Vanishing Fractality: A deterministic, endogenous, non-stationary S-adic model for the sieve of Eratosthenes

We present a deterministic, endogenous, non-stationary S-adic automaton thatmodels the Sieve of Eratosthenes as a dynamical system over a finite symbolic alphabet. Theautomaton 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-lettersubstructure {a, b, c, d} governed by an explicit substitution morphism and upper triangulartransition 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 immutableinterval in which prime candidates survive all prior filtering steps. Using a Frozen Windowtechnique, 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 primecandidate set within the Stability Zone. It begins near 0.92 and increases toward 1 asn → ∞, 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 aresearch instrument: a symbolic framework in which arithmetic properties of the naturalnumbers emerge from the internal dynamics of the system.