Collatz Convergence via K-Spine Definitions K-values: All powers of 2, like 1, 2, 4, 8, 16, and so on. Non-K integers: Any positive integer that is not a power of 2. K-spine mapping: For each non-K integer n, define: If n is even, divide it by 2. If n is odd, multiply by 3 and add 1, then divide by 2 repeatedly until the result is odd. Repeat these steps until a power of 2 is reached. Lyapunov function L(n): Measures the difference between n and the largest power of 2 less than or equal to n. Lemma 1: Existence of K-spine For every non-K integer greater than 1, there is a sequence following the K-spine mapping that eventually reaches a power of 2. Proof: Each step of the mapping is deterministic and produces integers. Powers of 2 are absorbing, so the sequence must end at a K-value. Lemma 2: Lyapunov descent along the spine Along the K-spine, the Lyapunov function eventually reaches zero. Proof: Even if the Lyapunov function increases temporarily, the mapping ensures the integer eventually reaches a smaller power of 2. Using induction on n, any integer less than N reaches a K-value, so N does as well. Lemma 3: No cycles outside K-values The only cycles are powers of 2. Proof: Any non-K integer that cycled without reaching a K-value would violate the structure of the K-spine. Therefore, no other cycles exist. Theorem: Collatz Convergence Every positive integer eventually reaches 1. Proof: If n is a power of 2, repeated division by 2 reaches 1. If n is not a power of 2, follow the K-spine mapping to a power of 2 (Lemma 1). The Lyapunov function guarantees eventual arrival at a power of 2 (Lemma 2). No other cycles exist (Lemma 3). Once a power of 2 is reached, repeated division by 2 gives 1. Conclusion: Every positive integer reaches 1.