The Alignment Limit for All Primes
The gap
The first paper proved that the alignment formula α = (2m − 1)/(3m − 1) holds for p = 3 and that the golden ratio selects it uniquely. The second paper extended the formula to all digit-partitioning primes (p ≤ b + 1) and showed that the digit function's injectivity is the mechanism.
A gap remained. For primes p > b + 1, the digit function is not injective. Fractions from different residue classes can share digits at common positions, and the alignment exceeds the baseline (2m − 1)/(pm − 1). How much does it exceed it, and does the golden ratio's selection of p = 3 survive?
The digit function and its bins
The digit function δ(r) = ⌊br/p⌋ maps the p − 1 nonzero remainders into b digit values. When p ≤ b + 1, every remainder gets its own digit (injectivity). When p > b + 1, some remainders share a digit. They fall into bins:
$$B_d = {r \in {1, \ldots, p-1} : \lfloor br/p \rfloor = d}$$
The bin sizes follow a simple pattern. Let q = ⌊(p−1)/b⌋ and r = (p−1) mod b. Then r bins have size q + 1 and b − r bins have size q.
The alignment limit
For any prime p, the repetend alignment of n = pm converges as m → ∞ through b-smooth integers. The limit is
$$\mathcal{L}(p,b) = \frac{L + T}{pL}$$
where L = ord_p(b) is the multiplicative order and T is the orbit bin sum: the total bin sizes encountered as the orbit ⟨b⟩ traverses the digit function.
When the orbit is equidistributed across bins (which covers most primes), this simplifies to a closed form depending only on the bin sizes:
$$\mathcal{L}(p,b) = \frac{p - 1 + S(p,b)}{p(p-1)}$$
where S(p,b) = Σ n_d² is the sum of squared bin sizes, explicitly computable as
$$S(p,b) = bq^2 + r(2q+1)$$
For digit-partitioning primes (p ≤ b + 1), all bins have size 1, so S = p − 1 and the limit reduces to 2/p, recovering the earlier results.
The golden ratio bound
The alignment limit is bounded above by (p + 1)/(2p) for all primes p ≥ 5 in base b ≥ 2. For p = 5 this gives 6/10 = 0.6, which is strictly below 1/φ ≈ 0.618.
Even with cross-class corrections included, no prime beyond 3 produces enough alignment to reach the golden threshold. The golden ratio's selection of p = 3 is not an artifact of the digit-partitioning restriction. It holds universally.
In base 10
| p | L | S | Limit | Decimal |
|---|---|---|---|---|
| 3 | 1 | 2 | 2/3 | 0.6667 |
| 7 | 6 | 6 | 2/7 | 0.2857 |
| 11 | 2 | 10 | 2/11 | 0.1818 |
| 13 | 6 | 16 | 7/39 | 0.1795 |
| 17 | 16 | 28 | 11/68 | 0.1618 |
| 19 | 18 | 34 | 26/171 | 0.1520 |
| 23 | 22 | 50 | 36/253 | 0.1423 |
| 29 | 28 | 80 | 27/203 | 0.1330 |
Only p = 3 exceeds 1/φ. The correction terms enrich the alignment for larger primes but never enough to close the gap.
The program so far
- Paper 1: the golden ratio's minimal polynomial divides the self-referential cubic only at p = 3, producing the identity τ⁴ = 2 − 3τ.
- Paper 2: the alignment formula (2m−1)/(pm−1) holds for all p ≤ b+1, characterized by injectivity of the digit function.
- Paper 3 (this paper): the alignment limit extends to all primes via the bin-sum formula, and the golden bound holds universally.
The next layer is the alignment of composite denominators with multiple prime factors, where the Chinese Remainder Theorem governs the interaction between prime contributions.
The paper
The Alignment Limit for All Primes (PDF)
All three papers can be verified computationally using nfield:
./nfield verify # paper 1
./nfield verify2 # paper 2
.:.