We apply the cross-bracket and the Torrado Classification Theorem (Ar-ticle IV) to derive the critical exponents of the Ising model in two and three dimensions. In 2D, the containment chain has length one and all exponentsdepend on C(1) alone, recovering the exact Onsager solution. In 3D, each ofthe three independent exponents (η, ν, β) is derived independently from theHilbert–Schmidt trace of the cross-bracket evaluated over the participatingfolds. The powers of C(n) in each formula are the Hausdorff dimensions of thefolds — three for H3, two for S2 , one for S1 — as they emerge from expanding the trace over the Killing directions of the containing fold (Diary 12). The nor-malisation of β via the intermediate-node mechanism of the compound bracket B(134) is proved without ad hoc postulates (Diary 13). The TCT proves thateach exponent has a unique representation in L(M) (Diary 13). The scalingrelations are verified as algebraic identities, not input constraints. The results are η = 9/250, ν = 63/100, β = 16/49, in agreement with conformal boot-strap to 0.86%, 0.005%, and exact within uncertainty. The Ising 3D problem is closed.