In last passage percolation models lying in the Kardar-Parisi-Zhang universality class, maximizing paths that travel over distances of order $n$ accrue energy that fluctuates on scale $n^{1/3}$; and these paths deviate from the linear interpolation of their endpoints on scale $n^{2/3}$. These maximizing paths and their energies may be viewed via a coordinate system that respects these scalings. What emerges by doing so is a system indexed by $x,y \in \mathbb{R}$ and $s,t \in \mathbb{R}$ with $s < t$ of unit order quantities $W_n\big( x,s ; y,t \big)$ specifying the scaled energy of the maximizing path that moves in scaled coordinates between $(x,s)$ and $(y,t)$. The space-time Airy sheet is, after a parabolic adjustment, the putative distributional limit $W_\infty$ of this system as $n \to \infty$. The Airy sheet has recently been constructed in [15] as such a limit of Brownian last passage percolation. In this article, we initiate the study of fractal geometry in the Airy sheet. We prove that the scaled energy difference profile given by $\mathbb{R} \to \mathbb{R}: z \to W_\infty \big( 1,0 ; z,1 \big) - W_\infty \big( -1,0 ; z,1 \big)$ is a non-decreasing process that is constant in a random neighbourhood of almost every $z \in \mathbb{R}$; and that the exceptional set of $z \in \mathbb{R}$ that violate this condition almost surely has Hausdorff dimension one-half. Points of violation correspond to special behaviour for scaled maximizing paths, and we prove the result by investigating this behaviour, making use of two inputs from recent studies of scaled Brownian LPP; namely, Brownian regularity of profiles, and estimates on the rarity of pairs of disjoint scaled maximizing paths that begin and end close to each other.