We establish a structural resolution of the Collatz Conjecture. The proof proceeds through a strict interception of the scalar cycle equation over the Syracuse dynamics. We demonstrate that any hypothetical cycle defines a forced mixed-radix expansion satisfying the identity n * (2K - 3ᵐ) = S, where the accumulator S is strictly positive. We introduce the Diophantine Sign Lemma, which proves that if the average of even projections falls below the viability threshold (K/m < log₂ (3) ), the fraction collapses, forcing the cycle into the negative integers. Conversely, if the cycle operates in the strictly positive regime, we apply the Schmidt Subspace Theorem for S-unit equations to prove the algebraic disjointness of the accumulator, rendering the space of integer solutions empty. Furthermore, divergence to infinity is analytically ruled out via a lexicographic measure over the binary topology, V* (n) = (MSB (n), Ω (n) ). We prove that the Syracuse operator acts as a strict contraction: the carry propagation induced by block fusion triggers an Avalanche of Zeros, forcing mandatory even divisions that algebraically overpower the multiplicative expansion and guarantee the ultimate collapse of the Most Significant Bit (MSB). The foundational Diophantine bounds and collapse lemmas have been mechanically certified in Lean 4.
Garcia Lecaros Eduardo Andres (Thu,) studied this question.