Both I and another local GM have been scratching our heads at trying to figure out how to work the hd of Bestiary 3's Unfettered Eidolon, and we must be missing something, because I don't see any other rules questions or examples of people having trouble with it (so I can only assume that either everyone else understands how this works, or that the unfettered eidolon has been largely unused and most people haven't looked too closely into building one).
So far all we've determined from the example one in B3 and what's written under building an unfettered eidolon is that they're supposed to have 1hd, plus 1 more for each EP over 8. The example unfettered eidolon is stated to have 15 EP which is 7 over 8 and should give it 1+7 hd. 8. It actually has 16 EP in spite of being listed as having 15, which should give it 1+8 hd, or 9, instead. Its actual hd is 10, however, and we can't seem to figure out why.
Our best guess so far as to how it actually works is: The 'building an unfettered eidolon' section is mostly-wrong. An unfettered eidolon's actual hd is equal to its total EP -5. The minimum EP an unfettered eidolon can have is 8, 5 of which must be spent on the base form's default evolutions (which is in line with the lowest EP a normal summoner's eidolon could have). CR is calculated as normal based on hd using table 1-2 in Bestiary 1.
Does anyone understand this any better? Barring that, does anyone know any AP or module where another unfettered eidolon (one different from the one in B3) has been used? It'd likely help a lot to have a second stat block to compare to. (Our potential answer explains the errors in the example unfettered eidolon, but that's not particularly useful unless we know that it can work on multiple examples.)
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I counted 15 EP (the starting evolutions are 5 points, with energy attacks at 2, flight at 4, gore at 2, and improved natural armor at 2). That would give it 7 more Hit Dice, for a total of 8 according to the text.
I'm thinking, however, that this unfettered eidolon was given two additional hit dice beyond the eight that were used to build it to show advancement which doesn't add anything else beyond hit points, base attack bonus, a feat, and more skill points (it does say that they continue to both evolve and advance in this unfettered state).
Hope that was helpful. :)
It's also got slam listed among its evolutions, which is another 1 EP (though come to think of it, slam isn't legal on an armless eidolon, so if that's taken out, it would be the 15 EP creature that it's stated to be).
Advancement is possible I suppose, though a bit of an awkward answer. There are other creatures with varients made using additional hd, but I can't think of any other instances where a bestiary entry gives us an extra-hd monster as the statted example without providing a 'basic' one.
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The example unfettered eidolon is stated to have 15 EP which is 7 over 8 and should give it 1+7 hd. 8.
The version in the PRD says it has 17 EP... which lines up correctly with 10 HD.
That said, it looks like it actually has 18 EP;
"bite, climb, energy attacks, flight [2], gore, improved evasion, improved natural armor [2], reach [bite], tail, tail slap"
Bite, climb, reach, tail, and tail slap are 1 EP each = 5
Energy attacks, gore, and improved natural armor [2] are 2 EP each = 6
Improved evasion is 3 EP
Flight [2] is 4 ep
That's 18 total. So they're still off by one.
I'd take the HD calc as stated and just assume they had trouble with the EP count.
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I failed to notice the slam in its attack routine. I think that might be where the problem lies. If it didn't have the gore attack, then that slam would come from Multiattack to give it three attacks. It has three attacks with the addition of the gore evolution, however, and so it shouldn't have the extra attack.
So it looks like the sample unfettered eidolon was advanced 2 Hit Die and given an attack that it shouldn't have. All else checks out.
EDIT: I was looking at the one on the PFSRD that says it was built using 15, which is what I counted. Now the one on the Paizo PRD says it was built using 17. Slam wasn't added to the list of evolutions there either, but it was given improved evasion. Going by the PRD, that would still leave one EP unaccounted for.
EDIT for the EDIT: This was pointed out in an errata thread, with the suggestion to give it improved evasion in place of the slam. But that would indeed put it at 18 EP, not 17 as listed on the PRD. Either way, it shouldn't have a slam attack, whether it be due to a lack of arms or gore giving it three attacks so that Multiattack gives the tail slap a -2 rather than -5 on its attack.