The Characteristic Ideal of the Collatz–Iwasawa Module and the Character Decomposition Bridge

This work introduces the Collatz–Iwasawa module X₂₎₋₋, a finitely generated torsion module over the Iwasawa algebra = Z₂[T], designed to quantify the algebraic deviation between the Syracuse return map and pure tripling on odd 2-adic integers. At each finite level of the 2-adic residue tower, the module is defined as the cokernel of the operator difference S^* - ^*, where S is the Syracuse map and is the multiplicative action by 3. By explicitly constructing the induced 2 2 transition matrix on the two -orbits, we compute the finite-level Fitting ideals and show that they factor as Fitt䂷 (X₂₎₋₋, ₌) = (T 2^ (m) gₘ (T) ), where T = - 1, the factor T corresponds to a simple trivial zero at the trivial character, and gₘ (0) Z₂^ is a 2-adic unit for all computed levels m 13. This establishes unconditional finite-level nonvanishing of the characteristic ideal at the trivial character after removing the trivial zero. The extension of this nonvanishing property to the inverse limit is formulated as a conjecture, contingent on a suitable control theorem for the module. To connect the algebraic structure to Collatz dynamics, we introduce a distinct orbit module Yₘ = coker (S^* - I), through which orbit evaluators factor. A character decomposition bridge is established via Fourier analysis on the cyclic -action, yielding a decomposition of the orbit drift into a universal negative leading term and higher-character corrections. Finite-level spectral evidence supports this framework: maximal spectral gaps at ghost-free levels, strict 1/L projection norm decay, and convergence of positive-drift fractions toward 1/2. These results collectively reduce the Collatz conjecture from an infinite-dimensional dynamical problem to a finite-dimensional spectral bound, while isolating the remaining obstruction as an inverse-limit arithmetic problem. This note represents a rigorously grounded algebraic component of a broader (2, 3) -adic framework for the Collatz map, with clear separation between proven results, computational evidence, and conjectural extensions.