity. The organization is: none. At $p = 13$, two bright blocks on the diagonal with dimmer entries between them. The organization is: two orbits, mostly independent. At $n = 12$, three solid blocks separated by voids. The organization is: three groups that internally agree perfectly and externally share nothing.
If you could see the matrix as a grid of light, the diagonal would be a clean bright line (every fraction knows itself). The blocks would glow where fractions recognize each other. And the voids between blocks would be dark, marking the boundaries between strangers. The scalars average that entire landscape into a single brightness value. The matrix keeps the landscape.
Try it yourself
Three fields, three matrix types:
$ ./nfield field 7 # identity: all strangers
$ ./nfield field 13 # two orbits: two blocks, off-diagonal texture
$ ./nfield field 12 # composite: three clean blocks, voids between
At 7, read down any column of digits in the repetend. No matches between fractions. At 13, look for shared digits across fractions and notice they cluster within the two groups of six. At 12, sort the fractions by their repeating digit (3 vs 6 vs terminating) and you will see the blocks.
Code: github.com/alexspetty/nfield
Paper: The Cross-Alignment Matrix
Alexander S. Petty
August 2022 (updated March 2026)
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