  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, in the context
of pure-glue QCD. 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.
