The Coherence Decomposition
Beyond alignment
The previous papers measured one thing: how coherently the fractional field {k/n} aligns with 1/n. That measure, α(n), produced the three-tier theorem and the golden ratio's selection of p = 3.
But it left a question unanswered. When two fractions in the field share a digit at the same position, is that because they both align with 1/n, or because they align with each other independently?
This paper separates the two.
Two new invariants
The pairwise alignment σ(n) measures background coherence: the average digit overlap between all pairs of fractions in the field, without reference to 1/n.
The focused alignment F(n) = α(n) − σ(n) measures how much more coherent the field is with 1/n than with itself. It isolates the structural identity of 1/n within its own field.
The decomposition
For digit-partitioning primes (p ≤ b + 1), the three invariants have exact formulas:
$$\alpha(ps) = \frac{2s-1}{ps-1}, \qquad F(ps) = \frac{s}{ps-1}, \qquad \sigma(ps) = \frac{s-1}{ps-1}$$
As the smooth factor s grows, all three converge:
- α → 2/p (total coherence)
- F → 1/p (focused coherence)
- σ → 1/p (diffuse coherence)
At infinite resolution, total coherence splits equally into focused and diffuse halves.
The golden gap
The golden ratio 1/φ ≈ 0.618 lies between F → 1/3 and α → 2/3 for p = 3. It sits in the gap between focused and total coherence.
For any other prime, the gap is too narrow. For p = 7: F → 1/7, α → 2/7, and 1/φ > 2/7. The golden threshold overshoots the total coherence entirely.
The condition for 1/φ to fit inside the gap is 1/p < 1/φ < 2/p, which simplifies to p < 2φ ≈ 3.236. Only p = 3.
The golden ratio selects p = 3 because it is the unique prime whose diffuse coherence is large enough to carry the total above the self-referential boundary. The focused component alone never reaches 1/φ for any prime. It is the accumulation of pairwise background coherence that pushes the total across.
Sigma and digit-partitioning
The pairwise alignment σ(p) = 0 for single primes provides an equivalent characterization of the digit-partitioning property: a prime is DP if and only if no two distinct fractions in its field share a digit at any common position. This is a cleaner statement than the injectivity condition of the earlier paper, and it extends naturally to all n.
Three invariants
The fractional field now carries three measurable quantities:
- α(n): total coherence
- σ(n): diffuse coherence
- F(n): focused coherence
These are distinct, computable, and structurally meaningful. Alpha connects to the golden ratio. Sigma characterizes digit-partitioning. F measures structural identity.
The paper
The Coherence Decomposition (PDF)
All five papers can be verified using nfield:
./nfield verify # paper 1
./nfield verify2 # paper 2
./nfield verify3 # paper 3
.:.