When Arcane pcs level up their master’s teach them new spells for free.
I assume that if a pc cannot travel to their master they simply do not learn anything.
What happens if the master dies or disappears? Can they get new one?
When a pc has an apprentice and they level uo does the pc pay to teach them their new spells?
I read it as “when a PC mage levels up, please let them learn new spells and don’t give them too much grief about, OK?” 
If they don’t have a master handy, maybe:
- that old spellbook they could not understand earlier suddenly makes sense?
- they’ve been working on some ideas all along, and are finally able to make them work?
- there’s a local mage who provides them with a couple of new formulas for money for future favors?
- they’ve had the formulas for their new spells all along, but have just been unable to cast them?
Well that makes sense but since spells cost so much even when found and ACTS is old school I didn’t think it was meant to be that forgiving. 
Re-reading the bit about leveling up and learning new spells, I may have been overgenerous in my first interpretation.
Aside from possible research, I don’t see a cost assigned to learning new spells – only to switching out old spells in favor of new ones.
I still stand by my take, though. In most cases[] when a master or local guild is not available, it’s because the Judge as said it isn’t. If the explicit purpose of this is to make leveling up harder or more expensive for mages… fair enough. If that wasn’t the purpose, though, saying “no new spells for you unless you pony up” is skirting close to douchbaggery.
[ Sure, after 9th level you don’t get teachers, and if the players are in a boat in the middle of the ocean, or somesuch – that’s one thing. “No, your master is not available, and neither is anyone else.” is another.]
THAT said, I do note that there are no guidelines for buying or selling spell formulas in the book. Perhaps because they aren’t typically sold, but taught in confidence – or traded against other formulas or favors?
What happens if the master dies or disappears? Can they get new one?
APM: If the master dies or disappears, that would be the result of the Judge’s decision in his campaign world. It would be up to the Judge to determine whether they could get a new one. For example, if the master dies because The Dweomer Plague has slain all mages of 5th level and higher, then no, there’d be no new master. If the master dies because he got old, but there’s a Mage’s Guild with other mages, then sure. Judge’s call.
When a pc has an apprentice and they level uo does the pc pay to teach them their new spells?
APM: No.
Ok I guess the part that throws me is that it takes lots of money to scribe a spell from memory to a fresh book say if a caster lost their spell book. This implies making a clean copy also costs money nd would therefore cost either the student or the master something.
If a master can teach someone something for free what is different about this process? Also does this mean pcs can swap spells for free if they wish? And if not why not?
It’s not that the cost is free, it’s simply subsumed within the abstractions of learning spells in the game world.
Consider the 1,000gp cost for scribing spells an abstraction. The mage doesn’t literally heat the gold into liquid and then scribe the spells with gold ink. Rather the gold is spent on scribes, special quills, one-time grisgris charms, lots of scratch paper, fancy inks, strange chemicals to induce trances, massage therapists to ease aching hand-muscles, sacrifices to spirit powers, tithes to the god of knowledge, and so on.
As part of a mage gaining a level, some of these abstract things are assumed to have occurred in the background that justify gaining the new spell. Perhaps the mage had an “ah-ha” moment and was able to scribe the spell without needing lots of scratch paper. He pricked himself and used his own blood for ink. He achieved a meditative state without needing exotic yellow lotus powder. Etc.
Well I never assumed they actually wrote things in liquid gold but rather expensive inks and papers. However if there are non-expense elements involved then that kind of works for me.
What about Pcs teaching each other?
PCs can teaching each other spells at the normal time and cost.
But just swapping formulas doesn’t cost anything, right? (Ie. not learning the spell, but recording it in order to possibly learn it later.)
See you had me until PCs aren’t able to swap. Surely if they’ve abstractly achieved all that stuff it shouldn’t matter where the formula comes from…they should just attune to one that’s available.
Tywyll, I have to apologize here, and do a major face-palm, because I’m afraid I mis-remembered my own rules.
PCs can learn spells from each other’s spell books at a cost of 1 week.
To summarize the rules on Learning Spells:
- There is no cost to add a spell to your repertoire from your master if you can still learn new spells of that level. It takes 1 week per spell.
- There is no cost to add a spell to your repertoire if you can still learn new spells of that level, and you have access to a scroll or spellbook. It takes 1 week per spell. [Note that this is functionally the same as #1]
- If you cannot access a spell from a master, scroll, or spellbook, you must research the spell first. This cost 1000gp/level and takes 2 weeks/level.
- If you need to replace a lost spellbook, it takes 1000gp/level and 2 weeks/level. This is basically a penalty cost. It’s for strange chemicals to induce mimetic trances, massage therapists to ease aching hand-muscles, sacrifices to spirit powers, tithes to the god of knowledge, one-time grisgris charms, etc.
- If you need to replace one spell in a repertoire with another, it costs 1,000gp and 1 week/level. This is, also, a penalty cost. You must already have a copy of the spell formula either from a master, a spell book, a scroll, or research.
Sorry for the confusion - I hope that clears it up.
Yup that total clears it up. Thanks for that!
Ok a player of mine just asked…what happens if you lose your spellbook but have an enemies?
I think it would let you function normally with any spells you shared in common but you’d need to pay normal swap or rewrite costs to fully replace your own. I.e. you would save money on replacing your book basically. Is this right?
Your solution sounds elegant so let’s go with it.
Assuming you can read the language it’s in, of course.