On The Nature Of Primes: A deterministic, endogenous, non-stationary S-adic Automaton for the sieve of Eratosthenes

We present a deterministic, endogenous, non-stationary S-adic automaton that
realizes the Sieve of Eratosthenes as a symbolic dynamical system over a finite alphabet.
Its evolution is governed by three operators—shift, expansion, and filtering—acting on a
growing symbolic tape, and it reproduces the classical prime–composite classification for
every integer n ≥ 2.
A central result is the Stability Zone
SZn = [n + 1, 2n − 1],
an interval in which the symbolic state is provably immune to all later filtering steps. This
yields pointwise stability of the prime encoding and makes large-scale experimental verifica-
tion possible through a Frozen Window technique, which we implement up to n = 250,000.
The tape also exhibits a canonical four-letter substructure {a, b, c, d} governed by explicit
prime-dependent substitution rules and an upper-triangular transition matrix Mp.
The automaton is not intended as an efficient prime generator, but as a symbolic re-
search instrument in which arithmetic properties of the natural numbers become accessible
to combinatorial and dynamical analysis.