Structural Minimal Conditions for a Complete and Verifiable Proof of the Collatz Conjecture

Structural Minimal Conditions for a Complete and Verifiable Proof of the Collatz Conjecture   This work defines the minimal structural conditions that any complete and verifiable proof of the Collatz Conjecture must satisfy — independent of analytic, algebraic, or automaton-theoretic approach.   The framework introduces a structural triad of logically necessary closures: (I) Integrality Closure — integer orbits are preserved. (A) Aperiodicity Closure — bounded integer invariants forbid non-trivial cycles. (G) Global Inductive Closure — all orbits descend into a finite inductive kernel.   Together, these closures are jointly necessary and structurally sufficient for total convergence.   The framework completes the trajectory initiated by Tao (2019), unifying deterministic induction and probabilistic analysis within a verifiable formal schema.   Author: Moon Kyung-Up (RehaBrain Development Center) Version: v2.6 (Referee-Edition) DOI: 10.5281/zenodo.17591580 License: CC-BY-NC-SA 4.0   Keywords: Collatz Conjecture · Structural Invariants · Aperiodicity · Global Induction · Proof Theory · Formal Verification · AI No-Go Theorem