(Primes) Rail Arm Theory

We investigate prime-generating behavior in concatenations of prime blocks where the terminal block is transformed by an operator F acting on primes. We show that such constructions are governed by deterministic rail structures induced by base-dependent modular quantization, which route candidate integers onto a subset of prime residue classes modulo 30. For quadratic operators such as F(p)=2p²+1, residue-image collapse restricts reachability to specific spiral arms, producing operator fingerprints that are invariant across prefix generators and digit lengths. Within the reachable arms, survival under small-prime sieving induces stable modular attractor wells modulo 210, which sharpen under wheel refinement to mod 2310 and mod 30030. Control experiments using random tails demonstrate that these wells are operator-driven rather than prefix artifacts. We further show that operator viability is base-conditional: in base 12, F(p)=2p²+1 collapses all rails to multiples of 3 and produces no primes beyond trivial cases, while the repaired operator F(p)=2p²−1 restores reachability and predicts viable arms {1,7,13,19}. The results establish a deterministic routing–survival geometry for prime-block constructions, yielding predictive residue classes and a taxonomy of operator behavior.