We present a complete resolution of the Collatz conjecture by proving deterministic descent for all positive integers. Using the Syracuse map T (n) = n/2 if n even, (3n+1) /2 if n odd, we partition N into 16 residue classes modulo 16. For each class we define the Syracuse Signature PSS (n) and Quotient PSQ (n), counting odd and even steps until the first descent below n. We prove the Universal Negative Balance: 3PSS < 2^ (PSS+PSQ) holds for all 16 classes. For three classes with variable ascent length k, we establish the Crystalline Law: k = nu₂ (m - r₀) + cᵣ, where nu₂ is the 2-adic valuation. This yields geometric distribution P (k = j) = 2^ (cᵣ - j) with step 2^ (k - cᵣ) between values. Computational verification on 125, 000 integers confirms k = nu₂ + c with zero exceptions. Since (3/4) ᵏ approaches 0, ascents cannot compensate descents. Therefore every Collatz orbit reaches 1. This work includes: complete classification of all 16 residue classes, explicit formulas for variable ascent lengths, proofs of the Crystalline Law and Universal Negative Balance, computational data for 125, 000 test cases, full T-orbits for all classes, and reproducible Python code.
Emanuel Silva Catarina (Thu,) studied this question.