A Parity-Progression Framework for the 3x+1 Problem: Structural Analysis, Common-Difference Growth, and the Remaining Bridge

This paper develops a systematic algebraic framework for studying the Collatz map based on parity-induced arithmetic progressions. The central innovation is the introduction of arithmetic-progression triples (a,d,w) — comprising first term, common difference, and inherited depth weight — and the formulation of parity-splitting, dyadic reduction, and affine transport as exact operators on these triples. This operator formalism organizes the positive integers into a parity-layered tree and transforms the study of Collatz orbits into the analysis of state evolution in an algebraic dynamical system. Within this framework, we establish several rigorous structural results: · A forward-invariant inequality d > a governing the relative sizes of first term and common difference.· Exact local evolution laws for the pure-odd operator Φ, including branch formulas for faithful and mixed steps.· A complete theory of auxiliary pure-even paths: constant common difference D = 2 · 3^n, finite-state first-term dynamics, eventually periodic odd outputs, and automatic regeneration of the next-level pure-even mechanism.· A strong jump theorem (Theorem C): every genuine Collatz orbit enters arithmetic progressions with attained common differences D_n = 2 · 3^n and hence unbounded. We also identify two fundamental structural obstructions: the present operator is not fully first-term faithful to the odd Collatz map, and single arithmetic progressions are not closed under exact odd Collatz image. The paper concludes by formulating a precise Candidate Lemma D whose proof would, together with the strong jump theorem, complete the proof of the Collatz conjecture via an infinite descent argument. The current state of the program is thus: Theorem C proves that non-convergent orbits must attain unbounded common difference; what remains is to prove that unbounded common-difference growth forces the orbit to encounter 1.