Submitting more than three letters of recommendation to a math PhD program
The first criterion, of course, is that if a particular institution's application intsructions communicate that only three letters should be sent, then abide by that. For the ones that say &quot;at least three&quot; or the like:
To be honest, in practice it really depends on the internal details of each university's application storage system and how it lets reviewers view applicants' files—and this can vary wildly from place to place. At my institution, if you have four letters of recommendation, they will all be visible when I look at your file. In other places, their online portal might have specific spots for three letters and no more (I would hope such institutions would make it clear in their instructions that only three letters should be sent).
As someone who has reviewed graduate applications for nearly twenty years, I can tell you that if my file system allows me to see the letters, I will read all the letters if I am interested in an applicant. (Well, I probably wouldn't read ten letters … but certainly four.) If you have reasons to believe that the letters will offer complementary perspectives on your excellence, I would use them all.