The Triple Equivalence and the Metric of Survival

We establish the Triple Equivalence: for any typed system S, the conditions (I) S is sound, (II) π₁(Γ(S)) = 0, and (III) H(S) = H_min(C) are not correlated properties but identical conditions measured in three formal registers. This identity constitutes a single diagnostic invariant. We prove that failure of any one condition entails failure of all three, that remediation requires discrete crossing changes with a computable unknotting number, and that systems exceeding the trust horizon unknotting budget are thermodynamically terminal. The unknotting number is anchored to a time-complexity constant via Theorem U, establishing that the cost of simulating soundness when π₁ ≠ 0 grows exponentially with depth. The Selection Functor is identified as the Second Law of Thermodynamics operating on typed structures. We provide empirical verification from a live diagnostic.