We present a computer-assisted proof that every positive integer orbit under the Collatz map $T(n) = n/2$ (if $n$ is even), $(3n + 1)/2$ (if $n$ is odd) eventually reaches 1. The argument consists of two independent parts, H1 and H2, each of which reduces an originally infinite verification problem to a finite exact certification step. The mathematical reductions are traditional proofs; the certifications are finite, deterministic computations using only exact integer and rational arithmetic, whose complete specifications (input, algorithm, output) are given in the appendices.For H1 (divergent-orbit exclusion), a deterministic normalized block-drift analysis establishes an exact affine envelope that rules out all parameter regimes below an explicit threshold $V_{env} approx 1.5990$. The remaining oscillatory regime is reduced to a finite certification problem over a quotient-state automaton whose mathematical soundness is guaranteed by a chain of five theorems (Theorems 12.8—12.14). The certification is discharged by exhaustive exact computation (Theorem 12.17).For H2 (non-trivial cycle exclusion), a gate-based reformulation converts the cycle problem into an explicit inequality system. Discrete convexity arguments and a two-value support reduction compress the problem to a single-parameter bound, which is then verified by exact rational arithmetic over a rigorously delimited feasible range. The computational components are finite, exact, and logically isolated. Their role is analogous to the finite case verifications in other computer-assisted proofs such as the Four Colour Theorem and the Kepler Conjecture: the mathematical argument provides a complete reduction to finite exact certification problems, and the computations discharge those problems. The complete source code for both certifications, together with their full execution logs and exact rational outputs, is included in the appendices (Appendices E—G).