The Cross-Alignment Matrix

The scalars $\alpha$, $\sigma$, and $F$ from The Coherence Decomposition are averages. They compress the fractional field into a single number. A number is useful. But something is lost in the compression.

To see what, build the full grid. Compare every fraction of $n$ to every other fraction, digit by digit. Record the proportion of matches. Lay the results in a table with fractions on both axes. That table is the cross-alignment matrix $\mathbf{A}(n)$, and it has a shape.

Three primes, three completely different shapes.

At $p = 7$, the matrix is the identity. Six fractions, six distinct six-digit patterns, no two sharing a single digit at any common position. Write the grid down and it is white everywhere except the main diagonal. Nothing in that field talks to anything else.

At $p = 13$, the isolation breaks. Twelve fractions, and they are no longer all strangers. They fall into two groups of six. Within each group, fractions share digits at some positions. Between the groups, nothing. Two bright blocks sit off the diagonal, one for each group.

At $n = 12$, the field fractures into three families. Terminating fractions, fractions repeating 3, fractions repeating 6. Each family agrees only with itself. Three solid blocks, different sizes, sharp edges.

Same measure, three completely different pictures. The averages cannot tell these apart. The matrix can.

The eigenvalue spectrum

A matrix has eigenvalues. They measure the independent directions in which the matrix acts. At $p = 7$, every eigenvalue is 1. The identity matrix has no internal structure to reveal.

At $p = 13$, two distinct eigenvalues appear, $4/3$ and $2/3$, each with multiplicity 6. Two groups, two levels. At $p = 17$, nine distinct eigenvalues spread between $0.25$ and $1.75$. The spectrum fans out as the prime grows.

Now the observation I did not see coming.

For primes where 10 is a primitive root, the rows of $\mathbf{A}(p)$ are cyclic shifts of each other. The second row is the first row shifted one position. The third row is the first row shifted two. All the way down. Mathematicians call a matrix with this structure a circulant, and circulants have a clean spectral theory that applies to all of them at once, regardless of what their first row contains. The eigenvalues of any circulant are the discrete Fourier transform of its first row.

The first row of $\mathbf{A}(p)$ is the autocorrelation of the digit sequence of $1/p$.

That digit sequence has been sitting in arithmetic tables since before Euler. Gauss computed these. Every schoolchild who has done long division has produced a piece of one. The cyclic structure of the repetend of $1/7$ has been marveled at for centuries, usually without explanation.

Here is the explanation. That digit sequence is the first row of a circulant whose eigenvalues are its own Fourier transform. The pairwise agreement structure of the entire fractional field, every fraction compared to every other fraction at every digit position, is encoded in the Fourier transform of the sequence you have been looking at since Why the Golden Ratio Selects the Prime Three.

It was always there. Nobody asked the right question.

This is where the work moves from digit counting toward harmonic analysis on the field.

The matrix is the object

$\alpha$, $\sigma$, and $F$ from The Coherence Decomposition are averages of this matrix. Average the off-diagonal entries and you recover $\sigma$. The total alignment $\alpha$ still needs the terminating-fraction convention from The Coherence Decomposition, and then $F = \alpha - \sigma$ measures the excess coherence of $1/n$ over the pairwise background.

The earlier scalar invariants were shadows of a larger object. Once the matrix is written down, the geometry of the field is no longer hidden inside an average.

A note from 2026

April 2026

The matrix held up. Once it was written down, the rest of the spectral side of the work had somewhere definite to live. The Spectral Structure of Fractional Fields reads its eigenvalues, The Spectral Power of the Digit Function isolates the magnitude side of the same Fourier data, and The Autocorrelation Formula closes the loop by recovering the self-match counts.

The same grid idea survives into the collision work. The Collision Periodic Table keeps the finite table but changes the signal being written into its cells. The complement symmetry remains. The block structure remains. What changes is what is being measured.

It is the point where the geometry of pairwise agreement becomes explicit enough to support everything that follows.

.:.

Try it yourself

Three fields, three matrix shapes.

$ ./nfield field 7     # identity matrix: no off-diagonal agreement
$ ./nfield field 13    # two orbits: partial block structure
$ ./nfield field 12    # three solid blocks: composite case

At $p = 7$, no two fractions share a digit at the same position. At $p = 13$, shared digits cluster within each orbit of six. At $n = 12$, sorting by repeating block makes the three families visible.

Now look at the spectrum.

$ ./nfield spectral 7    # all eigenvalues = 1
$ ./nfield spectral 13   # two eigenvalues: 4/3 and 2/3
$ ./nfield spectral 17   # nine distinct eigenvalues

At $p = 7$, the spectrum is flat. At $p = 13$, two levels. At $p = 17$, many more. The Fourier transform of the digit sequence of $1/p$ is sitting there in the eigenvalues the whole time.

Code: github.com/alexspetty/nfield
Paper: The Cross-Alignment Matrix

Alexander S. Petty
April 2021
.:.