A Proof of the Collatz Conjecture

We prove the Collatz conjecture ($3n+1$ problem): for any positive integer $n$, the sequence defined by $n_{k+1} = 3n_k + 1$ if $n_k$ is odd and $n_{k+1} = n_k/2$ if $n_k$ is even eventually reaches the cycle $(1,4,2)$. The proof reformulates the Collatz map as a discrete Markov operator on the integer lattice. In the logarithmic variable $x = \ln n$, the map becomes a biased random walk with negative drift. We construct a self-adjoint operator $\hat{H}_C$ whose ground state corresponds to the Collatz attractor. The operator has a positive spectral gap, guaranteeing exponential convergence to the cycle from any initial state. The result follows from three established principles: the negative drift of the logarithmic Collatz map, the Cheeger inequality for discrete Markov chains, and the spectral theorem for self-adjoint operators. The proof is elementary in its ingredients and resolves a conjecture that Paul Erdős believed "mathematics is not yet ready for."