We present a deterministic, endogenous, non-stationary S-adic automaton that models the Sieve of Eratosthenes as a dynamical system over a finite symbolic alphabet. The automaton operates through three operators — shift, expansion, and filtering — applied sequentiallyto a growing symbolic tape, and provably reproduces the classical prime-compositeclassification for every integer n ≥ 2. Unlike algorithmic sieves, the automaton generatesan internal symbolic representation of the number line whose structure can be analyzed atevery step. Our first focus is: Can this new framework reproduce known mathematical knowledge? We demonstrate that this representation is not arbitrary: the tape exhibits a four-letter substructure {a, b, c, d} governed by an explicit substitution morphism and upper triangular transition matrix Mp. The dominant eigenvalue p − 2 controls the population dynamics oftwin prime templates, yielding a recursive growth formula consistent with OEIS sequenceA059861 and consistent with the combinatorial factors underlying the Hardy-Littlewoodk-tuple conjecture. A central structural result is the Stability Zone [n + 1, 2n − 1], a provably immutable interval in which prime candidates survive all prior filtering steps. Using a Frozen Window technique, we verify the persistence of symbolic structure experimentally up to n = 250,000. Our second focus is: Can this new framework lead to new mathematical knowledge? Finally, we discuss a new way of fractal dimension (self similarity), fitting for the prime candidate set within the Stability Zone. It begins near 0.92 and increases toward 1 as n → ∞, following D = ln(p − 1)/ ln(p). This process — vanishing fractality — unfoldsdynamically inside the growing, advancing Stability Zone as it travels through the numberline, and provides a structural perspective on the transition from the ordered structure ofsmall primes to the apparent randomness observed in large-scale prime distributions. The automaton is offered not as a computational tool for generating primes, but as a research instrument: a symbolic framework in which arithmetic properties of the natural numbers emerge from the internal dynamics of the system.