What is non-hückel double bond of Aromaticity?
In the given molecule Pyrene , the π  bond shown in green is in cross conjugation (shown in green in figure 1). This bond( figure 2) is excluded from interaction with rest of conjugated system. Hence , the central $\ce{C=C}$ isn't hosted by the largest p orbital loop, so it's not counted towards Hückel's rule for aromaticity.

The outer periphery (shown in blue in figure 3), is a loop of $\ce{14 π}$ electrons , therefore aromatic. This may be possible hückel double bond you are referring to.

The π bond in cross-conjugation (shown in green) maybe non-hückel double bond.