Accepted answer
four weeks is 28 days, which at 2–8 °C is on the order of 28 refrigerated days. 2–8 °C is the condition the rule of thumb is anchored to, so it is the baseline rather than a multiplier: everything else in this thread is quoted relative to it. In solution the routes that matter are hydrolysis of the backbone, deamidation at Asn, and physical association — the first two cost content, the third costs neither until it precipitates. Over 28 days at 2–8 °C you should expect all three to have moved, and a purity figure to have noticed only some of them. Reconstituted material has no certificate; the one in the box describes the powder.
Start with the sequence, because which pathways are available depends on which residues are present.
Oxidation targets methionine, cysteine and tryptophan, adding sixteen daltons per oxygen. It is catalysed by trace metals and promoted by dissolved oxygen and by light.
Aggregation is physical: peptides unfold at air-liquid interfaces and associate. Shaking maximises that interface, which is why swirling and shaking produce visibly different outcomes on the same vial.
Metal-catalysed oxidation of methionine is documented across peptide and protein formulations and is why chelators appear in some formulations.
Sequence determines which pathways apply, so general statements are general.
Swirl, never shake. Aggregation is a handling problem more than a time problem.