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 thatrealizes 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 agrowing symbolic tape, and it reproduces the classical prime–composite classification forevery integer n ≥ 2. A central result is the Stability ZoneSZn = n + 1, 2n − 1, an interval in which the symbolic state is provably immune to all later filtering steps. Thisyields 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 explicitprime-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 accessibleto combinatorial and dynamical analysis.