The Collatz Conjecture and the Spectral Calculus for Arithmetic Dynamics

We develop a complete operator-theoretic and spectral framework for the Collatz map by analyzing its backward transfer operator on weighted Banach spaces of arithmetic functions. The associated Dirichlet transforms form a holomorphic family that isolates a zeta-type pole at s=1, while on a finer multiscale space adapted to the dyadic-triadic geometry of the Collatz preimage tree we establish a two-norm Lasota-Yorke inequality with an explicit contraction constant, yielding quasi-compactness, a spectral gap, and a Perron-Frobenius theorem in which the eigenvalue 1 is algebraically and geometrically simple, no other spectrum meets the unit circle, and the unique invariant density is strictly positive. The fixed-point relation is converted into an exact multiscale recursion for the block averages c_j, revealing a rigid second-order coupling with exponentially small error terms and asymptotic profile c_j~ 6^-j. This spectral classification forces every weak* limit of the Cesàro averages derived from any hypothetical infinite forward orbit to be either 0 or a scalar multiple of the Perron-Frobenius functional, with convergence to 0 occurring precisely under the Block-Escape Property. Since the forward map satisfies an unconditional exponential upper bound, whereas Block-Escape combined with linear block growth along a subsequence would impose an incompatible exponential lower bound, all analytic and spectral components needed for such a contradiction are complete, reducing the Collatz conjecture to excluding infinite orbits exhibiting Block-Escape without the supercritical linear block growth prohibited by the spectral theory.