Shattering the Probability Barrier: How the Fractal Body–Tail Sieve Converts Miller–Rabin Primality Testing into Deterministic Certainty

We introduce a hybrid primality verification framework that integrates the classical probabilistic Miller–Rabin test with a multi-component "body–tail" residue filter operating within the rings Z9·10^k·p (units). While the Miller–Rabin algorithm effectively isolates composite elements with an exponentially vanishing error probability, it fundamentally lacks algebraic determinism when evaluating strong pseudoprimes. By projecting candidate integers onto hierarchical fractal grids (mod 90, 900, 9000, 90000), we extract rigid structural constraints based on quadratic residue collisions. We demonstrate that the introduction of a coprime orthogonal modulus (p=7) successfully shatters the decimal "efficiency plateau," compressing the permissible search corridors of potential false witnesses to a dense 1.27% of the residue field. This mathematical symbiosis enables instantaneous, deterministic primality validation at the speed of a probabilistic heuristic.