Transfinite Fixed-Point Collapse via the φ^(∞) Operator

Abstract

We introduce a transfinite fixed-point operator φ^(∞) that extends classical fixed-point constructions into the transfinite. While Kleene, Tarski, Scott, and Gödel established foundational fixed-point theorems in recursion theory, lattice theory, domain theory, and logic, these results face limitations in infinitary settings. By iterating a semantic operator through ordinal stages, φ^(∞) captures the eventual convergence or collapse of the fixed-point sequence. We formalize conditions under which φ^(∞) yields a well-defined fixed point or returns a failure symbol ⊥, a phenomenon we call fixed-point collapse. Two theorems relate this collapse to an entropy measure on transfinite proof trees: if a proof's semantic entropy exceeds a critical threshold, no stable fixed point exists. We interpret φ^(∞) as a colimit in category-theoretic terms, revealing that collapse corresponds to the non-existence of a universal morphism. Applications in proof assistants, AI logic engines, and λ-calculus demonstrate the relevance of transfinite fixed-point detection.