four weeks is 28 days and, on a weekly schedule, 4 stopper punctures out of one vial at 1 mg/mL. Set the chemical question aside for a moment, because the puncture count is the one with a convention attached: 28 days is 1 times the twenty-eight days conventionally allowed for a preserved multi-dose preparation once it has been entered. Chemically, 1 mg/mL is high enough that adsorption to the glass is a rounding error and low enough that it is not protecting you from anything. What 4 withdrawals do add is 4 opportunities to introduce air, 4 coring events on the same stopper, and a headspace that grows with every draw — none of which show up on a certificate and all of which are avoided by splitting into aliquots at reconstitution.
The honest answer is that most reported "degradation" is adsorption and dilution error rather than chemistry.
Deamidation converts asparagine or glutamine to the corresponding acid via a succinimide intermediate, adding one dalton. It is base-catalysed, accelerates above neutral pH and is the dominant aqueous pathway for many peptides.
Freeze-thaw cycling drives aggregation through concentration at the ice interface and pH shifts as buffer components crystallise out at different rates. Each cycle costs something.
The caveat is that none of these pathways can be seen by looking at a vial, and a clear solution can be substantially degraded.
Cold, dry, dark, still. Those four words cover most of the mitigation.