By Alex Petty — 28 Mar 2026

The Three-Tier Theorem

The same field, partitioned and weighted as before, now resolves into three regimes. In some cases the structure remains fully coherent, in others it holds at a critical threshold, and in the rest it disperses. The classification is not imposed from outside; it emerges from the way mass flows through the bins. The arithmetic of n determines which regime the field inhabits.

The conjecture

The first paper in this series proved the alignment formula for n = 3m and conjectured that the golden ratio threshold 1/φ classifies all positive integers into exactly three tiers:

Tier 1: n is 10-smooth (all prime factors divide 10). Alignment = 1.
Tier 2: n = 3m with m smooth and m ≥ 4. Alignment ≥ 1/φ.
Tier 3: everything else. Alignment < 1/φ.

This paper proves the conjecture.

The rough part

Every positive integer n factors as n = s · t, where s is the largest divisor whose prime factors all divide the base (the smooth part) and t collects everything else (the rough part). In base 10, the smooth part absorbs all factors of 2 and 5, and the rough part is what remains.

The terminating fractions in {k/n} are exactly the multiples of t: there are s − 1 of them. The rest have repeating expansions whose structure is governed by t.

The proof

The proof proceeds by cases on the rough part.

If t ≥ 7 (contains any prime factor ≥ 7): the alignment limit from the earlier paper gives α(n) ≤ 2/7 ≈ 0.286, well below 1/φ ≈ 0.618.

If t = 3, s ≥ 4: this is Tier 2, with α = (2s−1)/(3s−1) ≥ 1/φ by the golden threshold theorem.

If t = 3, s < 4: only n = 3 and n = 6. Direct computation gives α(3) = 1/2 and α(6) = 3/5, both below 1/φ.

If t = 9 or higher powers of 3: the alignment approaches 2/9 ≈ 0.222 or smaller. Far below 1/φ.

If t has two or more distinct prime factors: the alignment is bounded by the single-prime limit of the smallest factor, which is at most 2/7.

Every case is covered. In all of them, if n is not in Tier 1 or Tier 2, alignment is strictly below 1/φ.

What this means

The three-tier classification is a complete partition of the positive integers by decimal coherence. The golden ratio 1/φ is the dividing line between Tier 2 and Tier 3, and no other algebraically natural threshold produces a comparable classification.

The result depends on three ingredients developed across the earlier papers: the alignment formula, the digit-partitioning characterization, and the alignment limit for all primes. The three-tier theorem unifies them into a single statement.

The paper

The Three-Tier Theorem (PDF)

nfield on GitHub

.:.