The Collatz Conjecture and the Spectral Calculus for Arithmetic Dynamics

We develop a full operator--theoretic and spectral framework for the Collatz map based on its backward transfer operator acting on weighted Banach spaces of arithmetic functions. The associated Dirichlet transforms form a holomorphic family that isolates a zeta--type pole at $s=1$ and a holomorphic remainder, while on a finer multiscale space adapted to the Collatz preimage tree we establish a Lasota--Yorke inequality with an explicit contraction constant $\lambda<1$, yielding quasi--compactness and a spectral gap at the dominant eigenvalue. This proves a Perron--Frobenius theorem for the backward operator: the spectral radius is $\rho(P)=1$, the eigenvalue $\lambda=1$ is algebraically and geometrically simple, and no other spectrum lies on the unit circle. The invariant density is strictly positive with a $c/n$ decay profile. The spectral classification forces every weak$^{*}$ limit of the Ces\`aro orbit averages $\Lambda_{N}(f)$ of any hypothetical infinite forward orbit to be either $0$ or a scalar multiple of the unique Perron--Frobenius invariant functional, and we show that $\Lambda_{N}\!\Rightarrow 0$ occurs precisely under an extreme transience condition, the \emph{Block--Escape Property}. The forward map satisfies an unconditional exponential upper bound, while Block--Escape forces the block index to diverge and, if strengthened to linear block growth along a subsequence, would imply a contradictory exponential lower bound. Thus all analytic and spectral components of the proof are complete, and the Collatz conjecture is reduced to a single forward--dynamical assertion: that Block--Escape cannot occur without linear block growth.