The spelling of the word "stays treadmill" can be explained through its phonetic transcription in the International Phonetic Alphabet (IPA). The first syllable, "stays," is pronounced with the short vowel sound /eɪ/, represented by the phonetic symbol "e" with a horizontal line (macron) above it. The second syllable, "treadmill," is pronounced with the long vowel sound /ɛ/, represented by the phonetic symbol "e" with a vertical line (open epsilon) above it. Thus, the word is spelled according to the sounds of the English language.