This is the first of three papers on a recently discovered melt flow singularity during capillary flow of linear polyethylene, the ’extrusion window’ at and close to 150 degrees C. After placing the subject in its historical perspective, the various experiments are described by which the ’extrusion window’ is revealed. These are : a minimum in pressure at around 150 degrees C for constant material throughput and a maximum in material throughput at 150 degrees C when a constant pressure level is maintained. In the former case the effect sets in at a critical piston velocity v(c) (i.e. shear rate gamma(c)) and in the latter case at a critical pressure p(c). The v(c) value was found to be in a steeply inverse relation to molecular weight (M), obeying a power dependence of precisely -4, while p(c) was practically independent of M. The singularity in melt flow behaviour in question is inexplicable in terms of continuum theology and points to a thermodynamic origin, more specifically to a phase transition at the particular temperature. In line with previous work, the formation of a transient, hexagonal ’mobile’ crystal phase is suggested, here created through flow-induced chain extension. While seemingly the criticality in v(c) and the low end cut-off in M suggest the agency of elongational flow-induced chain stretching, such as is known to take place at the orifice entrance (a view held in the preceding publications), the newly found invariance of p(c) with M now speaks decisively against the orifice being the site of initiation of the window effect. Rather, it points to a yield phenomenon, such as wall slip in the capillary, which itself would be the consequence of a chain-extension-induced phase transformation at the sharply defined temperature of 150 degrees C, the plausibility of which is argued in this paper. Irrespective of the ultimate source of the effect, its potential advantage for certain melt processing operations should be apparent through requiring less energy for a given material throughput and by ensuring distortion-free extrudates under conditions where otherwise such distortions would be present. Other, more specific issues raised by the work include the relation of the newly found power law to the established M dependence of melt viscosity.