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The polymer poly(4-methylpentene-1), P4MP1, displays an unusual pressure−temperature (p, T) phase diagram raising also issues of wider generality for the phase behavior of single-component condensed matter systems. Previous exploration of this phase behavior through X-ray diffraction has been extended through high-pressure calorimetry, Raman spectroscopy, and time-resolved in situ X-ray scattering. As a result the p, T phase diagram, previously postulated on structural evidence is now supported by heat effects defining phase transitions of first order along the appropriate p, T phase lines with signs which are self-consistent in accord with the unusual nature of some of the transitions involved. The latter include (1) amorphization under pressure, (2) ordering on heating and disordering on cooling, and (3) sign inversion of the pressure coefficient of the melting point, dT m/dp, with increasing pressure. The newly recorded heat effects, consistent with the above, include exotherms on amorphization and endotherms on crystallization. The most salient features of the phase diagram involving effects 1−3 are as follows:  (α) a re-entrant melt phase region and (β) the assertion that the amorphous material can be compressed to smaller volumes than the crystals. While seemingly counterintuitive, pertinent historical precedents are being quoted. All the above conflicts with the tenets of the widely held Kauzmann restrictions (“paradox”), which it seemingly invalidates. A possible way to reconcile this conflict is by invoking a theoretical explanation on metals existing in the literature. In structural terms, high-pressure Raman spectroscopic and time-resolved X-ray scattering evidence have been obtained. Further issues arising such as the distinction, or otherwise, between fluid and solid “amorphous” phases and their incorporation into phase diagrams are being discussed, and the still open-ended question concerning the molecular nature of the pressure generated amorphous material is being raised.