Alright, after looking through the winning plan, I can put out some guesses as to what happens on the dice front.
Social Action with Kyouko: The Social Action with Kyouko should either unlock a Mission (which further Social progress would be locked behind), or get her up to Social Link 5.
-Should progress be made, Kyouko will also be getting two half Social Rolls from Requests, and we can expect between 27 and 126 Social Link progress towards the next threshold of 500.
Galahad: Between Homura (30 Investigation, 10 Madness Of Love) and Kyouko (10 Investigation), Galahad's Mystery should yield either a Clue, a Mission, or open up Galahad as a Social Action target. If it opens up Galahad as a Social Action target, and we get a half Social Action, we'll be looking at somewhere between 19 and 88 Social Link progress with them immediately.
Big Ben: Between Homura (30 Investigation, 10 Madness Of Love) and Kallen (15 Investigation), Big Ben's Mystery should yield either a Clue (not necessarily for it's own Mystery), Locations, Resource(s), Items, Character(s), or Mission(s). If it is a person up there, we'll be looking at somewhere between 37 and 95 Social Link progress with them immediately.
Buckingham Palace: Between Homura (30 Investigation, 10 Madness Of Love) and Kallen (15 Investigation), there is an 85.5% chance of making a Discovery. Should the Discovery be a target for a Social Link, we will be looking at somewhere between 26 and 85 Social Link progress with them immediately.
Study Emrys Tome: I suspect this will be a Magical Theory roll. With that in mind, Homura could be rolling with a couple of bonuses that I can find. Obviously, a +16 from Magical Theory [General] itself might apply. With Akemi Madness active, Madness Of Love will be providing a +20 on it's own. Should both apply for a total of +36, then Homura will be making between 37 and 136 progress on Emrys Tome.
Research Efficient Time Stop: Drawing information from
previous Research attempts, we know that Time Stop has a base Research DC of 15, and is increased by roughly 5 per point of Ambition. This results in a final Research DC of 65. Homura only has a Research Skill of 12, along with a +10 Madness Of Love, for a total of +22. That is a 58% chance of producing another Theoretical Spell. In the case that it fails, it would also show us what happens when a Research attempt at prototyping a Spell fails as opposed to one that fails to actually Develop a spell.
Investigation Training With Sherlock: Homura currently has Investigation at Rank 30, which means she will need 150 progress to rank up. If Sherlock Holmes was an ordinary teacher for Investigation (and not a Legendary one), he could grant +100 progress. With Akemi Madness active, Madness Of Love grants an additional 20 progress. All together, this represents somewhere between 121 and 270 total progress (outside of any other benefits from having
the Sherlock Holmes as a teacher).
Training Magical Theory[General]: Emrys Tome is going to be a massive boon here, both as a Library (providing it with many charges to be used for Training) and because of who it is written by. I would expect it to provide at least the standard +50 from a Training Book, aside from Madness Of Love granting an additional +10. With 61 to 160 progress expected, Homura is guaranteed to rank up at least once.