Recent research has explored the memorization capacity of multi-head attention, but thesefindings are constrained by unrealistic limitations on the context size. We present a novel prooffor language-based Transformers that extends the current hypothesis to any context size. Ourapproach improves upon the state-of-the-art by achieving more effective exact memorizationwith an attention layer, while also introducing the concept of approximate memorization ofdistributions. Through experimental validation, we demonstrate that our proposed boundsmore accurately reflect the true memorization capacity of language models, and provide a precisecomparison with prior work.