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I'm curious as to whether getting a permanent enhancement bonus to Intelligence (e.g., by wearing a headband of vast intelligence +2 for more than 24 hours) increases the number of 1st-level spells/formulae known (for, e.g., an alchemist). I'd like to assume it does not, as these spells are determined at character creation, and presumably use your (unmodified) starting Intelligence; but I'm curious whether anyone has a reference that might clarify this.

For example, the alchemist's Alchemy supernatural ability states: "An alchemist begins play with two 1st-level formulae of his choice, plus a number of additional formulae equal to his Intelligence modifier." To me, this sounds like my alchemist should gain a number of bonus 1st-level formulae/spells known equal to whatever his presumably unmodified Intelligence bonus was at 1st level. However, I know Pathfinder mechanics favor retroactively gaining, e.g., skill ranks, when your Intelligence increases, as stated here and here. James Jacobs doesn't explicitly mention spells, but maybe they're included in "... all of them." Is it the case then that, having acquired the headband (at 9th-level), my alchemist also gains an additional 1st-level spell/formula?

Thanks in advance for any input!

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Short answer

The rules are not completely clear on this one, but in the absence of an exception, I would say that your Alchemist does get additional formulae if his intelligence increases permanently, and the first James Jacobs' quote you cite would seem to support this. This is no less 'illogical' than the fact alchemists get new formulae when they go up a level.

Long answer

The closest we get to RAW on this is under Ability Score Bonuses / Permanent Bonuses:

Ability bonuses with a duration greater than 1 day a­ctually increase the relevant ability score after 24 hours. Modify all skills and statistics related to that ability. This might cause you to gain skill points, hit points, and other bonuses. These bonuses should be noted separately in case they are removed. (PCR 555, my emphasis)

This is supported by James Jacobs' comment in the first post you mention:

All bonuses are retroactive when an ability score increases, be they bonuses to damage, to skill ranks, to hit points, to saves, to skill checks... all of them. Skill ranks not being retroactive are a 3.5 convention we specifically removed from the game because it was a weird exception to the rule, and since now there are no exceptions to this rule, there's no need to specifically state that skill ranks are retroactively granted if your Intelligence goes up.

Unless he forgot about spells/alchemists' formulae etc, no exception means no exception.

If you are looking for rules-based logic, I would say it is no more illogical for your Alchemist to gain a formula because his intelligence has permanently increased than it is when he goes up a level:

An alchemist begins play with two 1st level formulae of his choice, plus a number of additional formulae equal to his Intelligence modifier. At each new alchemist level, he gains one new formula of any level that he can create. (http://www.d20pfsrd.com/classes/base-classes/alchemist)

In terms of real-world logic you could just say that your alchemist had struggled with understanding the new formula, and it wasn't until his intelligence increased that 'the penny dropped'.

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    \$\begingroup\$ Modelling/playing it as a Eureka moment is probably the smoothest approach. "Ha! I can preserve the property of A by doing B instead of C, which allows me to combine it with D, stabilizing the E of F!" looks at confused party members "That means I don't kill myself when ingesting this extract I've been thinking about for years." (insert appropriate magibabble). \$\endgroup\$ – MrLemon Mar 30 '15 at 9:38
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The rules say that you “begin play” with X number of spells. That is a singular, one-time event, not a continuing statement on the status of your spell book, formula book, or familiar. It defines the initial state of the book, and no more.

Regarding James Jacobs’s claims about there not being any exceptions to the lack of retroactive effect regarding skill ranks, he is, quite simply, wrong. His claim that it is unnecessary to specify that skill ranks increase retroactively is incorrect, and we see that Intelligence bonuses do explicitly point out that skill ranks increase. This is necessary from a rules standpoint. And since James Jacobs is Paizo’s creative director, rather than any kind of rules authority, and he furthermore gets things wrong all the time, I see no reason to give his comments any more consideration.

If Paizo intends for Intelligence bonuses to retroactively increase the number of 1st-level spells in spell books, formula books, or familiars, they need to say so, preferably in an errata that eliminates the problematic “begin play” wording and spells out how this works.

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No, your "starting spells" are one of very few things that aren't retroactively raised when their setting stat goes up. They're a measure of how much stuff you managed to figure out before you started adventuring, and so only your pre-career Intelligence matters. Everything else that isn't explicitly called out as not changing do; skill ranks, bonus spells, hp, etc.

As far as rules references, I think this is going to have to fall into common sense; you start with a spellbook or formula book with so many spells in it. If you can make a case to your GM that putting on a new hat (even if it makes you smarter) suddenly writes new pages into your book, feel free to convince him (or her) to let you have the extra spells. For me, though, I think it's just going to make it easier for you to scribe new spells/formulas into your book yourself.

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