RCX: A Recursion-Only Structural Foundation for Logic, Computation, and Paradox Containment

RCX is a minimal formal system in which logic, operators, and structure emerge solely from recursive transformation beginning at the empty set. No external logic, axioms, types, or semantics are assumed. The RCX kernel permits recursion only when identity is novel and structure traceable to ∅. Fixpoints emerge only when recursion stalls, and operators arise structurally, not symbolically. The system halts on contradiction but survives paradox through recursion sealing (Ω′). Shadow Calculus is introduced as an extension domain for identities that fail to flatten under structural recursion, allowing formal reasoning near paradox without explosion. This paper defines the RCX kernel, demonstrates complete derivation from ∅ to operator, and proves minimality, consistency, and expressive completeness.