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 λ_{IR}=0, the singularity of spectral density defining the IR phase, and the upper edge with λ_{A}, the previously identified Anderson-like nonanalyticity. Details near λ_{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 a topological origin. We find similar structure at λ_{A}, and associate its adjacent thin layer (d_{IR}⪆2) with Anderson-like criticality. Our analysis reveals the manner in which nonanalyticities at λ_{IR} and λ_{A}, originally identified in other quantities, appear in d_{IR}(λ). This dimension structure may be important for understanding the near-perfect fluidity of the quark-gluon medium seen in accelerator experiments. The role of λ_{A} in previously conjectured decoupling of IR component is explained.