 Hello and welcome, everyone, to Act In Flab. Today is May 6th, 2021, and we're really excited to have Shauna Dobson back for MathStream 2.1. This is gonna be a fun discussion, Diamond Holographic Principle, and D Square, Immersion Time. Maybe you'll teach us how to pronounce it and what it is. So thanks again for coming on this stream and really looking forward to it. So please take it away. Thank you so much, Daniel. I just wanna say your community is so supportive and I'm so happy to be back for a second talk. You are all working on some of the most, you know, profound ideas here. And so I'm really excited and thank you for having me. Yay, all right. So this talk, I'm gonna go a little bit into the more like the diamond part because I got some feedback on one of these diamonds and what was she talking about? And so I was really excited that D Square is actually a D diamond for emergent time. So if I, I'll try to do my best to lay all these out. Please ask, like Daniel was saying, questions every 10 slides or something like that. I'm happy to answer it. I'm also trying to constolate and weave together this sort of kind of grand unified picture. So there's some drawings in here and I hope that it's fun. Okay, so let's go. I'll be talking about the Diamond Holographic Principle and D Diamond Emergent Time. Yay. Okay, so the goal, I sort of talked about this last time, but not too, too in detail. Today, I'm gonna actually weave together this kind of new math with active inference and let's try to put it together and all work together. So the goal is like to take the Holographic Principle, which we're, you know, all aware of, which was like be gotten by Maldecina's, you know, beautiful gauge gravity duality between anti-dissider space and conformal field theory. I have an idea to sort of reinterpret it in terms of continuous K theory and something I'm gonna call pro-emergence. So a pro-object is a specific object kind of in category theory that's being used everywhere. And it's sort of this idea of sewing together a bunch of kind of smaller things that are self-contained to get a grand object. And it's sort of built into the very structure. I thought, oh, this is a great idea. So I talked about K theory a little bit last time, which is this beautiful construction by Grafandeek to actually look at classifying very difficult objects using isomorphism classes of the spaces. So it's like, oh, we can tell if two spaces are equivalent based on their equivalence relations. And I thought that's a very, very, very nice, higher way of looking at things. And there's a new work by Efimov talking about continuous K theory. So again, this idea of sort of sewing together maybe smaller K theories to build a bigger object. So for this pro-emergence, right, I am fascinated by time and I want us to kind of get into it, right? How does active inference see these sort of time streams? Like, are there little local times with little thicknesses around them? What is this? So in this idea of mine, this pro-emergence, you have a doubly emergent time that's emergent on two fronts. So you have a local singular condensed time, which is sort of the time that we all experienced is very like, you know, I'm presenting to your lovely community, 9.06 a.m. This is like Shana's now, right? There's also like, you know, Alpha Centauri now and the unicorn constellation now, right? So there's all these different nows and whose now is what? So you have these like local singular time and then you're gonna actually have this global what I'm gonna try to push for is a non-locality in the diamonds. So to rephrase the holographic principle and kind of upgrading ADS, which is anti-decider space, which is the bulk, which has gravity in it, I'm gonna try to actually rebuild that from the six functor formalism. So that's what you see in this figure with these sort of six columns is the six operations from Peter Schultz, the telecolonology of diamonds. It's so beautiful. So imagine building space from a functor formalism, which is very, very specific. And then if you're gonna get the conformal field theory, which has no gravity, I was thinking, okay, so the gravity list could be in the profinite condition of the diamonds. So you see, I have a bunch of diamonds kind of glued together. And so maybe this gluing in the diamond condition can kind of replace the conformal field theory. So in this, you can actually get a holography. And it's really interesting to think what active inference thinks about a holography, like what's the actual reflection? What's the actual object? So you can relate this to strong and weak coupling, whichever theory is strong, right? Has a strong coupling, which is like the tension in the interaction terms. So you can relate whichever one has maybe a weaker coupling, you normally use like a perturbation theory. The stronger coupling, you have to use the non-perturbation methods. So it's very physics oriented. And the beauty and the duality is that if one space is too hard, you go to the other space. So this is really fun. And this is in the, I sort of have this grand opus paper, the DOP 21. You're like, she refers to it all the time. Okay, so the goal, right? So I said, you're gonna have this pro-emergent thing, which is like has multiple emergent times going on. It's not just emergent time, but I think they're all over the place. So in that is what I'm gonna call the stack time. So a stack is going to be this particular object that takes values in categories, not just sets. So we're not looking at like the set of continuous functions, but maybe you're looking at the category of continuous functions and every object in the category is a continuous function. So you're getting this sort of grand unified view of objects, which is really, really neat. I say V because it's gonna be a stack with a V topology on it. I think I spoke last time that you can actually put a topology on categories and that's what I'm really interested in. So the V topology is Peter Schultz's topology that he's using for the diamonds and it's a nice fine topology and I'll go into it in a little bit. So the big questions that are driving some kind of wild diagram that I have here in the middle is con D diamond. It's like, what is the mathematics of simultaneity? So I'm really interested in what we mean by simultaneity when we say two events happen quote, at the same time, what time is that? And is there a difference and what is the difference between simultaneity and non-locality, non-locality, you have spatial non-locality, maybe the view from a pair of entangled photons. And then what does temporal non-locality mean that time is actually everywhere, that there is no difference in time between events, right? You have events that are maybe spread out into space but maybe they happen at the same time. So I'm really interested in what kind of mathematics would be able to hold something like that, right? Maybe you have a bunch of multiplicities going on but they're all at once. So I'm asking what is embodied cognition in non-locality? How could you actually reframe one of the fundamental tenets of active inference based on, like reframe it in non-locality, not a progression but what would actually happen all at once? So this is where I think the mathematics would actually help like infinity categories and this double emergence that I'm talking about. So I talked about infinity categories last time. These are categories that have a lot of rich structure in them and you're able to create morphisms between morphisms and between morphisms all the way up to infinity morphisms which is really, really beautiful. So maybe the claim, maybe it's kind of a wild claim is that embodied cognition like is emergent time, right? So what am I gonna, hopefully that's something that I can kind of show throughout these slides. Embodied cognition like is emergent time. So again, the basis behind emergent time is was it always here? I don't know, was time always here? Who knows? And that's what I'm kind of after. So in this diagram you see I have this spot FQ which is the added spectrum I'm gonna go into on the other side. You have spot QP which is the added spectrum of the P addicts, the diamond interpretation with the little diamond on top. You've got these two factorization maps, these XBs from Farg's work on his geometrization of the local Langland. So heavy math in this, which is really beautiful. On the bottom, you've got this Y diamond SE. These are, this is called the diamond interpretation of the Farg fontane curve. This is an addict curve that has an actual Frobenius action on it. And so this curve is very important. It's used in the Langlands program, which is a V-top, a gran unified theory of mathematics right now. So then you have these LN maps that are mapping up, whatever up means, into what I call condensed T diamond. So the condensed category is Peter Chulso's construction about category of condensed objects. I'm gonna go into those. It's a beautiful idea about sheaves over points. So anybody in the crowd who knows what a sheaf is it's like an awesome topological way of keeping track of local data, local data, local data, local data, like a little bag. And so I can't like, normally, you don't, you have sheaves over spaces, but here's an actual sheaf over a singular point. And so he's, he and Dustin Klassen have been like putting together this really awesome lecture notes on this topic. So it's great. It's in its infancy and I think it's great. So that D diamond is a conjecture of mine that you take the diamond I talked about last time and you can actually extend it to the infinity category of diamonds. So I call that D diamond. A lot of stuff in that diagram, but this is what I think is actually the aesthetic because it, what if we could actually map this to like a string theoretic interpretation or something. Then you get the grand unified theory of math and physics. Boom. Okay. So let me open that just a little bit. So if you take, let's see be the category of diamonds let condensed C be the category of condensed objects in C. Objects are condensed diamonds, right? That's what's in there. Let R be the reflective full subcategory of condensed C. This is a subcategory where all objects in R actually reflections of objects in the bigger category. So it's very neat. You have diamonds in their reflections. So objects in our reflections. So if you take D diamond to be the infinity one category of diamonds you can extend this to con D diamond. So condensed D diamond and you get this actual object. So following precedent from Fargg's amazing work the fiber product of this diagram has claimed a modular space and a diamond. That's very hard, right? So that's hard. So modular V stack time. So let's take this condensed diamond and just up it just a little bit. So I have this other conjecture, right? Let's take, so the way that you get these diamonds I wouldn't have to figure out I have to figure of a diamond very, very soon. The way you get the diamond is by taking equivalence classes, certain pro etel equivalence classes of these objects called perfectoid spaces which are like Berkovich space is kind of like cancer objects. And so you glue those together and you get a diamond. Well, why don't we glue diamonds together and get something else? So you can do that. So if you can take diamond equivalence relations of this diamond curve then you can get a stack and the V topology. And then why don't we work in the modular space of V stack? So modular space is this awesome parameter space where each point represents an isomorphism class of objects. So you're looking at, so each point like in this little figure below is an isomorphism class. And so the question is like, you know how do you move in a diagram like that? Is it a chess board? Is it can I only move this way and this way and this way? So I'm really interested in like what it means phenomenologically to have something like this and to start in the upper left like dot and jump to the like the bottom right dot. So how do you actually move through this thing? So I'm thinking that every point of that modular space would actually be an incarnation of the condensed D diamond diagram. So that huge factorization with these maps would actually be each point. And I think that's actually really beautiful. So that's the type of like grain unified theory I'm kind of thinking of. Yeah, so these LN maps play a dual role of what's called topological localization. Localization is a beautiful way of actually constructing equivalences between objects. And so topological localization passes through this reflective subcategory. So objects are being sort of stared at by their reflections and it's a way of getting more equivalences in maps. So I'll just leave that at that. So this pro terminology, this is a little mathy but I thought, you know, let's do it. I'm not gonna go super mathy but I wanted to give you enough terms so you know, like what I'm talking about. So for this pro, which I think is very cool, the idea is that emergence is built into the very space as a pro object and not as additional counter terms. I think the AI community, the active inference community does that as well. You want everything sort of built into the structure as opposed to adding these counter terms. There's a lot of beautiful like Feynman path integral work that has all of these additional counter terms added to it. And it's like, I wish it could just be built into the actual space. So you can actually do that with this pro terminology, right? So a pro object, all this is, there's a great reference called in cat lab that has all these definitions. So I'm pulling all these and lots of great lecture notes on any kind of pro object. So what is a pro object? A pro object of a category C is a formal co-filtered limit of objects of C. So basically roughly, what we mean by this co-filtered limit is that for every two objects, C1, C2 and C, you have a third object, C3 and C that they both get mapped to it. So it's sort of self-contained. You see it's built into the structure. You don't have to go off and leave the category to find a limit. So you have a hom-functor also. So what do you do? You take S to be a closed symmetric manoidal category. Closed, I have the definition for both of these. Symmetric manoidal means it has a tensor product and you take C and S enriched category, which means that it takes values. The hom set of C takes values in S. So you have this enriched hom-functor, which is written like this. C takes in two pairs, going from the product of C opposite with itself, where C opposite is the same category. The arrows are just reversed. Two S that sends a pair of objects in C to this hom object, which is a collection of morphisms. So look at that. You're actually mapping each pair of objects to where it can go, right? To all the possible paths. And I think this is very nice. So a closed category means for any pair, CC prime of objects, the collection of morphisms is an actual object of C. So everything is inside of itself. And it's sort of like a categorical closure property, right? Many of you are familiar with just the closure property that you learn in like abstract algebra. So this is called the internal hom object, right? So five, the pro-functor. So with the pro-term analogy, you can actually have a pro-functor, which generalizes the word term functor. So let's C and D both be enriched over C, a pro-functor from C to D is an S-functor. So it's enriched with D op, tensor C over S. So again, taking the pairs and mapping them over. So you can also make this into, like I told you, this modular space, right? Right here with the points where isomorphism classes are each point and they can vary geometrically. And I'll go into what that means. And you can get stacks from that. So I think I'll finish the pro and then ask if there's any questions, Daniel, just to wrap up this pro. Okay, before, yeah. Let's get the super techy stuff out of the way. All right, so what's also beautiful and all over the Langlands program is the idea of a pro-finite group. So you can take this pro language, this kind of multi-gluing and self-containingness and extend it to a pro-finite group, which is a topological group that's isomorphic to the inverse limit of an inverse system of discrete finite groups. So you have this inverse limit over the discrete finite groups, these are finite groups that they all have the discrete topology and you have a collection of homomorphisms that satisfy nice relations. So my idea is like, okay, let's up the game even more. There's a lot of work about spectra and prime spectra and this is also Roth and D where the spectrum of a ring, I think I talked about it last time is a certain, the spectrum of a ring refers to the space where the actual points of the space refer to prime ideals. So again, you have this like point referring to something higher than a point. Okay, so you can extend that to what are called a formal spectra. So Roth and D had this great idea of like including infinitesimals in things. So it's not like you have a singular now, but what if we had a singular now with a little bit of infinitesimal thickening on it, right? Like every now or every like local time of ours had a little bit of thickening and we could call that an infinitesimal thickening. And so he actually has the same idea and he calls it formal spectra. So we say the formal spectrum, spuff R of a commutative ring R and it's an ideal that obeys certain properties. It's actually the inductive limit of the prime spectrum. So inductive limit meaning like this. So the pro refers to a projective phenomenon and the end refers to an inductive. So it's like a code limit. So spuff R is the code limit of spec R in, which is where R is complete with the I addict topology. There's various ways of completing rings in the I addict. So where the connecting morphisms are these closed nil potent and immersions, spec R I N immersed with spec R I N plus one. So these are of affine schemes and the code limit is taken in the category of topologically ringed spaces. So what is a closed nil potent immersion? So a nil potent phenomena is some property that you know, take an element, raise it to a certain power and eventually go to zero. So it's a way of canceling out elements. The affine scheme is a ringed space. It's a topological space. And it has a structure sheaf, which is again, this kind of thickening thing. So imagine a space with a little bit of like skin on it. And so I'm thinking that, okay, it's not just a pro emergence of points. It's a pro emergence of these formal spectra, which are like times intervals with little thickenings on them. So nil potent elements, I say elements, nil potent elements are infinitesimal neighborhoods. And so a beautiful way to complete, again, to complete the ring, having convergence issues like settled is to actually take attic completion. So attic completion is to actually have all the infinitesimal neighborhoods at once. And I think that's a grand goal. And anybody after attic completion to have all the thickenings, right, is probably what we should be after. Any questions on that so far before I get to the fun? Just a one. Yep. Okay. Thanks for this interesting intro. I think one thing that's really gonna resonate with people hopefully of all different backgrounds is what you said about, we want emergence built into the base structure of our model, not a counterterm, which isn't a term that's familiar to me, but it's kind of like magic shouldn't be added at the higher level. We should have a model where at the base level we have emergence. And then you mentioned the pairs of objects and a few different kind of pair operations like inverses. And so I was just wondering, how should we think about these pairs of objects? For example, are they organisms in the niche or are they two organisms? Because if so, then we can kind of think, every time you say pair, we might be able to like map it onto something that we've talked about from the embodied cognitive perspective. Oh, I like that. Yeah, go ahead, try that. Try that, you know, when you could take, you know, the category C of organisms and then it can be any two organisms in the actual category. If you wanted the category, you know, to be, you know, pairs of organisms like the organism Daniel, the organism Shanna or something different, then you could actually construct the category to be that. That's the beauty of category theory. You construct it to be whatever you want because structurally, you know, the goal, the goal is to get everything built inside the actual space and to have these consistency issues that you're not like, oh, well, yeah. You know, so like we come online and then there's this counter term. Oops. So you actually have it where organism, you know, is relating to or interacting with like organism, right? Or you could actually have a category B just the way that my organism is identifying and interpreting information. Like whatever you want those objects to be, it's completely loose. Cool, thanks. And I'll relay any questions from live chat in another 10 slides. Yay, awesome. Okay, this is so great. Yay, so we're all on the same page. You know, built it into the structure, you know? And if you want the structure centered around, you know, you can actually have this C3 being like the organism Daniel, the organism Shanna, that everything maps back to me because that's what we're doing, right? Inactive, everything comes back to me. I'm trying to process. Okay, cool. All right, so, okay, this is a, I don't know, these ideas may just all be wild, but I'm glad, you know, I'm here to help. So, I don't know, this is where my brain thinks. Okay, so we have the idea of an infomorphism, which is an actual pair of, an adjoint pair of functors that's on a chew space. And my colleague, my great colleague Chris Fields has done so much work with that. You know, Chris Fields pairs with Levin all the time. They come up with this great stuff. So an infomorphism is one pair, one set of adjoint functors, which satisfies some information. My idea is to make a two infomorphism from these six functor formulas. So in these kind of wild equations down here, the two tails is a structural property, an internal hum and an actual tensor product, how you actually combine things. But the four in the middle represent two pairs of adjoint functors. So if one set of two pairs is an infomorphism, I just want to actually construct a two infomorphism, which would actually be a morphism between morphisms. So that's a categorical thing at this point. We're working on the semantics of what that actually means right now. And so bear with us, that should be out very, very soon. So what is this? The idea is the incarnation of global time emerges from diamond non locality, what I just tried to say. Local time emerges from these singular condensed sets. The two infomorphism is going to connect the global time with the local time. That's the whole point, right? That's how your organism evolves, right? That's how you deduce, is to connect your local time with what's actually happening globally, right? These cannot be singularly, they have to intersect at some point. So if this two infomorphism can actually do that, then you can actually get this temporal simultaneity non locality. I'm still not convinced that there is a difference between simultaneity and non locality. I would love it for anyone and you know, if anyone's like interested in that, that's really, that's tricky. Because maybe the instance of simultaneity is a slice of the non locality, or maybe we were actually in tune enough with what's happening, like embodied cognition wise or whatever, that we were actually able to access for just a second, maybe the true non locality of everything that is, the non locality of information, whatever you want to call it. What is the difference? I don't really know. People normally refer to like special relativity and reference frames. And I have a couple of questions about what reference frames are. So anytime, I think I'm gonna tackle a new subject, I invent a new dictionary. I'm like, okay, let's start with a dictionary to actually translate what I'm talking about. Okay, so the notion of a site in category theory is where I can put a topology. So I'm thinking, if I'm, for instance, looking at the site over, organism investigating its environment, something like this, the site of Shana, that's a categorical way of actually referring to the localization phenomena in the collapse, right? So quantum mechanics is saying you go from, the collapse of the wave function is great. I go from a simultaneous, like superposition state to like a localized now. Of course, there's controversy about the collapse and things like that. That's why I'm using the word localization. You localize in the now. So coupling perturbation, right? So again, strong coupling, you have to use these non-perturbative methods, weak coupling, you can use perturbation. There's a, I think there's a link between perturbation and this perfectoid stuff that I've been looking at. So there's a way called tilting to go from characteristic zero, if people are familiar with that, to characteristic P. You can also add these beautiful objects called P-divisible formal group laws, which are once again, I would say, ways of connecting like groups. The kernels of these actual maps are P to the N torsions. So the torsion element actually annihilates any other element. So an element is torsion, if it actually is annihilated by another element. So you can see that you have these P to the N sort of phenomena that are canceling terms. So the whole idea is that you can get this cancellation and sort of a sequence phenomena also in perfectoid spaces. So there's also torsion in these objects called shimmer varieties, which we'll talk about later. That's in the Langlands conjecture. Strong coupling, maybe it's strong. Maybe in the diamond, we could say strong coupling is in the diamond side of the holography because there's many incarnations of these covers, spa CD, and I'll go into this a little bit later, just trying to put the ideas up front. Weak coupling, right? So maybe you can do perturbation phenomena in the condensed setting, in the interior. So maybe that's in the singular perfectoid time. It's weak because it's condensed. Maybe it's easier to work with pro-atel covers in the condensed setting. So it would make you work in the conformal field theory, which is the diamonds. So the whole relation to the diamond holography is that if you have these six operations right in the bulk, then you're gonna have to go from the bulk to the conformal field. So maybe the inside is the six operations, whichever one you wanna call strong and weak coupling. So what do I need to finish? Well, I need to actually take the six operations and put them in the condensed setting and then link the diamond profinite conditions with non locality. So you get what's called diamond descent. Descent is another way, like Daniel was saying, about keeping the system contained. Descent is a way of actually making sure that the covering spaces can reconstruct the lower space. So my local space is actually constructable from my global space and vice versa. This is another way of keeping the structure built inside the actual space. Okay, and we'll talk about diamond localization, which is gonna be a way to make relations between the various diamonds. All right, so let's try to link this, right? So recall an infomorphism, it's an adjoint pair of structure preserving maps for a two space. The beauty of that two space is that it's got a very, very nice construction categorically. The two constructions build what's called a star autonomous category. This is a minoidal category, it has a tensor product in them to help you move now objects around where all objects have duals, right? So I thought this might be interesting from embodied cognition, because maybe you're looking around at your reflection and learning off of your reflection, right? And where are these reflections everywhere? So if all objects have duals, that's a very strong property for a category, which is why there's a lot of type theory and stuff that works in the two space. So you also have a category of perbinius algebra. This is an associative algebra with a linear form on it. I told you, perbinius was even in the Langlands, right? So perbinius is powerful. In body cognition, everything is related, right? So you want everything to be related. So what if everything was a star autonomous category? So I think the two construction is actually very pertinent here. So if the hidden states, what if those are like a profinite condition or a pro-object that maybe just only become visible upon a certain pullback? So in the diamond construction that I gave you, I said those geometric points were mathematical impurities, yeah, that they were not visible, unless you pulled them back over these covers. Once again, a lot of work about neurology acting topologically, meaning that your appropriate exception is gotten by these topological covers where portions of the mind or the brain are actually working together and where portions sort of intersect and overlap. That's a covering phenomenon and that's how you actually deduce and things like this. So shared action. Okay, so if we're after shared action, what if we start looking at relations between relations? That's where the shared action is. We wanna think collectively, yeah, in the face of uncertainty. If we start thinking in the framework of two morphism, then you're already in the global space and I think there's nothing more shared than working in the relation of the relation space instead of being so siloed. You can be siloed in your own localization, but that's not gonna help for shared action. You need to actually, so if everyone can get a hold of everyone else's two morphisms, what does that mean? And that means that you're actually all intersecting and interacting. You can take that and go to in awareness, which is what my awesome colleague, Robert Prentner and I have been working on as well. That I'm not sure, I think I said that last time that I do not believe that there's only one channel here. I feel like through shared action, you could access more of these sort of consciousness channels. And so the in awareness is actually a categorical phenomenon that I can show here. So I think I showed this last time, but I'll just show it again. So on the left, you have a basic depiction of a category, this one right here. My little mouse is not working. All right, so the one on the left, top row left. So you have two objects here, A and B, F1 and G1 are morphisms, right? You have two objects, C and D, F2 and G2 are, those are morphisms, we'll call them one morphisms. Let's go to the category in the middle. So the category middle is a two category. Oh, and I'm sorry, the category on the left is a one category. So if you go to a two category in the middle, you have the same objects, A, B, C, D, and the same one morphisms, but look at these wild maps on the side. So you're mapping, look what we're mapping on the left. You're mapping the morphism F1 to G2 by that map phi. And then you're also, by gamma, you're mapping G1 to F2. So this is awesome. So look what we're doing there. Now you're mapping the morphisms between the morphisms and this is where I think shared action lives. You can go even wilder, because you know, it's me. So the third, you can have a three category, which is the top right piece. And look at that. A three category consists of objects, one morphisms between the objects, two morphisms between the one morphisms, and then look at the three morphism you have going on. It's a big double bar, triple bar horizontal bar that's connecting phi and gamma. Is that right, see? So you see, you can just keep going. And why not? That's where collection, yeah, go ahead. Oh, just wanted to give one little thought on this. It's almost like, let's say A, B, C, D are people. And then the two mapping is kind of like best friends or marriage. It's like two people who are married. There's some parallelism between couples of married people. And then the three is like the mapping between marriage and being at best friends. And so it's like saying the relationship between person, A and B in the marriage is like the relationship between A and B in the best friend relationship. So it's like, we can put the relationship and almost objectify it, but not in a demeaning way, just in a containing formal way, and then use that to build higher order relationships. So this is like very interesting. Look at that. Yeah, you say it, I should just give you my slides and you present. It's a co-presentation and we as a dyad are interacting with the material you've prepared as well as any questions that anyone else brings. Wow. And so does everyone see? So like collectively, oh, that makes me excited. So collectively like even this interaction is like a three morphism right now. And so that's actually really cool. And you see, I like the, yeah, with like marriage, maybe if you stayed in the one category, you would have just been friends and you would have never actually tried the other relationship or the other relationship or you can go like twin flames and all this kind of stuff. And so it's actually really exciting. And so you can see just like Daniel's saying, that's shared action and let's take shared action to infinity and then what a beautiful place this would be. You kind of have like the people like me that are related to animals too. So it'd be like, oh, okay. So then you have the morphism between like the theta wave of the animals talking to like the humans and then like oceans and things like that. And then you might start relating on the higher like higher connection level, right? Not hierarchical, but like a higher connection level. And so you can see that Robert and I were trying to think, okay, let's get weird and like take, you know, the one level of awareness which is like, you know, Daniel and I co-presenting here because he says my slides better than me. So the co-presentation, right? Is maybe like a one awareness, okay? And then two awareness, let's just bump this up a little bit would be that, you know, he and I are presenting here and then simultaneously like we're both Cambridge apostles. What? Yes. That's the level that I'm willing to go, right? Because if time is non-local, which I sort of think it is why can't we map to it using these morphisms, right? Of course we'd have to do, I don't know some kind of partition in the brain or whatever, but we'll get to that later. But that's what we're saying about this nested hierarchy. So use the same, everything that Daniel's saying about the categories, right? One category, two category, three category and bump it to like awareness, meaning actual interaction. Like let's take embodied cognition and bump it. Like what would a two embodied cognition be? A three embodied and let's keep going up because that would require higher, higher levels of this awareness, this stacky thing. And again, the beauty about this is that look, we didn't add some counterterm, right? We didn't just add, the whole thing is magic, of course. And I love magic, but we didn't just add it out of nowhere. This was like built into the actual structure, which I think is cool. So again, most people maybe intimidated by category theory or whatever at the first, but it's like, I think there's a payoff. It's like, let's put a little bit of work into that structure and then you can make this aesthetic and it actually, I think it helps, you know? So just an overview real quick, I'm gonna try to talk about this emergent time a little better and a concomitant theory of condensed types, which are supposedly, they're supposed to be immediate from the recently introduced diamond holographic principle. You've seen me already kind of introduced that. So if you're connecting this beautiful mathematical gran unified theory of geometrized local language in the stack language of emergent time, that would be a new incarnation of a new reciprocity law. You see, I'm using the word incarnation to refer to diamonds and reflections, yeah? Also two spaces, infomorphism, star autonomous. All of this star autonomous, objects having duals, reflections, reflective subcategories, profinite condition, it's all about the mirror phenomenon, which is in embodied cognition, I think. Of course, there's problems everywhere, which is why I'm excited about this, right? Problems everywhere, you know? You're gonna talk about a theory of emergent time. How do you measure that? How do you actually measure time without constructing it? And that is like, that's huge. That'll stump me, you know? I talk fast, I'm like, wait, wait. So how do you measure a double emergent? Something as crazy as I'm talking about. How do you measure a pro-emergence? How do you measure like, you know, if you wanna do this categorical construction like we're doing, a two emergence, a three emergence? I don't know. And that's what I hope to, so like, we can all work together to figure this out. How do you actually measure emergence? It's like, do you collapse emergence? Are you too late? If you measure it too early, is it a partial emergence and you've missed it? If we sneeze, do we miss it? Is it fast, you know? So our model gives levels of non-locality, levels of non-locality. Not just saying like, you know, there exists a spatial non-locality. No, if you can put them together in that formal spectra, what if you had levels of non-locality everywhere? And that's sort of what I'm into. So I call it like a stacking non-locality and stackification. I think stackification is a word like spaghettification. It does exist. I may have misspelled it, sorry. All right, so you can do local, this, you know, I'm interested, one of your awesome members of the community was asking me about like, what are you talking about this local global? Meaning like the local singular now is like, where I'm existing now. And then of course, you know, you have your unicorn constellation time on a seros and all this other stuff. So that's what there's a play here between taking in information and then like existing and stuff. So you can do this by this diamond descent. I told you this gluing and that's gonna be an infinite one version of spatial Vedicin, which is at Peter Schultz's work, diamond localization. So what's already neat is that once again, when I talked about the emergence is built into the structure, this localization, those L maps in the reflective subcategory, it's already a descent condition. Whoa, so let's use these beautiful theorems, right? And actually use them for phenomenology, for phenomenology. And so that's already a descent condition. So we're on our way, you know? So the goals to actually link like strong and weak coupling, how do you actually construct this diamond non locality for a temporal multiplicity? I think that's really in tune with embodied cognition is to have a multiplicity time. Meaning like, there is not just one, but it's what, how could you actually model time acting globally, non locally, but also in a multiple way. And that's super French continental philosophy at that point. And I, you know, that's like my upbringing. So I'm really interested in this phenomenon multiplicity because it lets the individual be and it makes like what embodied cognition is like what I feel it is. So I'll have to build this diamond descent. And then again, these V stacks are gonna have to satisfy this gluing condition. That's fine. Our model is a double emergence, but it's profinitely many copies of emergence. That's what's strange. I told you last time a profinite set was a totally disconnected compact Haasdorf space. So that's, that's a, what? Like what is that? You know, not Haasdorf, what? A totally disconnected compact space, not Haasdorf. So, but so the profinite is very strange. It's like, what is that? If you have profinitely many copies of something, what is that actual space? So if you have this double emergence and it's very kind of fractal like, I'm really interested in that. So again, maybe the strong coupling has to do with irreversibility and reconstructibility. I'm very, very interested in what we mean by, you know, something is irreversible, right? I think we need to go into that just a little bit more. And is there a play between irreversibility and reconstructibility, right? The descent condition says you can reconstruct this higher space from this global space. What does it mean that if you like fail to reconstruct? So we say as a diamond X, like reconstructable up to irreversibility, you've seen the up to condition in mathematics, which is like A is sort of equal to B up to some condition, meaning can considering that condition. So the hope is that yes. So if I can get this play between reconstructibility and irreversibility, maybe they don't make more sense mathematically for what it means to be irreversible. So coupling, the strength of the coupling would be in the levels of the profiniteness and the non-locality. How many levels do you actually have there? What's the strength? So coupling by perturbation, I already said that's perfectoid and tilting. So big questions, what would actually non-locality be an active reference? I wanna leave these rhetorically because I wanna get to this, that pretty image, Daniel, before we wait for pause to the next question. So, and I think it's like coming up. Yeah, it's right there, okay. So what is non-locality and active inference? What is holography and active inference, right? Which is this very awesome dimensional phenomena where there's a strong interaction in, say dimension N, and then maybe there's like a weaker interaction in N minus one, what does that mean? And that if one space is too difficult, you go to the other space, what does that actually mean? And they're actually related sort of like a hologram. So hologram is this 2D kind of scrambled image that maybe you bounce light on and you get this 3D phenomenon. So in one sense, the 3D is redundant because you already had it in 2D. So what is the notion of redundancy and active inference? That's what I'm interested in. So what is a diamond equivalent of profinite and active inference? Where's the fractalness that's sort of happening, right? The fractal multiplicity. What is object persistence and active inference? That's tricky, right? So, when we say, the observer measures the electron and things like this, the pair of integral photons acts non-locally or something like this. What does that actually mean? Why is it the same photon? And so Chris Fields and I are trying to work on this awesome idea of object persistence and what that actually is, right? I mean, you saw me a couple of weeks ago, right? But is it the same Shanna? How does that work? How does objects persistence work when you're going through like a multiple time, you know? What is emergent time and active inference? Where is it more likely? Where in embodied cognition is emergence? So what is pro-emergence there? So I want you to think about these as we're kind of going through. What is a singularity, right? In active inference, like a singular event. What is coupling, right? What is the actual coupling? What is the strength of an interaction mean? So, if you want to play it for shared action amid global uncertainty, maybe we should work in discrete time. Operating grain unified theories. Master this local global localization by these higher order awarenesses. Machine learning, right? Operating grain unified theories. Let's go higher, right? Let's not work in networks anymore. Let's work in three morphisms. Medical, okay. Let's model the brain as a hologram. Maybe these thoughts or these certain chromatic towers. I talked about the chromatic homotopy theory last time. It has this beautiful notion of equating spectra. You have this notion of this theorem called chromatic convergence theorem that says that you can reconstruct a certain object from its towers of spectra. So you have a boost field localization, which is sort of like the idea of the topological localization I referred to, which is a way of sort of connecting spaces. And again, if in certain neurological maybe degeneration, maybe the actual morphisms are cut, you can think about things like that. You're no longer able to recall because maybe all the maps are gone, right? So maybe you could more aptly model things like enterograde amnesia and early generation. So medical level and leaven and fields have this awesome kind of new workout about entanglement across daughter cells. What is holography in daughter cells? At least we can measure entanglement in something besides photons, you know? So you need this particularity. Is temporal non-locality, is it like instantiation without reciprocidus advent, right? This is me going like wild here. So advent means that you've arrived, but in body cognition, if the environment is arrived, I've arrived, this is double play, right? So is it an instantaneousness without a sort of two-way interaction? What does that actually mean? What does temporal non-locality mean? I don't really know. Can we actually construct, for instance, a profinite version of that? Which is again this kind of super disconnected fractal space. Is temporal non-locality dialytheism time? Everyone knows I'm a super fan of dialytheism. This is a construction of logic where you can have both A and not A at the same time. Normally you cannot, right? You have A or B either, not A there for B. Why can't A and not A obtain at the same time? So you get this beautiful construction called dialytheism. We may start messing around in that. So it is, does a diamond construction of temporal non-locality fail time as it fails all pro incarnations of duration? So you can get a lot of interesting, heavy questions with this. If so, what are the actual conditions for object persistence in a non-locality that is profinite? How do you actually do that, right? How does something persist in those conditions? How does persistence work in a singular case, such as anterograde amnesia? I spoke of a Clive wearing last time. Was a beautiful pianist. He's still living, which is great. But he got an anterograde amnesia and has no more personal memory, but has a motor memory. And so there's some difficulties there, but it's very interesting how the memory levels work. So the question is multiply profound. So at least try to hold on to it, you know, with a small assessment. Maybe this has to do with like, with thoughts and memory recollection. We can go into that. But first a dream. Oh, go ahead. Can I just make a few active inference? Yes, do it. Okay. So I'm recalling my thoughts here by looking at the paper, which I took notes on. First, you mentioned the dynamic persistence and a multi-scale persistence. And that reminded me of the ship of Theseus, which is like, if you replace all the planks of the ship, is it the same ship? So it's a philosophy styled question. And then we're approaching it in a formal way. And then it comes very close to active inference when you had the slide, a needed particularity. That reminded me of Carl Friston's recent paper, which was called a free energy principle for a particular physics. And particular is a pun because it means specific, but also it means particle. And so when we say a particle or a point, it's like that is the unit of analysis that we can pick up. We can make edges between points and we can engage in a higher level of network mapping. So when we're thinking about particles here and this way that particles can be related in nested models or nested categories, sure what technical it would be, we're kind of talking about a particular approach. Oh, I love this. Yes, yes. Just trying to give a little, little active inference, you know, throw out some red. No, please. Yeah, every 10 slides, every five stopped me. Yes, I think, wow, how neat. You see, and that's like an incarnation. It's like, this is exciting. Yeah, what a play. That's perfect. I'm curious to see what you think about the next couple of slides. Okay, I say, but first, right, so you have all this math-y stuff. We're gonna do this, we're gonna do this. You know, like Daniel said, we're pulling in frist and stuff, pulling in everybody's stuff. Schipathesias, but first a dream because I wanna get like a little pretty. So we are interested in the relation between diamond descent, the stop mechanism. So I know fields and leaven and everyone has been like baffled by that, the stop mechanism that, you know, doesn't work in something like cancer cells. Yeah, when do things stop? I'm actually very curious about that. And the storage and reflection and recollection of profoundly many copies of information in the form of mathematically impure geometric points, right? If you're modeling, you know, neurons as diamonds, what is, whoa, okay, you have covers of diamonds. Like I said, you know, if the proprioception is topological and in the covers, then maybe you could actually model neurons as these diamond phenomenon. So someone like Clive Waring has all the diamonds, but the covers are cut. So it's almost like you cannot access that, right? Very interesting when the actual, you know, balloons are cut, you cannot get the balloon anymore if they're attached to you. So we've posited a model of the brain that's actually allowing for various mathematical partitions in the brain in the form of profanity and many copies of the diamond, right? So the neurons are geometric points, which are morphisms of schemes, like I said, then maybe anterior grade amnesia would resemble some kind of sustained truncation of diamond descent. Why are those maps continually, continuously cut? Which is why I'm very interested in this object persistence. So considering temporal non-cality, can we model thoughts as profinite reflections of pro-Hotel topological covers? Oriented and non-locality, right? Someone from your community also was asking me about thoughts and I was like, that's such a great question. What, you know, clearly there's some, like there's an EM field going on, but it's something more than that. And then the brain seems to operate like a hologram as well. So what is that, right? Are these actual, the profinite reflections? You know, like if you cannot organize your thoughts or things bouncing all over the place, you know, there's a lot of people that are internal or the, you know, is the diamond super internally reflected at that point? So this may help. I said, you know, to model the state of being fully conscious, right? During dreamless sleep, no one has that. Fully conscious during the dreamless sleep, what would that actually be? Which perhaps takes the form of sleeping in a diamond hourglass. So, you know, I write like kind of fantastical mathematical fiction books, and this is my second one that I'm working on about, you know, sleeping in a diamond hourglass. What does that actually mean? So you're working in diamond time at that point. What does that mean? Profinitely many reflections of time, which probably takes some kind of image like this, right? So I always have unicorns ever because I'm like, oh, it's magic, right? And what does that actually mean to sleep inside of anything hourglass? I'm very, very curious because, you know, when we go to sleep, nothing is really asleep. Your system is still functioning, right? But quote, where do you go? And I'm not sure what, you know, I'm gonna put that out there to your community. What is that, right? Why can you not be fully conscious during dreamless sleep? Not the little brief periodics, right? Of dream sleep that you get, but where is that? So it'd be something like sleeping in an hourglass, which is actually very strange. And you think, oh God, what if somebody takes the hourglass and turns it upside down and shakes it, you know? But this would almost be an hourglass without cardinality. So there is no up or down here, you know? And so stay tuned, that book is coming out soon, okay? Which is cool, but it's all, it's nested in the profinite condition. If you're gonna say object persistence, that happens when you sleep as well. So to get to some of the mathy stuff, there's some main conjectures. I'll talk about the Ephemoth K theory of diamonds. Let's actually look up that diamond shape again. And let's get into the diamond holographic principle, the Fart Fontaine curve, why we're using a telcoemology, and let's go to emergent time, you know? Or as far as many of these as we can. So I may start skipping around, Daniel, in the interest of time, you know? So that we can get to the good stuff. So I introduced this kind of wild conjecture about this Ephemoth K theory of diamonds in this localization sequence, which is a cofiber sequence. And a modification of something called continuous K theory. So again, I told you earlier that maybe continuous K theory is the way to go if we're gonna be modeling emergent time, because you wanna kind of stick together small pockets of local emergence. So this number two localization sequence is a cofiber sequence. And you've got the K theory acting on the little sub-diamond and then K Fmoth of the diamond Fart Fontaine curve, mapping to the Fmoth K theory of the non-diamond Fart Fontaine curve with a modification of this F continuous working in the category of sheaves. And this is actually this third left hand side is actually gonna be corresponded to the actual, that omega to the n is an actual continuous K theory spectrum of a D diamond. So again, it's a way of actually gluing together and looking at the like isomorphism classes of what's happening in these diamonds. So again, my idea is this, you have these profiling many copies that of the actual geometric impurities. So you already have those and we actually wanna figure out what's happening between those. So like what's a two morphism between the profiling copies? That's what my interpretation of the K theory is going to be. So you have this level, let's go up a higher level. K theory is actually looking at what makes things like unified at that higher level. So it's cool. So this D upper diamond is a stable dualizable, presentable infinity one category of diamonds. D sub diamonds, actually the complex of V stacks of locally spatial diamonds where complex is a sort of like a collection. And then you have the actual relative Fart Fontaine curve and the diamond. So I don't wanna talk about that too too much. Let's get to the actual diamond holography principle. But the diamond holography principle should fall out and be immediate from this localization sequence. So once again, our diamond holography principle is gonna use these two adjoint pairs where these, you have a direct image and inverse image functor and an exceptional and an inverse exceptional functor which is this F lower shriek and F upper shriek. That's the kind of cool way of saying exclamation point. What's the main point is that, again we wanna take these double adjoint pairs and make a two infomorphism out of them. So these are from Schultz's six operations at telecomology of diamonds. We'll actually wanna condensify this. So how do you rephrase all this in terms of a sheaf of sets like on a certain site? I'm gonna go into what that means. And then a sort of, how do we think about holography in terms of a mathematically, mathematical impurity geometric point, the spa CD, that's the attic spectrum of this closed algebraic curve mapping to the actual diamond. So again, try to think of, when we think about reflections we're trying to mathematically interpret that. So let's actually discuss emergent time from that condensed setting and investigate a condensed version of the actual continuous K theory for pro-emergence, right? That's wild. So the continuous K theory, right? So let's use up a continuous K theory and then actually try to put the pro-object together as a pro-emergence. Okay, so here's this image I think I showed it last time. So the one of the main goals is to actually have an infinity one categorification of geometric Langlands. The Langlands program is so beautiful. What is it trying to do? It's a mathematical interpretation of shared action. You are working across very difficult, disparate fields with lots of uncertainty. And so you have these beautiful phenomena called Langlands reciprocity, Langlands functoriality, where actually you're creating a certain situation in which certain representations of objects are actually automorphic forms. And so different incarnations are being shown to be the same thing, which is like a version of the same thing, which is cool. So in this model that I have here, so the diamonds in the middle, and again, you have the FM off K theory of the actual diamond curve. And at the bottom, I said, oh, this structure is possibly an infinity one topoi, which goes back to Daniel's point about bringing all the structure in. So an infinity one topoi is an infinity one category. It satisfies a descent condition. It's locally Cartesian closed, so everything maps to each other. It's got a Kappa filter on it. So everything is nice and contained, which is really nice. All right. And so you can also, these LNs can also be interpreted in terms of, I have a little note there about a Lubentate formal sheaf module. This is a law to link, once again, one of these higher grain unified theories. You could actually maybe link the geometrization of local Langlands with FM off K theory. So I'm just putting some super mathy things because some people wanted some mathy things. So that's Weinstein's work, which is really pretty. So let me actually formally define diamond. I don't think I did last time. Yeah. And then we get to the actual holography part. Okay. So let Perf D be the category of perfectoid spaces. So again, these are attic spaces. They look like Huber spaces and they are fractal and strange, and they're gonna look like Berkovich spaces. So let Perf D be that category and Perf be the subcategory perfectoid spaces of characteristic P. So no longer zero P. So a diamond is a pro-atel sheaf. That's a sheaf in the pro-atel topology, which I told you is a very fine topology, which can be written as a quotient. You know, X mod R. So it's of a, you take the perfectoid space and you mod out and you glue by these pro-atel equivalence relations. Boom. You get a diamond. What a beautiful way of doing that. So we say let C be an algebraically closed aphanoid field. It's a certain construction and the Ubersettings. And then we say geometric points. This is the thing that I think would be beautiful to actually the thing. This is the actual object that could actually model neurons very nicely. Geometric point, which is the attic spectrum of that C mapping to the diamond is made visible by pulling it back through a pro-atel cover X to D resulting in profiling many copies of spa C. So this is again, huge relation with like proprioception by topology, right? How do you actually visualize? How do you interpret things like this? So diamond is an actual algebraic space for that beat topology. And the beat topology, anyone wants to know is this actual cover F I X I X that consists of any maps X I to X such that for any quasi compact open subset U and X you have final indices I so that these open subsets jointly cover U. So X there is an actual V stack. Okay, so here's the diamond again, right? So we can think what are these actual geometric points in terms of like embodied cognition, in terms of shared action, whatever you wanna think. If I'm a diamond Daniels and I how do you actually relate those two diamonds by these like big maps? And so you can see that right here you have this diamond. So the actual footnote is calling it what it is. So you take the P addicts, right? Which is the field of P addicts QP. We can, if we wanna find the diamond spectrum of that which is that spod D. So the diamond spectrum that's akin to the actual prime spectrum the addict spectrum I showed you. So you can actually only ever get the addict spectrum of QP cyclotomic which is the cyclotomic field of the P addicts. You don't actually get all the P addicts. So this modded out by this ZP cross which is a Galois action. It's underlined because it's profinite. So how interesting that you never actually get you want the left hand side spot a QP but you only get profinitely many copies of the cyclotomic. So you never actually get what you're after and this is actually has a lot of phenomenology behind it because like again, Robert and I really and Chris and I are really fascinated by the phenomenology of what is. So if we actually have, you know if you wanna think interface or if you wanna think whatever it is in our organism are we actually ever truly interacting with the thing or just our representation of the thing? You know, you've got me at the level going, well, you know, if interaction is just me interacting with a reflection of my environment if it's actually me interacting with hidden states that were probably my reflections the whole time. What is it? Are we just sitting around with the profinitely many copies of the thing and we never actually get the thing? Any questions for, I feel like that's a lot of math. A lot of math is pretty pictures. There's a question in the chat. Oh, wonderful. Yep, CB writes, according to this fascinating framework and from an awareness perspective is the dark room problem still considered as a problem. Grateful for any elaboration on that and thanks to the speaker. So the dark room problem being the philosophical thought experiment where what does an agent in a dark room do? And in the free energy world some people initially several years ago brought that up as something like a critique. They'd say, well, aren't you very precisely sure that you're in a dark room? So if active inference is about reducing uncertainty then isn't that the perfect state? But of course the pragmatic answer is that while you're in that dark room your uncertainty about the world is increasing. So I'm wondering if that maps on to a multi-level uncertainty concept because it's almost like you're getting precise at the visual level that you're in a dark room but you're getting increasingly imprecise at not just a different lateral level like you're getting imprecise on smell while you're getting precise on vision but actually at a higher level you're getting imprecise on visually what's outside. Oh, wow. Well, yeah, keep going. What do you think? What a beautiful question. Thank you so much. Oh, yes. More questions and less me talking. Yeah. I'm just, I'm thinking of it as there's three roads meeting in some ways here. There's philosophy. You know, we've brought up the dark room, the ship at these years. Then there's a lot of the formal mathematics which certainly I'm scribbling down but are just beginning to learn. And then there's the active inference community who might be familiar with this idea of multi-level uncertainty but not have connected it to, for example, philosophy or math. So we got people coming and going on all those roads. No, this is beautiful. So it's like, yeah, yeah. Okay, so let's not take one single dark room. Let's expand it. If every organism had their own dark room, yeah. Let's take that question and accelerate it a little bit. Would a dark room survive in a two-infomorphism, right? Let's go a little, let's say on an information perspective, is it dark? Probably not. It's just on the organism perspective, right? So that's actually the dark room is a very beautiful, I think instantiation of what I'm actually saying here with the diamond. So in the dark room, you may just have profiling many copies of the dark but as Daniel's saying, like all your other senses are being very like perfectoided. I'm just saying that, you know, other senses are being more fine-tuned at the expense of something else. So my question is this. It's like, why don't, why can't we increase the sort of bandwidth to be able to be precise everywhere? Like why is it dark to the extent of something else, right? And how do we actually do that? That's very interesting. I think dark room is still a problem. Does anybody solve that? It's still a problem, right? I guess a problem for a philosopher but not a problem for the human in the dark room. And let me ask one question on the operations on diamonds. People might be familiar with the operations on numbers like addition, subtraction, multiplying, dividing. So is it fair to say that the operations that you're gonna be describing, they're like ways to work with diamonds, just like plus and minus are ways to work with numbers? Oh, yeah, okay, it's a very good question. So we can answer that very abstractly that there really is no intrinsic definition of a number. You can say the number of five, well, what is it, five times one, six minus one. You have so many ways of going at it that a number is actually just an equivalence class, right? So if a number is just an equivalence class, then you can think of the same thing as a diamond. Diamond is still an equivalence class, right? A certain equivalence class between, but you're in a different structure. So sheaves, sheaves don't really add like one plus one. So there's a little more, and you can, is there an algebra of sheaves? Oh, absolutely. And the beauty is that sort of sheaves are this like way of keeping track of local data, like a little bag that's associated to every space. There are ways of gluing those together. You don't really add them, but you can glue them and start gluing them and start gluing them and start gluing them. So in a sense, they are number theoretic because if you're defining a number as an equivalence relation. So you can think, yeah, one plus one equals two, whatever that means, or you can say gluing and then gluing and then gluing and then gluing. So you can actually get an infinity category of these sheaves. So I recommend anyone looking at like, you wanna learn like some of the best global mathematics and like learning it conceptually first is to look at the sheaf interpretation. And you're like, wait, was addition a sheaf the whole time? Yeah, talk about darker than mathematics. I'm just funny. Yeah, so operation, and so, you know, bless Peter for being so amazing in coming up with this idea. It's like, oh, not only can you put a sheaf on a topological space, you can put a sheaf on a category. That's the sort of interesting thing that I'm into. So take your idea of operating, of like adding numbers and just go to the category of numbers. And then if you start working categorically, that's already shared action. You're in this higher space. Yeah, so operations with diamonds on diamonds, showing something is a diamond. It's all just a little rigorous, but at the end, it gives you such an aestheticism because I think that, you know, all of this work should be pulling at everyone. You need the philosophers. Make sure we're asking the right question. You need the, you know, crazy math people like me to come up with like, well, I don't know. Let's try like an infinity category. And then you need, you know, you need active, like we need people actually, you know, working on how do you relate all this? So you need active inference. You need the philosophy and you need to put everyone together. Yeah, so in the dark room, my question is, it's like, you know, at what point was dark like light? You know, so I think also the darkness is like maybe a profilinite condition. The question is, if we sat in the dark room forever, would it eventually become light? You know, and then you bring in time. So dark room is really interesting and try to think of dark room in a temporal non-locality. That's how I would take that problem and kind of accelerate it just a little bit. Dark room in terms of a diamond, dark room in terms, change the time interpretation. Like in all these models, you know, time is always modeled as a total order. That's another thing I would like to hear from your community. Like, you know, total order has this like connexity property that every two objects in the total order are comparable. Why? Why do we model time as continuous? Why do you model time as a total order? I think that, I think that should be changed just a little bit and move it like non-archimedean. Quantum Mechanics is gonna work non-archimedean with a very specific cutoff, right? The plank length. And I think you should also be working with non-archimedean time. So the awesome person who asked the question about the dark room, please try to think about dark room like in some kind of temporal non-locality if time was everywhere. I'm really interested in that. I make one more comment here is quite early you mentioned how a stack, you said it takes in categories not for example sets. And connecting that to what you said about operations, this might be quite familiar to a computer scientist for whom the function can take in different things. So we know there's functions that take in numbers like two plus two, but then there might be a function that takes in a large data set that has a diamond description. And so computer science and engineering kind of the application, it's really easy sometimes to apply these recursive definitions recursion central to computer science, but you're right when you go to the infinite level that's kind of where the theory builds beyond any real implementation of a computer program. So we're on the bottom looking up at the computer science implementations of nested networks. And then here's the hand coming from the top down saying there's a way to infinitely structure that and throw a sheath on it or I don't know. Oh, see that's what I'm saying. Yeah, it's like to upgrade a machine learning it's another project I'm working on. It's like categorify it. You know what I mean? Like the specific definition of stack is a two sheath. So it's a sheath that takes values in categories. So it's like, yeah, if you move from like an output is a weighted function or something, if you move from that to the category of all possible weighted functions, then your complexity has just grown probably like three fold. You see the output no longer needs to be something singular. I outputted a weighted function, particular to this distribution. Why? Why don't you output an actual category? Yeah. Here's another great question from the chat from Steven. He wrote, if these category theory and geometry approaches enable another multi-scale formalism that treats time differently, can this approach play nicely with current multi-scale active inference formalisms? Sounds like he summarized our conversation. Steven should present. Oh my God. Yeah, can he tell me a little bit more about the multi-scale active emcee? You're right on it. If you stackify things, right? Think of it like this. Think that, you know, like Daniel was saying, if every, you know, if every like, think of every network being like a point in the modularized space, do you see? Then that whole thing is just a point. And then how do you actually get from point to point? So is that the sort of like multi-scale multi-class? I'm super into like multi-class classification, you know? I really like this question. Yes, switch it up to like pro-finite. If you categorify the thing, you're gonna get more connections, right? Then you're working in like categories of multi-class active inference, not just one. Why don't we split it up? Why don't you color it, call them incarnations and relate them if it's a powerful enough category, you can put a sheaf on it, put a sheaf on it and then start gluing them all together. And then I think you really have a multi-class shared inference. Does that make sense, Stephen? Wow, what a great question. Do you have any thoughts on that on the multi-class? I mean, I think that's what, you know, we're saying. One thought and the way that we've been talking about multi-scale active inference is like, let's say the brain is the internal state and the body is external to it. So then you have the blood-brain barrier. And then at the next level out, you have the brain and the body interacting with the social niche. And then you can go up and you can have societies or groups that are interacting with each other. And it's a debate in active inference. Does the group really have a blanket or what kind of blanket does it have? And here we're not gonna speak directly to that, but we can think of quote, American society as a point. It's a point and then it has internal structure. But by using that basically linguistic token by saying it is a point, it's literally one term that you can pass, then it abstracts over lower levels. And I'm not exactly sure on what is strong enough to have a sheath or not or what that enables, but it sounds like it's making these nested structures and laterally interacting structures, making them composable and kind of compilable so that we can tractably zoom in and zoom out or work with guarantees or proofs on these types of structures. Oh, I love that. Yeah, and so I encourage everyone to maybe as a challenge and Stephen will like this too. When we say like, okay, so thank you for explaining that for the multiclass, yeah, I agree. Super, like all these levels, yeah. To reinterpret that because it's always a boundary condition, right? Blood, brain, barrier, barrier between me and group, barrier between like collectivity and non-collectivity, things like this, right? What is a group, group action and things like this? So reinterpret that with a notion of a profinite boundary. I'm very curious as to what these boundaries are, right? They're not solid. They seem to be sort of like, are they hidden? So the very concept of a boundary is a hidden state is something that I was playing with. And I was like, you know what? At what point is the boundary there? Does that make sense? It's almost like, you know, boundary and hidden state is like this dark room phenomena that all of a sudden somebody turns on the light, boom, boundary is there, right? So when you have leaven and fields talking about like entanglement, like, you know, across boundaries and things like this, like, or maybe the only way to deco here is on the boundary, where is the information? And that's actually a really important question. But if you can reframe all of that, try to reframe the multiclass, like, you know, the multi-scale active inference in terms of a boundary that's like a fractal, that's very strange. You see what I mean, Daniel, right? Then it's like, then there's no clear defined boundary. I'm sure there's a lot of people listening, like anyone kind of super empath or if you have empath friends, I mean, I swear, like I can just be walking and like, you know, the tree goes into me or something like that. So I don't have very good boundaries like that, right? My mind thinks about everything and it swaps, you know, like I wish there was like a Jedi school, you know what I mean? I would totally be there like, how do I do this? And I think the active inference people like are the Jedi school. It's like, okay, let's figure this out. So if you do multi-scale any of this, play with that notion of a boundary, you know, just like you saw, I'm very, you know, math people can almost get kind of, you know, too much with it, but you want to be precise about every word and we're real careful when we define things like modularized space, like what a family of bundles means, what a family of isomorphism classes mean. So really think about what you're defining as boundary. And if that boundary was not solid, because they never are, if you have a profinite boundary, it just changes a little thing. It changes the way in which that scale is sort of hierarchicalized, I think. Yeah, do you have multiple levels of inference going on? Oh my God, yeah. If I can give a physical metaphor to what you said about these multi-scale. I'm getting excited, yes. So let's just say that you were looking at a textbook for anatomy and it would say what is the boundary of the lungs? What is the blanket around the lungs? Well, at that level, the gross anatomy level, it's the periplura, it's the membrane that just encloses it like one bag. And you can even have a point, just the lungs. But then if you really zoom in, it's actually truly a fractal with the alveoli and the way that actually the boundary of the lung, it probably looks like a piece of broccoli because are you looking at it from the outside with just a single sack? Or will you go into finer scale and see that the interface has a fractal dimension? Ooh, see, and that's what I'm saying, right? Like, you know, at what point was the room dark? Was it always dark and we just didn't notice it? That's what I'm talking about. And will the room always be dark? I don't think so, right? I think at some point of view when the dark room would change. And so just like what Daniel was saying, you zoom in on this stuff and you know there's more space between the like atoms if you actually want to use that approach, right? What is that? And that's what I'm really interested in. So you have all these different smears. It could be happening that like if all of these events actually happen and they are just temporally extended, what does that actually mean? It's almost like a time sheaf, you know what I mean? The interaction is like this, what, time spread? No idea, no idea. I think these interrogatories should direct all of the work. And that's why all my math is like super philosoph, like philosophally, I don't know. I did say elephants earlier now, I just said philosophally. So it's like, okay, I'm just getting excited. This is awesome. What is that? Yeah, one more on the philosophy point. Please. The dark room problem. So you just rephrase the dark room problem as being like, you know, was it room always dark? It's almost like, well, if there's no person in the room, what's the problem? Or it's thinking about the person's, the time that they're in that room. Because you're not appealing to an alternate universe where somebody has always been and always will be in a dark room. You're appealing to our everyday experience of being in a dark room. But that is only part of our regular behavioral sequences. So not exactly sure where that would take the dark room investigation, but it helps us see that it's part of a bigger set of experiences or of scenarios that the agent is experiencing. Yeah, and my question is like, if this boundary is like profinyte and fractal, what's to prevent us going from dark room and then maybe a white room, right? Dark room, white room, dark room, white room, dark room, white room, right? Or let me ask, if anyone knows what would be the actual antipodal reference of a dark room? Is there an actual opposite of a dark room? Would that be like, what would that be? If it were stochastic signal coming in to your eyes, so just like a white noise, then it would almost be like a dark room of a different kind, because sure there'd be light coming on your retina, but informationally, you're still not getting anything. So at the higher level, it's a similar problem, but it uncouples it from the photons hitting your retina, which actually doesn't matter. Right, yeah, and I think there was something, I was reading, I forgot what it was about, like synesthetes and how synesthetes store information, because I have sort of half of that, but I was like, maybe I should find out a little about myself. Anyway, there was some wild study that was like, if you put pure white light on the retina, I would love for your community to investigate this, like you actually can't see. It actually negates object vision. Pure white light, so unspectured on the retina, will actually negate object vision. Now, again, I saw this, it was in a philosophy work, which I trust, again, I'm not like a super scientist about that, but that's really interesting, right? So just like Daniel said, in a pure white room, you'd also be in the dark. So that's interesting. And so you see, you need this like messy. So pure white light, you know, would hurt us. You cannot see object vision. So it makes me think as object vision itself profinite, right? If it seems to be emergent, if it seems to be falling out, if you cannot see the actual, when pure white light, if it's a non-spectra is interacting with our system, we cannot take it. What are the actual implications of that? And can you actually scale that, right? So that's why I was saying with the dark, and with emergence, like at what point it is emergent, at what point of like, you know, photonic light hitting your eyes must it be non-white, like pure, you know? So that's actually interesting that the actual purest conditions of something may negate the organism's object functionality, right? And then we're actually hurting. So at that point, you may have to rely on the higher tumor fisms, and then you go join the Shana Daniel Jedi School. I'm just like, all right. So maybe I should talk just a little bit more about the time stuff, and then leave with like another fun drawing. This is an awesome conversation. Okay, so again, we all know that time is a problem. Why is it emergent? Why am I talking about emergent time? When we can go so far with these ideas, it's good to bring in an example, like Daniel was saying about, you know, alveoli and things like that. The actual, I take care of a lot of elderly family members. And so my earliest memories are hospitals and that might be why I kind of grew up so quick, just with all these like heavy questions. And I remember, you know, we were having a lot of problems with like the actual oxygen CO2 exchange, right? That's actually really tricky. So it's not like if someone, you know, is suffering, you know, in the like the pulmonary sense, you can't just, you know, and put an oxygen mask on and it's fine. So the actual body's way of absorbing that, I think is very, very, very tricky. The CO2, you know, exchanging with the O2 in this fractal web of broccoli, you know? So what is that? And that's happening 24 seven as I like, diggle and give this talk. It's like, my alveoli are excited. I think I just said alveoli, like ravioli. My ravioli are excited, right? I don't know what I'm saying. All right, so, but it's nice to, when you have all this high math to try to rein it in with something biological. So people like, you know, why are you talking about biology? And like, cause I wanna try to find something here that we can use to sort of justify temporal non locality besides entangled photon pairs. Everybody uses entangled photon pairs. Why don't we use like Daniel was saying, like the actual system? Why don't we use the organism to test on? So again, if there's actual entanglement going on in the daughter cells, I am excited about that and wanna think, wanna ask the community what we think about entanglement, which is again gonna mess with this idea of time, right? Time and boundaries. So we know that there's a problem of time. You're like, you know, your heavy weights in quantum mechanics have the Wheeler-DeWitt equations as the universe is static to an external observer. I'm gonna have some questions with like the external versus the internal observer. I don't know what that is because again, I think all boundaries are profinite. So it's a little tricky. Quantum mechanics is, you know, I'm gonna say that time is universal and absolute. General relativity is gonna say that it's actually malleable and relative, right? And then, you know, people like me, the category theory of people, maybe time is like a morphism, you know? So I always, you know, I ask back to the dark room question and things like this, even your local time, was it like, what is time? Was it always here? Where is not our comedian time? So if we're thinking about like hidden states, right? And any kind of embodied cognition, it's very interesting how your memory, it's not very solid, maybe people's earliest memory is like when they're six. My one friend, he's like, I don't know, like 12. I'm like, you don't remember anything until you're 12? Like, what is that? So when does your little system actually come online? Is this once again a play between irreversibility and reconstructibility? Is it also a play between your multi-scale own active inference? Have you stopped, is it possible that you were only in your own reflections until age seven? And then you were able to actually reach your reflections, right? Is this like people's earliest memory? So if you're thinking about like I am, that interaction and perception and all these things that are feedback loops with your own system, right? Anything that's feeding back into the actual kind of non-local organism for cognition, it's reflection based. So, but if you were only in reflection land, maybe your memory would never come online. So where is it? Like, where is time? Was it always here? Where's yesterday? Like, I just don't know. So we say these like, we have these existential qualifiers like there exists, right? But what are the conditions for all of this? We're here for such a short time and there's all these big questions, right? I wanna know what an emergent time asymmetry is. It's very interesting when I ask, why do we seem to go forward or something? You're almost, you're dealt with another answer that's super fuzzy and only defined in the negative, right? There seems to be like a time asymmetry. And like, did you just use time to define time? Like, isn't that, we can't do that, right? I mean, I guess we can, but what would actually be an emergent time asymmetry? At what point, if you think about that pro construction I was talking about, how do you actually get a time asymmetry in that crazy environment, right? Or just like, you know, the beautiful darkroom example, or even Stephen's example, the question, right? How do you actually get asymmetries on this emergent scale? Go even weirder with what I said. How do you get asymmetries in this doubly emergent scale? Can you actually get an asymmetry in just one of the morphisms? I don't know. These questions are once again, huge. How would you actually have the notion of an entropy, right? In an embodied cognition where you're, you know, exchanging with the environment, if time was discreet, this is what I wanna think about. So just like Daniel gave us the great example, on the outside, we're all so beautiful. We wear these like awesome clothes and we have these, you know, beautiful features. And if you actually zoom in like to the actual lung sacs, right? Very tricky fractal stuff going on there. Why don't we look at time the same way? Out here, you have this, you know, like, supposed continuum, which I don't really like believe in. What if you zoomed in? How would you actually figure, if time was actually discreet, like I said, of points like that modular space, is there a dark room in between each time? And that's why I'm really interested in this dark room. Who's to say that you're not blinking in and out of it, like every plank second, you know? That's what I'm really interested in, is that, you know, your eye seems to be doing an interpolation anyway, right? They've done those studies about like eye tracking. And when you're actually looking at someone, it's just like eyes and mouth or something like this. You don't really look at the ear. So it's actually, you're putting together an image of me based on your tracking, yeah. And so you're actually doing an interpolation, like 24 seven, even go down to like the cesium atom, like time. So are you not doing the same thing for time itself? That's where I'm really, that's what I'm interested in. Take the time to zoom in. If time is also this boundary, this fractal curve, you know, my awesome mentor, beautiful, amazing wizard mentor, Dr. Lapidus, Michelle Lapidus, you know, came up with like, you know, fractal curves and fractal co-emology and all this stuff. That's really interesting, right? Fractal's data functions. What if time is something like that? So what is structural causality? We've got Bertrand Russell and these great wizards, right? Wittgenstein, language is a, you know, a game and, you know, Bertrand Russell saying, you know, there's no, there's no structural relation between now and actually assuming that the world was created, you know, five minutes ago. So there's last Thursday-ism and then you get these fun things. All right. Once again, what is object persistence? How do you actually do that? Who's to say that the dark room is not flickering in and out like every, you know, second. And once again, the stop mechanism really interested in what your community thinks about that, right? So again, in something like, you know, horribly sad, like the proliferation of cancer cells or something. Why, how does that act so locally, right? You could just have that in a particular area and it starts like spreading through like, you know, metastasis and stuff like that. But it originates locally. It doesn't originate as a systemic thing. And that's what I'm very interested in. Seems to be these pockets, right? Pockets of the stop mechanism failing can destroy the whole organism, which is back to that torsion thing I was talking about. You can have these few elements that annihilate, you know, these torsion elements that are annihilated, right? At the expense of the actual organism. So if we actually say, how does causality actually change per your turing degree? Remember, I sort of think that we can, you know, in your multi-scale active inference that it would be nice to classify organisms maybe based on their problem solving capability. So I really think there are no ontic boundaries. It seems to be these profinite boundaries. Then you're going to ask ontic boundaries of what? Ontic boundaries of the organism, which you zoom in and find that it's like lung sac, like fractal broccoli, you know? And then, you know, just to keep going, why is space time only a manifold? I showed you these other structures. What if we reinterpreted any of that GR stuff as like space time being a diamond, you'd be allowing the multiplicity, which is the power of the organism, right? You are, it is not top down. You are learning from everything, right? So what would actually like a number theoretic time look like? So let's go just a little bit deeper into emergence, right? Assuming it's emergent, what is an emergent from? What is the actual eschatology of time? That's what I'm really interested in. Like, is it going to be here? What if it just left? You know, what if time just walked off and left, okay? I'm really interested in planarium. So Chris Fields has taught me a lot about the planarium. So these are the little worms that if you actually cut off a piece of that, again, I'm not into like animal cruelty, but if somehow a piece of them gets like removed, that little piece can regenerate the whole organism. And so Daniel was going to ask like, do you all have any thoughts about that? Cause it seems to be that the nervous system is acting non-locally and it seems like, or the planarium like, can they, is that like unemergence? Like, what is that? You know, so how can you actually have a piece reproduce the whole organism when we can't do that? So is there any thoughts about that? Like we lose a finger and it's very sad. We have a stop mechanism that actually prevents that. So I was gonna just ask you all, since you're also complexity theorist, right? Do you also think that because our organism is more complex, we have a stronger mark of irreversibility, stronger cutoff of reconstructibility. My, like if we all lost an arm, our arm is not reconstructible given our system, but it is for a planaria. So is there a play between complexity theory, hidden, do we have more hidden states and then the actual planaria? Just one of Joel's thoughts on that. Awesome question. I'm gonna address it with an empirical observation rather than an appeal to theory. And just as you said, if we lose part of our body, we can't grow back, but maybe, you know, future technology could change that. But if you go to an ant colony and you scoop up the ants from a certain location, a few minutes later, other nest mates will have functionally replaced the nest mates that you had taken because they're all diffusing around in the space. And so it's almost like the ant colony is more resilient to perturbation, including extreme perturbations. However, it has less morphological complexity in the interactions than, for example, a synapse. So it's like, if you wanna have the library with a lot of books, it's gonna take a lot of space. If you wanna have a really precisely specified topology of interactions like neurons do, you're gonna have limited, not zero, but limited flexibility relative to a sort of distributed, totally fluidly mixing system like an ant colony. So there's definitely something there with a trade off on the complexity frontier of how much resilience do you have and ability to regenerate or maybe even to perpetuate indefinitely, like a sponge or a coral or something like that versus irreversibly moved towards order, which comes with the cost of building tall buildings that can fall down. Right, right. And would we rather have like our sugar cubes as an ant colony or the tall building? I don't know. Right, and my weird theorem would be able to do both in awareness, we could be ants, you know, just play. Oh, that's very remarkable. So that example is just as profound as these questions that I'm asking, right. There seems to be so many things happening at once and thinking about it in terms of perturbation, which again, I linked perturbation to the diamonds as well, right. So you can still do that. And again, so that's an excellent question and with like a more excellent answer, right. Like how do you actually talk about emergence and also emergence and perturbation, right. Those go hand in hand, but also emergence and irreversibility and just like you said, a complexity. So thinking about complexity in terms of emergence and also, I like how you said resistance, but with the advent of stuff like epigenetics and things like that, I'm wondering, I don't know if you put planaria in a very horrible environment if they still continue. Seems like ants kind of continue, right. They're like little fighters. And again, who's to say, I don't want to be speciest. Who's to say that we are more complex than the ants because they have a collective phenomena which is like super profound, right, yeah. And so it's like, are we building tall buildings to the extent of like not really connecting with each other, right. We need to think about stuff like that. Okay, so that's the question. Yes, please. Okay, okay, just CB has returned with some funny and interesting questions. So there's two questions. There's a tiny one and then a big one. The tiny one, no pun intended, is can you elaborate on the idea of the infinitesimal thickening? So what does that mean? What is the infinitesimal thickening? Yeah, sure, say you had a space, right. I have a space or I have a pin. This is my pin. So we would actually say that the pin is carved out and it has a specific boundary, right. Outside of the pin, we would say is like a measure zero. There's nothing else outside of the pin. The pin stops and then you have like space, no pin. Do we all agree? There's a very strict boundary line between pin and no pin. So in infinitesimals, these are, these are, they're like collections of infinitely small pieces. They're not identically zero. They are, they still, they exist, but they're almost zero, but they're not quite zero. So surrounding the pin, we could say is an infinitesimal thickness, which is not all the way to zero, but there is like a little bit of wiggle. Does that make sense? So that's what we're saying that there is a wiggle, a space around spaces. So just like I was saying, it gets, you know, it gets rid of notions like singular points. We know there's, there was a lot of problems and, you know, part of the standard model is great, but in particle physics, there was problems with having points interact in these Feynman diagrams, remember everyone? And so you actually made the point a one, a one dimensional object called a string to avoid the complications of just having a point. So in math, you can do the same way instead of having a singular condition like this is my now, there's problems with that. What if you actually had a now that had a little infinitesimal? So surrounded by not almost zero. So think about the smallest numbers possible, but they go infinitely, right? So this is almost like, if we know the, you know, the Archimedean property that you can always make a bigger number, you can always make a smaller number, always. So imagine that negative Archimedean all the way down. And so you surround a space with those and then you don't have to sing another space anymore. Thank you for the answer. And the second question from Cambridge Braths was, also, is it accurate to say that every experienced now can reflect innumerable other nows? If so, is time travel possible? Oh, see? Oh, I love your community, you know? Well, you know, I'm gonna say yes, because I'm like the magic person here. Yeah. So again, what is a now? It's very interesting. You can have all the elaborate physics on the planet and then you ask, what is an observer and the room goes quiet, right? So when we say what is a now, that's what I'm saying. Like a now according to whom? We're all connected right now. I shouldn't say now. But at the same time, you're peripherally aware of everything going outside, like my animals, your next appointment, your digestive system, your autonomic nervous system. So when we say now, it's like, now according to whom are you neglecting your own hidden states? And that's why I'm very curious as to what we mean by now. Also, I'm very, you know, I'm very adamant about saying, you know, there is no synchronous reference anyone who thinks this now is actually happening now. You've neglected the neuronal processes that are actually happening. There's neurotransmitter signals, you know, everything happening on neurological time to actually get you to have this now. So you're always in the past, does that make sense? Right? You're always in the past, but you're also not in the past. So you can get very like Alan Watts and Eckhart Tolle here and say, you really only have now, it is a hallucination to think that there is like a tomorrow. Because even when you actually remember the past, you're doing it now, we all agree, you're always in this slice, right? But if you, there's a little bit more, do you see that if you do a formal spectra with the now, if you put a little bit of thickness on it, it allows for you to actually have a little bit of more room in the now, which is just like what, you know, the question is asking, can you, you know, the now actually contains other nows, yeah? Because if the whole, if time is this pro construction, which is what I'm trying to say, then you were linking together all kinds of things, right? Your thoughts happening now, where are they coming from? I don't know. And so is time travel possible? Oh yeah, you bet. That's what I said. That's why I was saying, look at me, I'm all excited. You know, if you can actually partition the brain, okay, so let's get weird. I mean, not too weird. Don't do this to yourself, like right now, hold on. If you were actually to make the brain so that it was almost like a split screen, this is a very rough analogy. If you remember this, like, I don't know, older films that had like one panel going and the other one, I loved those just because, you know, I think like synesthete hyperactive, I need to be like doing a lot, like multitask. So imagine you had that kind of split screen, but in your own awareness, in the awareness, not the now, but if we were actually able to split the system, then you would actually be able to quote time travel. If time is non-local, and if it was always there, all you really have to do is try to get to your like, your own V-stack. And I think people are gonna be like, what is this girl talking about? But it's like, if you could actually get to where you had those higher morphisms, then that would be very, very exciting. And then time travel should be possible. When people time travel, it's like they normally, they normally like, then the object themselves doesn't travel, and they seem to think that, you know, perturbation doesn't exist. So it's like, if you're gonna time travel, you're gonna be perturbing time, you're gonna be ready for time to perturb you back. I'm not saying it will be like a very gentle experience, but I am completely all for it. There seems to be something in the system that keeps us on one channel. And I, you know, just to ask the community, those are great questions, and even ask the question, you know, if the questioner wants to ask again, what is their own thoughts about, why do you seem to be on this one channel? Everyone will say, well, because there's a time, there's a time asymmetry, and the past is irreversible. Well, we've seen how shady irreversibility is, right? It's a boundary phenomenon. And so you almost have to be like, somebody called me Ms. Frizzle once, like the magic school bus lady. I was like, okay, this is great. She's got like red hair and she's driving this school bus going into these things. I would totally take a school bus and go into the alveoli. Like, what is this place? If you were to actually drive the school bus into the actual boundary, you're gonna see it's not there. I mean, it is, because you can hear me like hitting the table. It is on this channel, but if we were actually like able to access the multi-channels, you know, I'm not sure. Is time travel possible? Yeah, you're gonna have to partition awareness. Not partition the brain, because you would be very sick and I don't wanna lose you. You cannot, you know, but try to think of why this whole interface and why embodied cognition is here for one channel. Like, can we actually have a multi, and that's what I was saying. I actually have like a free embodied cognition, you know. What do you think about time travel, Dan, or any of that? Sorry if that was too wild, everyone, but I'm getting excited, you know? Good questions. I have heard a few different ways it could happen. Yeah. And I would love to see evidence it could. Yeah, evidence it could. And that's why, you know, you can look to, so, you know, as free as I am in the math space and I'm open and I want to travel, because I want, you know, humans to like succeed and, you know, yeah, I want organic life to continue. Very worried about that. So whatever we can do, let's try to keep it going. And then, you know, something to help the questioner also is like, this great work, it'll be coming out soon by Levin and Fields about, you know, decoherence is actually localized to inter-compartmental boundaries, right? Where is it not? Okay, so if the internal bulk states of daughter cells may remain entangled for macroscopic times following the cell division. So you have little pockets of time travel, right? If you have entanglement. And so there is like hope if there is entanglement in the biological setting. Again, most people may think their work is like, you know, to future thinking or whatever, but I'm super future thinking, like let's try to figure this out, you know? Lots of ramifications, you know, this theory of extended minus philosophy theory I thought was actually like really great, Andy Clark, right? You know, if you're operating through time as these series of dots, you're gonna have something called a singular support. You've got, I'm really curious how you remember a memory, right? So again, if the whole point of shared action is to get these morphisms between morphisms, you know, when the questioner asked, is it possible that your now is a, you know, like a collection of other nows? Well, absolutely. You're probably operating in a nested space, right? Of like a perturbative, profinite memory of a memory, right? So you start having these meta memories, memory of memory. And it's very interesting. You're like, oh, I don't know what a cat looks like. And then I see it again, that is my cat. It's very interesting how you link that. So something as simple as object persistence, right? Is actually everywhere. Time is not a total order, right? And how do you actually relate all this stuff? Great. I should probably go in about like five minutes because I have a class, but I wanted to actually show, oh, I have unicorns everywhere, right? So there's magic. I'm like, how would you, so the, another main takeaway for the math, takeaway the diamonds and the sheaves and all this stuff is also the notion of a reciprocity law. These are really beautiful. Reciprocity laws connect seemingly different branches of mathematics. So like in the Langlands program, connecting these certain Galois representations of just one branch of math with automorphic forms, which is like, you know, harmonic analysis. Like that is beauty, right? That's actually kind of synesthetic at a certain point. And I think since I am kind of like synesthetic, but I align with these sort of grand unified theories, I'm like, oh, that feels like home. Like that feels right. To be siloed, like doesn't feel right, still be individual, but also be able to connect to these like reciprocity laws. So how would you actually have a reciprocity law of emergent time? That actually means you're able to link emergent time with Newtonian time, string theory time, Alveoli time, the time of CO2 and O2 exchange, right? How do you actually, how do you actually relate all of those times? Quantum time, everybody's time, no time, dark room time, the time before existence, that kind of like long like Christ consciousness time, all the times, right? That's huge. And that's why I said, well, I don't know, I put a big unicorn on there because it's gonna be some magic. So it'll take some magic to actually get a law of time, but you've got great heavy weights trying to work on that, you know, like emergent theories of time or like tricky and they're hard, you know? So if you have this like new reciprocity law, if you actually could connect the diamonds and like, you know, I'm really, everyone knows, I'm really interested in like the mirror framework. If mirror phenomenon are actually how you're interacting. So when people say, well, what is the now? Is it just your mirror system imitation? And then Daniel, I wanted to ask your community and you, what is your interpretation of like, if you believe in mirror neurons or the actual mirror system that learns by imitation? Cause that's very, very powerful, right? We were talking about stuff like that. So just wanted your thoughts on the mirror framework and the mirror framework in terms of like, these hidden states as well. It's a great question. I'd be curious to know what different hypotheses the mirror neuron helps us get at, what different observations in the world we could actually test. For example, does somebody who's not looking but they're feeling someone else move, maybe does that? I'm just curious about it. Don't have the answers. Although in general, I'd say adding an adjective, like, oh, it's the Jennifer Aniston neuron. It's the mirror neuron. It's like, okay, but that's a reductionist take because it's labeling the neuron. So if it turns out that we have a dynamic distributed function in our brain that does have action models of ourself as well as action models of others. And maybe there are some firing patterns that are overlapping, but there's probably some action patterns only if you're watching someone else and only if you're moving every other combination. So I just think it's one of the facets of the brain diamonds and it's not the whole picture. I like that. Yeah, see, and once again, right? Even like, even having this very strange like little piece that's this mirror thing. I just know I was watching someone, you know, like another heavy weight, get this talk about this and the sort of the phantom limb syndrome and stuff like that, right? If your limb is lost and someone else is maybe like, touching the same limb that you lost, do you sort of feel that? And there's like, and these people aren't even kind of empath like, like me, you know, they're not like me, come up with stuff like this. They're like, less hyperbolic, I would say. So it was interesting than the neurologist. I was actually happy to hear him kind of go, do a shared action kind of thing. Said, you know, it seems that the only thing that separates us is the skin because if I can feel what you can feel, you can feel what I can feel. Everything is imitation. Then it seems to be like the skin boundary. And then again, boundaries are profinyte not ontic, right? So you know that had me perplexed. So I appreciate your answer there. Yeah. Okay, I think I've already talked about most of this, the topological localization. So when I said, you know, you can put a, you know, a topology, I said you could put a sheaf on a category. Well, everyone should also think how you actually could put a topology on an infinite one category. So I'm working on something like that. That's actually really tricky. You have to figure out what the open sets are in an infinite category with infinite morphisms, you know? So this sort of condensed framework that I was talking about, I'll wrap up with this. So you have this notion. I'd said the condensed framework was a sheaf over a point. So if we talk about a site, so remember I told you the site is a place where you can actually put a topology. So we called Peter and Dustin called this the pro-itel site of a point. Is the category of profinyte sets S and with finite jointly surjective families of maps as covers a condensed set. It's a sheaf of sets on this site. So on this site, right? So it's a sheaf of sets over a point, but on the point is the category of profinyte sets. So you already have that kind of multi-scale in this sort of point. So you do have sheaves over a point, but the point has the site of profinyte sets. So you can do this to condensed ring. This is all about trying to do the same algebra and mathematics you've always done, but when your objects actually carry a topology, that's what's interesting, all of the objects. So reframe everything, like the six tuple and active inference, put a topology on it. Have all of your algebraic things, modules, groups, rings, carry a topology, how do you do stuff with that? Okay, so that's fine. That's fine. The way that you can associate this, you take T to be any topological space. To T, there was an associated condensed set, T underline. You saw the underline in the spot of QP, the spot QP, subatomic mod Galois action with the underline that I showed you earlier. This is defined by sending any profinyte S to the set of continuous maps from S to T. So that's what you're trying, that's how you can associate to any topological space a condensed set. So how beautiful is that? So you can reframe if you have stuff in terms of topology, we can bulldoze it, I shouldn't say bulldoze, that's a little extreme, reframe it in terms of these condensed if you want to get this very interesting sort of a sheaf over a point. And then you can do like the weird thing that I was saying, put a formal spectrum around it to make it a little like, a little thicker. Okay, so to actually consider this emergent time, what I said, let's consider the simplest case, an event, which is like a truncated perturbation, something like that. Topological localization of any particular reference frame. So that means I'm in the now, I am staring at my reflections, whatever that is. Let's take an event to be a point in the diamond topological space, T. We're gonna actually convert this to a condensed set. On that point is the pro etel site, the category for finite sets. So here we go. This global time maybe emerges as the set of continuous maps from all profinyte sets S to T because I said the individual time is diamond, which is profinyte. So if time, if global time is constructed as the sheaf of sets on this site as a condensed set, then emergent time is gonna result in passing to the larger category of sheaves. So we just keep talking about sheaves on categories. Let's talk about like the larger cat, let's just frame it all in the category of sheaves. And you can consider a condensed version of this continuous K theory. This is so hard and it'll take me a bit to finish it, but as soon as I make a little progress, I get excited. A second way to get emergent time is to take those like if condensed time as local time, let's take equivalence relations of those. Take it your multi-scale active inference again, and you can actually take diamond equivalence relations to get the D stacks and I already went over that. So I came up with this like dictionary between ordinary quantum mechanics and perfectoid quantum mechanics, upgrade the Hilbert space, which is a sort of like complete inner product space. Let's make it a perfectoid space, right? The sort of strange fractal space. Replace, you know, state vectors with the geometric points, the tensor product with the diamond product, the non-locality phenomenon, maybe it's the profoundly many copies of spa C. If you have that many reflections, you don't know which one is the reference point. You can, the way function collapses this tilting, the coupling behavior, right? Holographic principle, you know, quantum topology, replace it with this etel, comology of diamonds. Take your operator algebra, replace it with a non-notherian, complete evaluation ring. Unitarity, pro etel descent data, more of that diamond descent. That's great. Okay. So I will end there and say thank you. So, you know, I mentioned everyone working together and you see there's draft, elephant, baby unicorn, unicorns, everyone is working together here and there's some like zero sharp on the board and the spot of QP diamond. I should stop there. Yeah. And take other questions. Yeah. We'll let you head off to your class in just a minute, but this was very thought provoking and interesting. So I hope that people will ask you questions in the meanwhile and then I'm sure there'll be more math streams in our future. That's wonderful. And then does everyone have my email? I guess they do on the, I mean, I'm findable. Yeah. Anyone can always get in touch with Act in Flab to contact a guest or if you send us a link that you want us to put in the video description, then we'll add that there. But I think we're all going to go home, go back, hit the books and I really enjoyed doing it live. I used to, this was so fun. And I think like maybe the next talk would be like, it'd be fun to do these thought experiments, right? Like what if, what if we could actually pose the question of dark room and how we use active inference and this math to get out of it, you know? I'd suggest a event format where we have multiple different backgrounds and perspectives and then we give people the prompts and the thought questions and then we do have a spontaneous conversation about how different perspectives lead to different views on the thought questions. Oh, that sounds so great. Okay. Yes. Well, thank you. So whoever wants to join, whoever wants to join, you're always welcome. So thank you again and we'll see you later. Thank you so much.