As for wisdom: understanding, seeing, or knowing experience in terms of suffering (the four noble truths) is the definition of stream-entry.

MN: He trains thus: 'I shall breathe in contemplating cessation'; he trains thus: 'I shall breathe out contemplating cessation.'

BV: Now this is called the eighth jhāna.

H: "Again, by completely surmounting the base of nothingness, a bhikkhu enters upon and abides in the base of neither-perception-nor-non-perception.

BV: Now, your mind before, got very big. Now your mind starts to get small, and smaller, and smaller, and smaller, and it gets so fine, that it’s difficult to tell whether it’s there or not.
S:~
BV: Not like floating.
S: ~
BV: This is all different perceptions of mind. It’s not blank. Your mindfulness is still there. You have energy. You have joy. You have investigation. You have stillness. You have tranquility. You have equanimity. But it gets very fine, very subtle. That’s why this is called cessation, because you can’t tell, sometimes, whether there’s a mind there or not, or whether there’s a perception of anything or not.
Now -
MN: He trains thus: 'I shall breathe in contemplating relinquishment'; he trains thus: 'I shall breathe out contemplating relinquishment.'

BV: Now, I was talking about vibration before. Now your mind is vibrating at such a high speed, that you can’t tell whether it’s moving or not, or whether it’s vibrating or not, and eventually, it gets to a place where it turns off, and this is called, the cessation, of perception and feeling. You don’t have any perception; you don’t have any feeling. It’s just like somebody turned off the light. When you’re in this state, you don’t know you’re in this state. There’s no knowing – there’s no perception at all. Now this in Pāli, is called nirodha-samāpatti. Nirodha-samāpatti - cessation of perception and feeling. Now, this is not the state of nibbana, yet. Let me define the word “nibbana”, because there’s some mis-understanding, sometimes. The word “nibbana” means putting out the fire. Ok? That’s literally what it means, putting out the fire. The fire of craving.

Now, there’s two different kinds of nibbana. One is a mundane kind of nibbana – it’s very worldly. Now, every time you have an emotional state, and you finally let that go, that is a state of nibbana. But it’s mundane. The super mundane nibbana, that is other-worldly. That’s the one that we’re working for. Ok? Now the way you attain the super-mundane nibbana, is by having, this nirodha-samāpatti, occur. And then, when you come out of nirodha-samāpatti, what do you see? You see Dependent Origination, forwards and backwards. Dependent Origination has twelve links, and you see it very quickly, and you see, this is the cause of that, that is the cause of this, and you see the cause and effect relationships, but it’s fast. Bing, bing, bing, bing, bing, bing, bing. Bing, bing, bing, bing, bing. While it’s going up, you’re seeing the second noble truth. While it’s coming down, you’re seeing the third noble truth. This is how you experience the super-mundane nibbana.

When you get down to the letting go of ignorance, there is a change, that happens. There is a letting go, that happens in your mind, and this is how you get to become a saint. This is how you become a sotāpanna. Quite nice, actually. And there’s all kind of things that happen after that, but I’m not going to tell you about it. But it is a very, very, wonderful thing to happen. Very good.

MN: 22. "Bhikkhus, that is how mindfulness of breathing is developed and cultivated, so that it is of great fruit and great benefit.

BV: Now, if, you see Dependent Origination, the more clearly you see it, that depends what state you get to. If you see it, very, very quickly, that means that you’re a sotāpanna. If you see it, - more clearly, and not quite so quickly, that means that you get to the state of sakadāgāmi, that’s the second state. Then, if you see it, even more clearly for a longer period of time, that means you become an, anāgāmi, that’s the third stage of sainthood. If you see it very, very clearly, and you can contemplate it for a period of time, that, means that you get to become, an arahat.

btw, don't the suttas state that there is an experience of nibbana that leads to stream entry? i know thanissaro bhikkhu always mentions that there is the opening of the 'dhamma-eye' and the first sight of the deathless with stream entry, and he doesn't ever seem to claim anything unless it is directly stated in the suttas. not that what i just said constitutes proof, but does anyone know of the suttas making that direct statement?

Then to Sariputta the wanderer, as he heard this Dhamma exposition, there arose the dustless, stainless Dhamma eye: "Whatever is subject to origination is all subject to cessation."

Even if just this is the Dhamma,
you have penetrated
to the Sorrowless (asoka) State
unseen, overlooked (by us)
for many myriads of aeons.

Then Sariputta the wanderer went to Moggallana the wanderer. Moggallana the wanderer saw him coming from afar and, on seeing him, said, "Bright are your faculties, my friend; pure your complexion, and clear. Could it be that you have attained the Deathless?"

"Yes, my friend, I have attained the Deathless. "

The counter-intuitive part is that if you have sufficient faith or wisdom (along with the other faculties), and you attain a formless jhana, that implies you are (or causes you to be) at least a stream enterer...else, you would be excluded from the typology completely.

A logical oversight, or a surprising truth?

Sorry if I was unclear about that. I don't see any reason to think that faith is less important than any other characteristic. In some ways, it's more important.

Only when done at a sufficiently high level. This practice is endorsed as the way to stream entry, i.e. one will have to build one's understanding sufficiently in order to get there.