UF Operator and the Collatz Conjecture – A Unified Framework

This paper presents a comprehensive unification of the UF operator and the Collatz conjecture. The UF operator is defined as a new algebraic framework for implicit domain restriction, with a complete axiomatic system including undefined propagation (⊥) and a forbidden-simplification rule. We construct explicit algebraic filters for continuous intervals, discrete sets, modular arithmetic conditions, and non-elementary sets such as the integers using the Gamma function. We then embed the Collatz map into this UF framework, eliminating explicit branching. We prove a trigger lemma (by contradiction) showing that every orbit from n > 1 must eventually encounter a deep-plunge step with d = v2(3n+1) ≥ 3. We then derive a sufficient frequency condition for convergence: if the asymptotic frequency P3 of deep-plunge steps exceeds 0.2925, then the orbit converges to the 4-2-1 cycle. The remaining open problems—proving P3 > 0.2925 and excluding non-trivial cycles—are clearly stated. An extended LTE-based discussion is provided to illustrate that a single deep plunge cannot linearly compensate for a long surge; the original counterexample is corrected by constructing genuine long d = 1 chains via backward iteration. This paper is self-contained and includes full appendices.