Glad to hear this is helpful for you too :)
I didn't really follow the time-derivative idea before, and since you said it was equivalent I didn't worry about it :p. But either it's not really equivalent or I misunderstood the previous formulation, because I think everything works for me now.
So if we (1) decide "I will imagine yummy food", then (2) imagine yummy food, then (3) stop imagining yummy food, we get a positive reward from the second step and a negative reward from the third step, but both of those rewards were already predicted by the first step, so there's no RPE in either the second or third step, and therefore they don't feel positive or negative. Unless we're hungrier than we thought, I guess...
Well, what exactly happens if we're hungrier than we thought?
(1) "I will imagine food": No reward yet, expecting moderate positive reward followed by moderate negative reward.
(2) [Imagining food]: Large positive reward, but now expecting large negative reward when we stop imagining, so no RPE on previous step.
(3) [Stops imagining food]: Large negative reward as expected, no RPE for previous step.
The size of the reward can then be informative, but not actually rewarding (since it predictably nets to zero over time). The neocortex obtains hypothetical reward information form the subcortex, without actually extracting a reward—which is the thing I've been insisting had to be possible. Turns out we don't need to use a separate channel! And the subcortex doesn't have to know or care whether its receiving a genuine prediction or an exploratory imagining from the neocortex—the incentives are right either way.
(We do still need some explanation of why the neocortex can imagine (predict?) food momentarily but can't keep doing it food forever, avoid step (3), and pocket a positive RPE after step (2). Common sense suggests one: keeping such a thing up is effortful, so you'd be paying ongoing costs for a one-time gain, and unless you can keep it up forever the reward still nets to zero in the end)
Thanks for the reply; I've thought it over a bunch, and I think my understanding is getting clearer.
I think one source of confusion for me is that to get any mileage out of this model I have to treat the neocortex as a black box doing trying to maximize something, but it seems like we also need to rely on the fact that it executes a particular algorithm with certain constraints.
For instance, if we think of the 'reward predictions' sent to the subcortex as outputs the neocortex chooses, the neocortex has no reason to keep them in sync with the rewards it actually expects to receive—instead, it should just increase the reward predictions to the maximum for some free one-time RPE and then leave it there, while engaging in an unrelated effort to maximize actual reward.
(The equation V(sprev)+=(learning rate)⋅(RPE) explains why the neocortex can't do that, but adding a mathematical constraint to my intuitive model is not really a supported operation. If I say "the neocortex is a black box that does whatever will maximize RPE, subject to the constraint that it has to update its reward predictions according to that equation," then I have no idea what the neocortex can and can't do)
Adding in the basal ganglia as an 'independent' reward predictor seems to work. My first thought was that this would lead to an adversarial situation where the neocortex is constantly incentivized to fool the basal ganglia into predicting higher rewards, but I guess that isn't a problem if the basal ganglia is good at its job.
Still, I feel like I'm missing a piece to be able to understand imagination as a form of prediction. Imagining eating beans to decide how rewarding they would be doesn't seem to get any harder if I already know I don't have any beans. And it doesn't feel like "thoughts of eating beans" are reinforced, it feels like I gain abstract knowledge that eating beans would be rewarded.
Meanwhile, it's quite possible to trigger physiological responses by imagining things. Certainly the response tends to be stronger if there's an actual possibility of the imagined thing coming to pass, but it seems like there's a floor on the effect size, where arbitrarily low probability eventually stops weakening the effect. This doesn't seem like it stops working if you keep doing it—AIUI, not all hungry people are happier when they imagine glorious food, but they all salivate. So that's a feedback channel separate from reward. I don't see why there couldn't also be similar loops entirely within the brain, but that's harder to prove.
So when our rat thinks about salt, the amygdala detects that and alerts... idk, the hypothalamus? The part that knows it needs salt... and the rat starts salivating and feels something in its stomach that it previously learned means "my body wants the food" and concludes eating salt would be a good idea.
This might just be me not grokking predictive processing, but...
I feel like I do a version of the rat's task all the time to decide what to have for dinner—I imagine different food options, feel which one seems most appetizing, and then push the button (on Seamless) that will make that food appear.
Introspectively, this feels to me there's such a thing as 'hypothetical reward'. When I imagine a particular food, I feel like I get a signal from... somewhere... that tells me whether I would feel reward if I ate that food, but does not itself constitute reward. I don't generally feel any desire to spend time fantasizing about the food I'm waiting for.
To turn this into a brain model, this seems like the neocortex calling an API the subcortex exposes. Roughly, the neocortex can give the subcortex hypothetical sensory data and get a hypothetical reward in exchange. I suppose this is basically hypothesis two with a modification to avoid the pitfall you identify, although that's not how I arrived at the idea.
This does require a second dimension of subcortex-to-neocortex signal alongside the reward. Is there a reason to think there isn't one?
I'm not sure Level 3 is actually less agentic than Level 1. The Oracle does not choose which truths to speak in order to pursue goals; if they did, they'd be the Sage.