Is a woman chewing gum considered forbidden if it attracts men's attention and is misunderstood, or is it permissible due to the absence of a Quranic or Prophetic text prohibiting it?

Chewing gum is permissible for men and women, as the default rule for things is permissibility. There is no evidence to prohibit a woman from chewing gum in front of men unless there is a fitnah (temptation/strife). However, if a woman's chewing of gum is done in a manner that involves arousal and fitnah for men, then it is prohibited.