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 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 se-
quentially to a growing symbolic tape, and provably reproduces the classical prime-composite
classification for every integer n ≥ 2. Unlike algorithmic sieves, the automaton generates
an internal symbolic representation of the number line whose structure can be analyzed at
every 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 of
twin prime templates, yielding a recursive growth formula consistent with OEIS sequence
A059861 and consistent with the combinatorial factors underlying the Hardy-Littlewood
k-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 — unfolds
dynamically inside the growing, advancing Stability Zone as it travels through the number
line, and provides a structural perspective on the transition from the ordered structure of
small 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.