Support fibers commute with graded triangles

Restriction and the transition maps form a coherent diagram of arrows $F^{j}E(X)\to F^{j}E(U)$. Taking the pullback over zero at every stage is functorial, producing transition maps $F_T^{j+1}E(X)\to F_T^{j}E(X)$ with all composition coherences. The compatible augmentation maps to $E(X)\to E(U)$ likewise induce compatible maps $F_T^{j}E(X)\to E_T(X)$. This is the asserted augmented decreasing tower.

For a coherent square of arrows of spectra, taking fibers preserves its finite limits: each fiber is itself a pullback, and the universal property of limits identifies an iterated finite limit with the limit over the corresponding product diagram. Limits of arrow diagrams are computed at their source and target. The zero arrow has zero fiber. In particular, if a square of arrows is pointwise a pushout, stability makes it pointwise a pullback; after taking fibers it is again a pullback and hence a pushout. Thus taking fibers carries pointwise cofiber squares of arrows to cofiber squares of spectra.

Apply step 2.1 to the map of arrows from $F^{j+1}E(X)\to F^{j+1}E(U)$ to $F^{j}E(X)\to F^{j}E(U)$. Its pointwise cofiber is the arrow ${\mathrm{gr}}^{j}E(X)\to {\mathrm{gr}}^{j}E(U)$. The cofiber of its map of fibers is therefore naturally equivalent to the fiber of that graded arrow. By the definition of the supported graded spectrum, this is precisely the displayed graded fiber sequence. The construction uses the coherent diagram and universal properties, so it is natural in maps of such filtered presheaves and open pairs.

The definition of $F_T^{j}E(X)$ gives a fiber sequence $F_T^{j}E(X)\to F^{j}E(X)\to F^{j}E(U)$. Applying [F3] yields the displayed long exact sequence. Exactness at ${\pi }_i{F}^{j}E(X)$ says that its restriction kernel is exactly the image of ${\pi }_i{F}_T^{j}E(X)$, proving the first lifting claim.

Suppose the stated map on $U$ is injective. Choose a stage representative $a\in {\pi }_i{F}^{j}E(X)$ of an ambient element $x$ whose restriction to ${\pi }_{i}E(U)$ is zero. Compatibility of restriction with augmentation says that the image of $a{\mathrm{\mid }}_U$ in ${\pi }_{i}E(U)$ is zero; injectivity therefore gives $a{\mathrm{\mid }}_U=0$ in ${\pi }_i{F}^{j}E(U)$. Step 4.1 lifts $a$ to ${\pi }_i{F}_T^{j}E(X)$, whose image in ${\pi }_{i}E(X)$ is $x$. The argument explicitly uses injectivity to pass from the ambient kernel to the stage kernel. This proves all claims.