Théorème

N2 — Sieve self-consistency

The Eratosthenes sieve is the unique self-consistent multiplicative sieve on $\mathbb{N}_{\geq 2}$.

Statement

The set of primes $\mathbb{P}$ is the unique self-consistent multiplicative sieve on $\mathbb{N}_{\geq 2}$ . Self-consistent means $n \in \mathcal{S}(\mathbb{P})$ (i.e. $n$ survives) iff no prime $p < n$ divides $n$ .

Théorème

Plain reading. The Eratosthenes sieve has a special property: what it keeps matches exactly what it was supposed to keep. No other multiplicative elimination algorithm has this consistency. The sieve defines itself.

Why it matters

N2 closes a subtle question: the internal consistency of the sieve. One might imagine another sieve (eliminating by other rules, keeping different residues) that would return its own result. N2 shows that the unique sieve whose output is consistent with its input rule is Eratosthenes’.

This self-consistency is what allows the cascade T0 → T6 to close without introducing an external parameter.

Proof — outline

Existence: show $\mathbb{P}$ is self-consistent (a composite $n = ab$ with $a \leq \sqrt{n}$ has a prime factor $p \leq a < n$ , hence $n$ is eliminated).
Uniqueness: assume a sieve $\mathcal{C}$ self-consistent and different from $\mathbb{P}$ . Derive a contradiction.

Detailed proof

Part 1 — Existence: $\mathbb{P}$ is self-consistent

Let $n \in \mathbb{N}_{\geq 2}$ . If $n$ is prime, by definition no prime $p < n$ divides $n$ , and $n$ is kept by the sieve.

If $n$ is composite, write $n = ab$ with $a \leq \sqrt{n}$ . Then $a$ has a prime factor $p \leq a < n$ , hence $p$ divides $n$ . The Eratosthenes sieve eliminates $n$ as a multiple of $p$ . Self-consistency holds.

Part 2 — Uniqueness

Suppose another multiplicative sieve $\mathcal{C}$ is self-consistent. Let $\mathcal{C}_{\rm out}$ be the set of kept integers.

If $\mathcal{C}_{\rm out} \neq \mathbb{P}$ , let $n$ be the smallest integer where the two differ: $n \in \mathcal{C}_{\rm out}$ but $n \notin \mathbb{P}$ , or vice versa.

Case 1: $n \in \mathcal{C}_{\rm out}$ , $n$ composite. Then $n$ has a prime factor $p < n$ . If $p \in \mathcal{C}_{\rm out}$ ( $p$ kept by $\mathcal{C}$ ), then by self-consistency $\mathcal{C}$ must also keep $n$ (since $\mathcal{C}$ uses multiplicative rules). But then $\mathcal{C}$ keeps a composite — by construction of a strict multiplicative sieve, contradiction.

Case 2: $n \in \mathbb{P}$ but $n \notin \mathcal{C}_{\rm out}$ . Then a multiplicative divisor eliminated $n$ . But $n$ prime has only 1 and $n$ as divisors. None is strictly in $\{2, \ldots, n-1\}$ — contradiction.

In both cases, contradiction. So $\mathcal{C}_{\rm out} = \mathbb{P}$ .

Link to PT

Self-consistency is the property that lets the sieve dynamics close on itself. It gives the fixed point $\mu^* = 15$ its uniqueness (T5): it is the unique self-consistent solution of the cascade equation.

For the complete derivation, see chapter 2 of the monograph.