The canonical pair is a t structure

Replace $Y$ by ${\tau }^{\ge b}Y$ and in any left roof out of $X$ replace its vertex $L$ by ${\tau }^{\le a}L$. These replacements are quasi-isomorphisms, including when all cohomology vanishes. A map $L\to Y[n]$ is zero if $n<b-a$, since every degree has either zero source or zero target.

At $n=b-a$ only degree $a$ can be nonzero. The chain-map equations say exactly that this component kills $\mathrm{im}{d}_L^{a-1}$ and lands in $\ker d_{Y[n]}^a$, so it is a map $H^{a}L\to H^{b}Y$. Homotopies cannot alter this component: their possibly contributing terms have source above $a$ or target below $a$. A denominator induces an isomorphism on $H^a$, so roof refinements give the same map $H^{a}X\to H^{b}Y$. Conversely such a map defines a chain map from ${\tau }^{\le a}X$ into $({\tau }^{\ge b}Y)[b-a]$ by quotient then inclusion. The two constructions are inverse, by replacing a roof vertex as in step 1.1; all maps are the specified cohomology maps, hence natural.

The identity $H^i(X[1])=H^{i+1}(X)$ gives the two shift inclusions. Step 1.1 with $a=0,b=1,n=0$ gives orthogonality. The triangle ${\tau }^{\le 0}X\to X\to {\tau }^{\ge 1}X\to$ gives the required decomposition with the prescribed cohomology supports. These verify all axioms.

Canonical truncations preserve every one-sided or two-sided cohomological boundedness condition. The bounded embeddings are full and exact, so the same Hom vanishing and the same decomposition triangles lie in each bounded category. This proves the restricted t-structures.