Rail-Arm Theory: Type Layer Preference

Rail-Arm Theory [1] establishes two filtering layers for prime-block concatenation: rail quantization (base-dependent prefix routing) and arm reachability (operator-dependent residue injection). This paper introduces a third, finer filtering layer—the Type Preference Layer—which determines which mod-6 prime type classes receive candidates. This layer provides predictive power beyond reachability alone: it explains why certain reachable arms exhibit systematically different prime densities. Our central result is the Type Selection Theorem: the algebraic interaction between the rail subgroup Rail₃₀(b) and operator image Im₃₀(F) determines whether candidates access both mod-6 prime types or are restricted to a single type. We prove that base-12 concatenation with the repaired operator F₋(p) = 2p² − 1 confines all candidates to Type I primes (r ≡ 1 mod 6), whereas base-10 with F₊(p) = 2p² + 1 accesses both types. This single-type restriction represents a structural constraint that exists prior to any sieve application. We formalize the survival weight function w(r) quantifying differential elimination rates, and explain well sharpening under modulus escalation via the nested coset structure of the primorial tower. The complete three-layer fingerprint (Rail Reachability, Arm Reachability, Type Preference) enables predictive filtering of prime candidates before primality testing, transforming concatenative prime generation from exploratory search into structured algebraic procedure.