The Three-Tier Theorem
Divide $1$ by $7$ in base $10$ and you get the famous six-digit repeating loop. Divide every other fraction of $7$ by hand and you get the rest of the loop's rotations. The nfield tool prints this as a small table, with the repeating block of each fraction marked by pipes:
$ ./nfield field 7
1/7 => 0.|142857|
2/7 => 0.|285714|
3/7 => 0.|428571|
4/7 => 0.|571428|
5/7 => 0.|714285|
6/7 => 0.|857142|
The pipes are the program's way of saying "the digits between these marks repeat forever." The six fractions of $7$ are not six unrelated decimals. They are six rotations of the same six-digit string. Every prime in base $10$ has this property in some form. The fractions of a prime are not strangers to one another. They are parts of a small organized family.
That observation is the starting point. If the fractions of a prime form an organized family, then the family has a degree of organization, and the degree of organization can be measured. The simplest way to measure it is to compare every fraction $k/n$ to the special fraction $1/n$, position by position, and ask how often their decimal digits agree. Average that agreement over all the fractions and you get a single number between $0$ and $1$. Call it the alignment of $n$, and write it $\alpha(n)$. The alignment of $7$ in base $10$ is about $0.167$. The alignment of $12$ is about $0.636$. The alignment of $77$ is about $0.088$.
The alignment is a number you can assign to every positive integer. It is not the integer's size and it is not its prime factorization. It is a measurement of how organized the integer's family of fractions is, computed by counting agreements between repeating decimals. Earlier papers in this series (the digit-partitioning paper and the alignment limit paper) defined this number formally and computed its closed form for primes. The present paper takes that number as input and asks the next natural question. When you compute $\alpha(n)$ for every integer, what does the resulting list look like?
The answer is the three-tier theorem.
When you list every integer with its alignment, and you draw a horizontal line at the height $1/\varphi \approx 0.618$, where $\varphi$ is the golden ratio, every integer falls into one of three tiers. Tier 1 is integers whose alignment equals exactly $1$. Tier 2 is integers whose alignment is strictly between $1/\varphi$ and $1$. Tier 3 is integers whose alignment is strictly less than $1/\varphi$. The classification is sharp. The three tiers do not blur into each other. The line at $1/\varphi$ is not arbitrary, and its place in the story is something I will explain in a moment. The theorem is that this line sorts every integer cleanly into one of these three categories with no exceptions and no edge cases. This is proved.
Tier 1: the smooth numbers
Some integers produce only terminating fractions. Divide anything by $8$, or $25$, or $200$, and the decimal stops. No repeating block, no cycle. These are the smooth numbers, integers built entirely from the prime factors of the base. In base $10$, smooth means a product of $2$s and $5$s.
$ ./nfield field 8
1/8 => 0.125
2/8 => 0.25
3/8 => 0.375
4/8 => 0.5
5/8 => 0.625
6/8 => 0.75
7/8 => 0.875
No pipes anywhere. Every decimal stops. There is no repeating block to disagree about, so by convention every fraction is treated as agreeing with $1/n$ at every position. The alignment is $1$, the maximum. Smooth numbers have perfect coherence in a trivial way. There is nothing to mismatch. They sit at the top of the classification, quiet and complete, like silence that scores perfectly on a harmony test.
Tier 2: the golden family
Tier 2 is the place where the golden ratio actually lives. These are the integers of the form $n = 3m$, where $m$ is smooth (built from $2$s and $5$s) and at least $4$. They have repeating fractions, real structure, and genuine disagreements between their decimal patterns at certain positions. And yet, against expectation, their alignment number stays above $1/\varphi$.
The smallest example is $n = 12$. Its field has three flavors of fraction at once: terminating, mixed, and pure repeating. The pipes mark the repeating block where there is one.
$ ./nfield field 12
1/12 => 0.08|3|
2/12 => 0.1|6|
3/12 => 0.25
4/12 => 0.|3|
5/12 => 0.41|6|
6/12 => 0.5
7/12 => 0.58|3|
8/12 => 0.|6|
9/12 => 0.75
10/12 => 0.8|3|
11/12 => 0.91|6|
Four of the eleven fractions repeat the digit $3$ inside their pipes, four repeat the digit $6$, and three terminate. Most positions agree. The alignment of $12$ comes out high.
$ ./nfield align 12 # 0.636
$ ./nfield align 24 # 0.652
$ ./nfield align 60 # 0.661
$ ./nfield align 120 # 0.665
Each value above $0.618$. Each one climbing slowly toward the limit $2/3$. These denominators are governed by the prime $3$, and the prime $3$ is the one prime that the golden ratio singled out as its own. The reason is something the first paper in this series made explicit. The cubic equation $x^2 + x = 1$ that defines the golden ratio is the same equation that selects the prime $3$ as the unique prime whose alignment limit $2/3$ exceeds $1/\varphi$. The arithmetic of the prime $3$ and the algebra of the golden ratio meet at this number, and the meeting is what fills Tier 2.
The golden family is small. It is $3 \times 4$, $3 \times 5$, $3 \times 8$, $3 \times 10$, $3 \times 16$, $3 \times 20$, and so on. Just the integers of the form three-times-a-smooth-number-at-least-four. Nothing else. But it is the only family of non-smooth integers above the line.
Tier 3: everything else
Everything else lives in Tier 3. Every prime from $5$ onward. Every composite whose rough part (the part not coming from the base) is anything other than $3$. Every number that carries a prime factor the golden ratio cannot reach. There are infinitely many of them, including all the primes beyond $3$, all their multiples, and all their products.
$ ./nfield align 7 # 0.167
$ ./nfield align 13 # 0.111
$ ./nfield align 77 # 0.088
$ ./nfield align 91 # 0.100
Look at the numbers. $0.167$, $0.111$, $0.088$, $0.100$. None of them is close to $0.618$. They are not climbing toward it. They are not even in the same neighborhood. These values live in a different country from Tier 2. The gap between the lowest Tier 2 alignment ($0.636$ at $n = 12$) and the highest Tier 3 alignment is not a crack. It is a canyon, and $1/\varphi$ stands in the middle of it.
The proof, in plain language
The proof rests on a simple symmetry of repeating decimals. Every non-terminating fraction $k/n$ has a partner, namely $(n - k)/n$. The repeating digits of $k/n$ and the repeating digits of its partner sum to $9$ at every position. If $k/n$ has the digit $3$ at some position, then $(n-k)/n$ has $6$ there. If $k/n$ has $1$, the partner has $8$. They are mirror images in the digit system. Mathematicians call this the nines complement.
This symmetry has a strong consequence. At any single position, the special fraction $1/n$ has some specific digit, say $5$. A fraction $k/n$ matches $1/n$ at that position only if its digit there is also $5$. But its partner $(n - k)/n$ then has digit $9 - 5 = 4$ at the same position, which is not $5$. So at most one of the two partners can match $1/n$ at any given position. The fractions of $n$ pair up, and within each pair only one half can score a match.
This bounds the average alignment over the non-terminating fractions to $1/2$. Add back the terminating fractions, which always contribute the maximum alignment $1$, and the total alignment of $n$ satisfies the inequality
$$\alpha(n) \le \frac{t + 1}{2t},$$
where $t$ is the rough part of $n$ (the prime factors not in the base).
Now a fact specific to base $10$. The rough part of any integer is always coprime to $10$. The integers $4$, $5$, and $6$ all share a factor with $10$, so they cannot appear as rough parts. The smallest rough part strictly larger than $3$ is therefore $t = 7$. At $t = 7$, the inequality gives
$$\alpha(n) \le \frac{8}{14} = \frac{4}{7} \approx 0.571.$$
That number is below $1/\varphi \approx 0.618$. Every rough part larger than $7$ gives an even smaller bound. So every integer with rough part at least $7$ falls in Tier 3.
Three small cases remain. For $t = 1$, the integer is smooth, alignment is $1$, Tier 1. For $t = 3$ with $m \ge 4$, the alignment formula from earlier in the series gives
$$\alpha(3m) = \frac{2m - 1}{3m - 1} \ge \frac{1}{\varphi},$$
which puts these in Tier 2. For $t = 3$ with $m < 4$, direct computation gives $\alpha(3) = 1/2$ and $\alpha(6) = 3/5$, both below $1/\varphi$, so these go to Tier 3.
The whole argument pivots on the gap between $3$ and $7$ in the rough part. The golden ratio $1/\varphi$ sits above $4/7$ and below $2/3$. The prime $3$ reaches the threshold from above. Every rough part of size $7$ or more falls short from below. And nothing strictly between $3$ and $7$ is coprime to the base, so there is no integer that could land in between.
The rough part and the smooth part
Every positive integer splits uniquely into two pieces. The smooth part absorbs the prime factors that the base contains. The rough part is everything else. In base $10$, the integer $60 = 20 \times 3$ has smooth part $20$ and rough part $3$. The integer $77 = 1 \times 77$ has smooth part $1$ and rough part $77$. The integer $8$ has smooth part $8$ and rough part $1$.
The smooth part controls the resolution of the field. It says how many fractions share the same repeating block, how finely the field is sampled. The rough part controls the structure. It says which prime's digit geometry governs the field, which arrangements of repeating digits occur, where the symmetries lie.
The three tiers are determined entirely by the rough part:
- Rough part $= 1$. No prime structure at all. Every fraction terminates. Tier 1.
- Rough part $= 3$ with smooth part at least $4$. The golden prime governs. Tier 2.
- Anything else. Below the threshold. Tier 3.
That is the whole classification, in three lines.
What the classification means
Stand back from the details. Here is what the three-tier theorem says, in plain language.
Every positive integer has an interior. That interior is the family of fractions with that integer in the denominator, together with their repeating decimal patterns, their symmetries, and their disagreements. The alignment number measures how organized that interior is.
Now draw one horizontal line, at the height $1/\varphi$. This is not a line you chose. It is the line that the algebra of the simplest cubic equation puts in exactly that place, and the algebra is the same algebra that defines the golden ratio. That single line sorts every positive integer into one of three categories. Every positive integer. No exceptions, no edge cases.
The integers are not random when viewed from the inside. They are tiered. The tiering is controlled by one constant, and the constant is the golden ratio. Silence at the top. The golden family in the middle. Everything else below, separated by a canyon so wide that no borderline cases exist anywhere in the integers.
I did not set out to find this. I was measuring alignment because I wanted to understand repeating decimals, and the classification emerged from the measurements. The golden ratio was already there, waiting at the boundary, because it had been selecting the prime $3$ from the very beginning of the program.
A note from 2026
April 2026
The three-tier classification proved here completed the early alignment program. The two earlier alignment papers had defined the alignment number and proved the limit formula at every prime. This paper closed the gap by ruling out every integer above the golden line except the smooth numbers and the golden family. The proof's central tool, the nines complement bound on non-terminating fractions, foreshadowed the structural antisymmetry that runs through the recent work.
In the recent collision invariant program, the same nines complement appears as the collision periodic table's reflection identity. The pairing of digits at the alignment level (each pair of partner digits sums to the base minus one, so at most one can match at any position) is the local instance. The pairing of residue classes at the collision level is the global generalization. Both come from the same involution, and both produce a structural mean of exactly $-1/2$ that runs through the entire program.
The rough-part framing of this paper, splitting every integer into its smooth and rough components, prefigured the conductor structure of the recent work. The boundary alphabet at conductor $b^2$ classifies primes by their last two base-$b$ digits, exactly the slicing that the rough part defines for general integers. The "tier" of an integer in this paper is determined by its rough part. The cell of a prime in the recent collision invariant is determined by the same kind of arithmetic, one level up.
The closing image of this post, the line at $1/\varphi$ that emerges from the arithmetic of every $n$ and connects them all at the same height, is the same line the recent work has been climbing toward in different language. The collision invariant's centered sum cancels at the boundary $s = 1$ of the prime number theorem. The collision spectrum factors through L-function values at $s = 1$. The line is in different coordinates, but it is the same line.
.:.
Try it yourself
$ ./nfield align 8 # Tier 1 (smooth): 1.000
$ ./nfield align 25 # Tier 1 (smooth): 1.000
$ ./nfield align 12 # Tier 2 (golden family): 0.636
$ ./nfield align 120 # Tier 2 (golden family): 0.664
$ ./nfield align 7 # Tier 3: 0.167
$ ./nfield align 13 # Tier 3: 0.111
$ ./nfield align 77 # Tier 3: 0.088
$ ./nfield align 91 # Tier 3: 0.100
Each integer is a vessel. Its fractional field rises from it like a fountain of color, carrying the alignment upward. Some fountains are tall and coherent, their light concentrated. Others spray wide and dim. But at $0.618$ of the way up, something happens in all of them. A concentration, a brightening, a star-like point where the structure gathers. These points, one per integer, bridge together across the entire landscape and form a line. The line is not drawn. It appears. It is the golden threshold, emerging from the arithmetic of every single $n$, connecting them all at the same height. Above the line, silence and the golden family. Below the line, everything else, scattered but not random, governed by the same digit function, just too crowded to cohere.
Code: github.com/alexspetty/nfield
Paper: The Three-Tier Theorem
Alexander S. Petty
October 2020 (updated April 2026)
.:.