This short theoretical paper argues that the well known pseudogap phase in high‑temperature cuprate superconductors must be a precursor to the superconducting dome seen in their phase diagram. The authors show, with a macroscopic argument, that the pseudogap temperature T* as a function of doping cannot be a smooth, analytic curve across the dome. Instead T* develops a kink or singular behavior at the dome peak and becomes identical to the superconducting transition temperature Tc in the overdoped region.

The core of the argument uses two observable ingredients. First, the pseudogap temperature T* falls as doping increases. The authors attribute this to a shrinking of the size of extended, disordered preformed hole pairs as more carriers are added. Second, they introduce a configurational‑ordering rate R that measures how fast the random arrangement of those extended pairs can reorganize into the global, symmetry‑breaking pattern needed for superconductivity. The paper assumes the disorder measure (configurational entropy CE) falls linearly with temperature from each T* point, with slope R. If R grows with doping, the lines that connect T* to Tc retrace a dome shape; where R diverges the behavior becomes non‑analytic and T* and Tc merge in the overdoped side.

To explain why R should increase while T* decreases with doping the authors invoke a specific microscopic picture called entanglement and confinement hole pairing (ECHP). This pairing idea, developed earlier by Buot and colleagues, describes pairs of doped holes that are extended, entangled, and variably confined. Shorter‑range, more fluid pairs at higher doping are expected to reorder more quickly (larger R) while yielding a smaller pairing energy (lower T*). The paper cites angle‑resolved photoemission spectroscopy and other experimental phase diagrams as qualitatively consistent with this picture.