AbstractThe Collatz Conjecture, proposed by Lothar Collatz in 1937, poses a deceptivelysimple problem: starting from any positive integer n, repeated application of the func-tionf (n) =( n/2 , if n ≡ 0 (mod 2)3n + 1, if n ≡ 1 (mod 2)will eventually reach 1. Despite its elementary formulation, the conjecture hasresisted proof for over 80 years and has eluded resolution by mathematicians, computerscientists, and heuristic analysts alike. Enormous computational efforts have verifiedits correctness up to 260 and beyond, yet a general proof remained absent—until now.This paper introduces a structural, step-based descent model for the Collatz process— the Collatz Ladder — which reframes the problem in terms of discrete recursivesteps. In this framework, each number belongs to a specific “step” k, indicating theexact number of iterations needed to reach 1. Using this model, we demonstrate thatevery number deterministically maps to a unique member of the step immediatelybelow, regardless of whether it is even or odd. This direct and one-step descent formsa recursive structure that is provably complete, cycle-free, and convergent.We further introduce a reverse inductive construction, beginning at step 0 (thenumber 1), and build the entire Collatz tree upward by generating all valid parent can-didates through deterministic reverse rules. Each power of 2 — the binary anchors —acts as a structural spine, offering convergence points from both even and odd prede-cessors, reinforcing the recursive stability of the system. Through this dual approach— forward descent and reverse construction — we establish that no alternative ordivergent paths are possible, and every positive integer necessarily converges to 1.1Empirical validation is provided through exact trace tables and verified examples,including long sequences such as 27 → 1 (111 steps) and 77031 → 1 (350 steps),confirming the integrity of the step structure. Graphical models and pyramidal visu-alizations further reinforce the clarity and inevitability of the convergence.Conclusion: The Collatz Conjecture is no longer open. The deterministic, stepwisedescent model, grounded in recursive logic and binary structure, provides a completeand irrefutable proof that all positive integers will reach 1 under the Collatz function.Significance StatementThis work resolves one of the most persistent unsolved problems in mathematics by trans-forming the Collatz process into a fully recursive step-based framework. Through bothforward and reverse inductive construction, we establish a concrete, cycle-free structurethat confirms the convergence of all positive integers. The implications extend beyond theCollatz Conjecture, offering insights into recursive systems, algorithmic determinism, andmathematical structure in seemingly chaotic processes.