Handover Constellation in a Quasi-One-Dimensional Bose Gas: From a Single-Offset Crossover to the Five Characteristic Temperatures

A phase transition is conventionally described by a single critical temperature T_c. We analyze Bose–Einstein condensation of a quasi-one-dimensional interacting gas and find the transition presents itself as a cluster of separately locatable sub-events, quantified by a five-temperature constellation: condensate half-height T_R, peak-interaction handover T_I, cloud-size minimum T_B, momentum-width minimum T_D, and overlap peak T_S. Across 17 interaction strengths and three thermal-cloud models (Boltzmann, Bose, Hartree–Fock), we find: (i) the handover offset ΔT_I(g) is a monotonic crossover—hump at weak coupling (+0.15 T_c at g = 30), zero at g* ≈ 39 (Bose) or 44 (Boltzmann), deepening to −0.10 T_c—with a statistics double valley at the crossover; (ii) the constellation expands (separation 0.21 → 0.53 T_c) and its internal order reverses (T_I crosses T_R near g ≈ 32), these two markers bracketing the crossover region; (iii) the two cloud-sensitive dimensions carry different signatures (T_I: zero-crossing; T_S: deepening −0.13 → −0.18 T_c). The density profile remains center-peaked at all temperatures—an observed central dip would falsify the model family. Ground states use multi-start Krotov minimization (μ(g) strictly monotone, dominant-peak fractions > 0.994). All features map onto laboratory parameters (N ≈ 200–1600 for ⁸⁷Rb) and are falsifiable predictions for Feshbach experiments with simultaneous condensate-fraction and in-situ density measurements.