Polymers with no molecular mass distribution? What sort of ideal conditions are required - theoretically - to conduct polymerization reactions to make all the macromolecules end up with exactly the same molecular mass, just like the products in normal chemical reactions? How do living organisms manage to synthesize macromolecules, for instance proteins, different than humans do in the lab?
Living polymerizations can afford nearly zero polymer polydispersivity.  You can go anion (Kraton thermoplastic elastomers - and tell us about the one silicon atom in each polymer molecule), free radical (TEMPO or PROXYL capping of the free radical end, thermally activated), redox (copper halide), olefin metathesis polymerization (ROMP and ADMET), and other clevernesses.

Proteins are made from a DNA template via RNA, one AA residue at a time. There is no dispersivity.  Post-synthesis modification, e.g., glycosylation in erythropoietin, can be polydisperse.