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
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 sequentially
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 an 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 with the combinatorial factors underlying the Hardy–Littlewood k-tuple conjecture.
A central structural result is the Stability Zone SZn = [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 the symbolic structure experimentally up to n =
250,000.
Our second focus is: Can this new framework lead to new mathematical knowledge?
Finally, we discuss the local Hausdorff–Besicovitch dimension of the prime candidate
set within the Stability Zone. It begins near 0.92 and increases monotonically toward 1
as n → ∞, following D(p) = 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 deterministic, structural perspective on the transition from the
ordered regime 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.