The Prime Sequence as the Unique Fixed Point of a Causal Binary Sieve

This paper presents a causal binary sieve operator acting on sequences over the natural numbers and proves that the prime indicator sequence is its unique fixed point. In this formulation, the primes are not only the result of a sieve, but the unique self-consistent binary configuration that reproduces itself under a causal divisibility exclusion rule. The construction is then organized through a finite four-bit architecture. Every integer is written as n = 16K + r, where r is the local four-bit residue and K acts as an accumulated carry state across successive blocks. The initial cell detects exactly the prime residues below 16, while each later block is determined by the local residue, the accumulated carry, and the inherited finite sieve memory of primes up to the square root of the block endpoint. The paper also introduces a spectral refinement based on block fluctuations. The von Mangoldt function is used to connect the block architecture with Chebyshev's function and the logarithmic derivative of the Riemann zeta function. The resulting framework is not presented as a proof of the Riemann Hypothesis, but as a finite-cell and fixed-point architecture for studying prime fluctuations through a controlled logarithmic spectral program.