You've completely explored the library and rid it of threats. Now you can make the library checks.
Basically, each full day of research gives the participants a check. You can make this check untrained, but if you have one of the skills below, you get another +5. If the DC is beyond you, you can instead attempt a DC 10 check to assist another PC and grant them a +2 bonus to their own check.
If you succeed at DC 25, then roll for how many knowledge points (kp) you unveil that day. This is based on how scholastic your character class is, and reflecting campaign traits and so forth.
Iseret, Callum: 1d8+Int modifier
Kyra: 1d6+Int modifier
Selquet, Tabat: 1d4+Int modifier
Lastly, if you don't reach 25 kp in the first day, you'll have to either camp out for the night or return the next day for another full day of research to attempt to glean all of the pertinent details from the archives.
VAULT OF HIDDEN WISDOM
Complexity 25
Research check Knowledge (engineering), Knowledge (history), or Knowledge (nobility); Knowledge bonus +5
kp 25[ +- ] Full rules
Researching Hakotep I, the Mask of the Forgotten Pharaoh, and the Cult of the Forgotten Pharaoh in the Great Library is an important part of the first part of this adventure, but to find the valuable nuggets of information they seek among the library’s vast archives, the PCs must sort through the extensive collections of texts, investigating subtle clues and tracking down hidden leads.
Each library or collection of the Great Library has two statistics: a Complexity rating, which reflects the confusion of the library’s contents, and knowledge points (kp), which are an abstract representation of the amount of information a given collection contains.
A library’s Complexity rating acts as the DC for a Research check to unravel that library’s clues. A Research check is essentially a Knowledge skill check, though each library details the specific Knowledge skills that may be used for Research checks based on the nature of that library’s specific collection. A PC can attempt one Research check for each full day of research (from dawn to dusk). Characters who access the library at night can make one further attempt for the period from dusk to dawn. A character cannot take 10 or take 20 on a Research check, but Research checks can be made untrained due to the Great Library’s extensive collections. Each library grants a bonus on specific Knowledge checks (including Research checks) made within that library, and each full day of research (dawn to dusk) in a library grants a cumulative +1 bonus on Research checks in that library. In addition, up to two additional characters can aid another in research; if an assisting character rolls a 10 or higher on a Research check, the characters gain a +2 bonus on the primary Research check.
Succeeding at a Research check reduces a library’s knowledge points, similar to dealing damage to a creature’s hit points. As the knowledge points decrease, a library reveals its secrets. The PCs learn the listed information when a library’s knowledge points reach various "trigger points," as detailed in each library. How much a library’s knowledge points decrease depends on the character’s class. Bards reduce a library’s knowledge points by 1d12 + the character’s Intelligence modifier. Particularly scholastic characters, such as alchemists, wizards, and other characters at your discretion, reduce a library’s knowledge points by 1d8 + the character’s Intelligence modifier. All other characters reduce a library’s knowledge points by 1d4 + the character’s Intelligence modifier. For every 5 by which a Research check exceeds the library’s Complexity rating, the library’s knowledge points are reduced by 1 additional point. Rolling a natural 20 on a Research check acts like a critical threat. If the researcher confirms the critical hit by succeeding at a second Research check with all the same modifiers (this takes no additional time), the resulting knowledge point reduction is doubled. Conversely, rolling a natural 1 on a Research check automatically results in failure, and the library’s knowledge points increase by 1d8 as the library’s complexity causes a researcher to follow a wrong avenue of research.
When a library’s knowledge points reach 0, the PCs have learned everything they can from that library. To learn additional information, they must visit another library or collection and continue their research there.