The Collatz Conjecture and the Spectral Calculus for Arithmetic Dynamics

We develop a complete operator-theoretic and spectral framework for the Collatz map using 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 prove a Lasota–Yorke inequality with an explicit contraction constant λ < 1, yielding quasi-compactness, a spectral gap, and a Perron–Frobenius theorem in which ρ(P) = 1, the eigenvalue 1 is algebraically and geometrically simple, no other spectrum lies on the unit circle, and the unique invariant density is strictly positive with a c/n decay profile. The fixed-point relation Ph = h is converted into an exact multiscale recursion for the block averages c_j, showing that mass propagation along the preimage tree is governed by a rigid two-scale coupling with exponentially small error terms. The spectral classification then forces every weak-star limit of the Cesàro averages Λ_N(f) of any hypothetical infinite forward orbit to be either 0 or a scalar multiple of the Perron–Frobenius functional, and convergence to 0 occurs precisely under the Block–Escape Property, an extreme transience condition that drives the block index to infinity. The forward map always satisfies an unconditional exponential upper bound, whereas Block–Escape combined with linear block growth along a subsequence would create a contradictory exponential lower bound. All analytic and spectral components of the proof are therefore complete, and the Collatz conjecture is reduced to a single forward-dynamical problem: to exclude infinite forward orbits that satisfy the Block–Escape Property without forcing the kind of linear block growth incompatible with the spectral bounds.