01_THEORY

Establishing the mathematical isomorphism between the Sieve of Eratosthenes and the Symbolic Dynamics of the Logistic Map.

FOUNDATION

The Core Hypothesis

The central hypothesis posits that the Sieve of Eratosthenes—an ancient algorithm for finding prime numbers—is essentially a dynamical process. By encoding the "survival" (prime) and "sieved" (composite) states as a symbol sequence, we can map this process onto the symbolic dynamics of a unimodal map.

Key Proposition

"The chaotic orbit of the Logistic Map $x_{n+1} = 1 - ux_n^2$ at the band-merging point $u \approx 1.5437$ is topologically equivalent to the limit system of the infinite sieve process."

LEMMA 1

Symbolic Sequence Synthesis

We define the sieve operator $M_p$ for each prime $p$ as a periodic symbol sequence. The state of each integer position $n$ is defined as:

L (Left): Survival (Potential Prime)
R (Right): Sieved (Composite)

The cumulative dynamic sequence $D_i$ is the result of the first $i$ prime sieves acting together:

D_i = M_{p_1} \cdot M_{p_2} \cdot ... \cdot M_{p_i}

COMPOSITION RULES

L · L= L (Survival)

L · R= R (Sieved)

R · L= R (Sieved)

R · R= R (Sieved)

* "Destruction Priority": Once a number is sieved (R), it remains sieved forever.

MATHEMATICAL PROOF

Critical Lemmas

Admissibility & Truncation

For the cumulative sieve sequence $D_i$ , its subsequence of length $p_i^2 + 1$ constitutes a valid Kneading Sequence.

This lemma ensures that the sequence generated by the sieve can actually be produced by a unimodal map. It relates to Legendre's conjecture regarding prime gaps.