The Coherence Decomposition

Walk into a room where everyone is singing the same note. Now one person sings louder. Is that person leading, or just singing along?

That is the question this paper asks about the fractional field. The alignment $\alpha(n)$ measures how well the whole field matches $1/n$. But how much of that match is because $1/n$ is special, and how much is because the fractions already agree with each other, and $1/n$ is just one of them?

In a field where every fraction shares digits with every other fraction, matching $1/n$ is not an achievement. It is conformity. In a field where fractions are independent, matching $1/n$ means something. The alignment number cannot tell you which situation you are in. It conflates the signal with the background.

I wanted to pull them apart.

The decomposition

They come apart cleanly:

$$\alpha = F + \sigma$$

Two terms. $\sigma(n)$ is the pairwise alignment: the average digit-match proportion between all pairs of fractions, not just pairs involving $1/n$. It measures the background hum of the field. How much do fractions agree with each other, regardless of which one you compare to?

$F(n) = \alpha - \sigma$ is the focused alignment: the excess. How much does $1/n$ stand out from the crowd? How much of the total coherence belongs to $1/n$ specifically, rather than to the field's own internal harmony?

For digit-partitioning primes $p$ and smooth $m$, both have closed forms:

$$F(pm) = \frac{m}{pm-1}, \qquad \sigma(pm) = \frac{m-1}{pm-1}$$

As $m$ grows: $F \to 1/p$, $\sigma \to 1/p$, $\alpha \to 2/p$. At infinite resolution, the total coherence splits in half. One part focused. One part diffuse. Equal contributions. A clean symmetry hiding inside the alignment formula all along.

The golden gap

Now look at where $1/\varphi$ falls.

At $p = 3$: the focused limit is $1/3 \approx 0.333$. Below the golden threshold. The background limit is also $1/3$. Also below. Neither part, on its own, reaches $1/\varphi \approx 0.618$.

But together: $1/3 + 1/3 = 2/3 \approx 0.667$. Above.

This is the golden gap. The threshold sits between the individual contributions and their sum. The focused alignment cannot reach it. The diffuse alignment cannot reach it. But the two together, the specific agreement with $1/n$ and the general agreement of the field with itself, combine to cross the line.

Among primes $p \geq 3$, only $p = 3$ has this property. At $p = 5$, the total limit is $2/5 = 0.4$. Both contributions could be as large as they like, and the total would still fall short. The gap between $F$ and $\alpha$ is too narrow. The golden threshold does not fit inside it.

The three-tier theorem is this gap made structural. At $p = 3$, the background hum of the field is just loud enough to carry the focused signal above the threshold. At every other prime, the hum is too quiet or the signal is too weak. The golden ratio sits in the one place where both contributions are needed and both are sufficient.

The matrix

There is a way to see all of this at once. Take the field of 7:

1/7 => 0.|142857|
2/7 => 0.|285714|
3/7 => 0.|428571|

Compare 1/7 and 2/7, digit by digit. Position 1: 1 vs 2. Different. Position 2: 4 vs 8. Different. Position 3: 2 vs 5. Different. Every position, every pair, zero matches. Now do this for all six fractions against all six fractions. You get a 6 x 6 grid. The entry in row $k$, column $j$ is the fraction of positions where $k/7$ and $j/7$ share the same digit.

That grid is the cross-alignment matrix $\mathbf{A}(7)$. At the prime 7, it is the identity matrix: ones on the diagonal (every fraction matches itself), zeros everywhere else. The fractions are strangers. No background coherence. The room is silent until $1/n$ starts singing.

At larger primes, the matrix is no longer the identity. Try $p = 13$ in your head. Twelve fractions, but now some pairs share digits at certain positions. The off-diagonal entries are no longer all zero. The matrix develops structure: fractions in the same coset show higher agreement, fractions in different cosets show less. Bands appear. Blocks form.

The matrix encodes the full coherence geometry of the field. The diagonal carries the focused signal. The off-diagonal carries the background. The alignment $\alpha$ is the average of the entire row corresponding to $1/n$: diagonal entry (which is 1) plus all the off-diagonal entries. The pairwise alignment $\sigma$ is the average of the off-diagonal alone. The decomposition $\alpha = F + \sigma$ is not an abstraction. It is a statement about which part of the matrix you are reading.

What the paper proves is that this matrix, for digit-partitioning primes, has exact closed-form entries. The formulas for $F$ and $\sigma$ follow directly. And the golden gap, the fact that $1/\varphi$ sits between $F$ and $\alpha$ only at $p = 3$, is visible in the matrix as the unique prime where the off-diagonal is just strong enough to lift the row average above the threshold.

Try it yourself

The decompose command pulls the alignment apart. Start with the golden family:

$ ./nfield decompose 12
  alpha  = 0.636  (total)
  sigma  = 0.273  (background)
  F      = 0.364  (focused)
  1/phi in gap = yes

$ ./nfield decompose 120
  alpha  = 0.664  (total)
  sigma  = 0.328  (background)
  F      = 0.336  (focused)
  1/phi in gap = yes

Look at what happens as $m$ grows. $F$ drops from 0.364 toward $1/3$. $\sigma$ rises from 0.273 toward $1/3$. They converge toward each other. The focused signal weakens. The background strengthens. But their sum holds above $1/\varphi$ the entire time. The two components trade weight, and the golden threshold stays below them.

Now the prime 7, a digit-partitioning prime in Tier 3:

$ ./nfield decompose 7
  alpha  = 0.167  (total)
  sigma  = 0.000  (background)
  F      = 0.167  (focused)
  1/phi in gap = no

Zero background. The fractions of 7 are completely independent. The matrix is the identity. There is no hum. The only coherence is focused, and $1/7$ alone cannot carry the total anywhere near 0.618.

Now a composite, $n = 77 = 7 \times 11$:

$ ./nfield decompose 77
  alpha  = 0.088  (total)
  sigma  = 0.065  (background)
  F      = 0.022  (focused)
  1/phi in gap = no

Both components are present but both are tiny. The focused signal is nearly gone: $1/77$ barely stands out from its own field. The background exists but is faint. Together they reach 0.088. The golden threshold is seven times higher.

Code: github.com/alexspetty/nfield
Paper: The Coherence Decomposition

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Alexander S. Petty
June 2022 (updated March 2026)
.:.