This Labyrinth Lord supplement bills itself as Advanced Arcana for the Discerning Magic-User. Theorems & Thaumaturgy is available as a free PDF, but there are several Lulu print on demand options as well, for those that like physical books.
It contains three new specialist magic-user classes, the elementalist, necromancer, and vivimancer. There is also a fey elf class, which is presented in both “basic” race-as-class and “advanced” race with class format. In some ways, it feels sort of like an OSR equivalent to the old TSR Tome of Magic, though with fewer system-level changes.
Theorems & Thaumaturgy has what I would consider close to professional grade layout, and excellent, distinctive artwork that fits nicely with the Labyrinth Lord aesthetic without being exactly the same.
In addition to the new classes (which all contain full, custom spell lists), there are also new magic items, new monsters, and a collection themed books of magic. Some of these new spells are very creative, allowing things like detecting which spells another magic-user has prepared (spell reading), and manipulating those spells (for example, there is a charm spell spell). The fey elf presented is distinctive and much more thematic than the standard LL fighter/mage elf.
Gavin also did some great, practical work (like sets of prepared spells for magic-users of any level and an index of all the LL spells). This is the kind of effort that people rarely put into free resources, because despite being very useful it is often not as fun to put together as the parts where you make new stuff up.
There is a chapter on optional magic rules, including an awesome variation on at-will detect magic that functions like the search action (2 in 6 chance, takes one turn). I would be very tempted to use something like that (perhaps X in 6 chance, where is the the highest level spell that a magic-user was able to prepare). The d30 table of magic affinities looks good too. It offers a minor quirk/power for every magic-user, but the random determination makes it much more interesting than all magic-users expecting to be able to use mage hand or whatever.