Final answer:
The core value demonstrated by frequent messages in deaf culture is communication through ASL, reflecting the community's identity and the desire to preserve their unique cultural practices.
Step-by-step explanation:
The core value of deaf culture illustrated by the deaf man's frequent messages is the importance of communication and the preservation of deaf culture-specific traditions such as the use of American Sign Language (ASL). In deaf culture, the choice to communicate through signing, rather than verbal speech or lip-reading, is a reflection of the community's values. It underscores the desire to maintain a strong identity and pass down cultural practices, and it highlights the role communication plays in establishing and sustaining relationships within the deaf community.