A Formal Reduction of the Collatz Conjecture to a FiniteDiophantine System

<p dir="ltr">A new framework for a formal proof of the Collatz conjecture is presented. The method's novelty lies in its multi-stage reduction, which transforms the infinite, dynamic problem into a finite, static algebraic system. We first use a 6-state Finite State Automaton (FSA), justified by the Carry Uniformity Theorem (Theorem 1), to prove that the bitwise computation of (3n+1) is finite and predictable. This allows us to formally partition the infinite set of integers (Z+) into two distinct, exhaustive sets: the contracting "Strong Descent" set (S_strong, v >= 3), where a path is guaranteed to shrink, and the non-contracting "Trapped Set" (S_trap, v in {1, 2}).</p><p dir="ltr">This initial reduction narrows the entire conjecture to a single, focused question: proving that no path N>1 can remain in S_trap indefinitely. A path can only fail to converge if it stays in this set forever, which can happen in only two ways: the path grows without bound (divergence, N -> infinity), or the path falls into a non-trivial repeating loop (k-cycles, N1 -> ... -> Nm -> N1). This "infinite trap" is only possible in S_trap because it contains the T1(n)=(3n+1)/2 "ascent" function, which acts as the engine for all growth.</p><ol><li>First Reduction: We then prove that both of these failure modes (divergence and k-cycles) are governed by the stability of a single 2-adic mixed system. This system is a "pump" built from two "gears": the T1(n)=(3n+1)/2 recurrence (which acts as a 2-adic contraction around N=-1) and the T2(n)=(3n+1)/4 recurrence (a 2-adic contraction around N=1). The problem is thus reduced again: can this 2-adic "pump," which drains valuations at two different points, run indefinitely, or must it algebraically "leak"?</li><li>Final Reduction: We then formally reduce the stability of this 2-adic pump to a final, coupled system of exponential Diophantine equations. This reduction is based on "hops." A "1-hop" cycle (the simplest case, n0 -> n0) is formally disproven. Recognizing that "multi-hop" cycles (e.g., n0 -> n1 -> n0) are a known feature of similar 3n+k systems, we also derive the general Diophantine system for multi-hop paths. We formally disprove the simplest "2-hop" (k=0, k=1) base cases. This proves that any path that is <i>not</i> a non-trivial solution to this final system is algebraically forced to "leak." This leak is a Terminal Exit (n = 5 mod 8), a state that is triggered by a predictable failure in the 2-adic valuations, permanently ejecting the path from S_trap.</li></ol><p dir="ltr">By reducing the infinite problem of all integers to this finite, coupled Diophantine system, this paper provides a complete roadmap to a final proof. The Collatz conjecture is true if and only if this final, well-defined system has no non-trivial integer solutions. The paper provides the initial proofs for the base cases of this system, demonstrating its powerful algebraic constraints.</p>