Cool a cloud of atoms to a billionth of a degree above absolute zero and something profound happens: the atoms stop being separate and pour into the SAME lowest quantum state, overlapping into one giant matter-wave that acts as a single object. Two, then thousands, become one. Slide the temperature down through the critical point and watch the cloud condense.
For identical bosons below a critical temperature Tc, a macroscopic fraction occupies the single lowest-energy state — a Bose-Einstein condensate (Einstein 1925; realised in rubidium at 170 nK, Cornell & Wieman 1995). For an ideal gas the condensate fraction is N₀/N = 1 − (T/Tc)³⁄² below Tc and zero above: as T falls, more and more atoms share ONE wavefunction until, at T=0, all do — a single coherent matter-wave. A fail-loud self-check throws unless the fraction is positive below Tc, reaches 1 at T=0, and is exactly 0 above Tc. ◆ real quantum physics, node-verified.
The ideal-gas 1−(T/Tc)³⁄² law (harmonic-trap gives (T/Tc)³; interactions shift it); the sharp onset at Tc and the everyone-in-one-state limit are the exact, defining features.