A Unified Attack Framework for the Collatz Conjecture: KAM No-Resonance as a Conditional Cycle Proof and the 2-Adic Ergodicity Transfer as the Remaining Gap

This paper presents a unified framework for the Collatz conjecture with two primary results. First, the KAM no-resonance argument establishes a conditional structural proof that no non-trivial Collatz cycle can exist. The arithmetic core — that the KAM resonance condition requires 3^ (k-m) = 2ᵐ, impossible since powers of 3 are always odd and powers of 2 are always even — is unconditional and elementary. Three named formal verifications are required to complete the proof, each a specific technical problem in an existing mathematical field. Second, the single remaining step for the divergence problem is the transfer of the known 2-adic ergodicity of the Collatz map to ordinary positive integers. The Birkhoff ergodic framework and the Tao (2022) density result point directly at this transfer as the precise target. Closing both steps closes the conjecture entirely. The framework connects Poincare, Birkhoff, Kolmogorov, Baker, Tao, and Shakibaei (2026) into a single logical structure and presents an 18-loop constraint forcing any counterexample above n > 10⁵, 000, 000. Companion paper on the Riemann Hypothesis deposited same date. SHA-256: 2fc60833638a684b345cae0ca70b8d1622e24e8a747d16087c173e59bef37f55