A Finite-State Deterministic Approach to Resolve Collatz Conjecture

This paper presents a complete proof of the conjecture by demonstrating that both of the counterargument scenarios (non-trivial integer loop and unbounded integer growth) are mathematically impossible. Our approach is deterministic and algebraic, built upon two analytical frameworks. The first is a Perturbation Model used to prove the non-existence of non-trivial integer cycles. The second is a Modular Loop Framework used to prove the non-existence of divergent trajectories. The theoretical claims of both frameworks are supported by extensive, reproducible computational evidence. Lemma 0: This lemma serves as the logical foundation and theoretical preamble to this paper. The lemma uses modular arithmetic (mod 2v) to derive a predictable, deterministic distribution of the division exponents ai in Collatz sequence. This lemma allows us to construct specific, logically-derived models of hypothetical cycles and to analyze the dynamics of the Collatz function as a finite-state transition system.The Fundamental Diophantine Equation: An equation derived from multiple iterations applied on modulo residual representation of an odd integer. The equation acts as the heart of finite-state machine. Lemma 1A: This lemma proves that if all divisional exponents (ai) are equal, the only integer reappears in Collatz sequence is 1 when ai = 2 (equilibrium state). The Perturbation Model: The model assumes for contradiction, that, there exists at least one permutation of the divisional exponents for which the modified Diophantine equation from lemma 1A: R = has a non-trivial integer solution. The non-trivial permutation can be obtained by perturbing divisional exponents from the equilibrium state ai = 2 under certain mathematical restrictions. The model defines that a non-trivial integer cycle is possible when the hypothetical requirement reconciles with the actual structure of ∆N, i.e. ∆Nactual = ∆Nrequired.Lemma 1B: This lemma proves that purely positive perturbations of the divisional exponents from equilibrium state strictly leads to D > N violating non-trivial integer cycle condition. Lemma 1C: Analyzing 3-adic values of ∆Nactual and ∆Nrequired this lemma proves a fundamental mismatch that no purely negative and mixed (both negative and positive) perturbations result in non-trivial integer cycle. Lemma 1D: This is a supplement lemma that concludes the non-existence of non-trivial integer cycle by any valid perturbation of the last divisional exponent (an).Together, lemma 1B to 1D proves that the initial assumption of the perturbation model is false and hence no non-trivial integer cycle exists. Concept of Modular loop and Lemma 2A: Due to sufficient iterations, an integer belonging to a certain modular class revisits its own class creating a modular loop. The paper has identified this as growth/contraction engine for an integer. Lemma 2A shows by 2-adic analysis that such modular loops cannot recur indefinitely. Lemma 2B: This lemma generalizes the concept of lemma 2A and asserts that any modular path has the similar limitation as the modular loops – they do not have infinite recurrences. Finally, the theoretical proofs are supported by extensive, verifiable computational and empirical demonstrations. An exhaustive check of all 10.8 million unique exponent combinations for a mod 32 (n=16) cycle model confirms the impossibility proven in Lemma 1B. Similarly, the exhaustive solution of all 911 Diophantine equations for the mod 16 system provides concrete, reproducible evidence for the instability proven in Lemma 2A. Since both non-trivial integer cycles and divergent trajectories have been proven to be structurally impossible, the Collatz conjecture is affirmed.