  It was recently proposed that there is a phase in thermal QCD (IR phase) at
temperatures well above the chiral crossover, featuring elements of scale
invariance in the infrared (IR). Here we study the effective spatial
dimensions, $d_{IR}$, of Dirac low-energy modes in this phase. Our $d_{IR}$ is
based on the scaling of mode support toward thermodynamic limit, and hence is
an IR probe. Ordinary extended modes, such as those at high energy, have
$d_{IR}=3$. We find $d_{IR} < 3$ in the spectral range whose lower edge
coincides with $\lambda_{IR}=0$, the singularity of spectral density defining
the IR phase, and the upper edge with $\lambda_A$, the previously identified
Anderson-like non-analyticity. Details near $\lambda_{IR}$ are unexpected in
that only exact zero modes are $d_{IR}=3$, while a thin spectral layer near
zero is $d_{IR}=2$, followed by an extended layer of $d_{IR}=1$ modes. With
only integer values appearing, $d_{IR}$ may have topological origin. We find
similar structure at $\lambda_A$, and associate its adjacent thin layer
($d_{IR} >\approx 2$) with Anderson-like criticality. Our analysis reveals the
manner in which non-analyticities at $\lambda_{IR}$ and $\lambda_A$, originally
identified in other quantities, appear in $d_{IR}(\lambda)$. This dimension
structure may be important for understanding the near-perfect fluidity of the
quark-gluon medium seen in accelerator experiments. The role of $\lambda_A$ in
previously conjectured decoupling of IR component is explained.
