The Cross-Alignment Matrix
The complete object
The previous papers measured the fractional field {k/n} through scalar invariants: alignment α, pairwise alignment σ, and focused alignment F. Each captures one slice of the field's coherence. This paper introduces the object that contains all of them.
The cross-alignment matrix A(n) is the (n−1) × (n−1) symmetric matrix whose (i,j) entry is the position-wise digit-match proportion between i/n and j/n. It encodes the full pairwise coherence structure of the field in a single object.
The scalar invariants are recovered directly: α is a row sum, σ is the matrix average, F is the excess of one row over the average.
The identity at DP primes
For digit-partitioning primes (p ≤ b + 1), the matrix is the identity. Every eigenvalue equals 1. No two fractions share a digit at any position. The field is maximally non-degenerate: every fraction is structurally independent from every other.
Two levels at non-DP primes
For a single non-DP prime like p = 13 in base 10, the spectrum splits into exactly two eigenvalue levels: 4/3 and 2/3, each with multiplicity 6. These correspond to the two cosets of ⟨10⟩ in (ℤ/13ℤ)*. The coset containing 1/p gets the higher eigenvalue.
The rank theorem
The cross-alignment matrix has full rank if and only if n − 1 ≤ Lb, where L is the cycle length and b is the base.
When the number of fractions exceeds the number of digit-position slots (L positions × b digit values), the matrix becomes singular. Fractions can no longer be distinguished by their digit profiles. The null space dimension measures this structural redundancy.
| n | L | Lb | Rank | Null |
|---|---|---|---|---|
| 7 | 6 | 60 | 6 | 0 |
| 13 | 6 | 60 | 12 | 0 |
| 21 | 6 | 60 | 20 | 0 |
| 33 | 2 | 20 | 19 | 13 |
| 77 | 6 | 60 | 38 | 38 |
| 91 | 6 | 60 | 28 | 62 |
For composites with short cycle length (like n = 33 with L = 2), the digit resolution runs out quickly. For composites with long cycles but many fractions (like n = 77 or n = 91), the field outgrows its resolution capacity.
Resolution
The smooth factor s in n = ts increases the number of fractions without changing the cycle length. As resolution grows, the field eventually exceeds the capacity Lb, and structural redundancy appears. The ratio (n−1)/(Lb) measures how many fractions share each digit-position slot. When it exceeds 1, the null space is non-trivial.
The digit-partitioning property provides structural headroom: for DP primes, full rank is maintained across a wide range of resolutions.
The paper
The Cross-Alignment Matrix (PDF)
.:.