Accepted answer
eight weeks is 56 days, and at 2–8 °C the ten-degree rule of thumb makes that roughly 56 refrigerated days of equivalent exposure. 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. Compare that against what the certificate covers, which is the material as it left the laboratory on the date of analysis and nothing after it. That is short enough that a re-test is a stability study rather than a safety check — worth doing if you will publish the result, hard to justify if you will only reassure yourself. If you do re-test, send it for content as well as purity; the 56 days will have moved one of them further than the other.
The single most misleading statement on a research-grade certificate is a lot number with no statement of how many vials from that lot were tested.
If the lot was manufactured in multiple batches, testing vials from each batch separately establishes whether batch-to-batch variation is acceptable.
Acceptance Sampling by Attributes defines the number of samples you need from a lot to claim a specified quality level at a specified risk — it is in ANSI standard Z1.4.
Lyophilised peptide homogeneity studies show that vial-to-vial variation is usually small but occasionally large, depending on the distribution in the freeze-dryer.
If testing multiple vials, state how many you tested and why you chose those vials.