Well, I have to say that this is one of those special moments at UNESCO because it's not always that we go into a space that is particularly illustrative of perhaps giving us ideas, thoughts, reflections on how we deal with our everyday lives, on how we potentially find solutions to some of the most troubling problems and how potentially we also come together to look at the bigger problems that confront us, those challenges, and how we do them smartly, how we do them together. It is my pleasure today as we go live from UNESCO on our Facebook Live to introduce to you someone who you probably already know, someone who you've probably already seen on Facebook, and that is Sadguru. Thank you very much. Good morning to you as we sit in front of the beautiful artwork of Guam Niro here at UNESCO Headquarters in Paris, and we're going to have a conversation, I hope. We've already been talking amongst ourselves in ways that give me a really positive sense of the path that we're going to take over the course of the next few minutes. I happen to have the great, I would say, opportunity and chance to take advantage of watching you yesterday here at UNESCO for the conversation and discussion that you had, which was indeed a way of being both, I guess, unpacking life in a way and then at times very specifically providing a bit of a guide path. And you were doing this, of course, on Facebook Live. You're obviously you and your team see the wonders as we do. And so as we bring messages to people, and we do this every day sometimes from UNESCO or at least frequently through Facebook Live, I wanted to ask you, I'm talking to someone who has been ranked among the 50 most influential people in India by India today.  A few things people say. I didn't see that.  And of course, you also founded the Isha Foundation. And your work, your life is dedicated in a way. That is something that... That's a good change because I have no distinction between work and life. There are days when I don't either. It seems to be it. But I'm interested because, and you even discussed this yesterday at different points during the conversation, even when you were reflecting in your thoughts and in what you were saying about being here at UNESCO, an organization that was founded essentially from the ashes of World War II with a commitment to never again shall we see the world at war. And we struggle to fulfill this ideal and this mandate. And I'm curious in terms of the work that you do, the... what you represent, can we... can we look to the traditions of yoga to find a way for us to address the challenges that face us? I would say absolutely because the way yoga has come to United States during New York, well, it's... too large number of people has just come as a fad. It's about the Lululemon pants and I do yoga is kind of of whatever. It's a cool thing to do. But that's not what yoga is. The word yoga means union. Union means, see right now in individual experience, it's like this. There is you and there is the world. When life gets little acidic for whatever reasons, then it's you versus the universe. You versus the universe is a silly competition to get into, but almost every human being is in that state. Every human being is in that state, it is you versus the universe. So yoga means consciously you obliterate the boundaries of your individuality. Not just intellectually, not in thought and emotion, but actually in our experience. So right now your individual experience is a little concocted experience because of our strong identification with our own bodies and our thought structures, our physiological and psychological structures. Because of this you are an individual, but actually right now as we sit here and breathe what these plants are exhaling, we are inhaling. But it's not in human experience.  None of those thou shall nots will be needed for you. So essentially yoga means this, that you consciously obliterate the boundaries of your individuality. So your experience of life is just as large as the universe or as large as life is. So when this experience touches a human being even for one moment, even for one moment, if you experience all these people as you're experiencing the ten fingers of your hand, after that nobody needs to tell you to be peaceful, to be loving, to take care of them, not to harm them. This is simply not necessary for you because you have experienced what is around you as a part of yourself. Unfortunately, that experience seems to come to most people only when you bury them in the end that they are part of everything. Ashes to ashes, dust to dust. What when you were just describing this relationship that we have essentially with the universe, the universe begins in a way… It's not a relationship. It's a oneness. Relationship means you need to. It is a oneness. It is.  It's like a computer pop-up. You pop up for two seconds and you pop out. So then yoga is in many ways a reflection of also basic science because when you were talking about what the plants give off and what we inhale… I would say what you call as modern science is a small piece of what we call as yoga. That may sound because now modern science is big in the world and yoga is a small science fashion that you do, people may think that is a little arrogant. But I am telling you what yoga means is it doesn't exclude anything. In that what you call as modern science is study of physical material out of which we are learning how to reap many benefits in terms of technology. But even that knowledge which is so fantastic has turned against us simply because of the individuality that we have because our experience is not yoga yet. Our experience is not in union because of that even the knowledge that we have, scientific knowledge we have has turned against us. In a way science is causing more damage to the planet than anything else unfortunately, isn't it? Driven by our individualism. Because it is… see we must understand when you give an instrument, however wonderful that instrument is, it is the hand that handles it which decides which it is going to be destructive or constructive. And that hand is individualism. We have not fixed that hand. Yes. We have not fixed that hand to be inclusive. It's a very exclusive hand. We struggle here every day as I mentioned as we opened up our conversation and for those of our community that is just joining us in this Facebook live and those who are part of the Facebook family of Sadguru, we're coming to you live from UNESCO headquarters in Paris as part of a reflection, part of also an adventure perhaps that we will take in this conversation and an exploration. Be part of the conversation to whatever extent I'll try to keep an eye on… Why George, I'm becoming so difficult you're calling me an adventure. An adventure, an adventure. And adventures are worth exploring and so that is why we are together. But again at UNESCO is our Twitter and you can of course post your comments as we continue this conversation. Individualism is both… is it both a… is it more of a threat than an asset? Individualism means it's become like a religion of its own. That is very dangerous. It's a privilege that nature has given you an individual sense of experience though you're just a small pop-up for two seconds on this planet. A countless number of people have come and gone before you and me. Oh, where are they? Not a sign, they're all topsoil here. So this will also become topsoil but in the meantime I'm an individual which is a great privilege that I can experience life as an individual. I have my own experience of life, it's such a fantastic privilege. But this privilege we've turned against ourselves, we are using our individuality like a prison, a shell that we cannot break. We are in a shell, we are in an egg shell that we never hatch. If you hatch we call yoga. If the word flies then it's yoga, otherwise it's just a shell. So where do we begin to focus through yoga on addressing these issues, addressing these challenges? We live, it seems, in a world that is moving in a cycle that the human race goes through a cycle and some people have even posited that some of the extreme views that we see today, the lack of an appreciation for diversity, a lack of tolerance, a failure to see ourselves in the other is not unlike what we may have seen in the run-up to World War II. And so we have not only history teaching us lessons but potentially other resources and other ways of perceiving our place here in this world through yoga, for example. How do we change a course? How do we prevent ourselves from repeating our mistakes? You may not repeat those kind of mistakes like World War I and II because the soldiers on both sides may be Facebook friends. You never know. It depends on the depth of the universe. So, see essentially we are always trying to do repair jobs. We are not willing to attend to the fundamentals. When you say, how do we attend to this? Someone asked me, a very prominent person in Europe once asked me, Sadhguru, if there's one thing we can do for you which will transform this world, what is it? I just named, I said, I want twenty-five people for five days. You will see significant change will happen in the next two to three years time. He asked, who are these twenty-five people? I named the twenty-five heads of state of major nations on the planet. You give them to me for five days. There's the love of us coming back. Please continue. So, if these people who are holding positions of responsibility and power in this world, if their way of thinking and feeling and experiencing life changes even by ten to twenty-five percent within themselves, not hundred percent, if ten percent change happens, world will be full of solutions, not problems. For example, I'm talking about a very basic thing which is driving millions of people, they're estimating in the next ten years, some billion and odd people will be migrating. Some kind of estimate like that is there. Nourishment. Nearly twenty-seven to twenty-eight percent of the world's population is undernourished. But in twenty-fourteen, it seems we produced over eighteen point six billion tons of food. I'm sorry, we produced enough food for eighteen point six billion people. But our population is only a third of that. But still, nearly thirty percent of the people don't get to eat. This could be sorted out in a few months time if all the world leaders felt a deep sense of empathy and compassion within themselves. Empathy and compassion not as values that you practice simply because you experience another life as yourself. This is yoga. So if we can bring the experience of yoga, not the practice of yoga, the experience of yoga to responsible people, people who hold humongous amount of power and possibility. See for the first time, we have the necessary resource, capability and technology to address every human problem on the planet. This was never possible. You're talking about World War II, at that time it was not possible. Even twenty-five years ago we were not really equipped to address everything. This is the first generation of people that we have the ability to address every issue. Only thing is, there is no inclusive sense of consciousness among us. Still we are thinking in individuality or community or nationhood. We are not thinking, experiencing life beyond that is not happening. Thought cannot change without experience. Where, wherever your experience is, what is the level of your experience? That's what thought is. With a little bit of education, with interaction, you may extend it one step, but it doesn't go really beyond. When your experience changes of the world, if, if really as I was saying earlier, if you experience that your breath is connected to this tree right now, you will not need any ecological moments on the planet. Every human being will do what they have to do. At least if this happens to the leadership in the world, they will do what they have to do. Because the type of resources they have today is such that one nation can solve the entire world's problem of all kinds of problems that we have actually. Of nourishment, health, education, you can solve this. Just one or two nations can solve the world's problems, but we have still not experienced other life as my life. Why then are we at a time when we see ocean temperatures rising, which will have a significant impact on, on the environment? We see, and climate. We are witnessing climate change in what I would call humongous storms that have never been seen before. The, the planet is speaking to us. We don't seem to be listening. And we are, and I completely agree with you. I, I think we are one, one unit. And yet the message isn't, isn't The dead got the point that we are one unit. The living are yet to get the point. And so you mentioned the 25, but what about the six billion? The 25, are they really the, the lynchpin to change? Or is it the empowerment of, of the individual, in a sense, to understand and then to put that forward as a means of change? See, today, largely the world being a democratic process, these 25 represent those billions. These 25 are empowered by those billions. So they are not somebody who dropped from somewhere. They are people who have been pushed up by the other people. So these 25 are good enough because they can change a whole lot of things. The policy on the planet can change in a more human way. When we talk about ocean temperatures rising and many other ecological, you know, ingredients are working in a certain direction, this can be seen this way. See right now, everything that human beings have done on this planet is in pursuit of human well being. It's also written into the American constitution. Pursuit of happiness. Pursuit of happiness is a wrong direction to take because human happiness or misery both happens only from within you. If you think by getting this or that or ultimately conquering the planet or the universe, you're going to be happy, you're essentially destructive in your approach to life itself. Now, only when a small number of people were having such aspirations, it looked like normal. Now, six, seven billion people have those aspirations and it's just destructive. What destruction you see is just pursuit of human happiness, isn't it? Empowered by technology and science, of course. Pursuit of the individual's happiness, not necessarily human happiness. Oh, they're human beings, right? Yes, indeed. But I think when we speak about about pursuit of human happiness, I would take that more as a collective, if we're talking about everybody's... See, it is collective. It is all our actions put together that this destruction is happening, not just one individual. All of us, we want to build a certain kind of city. This is all of us, isn't it? This is we can't blame this on one person. All of us are enjoying the airplanes, all of us are enjoying the automobiles, all of us are enjoying various technologies. We cannot put this on just one person who is running a coal mine somewhere. No. He is only running the coal mine because we are using it. So all of us are equally involved in this. Some may be benefiting more, some may be benefiting less, that's a different matter. But essentially, we have taken the direction that you can extract happiness from the world. No, human experience happens from within you. So this is a fundamental, if this one thing is established, suppose you, if you sit here, you are absolutely blissed out by yourself, like me.  I look stoned all the time, never touched a substance, but blissed out all the time. So if you are blissed out by your own nature, you would not be in pursuit of happiness, you would do simply what's needed, nothing more, nothing less. Right now, all of us are doing excess because we are super, we have become super human beings with technology. What 10,000 men could do, one man can do in a day today, all right? In a given day, what 10,000 men could do in the past, today one man can do because we are empowered by technology. Once we are empowered like this, if we are still in pursuit of happiness, we're going to dig this planet through and through, all right? So temperatures going up is just a small reaction of the planet, bigger reactions will come inevitably. Where do people find courage? We'll talk about the 25 because if we're talking about the 25, they have been empowered by their communities to lead them. They have been in a democratic, but as we said, the democratic principled approaches to governing, their leaders have resulted out of a plebiscite. And yet the question is then, you're going to, we talked about the gold mine. Well, the guy who has the gold mine is going to contribute to the campaign of the person who's going to protect his right to, or protect his gold mine so that he can continue to dig. Gold mine, gold mine. Or a coal mine, gold mine, whatever it is, gold mine, coal mine, petroleum. Okay. Essentially digging the planet up. Essentially whatever it is to dig the planet up. And so there is a, there must be a place or there is a role for courage because there, one has to say no to one's self-interest and one's individuality in order for us to protect what we have. That is a failed idea already. Tell me. In the sense, see you're trying to control human aspiration. You will never do that. It's never going to work. All right? People are driving Tesla to their office, but they got a beast parked at home. I see this all over the place. So trying to control human aspiration is not going to work because it is within the human being too long to find his fulfillment. Unless we show him a method, a scientific method in a way. I'm saying scientific because it's repeatable and people can study it. To find that within yourself, this may look like an old thing. Oh, so many people said that that's not the point. The point is not a philosophy of turning inward. The point is of methodology of knowing. Suppose, see this is a fact that every human experience has a chemical basis to it, isn't it? So suppose I teach you a simple process with which you can concoct your chemistry in such a way that it's a chemistry of blissfulness. Now, there is no need for you to run in life unless you find a purpose that's worthy of you. You will not run. You will not run for the sake of winning a race because you are fine. You already won with life in many ways. Because you are living a very peaceful, joyful, wonderful life within yourself. Now you will do what is needed. This is a fact that with any human being, when they're feeling wonderful, they're naturally wonderful to everything around them. When they're feeling nasty, they'll of course share it with others. Right now, when you say we are in pursuit of happiness, you have a whole humanity who's largely nasty within themselves. They don't even know ease of life. Forget about the blissfulness of life or ecstasy of life. They do not even know how to be at ease. They're always fidgety. Either they're doing this or they're doing this. Something compulsive. Well, you raise another point. And again, for our communities that are joining us through Facebook, speaking here with Sadguru... No, no, don't tell them not to use their phone. They'll drop out. I want to see if we can have a conversation about these, this part of technology, which is changing the way I think that we live and potentially in ways that are counterproductive to our capacity to truly have meaningful change that is what we're striving for. Sadguru was here with us live from UNESCO headquarters as a part of our Facebook live. His community also on the Facebook page that Sadguru has cross-posting with us. And we are talking about where we are in this world and what our responsibilities are to this world as we try to address the challenges that are facing us. And a few minutes ago, this made noise. I was trying to do something to enhance our conversation. When you're talking to me, you're doing something else on the phone. And absolutely. This is called adultery. Oh, we've redefined the term. It may indeed be the case. But this has been, you know, I'm old enough to know the world before the mobile phone. And this is no longer a mobile phone. This is mobile technology that has given me incredible access to information at my fingertips in most places where I go in this world. It has allowed us in many ways to have this conversation transmitted effortlessly, transmitted in a way that is affordable for us to do as an organization. Same I would go for yourself as well. And to reach many people. And yet it also has a component to it that is to say that it's a distraction is perhaps using the most kindest terms. Maybe adultery is the way. So how do we manage this? I think in the very first part of the conversation we touched this, most beautiful things that are given to us, we will turn it against ourselves. Not because there's something wrong with technology. Technology is an enable, you know, it enables you, it empowers you in so many ways. Right now it's called a smartphone because that's the smartest thing about a whole lot of people. So this phone, is it a problem? No. What is a problem is compulsiveness. Do you know a whole lot of people have given up smoking because they're texting? Is that right? Are we going to call that a benefit? The problem is compulsiveness. If people start eating, they don't know when to stop eating. If they start drinking, they don't know when to stop drinking. If they smoke, they don't know when to stop. If they start using the phone, they don't know when to stop. We have to address the compulsiveness within the human being. This compulsiveness is coming because every action that you do is in pursuit of your happiness and well-being. So there's satisfaction in the 210 times that an individual checks his phone according to recent data. 210 times a day we pick up the phone. Not to answer it. Yes, if nothing to do. To do this and maybe not smoke. Because both hands are busy now, you can't smoke. But the mind is in thinking either. No, the important thing is this, we are in pursuit of happiness. All right? Now phone has given you a virtual means to pursue your happiness. Otherwise you had to go somewhere to meet somebody. You had to walk a distance. People were doing it then also. It's just that it was not so visible because now they're reaching out hundred people in a minute. Happiness, self-gratification, are these interchangeable? See, if you're really joyful, you will never see gratification of any kind. So you can put the phone down. Oh, that's not the point. Whether you use the phone or not depends on what kind of work and what kind of stuff you're doing. But the important thing is there is no compulsiveness about either the phone or the smoke or the food or anything for that matter.  See, it's only to a human being. This being is attached, all right? We don't say a tiger being or an elephant being or a crocodile being. Nothing like that. Only human being. Why this is so is we're supposed to know how to be. Supposed to know how to be means I can make my thoughts, emotions, my chemistry, my body the way I want. I know how to be. That's one thing, one description that most human beings are very far away from. They don't know how to be. So their only way is action, all the time action. Compulsive action because only through action they're trying to establish their being. But that's not how life happens.  If you know how to be, then you could have determined how your thought should be, emotion should be, action should be. This is what yoga means. First thing is you know how to be. Once you know how to be, how you conduct your life in a given time, in a given situation, taking, I mean considering the nature of your activity and life that you've chosen, accordingly you act to the best of your intelligence and capability. Sadguru, I think on that note we will bring this wonderful conversation to a close. Absolutely not. You brought it instead of saying anything wrong, you actually brought everything together. And I think it's always important, I think to provide people with at least, and myself included, to provide us with at least an opportunity to have a package that we can reflect on, in which all the components are blinking. You should stop looking at peace, love, joy, this, that separately. If we just know how to be, this means your body, your physiological structure, and your psychological structure takes instructions from you. If this one thing happens, you don't have to pursue any of those things, you will be just fine. And on that note, Sadguru, thank you very much. And to all of our Facebook communities, thank you for being part of this conversation.  We wish you an opportunity to reflect on these words. And hopefully that you also take them with you and make them part of your daily life. For now, with Sadguru, I'm George Papayanis, live from UNESCO headquarters, wishing you a great day, and perhaps a day for positive change.


Welcome to WSIS Forum 2018. I am delighted to be joined by Mrs Ursula Oussu-Ekufor, who is Minister of Communications in Ghana. Thank you very much for joining us. It's my pleasure. So to start with, I'd like to talk about the various ICT initiatives in place in Ghana to deliver the Sustainable Development Goals. I've just been in the job for a little over a year, but we've been busy doing a few interesting things to ensure that we deliver on the Sustainable Development Goals. And our president is one of the two UN co-chairpersons for the Sustainable Development Goals. And so it is something that is very, very uppermost in his mind that we take the lead and show the way in terms of what we do. And it is universally recognized that the ICT agenda can help us deliver on all of the sustainable development goals. And without that, it would be difficult to achieve much in that area. So if you look at education, we are piloting and rolling out a very robust e-education agenda. If you look at health, we're also working on e-health initiatives with some financing from the World Bank and some programs that are ongoing to ensure that we deliver health to all our peoples. In terms of even the way in which we live and work in agriculture as well, there are a few interesting initiatives going on to enable farmers get their goods to market using ICTs, which I think, and to even deliver market information to the farmers in terms of pricing for goods, accessing weather information and all that. We're piloting programs which are enabling them to also use ICTs to improve the ways in which they live and work. So in all sectors, we're working very hard to ensure that we really make it easier for people to live and work. We're very keen on promoting digital financial services as well. And so we're introducing very innovative ways of assisting people to move towards a cashlight society. We can't do cashless. And that's also helping to reduce corruption at all levels. We've introduced a paperless port system which has improved the turnaround time and efficiencies at the ports and reduced the frustrations that our people go through, where company registration is also benefiting from enhanced ICT applications. And even at the ports of entry as well, we have a new immigration project which is being rolled and actually tomorrow we'll be turning it on and opening it up for people to use at our ports. We have a new justice program going on to improve justice delivery systems as well. So in every area, we're working very hard towards improving efficiencies, output and the deliverables of all the work that we're doing. And we're quite excited about it. To do that, we also need to scale up our connectivity. And so one of our main priorities is to connect the 20, 30 percent of our country which do not have access to ICTs at all. And that's also very keen. That's also something that we're very, very keen on. And that our universal service provider, GIFEC, the Ghana Investment Fund for Electronic Communications, which I chair, is also very keen on that working in partnership with the private sector and our telecoms operators. And so we're rolling out aggressively and we hope to be able to connect the whole country by the end of next year. We're also looking at using fiber of other power lines, which is something that we haven't considered much in the country. So we're thinking about using that to reach the unserved and underserved areas and looking at last mile solutions to be able to deliver to the people. If what we plan to do would succeed, we need to make sure that every part of the country has access to high-speed data at affordable rates. And so we're also promoting infrastructure sharing amongst the telecoms operators to reduce their capital expenditure and make it easier for them to also deliver services to the people where it is necessary. The government is investing money in providing the infrastructure for them to locate their services. So we're quite excited about the many initiatives that are going on in the country. So it's obvious that Ghana really sees ICTs as a powerful tool for development and for inclusion as well. Absolutely. And I'm thinking youth and women's inclusion as well into the economy. Absolutely. It's indispensable. There isn't much you can do today without ICTs. And I use the mobile phone as an example. Ten years ago, some people thought they could live without the mobile phone. Today you can't. And it is the way the young people have embraced ICTs, which I find very exciting. We are actively encouraging startups, tech entrepreneurs, and we have a very vibrant tech community that we're providing some startup capital to. We have the Accra Digital Center. And the plan for that is to provide training to the people living in that area, particularly women and young people, to give them ICT skills, to enable them to also engage in the revolution that is currently unfolding before our eyes. The statistics are worrying in terms of the internet usage by women. I think it's just about 30% of those who own mobile phones. And that's a study that the GSM Association has done. So we're focusing particularly on giving women ICT skills to enable them to bridge the digital gender divide and let them reap the benefits of all that is going on as well. So every year we run a girls in ICT program in the country. And we train about 500 young senior high school students and junior high school students from selected schools and give them ICT skills and give them devices. And from the remotest parts of the country, and it is quite encouraging that after being exposed to that, the uptake is phenomenal. And so even in their homes, they're teaching their parents and siblings how to use the devices that we give them. And this year we're having it in Kumasi, which is the second largest city in the country. Last year when the Bronga Huffle region, so we move around the country creating a pool of young girls who are interested in ICT. Coding is also something that we're taking quite seriously. And so we're working with some private sector operators to be able to deliver coding education to primary school students, not just girls, but then young nine year olds, nine to 10 year olds and exposing them to it. And they're also doing some quite amazing things. We're going around the country as well to the markets and tests where women are concentrated and teaching them basic ICT skills to enable them to use their devices, to use mobile money features, and to even get them to sign up to the mobile money and platforms and to get them interested in it. The Ministry for Finance itself, seeing the potential for the sector as well, is just in the process of rolling out point of sale devices to all vendors across the country. And the majority of the market women, small entrepreneurs, are women. And so they are the ones that are going to benefit from that initiative. And it is going to be directly linked to the revenue authorities. And so it will be able to calculate the taxes payable on it and send the information directly to them without them having to worry their heads about it. So at the end of the day, it simplifies the way they work and enables them to be compliant as well with our laws and makes life easier for them so they can know the revenues that they are getting from their businesses as well. We're quite excited about that. And it just shows how ICTs have become pervasive. So in terms of transport, health, education, trading, e-commerce is also big. We're equipping our post offices to be able to engage in that space as well. We have a digital for inclusion project which is teaching ICT skills to rural areas and people who live in market centers and even the cocoa growing areas. Cocoa is one of our largest exports, exporter. And so even for those cocoa farmers as well, we're giving them ICT skills to enable them to also engage more and more fruitfully in this area. So it touches every aspect of the economy and society. All right, Mrs. Owisui Okufu, thank you very much for being with us today. It's been a pleasure. Thank you for having us.


President, I'm very honored that you could take me. I know you can tell me. I'll listen to you for taking me some. No, we're going to do the best job we can on that. Carry on the Reagan message to Latin America, which is the privatization and employment ownership message. But, you know, I had a story that I told not recently, but should tell more about this whole thing. Big Plantation, you're fine. Oh, yes. About Perlow Project. And the ownership things, and the guerrillas attack. That's right.  Well, that was... Because it was theirs. They weren't about to stand aside. No. It shows that Reagan Alex works. So we're going to carry that message. Okay, I appreciate it very much. I appreciate it very much, George. Thank you for having me on the short list of the role play. Thank you. I'm very grateful for that. I'll do my best to come that way. But thanks a lot, but appreciate it. Okay. Thank you. Thanks. You're so busy. I need to take your time. This is probably years here, and it's just going to be soon. You won't forget it. I was the author of this, and I just wondered why I didn't listen to that. I'm not assigning that particular time. I'm very pleased to. And to tell you that we have an extra verse where the stick suggests that we bought it. Would you like to hear what it was? Yeah. It was simply, how old would you be? I'm sorry. I mean, how many candles would be on your birthday cake if you told us how old you were for real? Too many for a cake. It was a fire, you see. They had to be blown out by a typical deal. And so I think it was best that we just left that out. And so... Let me just decide. That would be just good, sir. I know. You'll have a third last day if you're going to wait for it. I'll show you a thing for everybody to see you go on. I've heard another wonderful line by a spoken... several years ago. He's now 90, but by George Burns. He's 85th, and someone asked him, how does it feel to be 85 years old? Yeah. George said, when I feel like I'm 85 years old, I'll let you know. That's perfect. So it's been my... thrilled my life to work for you. Well, thanks so much. Thank you. Thank you. Say, about right here. Okay. Oh, that's going to background the place. Okay. You're not going to have any problems with all that light. Okay. Are you all set? Yeah. Ready? Yeah. Okay. Catch you later. The department did that. Well, it's fine. I was good to see you. And there's one who left the Democratic Party once before. Welcome aboard. Thank you. I'm glad that you needed more. I'm very glad. Do you ever get any concerns? I don't know if you've ever heard of this. Winston Church will be able to change the party's name. And you made a statement that I think fits our situation. He said some men change party or principle. Or some men change principle for party. And some change party for principle. I'm going to use that. All right, listen. It's just, I know you've got a final decision you have to make. I just want to tell you, I hope that the decision will be the one we want. We couldn't have accomplished any of the things we've done if we hadn't had a majority in the Senate. You need to keep it. I think that the decision to practice was made. Ah, good enough. I sure enjoyed the Senate in 1981 fighting the tax cut and budget battles at the House. And I was safe for us whole people. He has. And I thought you were right then. And we were right then. Good to meet you. I'll be there. I know you know I understand more about the battle than any president we've ever elected.  And before those days, he played Las Vegas twice. Two weeks once or two weeks. He did not know that. We'll be even for part of the day you can play Las Vegas again. I'd like that. I will do it. We were talking earlier, Jim, that Sigg-Rovitz, who helps a great deal with the presidential campaign on the Tuesday team, is also going to be helping you and the president. Delighted for that. We'll get the whole way to the end. I can't even, I can't ever have the example. Too far along with the likes of President Lincoln, Senator Icon, and Sigg-Rovitz. All right. This is J.D. Delighted to see you and ask you to have the image. I'm a private leader. I'm just happy to meet you.  You got to get here in front of the presidential. We'll have a little group photography here. Come over here. Are you smiling, Marcy? Thank you, sir. Rest assured. You know, reality could show up in the party. We're just going to the same direction. We're going to the right turn. I don't know any other way. It's very twisted. All parking, is there any time to work? There's no time. No exchange. And we're turning the car to the right. So don't. Is that right? That's what we do. We do. Turn around. And I'll be at the end. Come on, you're on your way. I'm sorry. I'm just going to capture by the institutions that have changed me instead of you. Yeah, that's unfortunate. I guess you can see in the audience. I'm just sitting there. That doesn't work. Does it resonate or agree with you? You have to. See you on the trail. Thank you. Thank you. Thank you, sir. Is that everything? Thank you, sir. Thank you, sir. Thank you, sir. Thank you very much. Let me break down the lights for a minute. Yeah. Yeah. We'll have a little bit of time to see the videos. Good morning, sir. Good morning. I'm always a little bit of a echo here. We could go out. Why are we going to have sort of individual phototaping up here? At which point we'll welcome back down here a group photo. That's because when the press is in here. You said very much taking your time to do this. Well, I'm busy or I'm pleased. I have to tell you that I say this to many people that I'd say you're going to press me to do this kind of a thing. I'd say that you're asking for kind of your religious statement about the last supper, the words you were referring to. Everyone wants to get in the picture.  Good day. Good day. Everybody in? All right. Well, good morning, Dave, and ladies and gentlemen. I appreciate this morning.  And I know you've all worked hard and very hard, as a matter of fact. And I'm glad that you report is bipartisan and unanimous. I've met with Dave over the past few weeks, and he's kept me apprised of your progress. And I'm pleased to say that your work may well be the most comprehensive bipartisan review of the defense establishment since World War II. And I plan to review your recommendations in detail, and I will act promptly and firmly to take appropriate implementation actions. And I hope that the Congress will join me and do their part as well. And now, Dave, the spotlight is on you. Well, Mr. President, it's a great honor for me to present this report to you that contains our first set of recommendation. And I speak on behalf of all of the commission to say that we've been very honored to have the opportunity to help you and Secretary Weinberger in this very important job of getting the most defense we can from the billions of dollars that are being spent. And we know that you and the Secretary have made a great deal of progress in the past five years in moving ahead toward that end. And I want to tell you that this commission has worked very hard. And that every single member of the commission has made an important contribution in developing these recommendations that we are presenting to you today. And we hope very much that these recommendations will help you and the Secretary continue this important job to improve the strength and the readiness of our armed forces so that they can continue to be the bulwark of freedom and peace in this troubled world of ours. Dave, I thank you very much. And as I said on Wednesday, the commission's recommendation points the way. And I'm looking forward to getting into this. And I'll implement the recommendations that are in here just as quickly as it can be done, even if they run counter to the will of the entrenched bureaucracies and special interests. And I will also urge Congress to read the commission's report and to remove those obstacles and the good management that the Congress itself has created over the years. And again, I thank you all for all the work that you've done. Mr. President, how much waste is there in the current system? How much waste is there in the current system, Mr. President? There is a lot less waste than there was when we came here. And there will be even less, as we go forward, as Packard has told you, with the recommendations that are in here and that he himself has said, follow the pattern of things that have already been started. Mr. President, there are reports that President Marcos has brought millions of dollars worth of currency and jewelry to Hawaii and the Philippines. Is that appropriate considering the economic problems there? I don't think, again, I think that there's no way for us to know anything about this. This is up to the government of the Philippines and the people of the Philippines. But I think now we've got to get on with some more meetings. Should Mrs. Aquino have released the Marxist leader in the Philippines? Sometimes. Thank you very much. Should Mrs. Aquino have announced the Marxist leader? Marxist. I have to say, of anything with regard to the meetings in Geneva, I learned there that General Secretary Gorbachev not only has the final approval of any quotation of his to be used, but also he goes over every TV shot and every still photograph and approves those. And they're the only ones going to be used. And I don't know why we can't. I should go up here. Yes. Thank you. Thank you very much. I know a lot of you are here to talk about it. You're doing a great job. Thank you. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. I'm going to say no to that. Can we ask y'all to move closer together, please? Let's see if the three MPs would be able to move over here to the final presentation. You just need a wider angle lens. I can't see you in the back row there. Everybody wants you to come down. We'll let you gel it, sir. We have to ask you to move a little closer. Judge Clark, you're fine because you're taller in the back row. As long as you can see me, as long as I'm clear of you, I'll get a clear view. Listen, we're going to have one more person. There you go. There's an empty spot right there. That's fine. You, sir, can you come on over? One more, one more, right there. Right in the middle. Let's raise.


He would be like, oh, there's Terry Siren. I need to get closer, so he'll see me and want to dance with me. Ha, ha, ha, ha, ha. Jen and I met at a church dance.  And we actually met again in college and started dating then. And never stopped dancing. We've known each other so long now. Probably more of our life than we haven't. We've known each other. We only wanted to do the math. In our relationship, reading out loud to each other is a way for us to connect. You know, so much of military life can be crazy. So much of it can be kind of unpredictable and stressful and all of that kind of stuff. Romance now is awesome. The idea of being able to spend that time in each other's company and have that focus without the distraction of something digital. It really creates a closeness. For my family, reading to each other is a way to connect even when I'm on deployment. I would call him up and I would recite Good Night Moon to him while Jennifer and he sat down with the book and would go through the pages. The biggest part about the military life is making sure that you create the bond with your kids that you would if you got to come home every day at 5 o'clock. We have at least six bookshelves in the house. And they're all separated by types of books and genre and then alphabetized because we're uber nerds. And it just creates a culture of imagination, I think. The idea that information is in a book is very tangible to my children. You just need a book and a library card. This is the most complicated portion that I've ever seen, so Hermione. And so that's who we are. We are the all-singing, all-dancing, all-traveling family siren and have been for probably 15 years. Thanks for taking the time to get to know the all-singing, all-dancing, traveling family siren, my family. Please take the time to check out some of these other stories and get to know your military.


So, we're gonna finish up these lectures today. So, it'll be a little bit less heavy-handed than yesterday. So, I mean, from a nicer point of view, it would have been good to switch the two lectures, but I kind of wanted you to see one thing before I did it at the end of today, because even if you kind of only vaguely understood the neck region of structures, I mean, I can at least pull back to something so we can sort of discuss morally what's gonna be happening. So what I wanna do today is discuss a little bit of the proof of Reifenberg itself. This is where I'm gonna start, even just the classical case. And for that matter, just for the classical case, what I'm gonna do first, actually, is simply write down again this example that we were discussing the first couple of days, and we'll prove it carefully there that we actually have a bi-holder equivalence between this wiggling example and an interval. But we're gonna do this not the way I kind of outlined it on the first day, but we're gonna do this in a somewhat different way, because I'm gonna do it in a way where literally, once we build the right structures, the proof of the classical Reifenberg will be verbatim with the verbatim estimates. It will be no different. So we'll get to seeing a nice clean example, how to actually, you know, how this looks and what it looks like, and then we can wave our hands through through what constructions are missing in the general case. So let's begin by recalling what this example looks like. So what we did was the following. So we had off our start with an interval and I did it from minus two to two. I really don't know why I picked two off hand. I did that so long ago and I've been writing with it for a while, I'm just kind of stuck with it. And then this is what we call our sort of S naught. And we started to build a sequence of guys, right? And then the first thing we did was we said, okay, we're gonna fix some background epsilon. And for us here, it's just some fixed background. Something, we're gonna build an assassin's triangle. Say this guy goes up, so if the length of this guy is the length of the single interval that sort of connects from from minus two to two, of course that's four, then this guy here is supposed to be epsilon times the length of the interval of the opposite side. And that was S one, which was itself a union of two intervals, right? So S one was union of, in general, SI, right, was a union of L, AI, J, and AI, J plus one. We'll recall by definition, these were points in R two and this was the segment connecting the two points by definition. And then the next step was that, okay, now we start to alternate. We went down by epsilon times the length of whatever each of these intervals are. And then we went up the next guy and this built us our nice S two, I guess, right? S zero, S one, S two, and so forth and so on. I mean, we kept alternating like this. And what you get out of this construction is a nice sort of Reifenberg set in the end and we sort of suggested why this thing was actually bi-holder to an interval and what I'd like to see is a little bit more carefully how we can construct this map to see that this thing is bi-holder to the unit interval. And the way I wanna do it now is the following. So the way I'd suggested before recall was that in the one dimensional case we can cheat and say that even though the lengths of all these things are going to infinity, we can parametrize each of them by arc length and try to start keeping track of what's happening with those mappings. But this is not at all how I want to do this. Instead of what I wanna do is the following. Let me make sure I have all my information recorded. So recall from our notation before, so our SI was our piecewise linear guy and the length of all these guys was the same because we always cut it up and their length was, because I want this down for later, the length of any interval in the ith segments was, so four was the original one and then every time it was what happened. It was the square root of one plus epsilon squared over two and then we multiply by ih time.  Okay, now here's how I wanna build the mapping. So here's our R as zero, here's R as one, here's R as two. Note the following and let me just actually draw the first two steps or one can view this as being any step somewhere in the middle. Maybe this is one of our LI, so this becomes LI plus one of length LI plus one. Then I wanna look at the projection map, pi, what's my notation? I, from SI plus one into SI. So there's a nice orthogonal projection here. So I can take any point here and look at the closest point, I just sort of boom projected, look at the closest point projection down. So if I'm here, that's literally just there. If I'm here, I'm literally just there. And I wanna build a map but what I wanna do is make the following sort of interesting observation. We're trying to eventually build a mapping from let's just say some SI. I want uniform estimates on Enei so I can pass it to the limit to get our wild sets into an interval. And what's one way of identifying the interval? The interval was actually our original S naught.  So the goal is the following. I'm gonna let, what's my notation? Those pies look exactly the same. I need different notation. Let's do phi. On the right up it's lower case and up a little bit who can tell here. I'm gonna get a mapping from SI. So after we wiggle on the scale two to the minus I roughly speaking into the interval which is just S naught so minus two to two by just composing. The goal is to show that this mapping is gonna be uniformly byholder independent of I. So in the end when we take our same limits to get our crazy set S we can just limit these maps right along with it. So we wanna get some nice estimates on this guy here.  We're just gonna play with these observations over and over again. So if we look at this mapping here which maps SI plus one to SI it always looks like this. I mean you've always got some interval on SI and then that's connected by two intervals in SI plus one and therefore the projection map from SI plus one to SI is you just look at that interval and you plop it down. So if you look at that this tells you two things right away. So let's call these estimates one. Minus pi I of Y on the one hand is actually precisely equal in this case but you can think less than or equal to for the context of the standard Reifenberg obviously. The square root of one plus epsilon squared times X minus Y. So Y is that. I mean note if I take two points even if they're far away here right I have a point X here and a point Y over here then the well what's the bilipsic constant of this mapping? If I take a unit curve here then it always or yeah we're not here. If I take a unit curve here then it always this length gets divided by the square root of one plus epsilon squared here. So if I take my unit curve connecting these two to get the length then they just get projected right in there and we actually get the precise quality up here. So that is to say these maps are bilipsics right? No bi-holder. There's a uniform bilipsic constant. If I only move one step right? You can't induct on that obviously. If I tried to use that forever you'd never get anything about your end map because all you'd be saying is you know after you do I steps to get down here you'd have a bilipsic constant that's one plus epsilon squared to the I which is useless information right? So clearly we need more than this but this is one thing we'll use. It holds for all X and Y. Well okay one should be a little careful about this you're right. If I'm using the intrinsic metric it's a quality. If I'm using actually Euclidean which is of course what I'm doing. Let's say that. It's actually less than or equal. I do not. I mean one step exactly. Yes. Cause if this was true for capital fee not only would we have bi-holder we'd have bilipsics right? So we definitely don't have bilipsics. So just one step nothing goes too wrong we have a nice bilipsics map. And the other thing I want to point out is the following. So what if I take some point and I just take it here and I project it down and I look how far does that point get moved in total? Now that point gets moved basically epsilon times the length of your side right? Cause that's the most distance it can be. So this is bounded by or we can say bounded by epsilon times the length of Li. Now the longer the two sides doesn't really matter. That's the farthest it can move is epsilon times this length right here cause by definition that there was epsilon times Li. Everything else is less. Again that's just one step one movement. Okay good. So the claim now though is that okay so bilipsics for one step not moving very much for one step. Actually if you combine these two in a clever enough fashion that's by holder. That's the claim. So this is worth absorbing a little bit cause this is the how does one prove by holder estimates? This is how one tries to prove by holder estimates. So the way it's gonna work is the following. So let's take our Si. Actually let's take three. Here's S naught our eventual goal. Now let's say we're on some Si which is what we're trying to study here to get our projection map down. So let me just wiggle this around.  It's really really small. Here's our Si. We've wiggled a lot at this point maybe. And what I wanna do is take two points X and Y over here. And I wanna introduce so we have two scales in front of us right. We have our I which is where we're at. We have the not which is where we're heading. And in fact we have an in between scale which is roughly the distance of these two points which may not be sort of comparable either to Li or L naught. So what we wanna look at is let's let X minus Y be approximately LJ. So let's find the J so that if we look at SJ the amount each segment is moving is roughly on the same order as the distance between X minus Y. And the precise way of saying this is just let's find the largest J say for which this is less than or equal to that. So that is more precisely we can find a J with the following properties. So it's less than LJ but maybe bigger than LJ plus one is the point so we find just that point. Okay great and let's look at that guy too. Maybe it looks something like that. Here's our SJ. Okay now here's what we wanna do. We wanna take these two points in SI right. We wanna move them all the way down to S naught. So I'm gonna break this into two steps. So let's define it in between map here which is I'll call it phi IJ that maps SI and do SJ basically by just stopping the composition at some point. So two that J or J plus one J great that's a zero. And likewise just by switching I and J it's the exact same definition but let's go ahead and point it out. I can let J naught be defined as J minus one composed all the way down to K naught. I'm gonna say Euclidean here and say Euclidean yeah. Oops, did I do those backwards? Do I have to compose? What do I have to compose these in? I'm sorry. My left and right are totally off right. You gotta do this map first and then you end with this map. You gotta do this map first and then you end with this map. We really need to write compositions the other direction. You think about moving in that direction unless you're Chinese, right? So you think about moving that way and it's actually moving this way, right? So I think I confused everybody except our Chinese students. Okay, great. So then what I wanna do now is simply write phi I as being the composition of these two guys. Let's make sure I do this one right. So that would be phi J naught composed with phi IJ. Okay, great. Yes. So, okay, goal is have a picture for what these estimates are supposed to look like. So my basic claim is that we're gonna use one of these to control one of these maps and the other one to control the other map, right? So we can try to end up with our bi-holder. So in some sense, J here basically just depends on the distance between X and Y. So what we don't wanna do just to get a hint of sort of what we might expect to be doing here. What we don't wanna do is apply this infinitely often, right? So in fact, we're going to be applying this to this map where we know we're only applying it a certain number of times based on the distance apart. That's almost the moral of what uniform continuity is. And if we check the actual constants, this is what bi-holder will be for us in the end. And what we can apply infinitely often is this because if we're applying this guy for all K in between I and J, it's like a geometric series. It's gonna end up not being that big of a deal to apply it infinitely often. So this is what we wanna think about. So let me talk about this map first for just a second. So let's write a goal, goal one I wanna show. So I'm interested in looking at, right? So what are we trying to show? Let's do our series of goals to each keep getting more and more refined. So our map was defined here, right? So let's just say, let's show one direction of the bi-holder estimate. The other is verbatim. Hold on to this board before it jumps away. So I wanna show there's a beta and we're gonna be able to take this beta explicitly and it'll go to one as epsilon goes to zero so that phi i of x minus phi i of y is bounded by something small. I don't care, something close to one, I don't care. Sometimes the distance of x and y to the power of beta, so this is our ultimate goal of what we wanna show. I wanna start with a weaker goal over here for this map. Basically I wanna say this map actually doesn't move a fixed point very much relative to the distance of x and y. C for me, by the way, always basically means some universal constant here because it's all one dimensional. If you were doing a standard right from Burguet we mean a dimensional constant, the same for y. So my first claim is that this map basically doesn't matter. When we do end estimates here, the amount that's gonna move the point x is gonna be small relative to the distance in x and y anyway. So in the end it won't count for anything in terms of bad estimates. So this is gonna be a good thing because this one definitely will count for bad stuff in the end. So we wanna see nothing moves much. And how do we see this? So let's just do it inductively. So let's let x k be phi of i k of x so I want k here to go from i to so phi i i is just x itself, right, nothing's happened and then I want k to go from i to j. And all I wanna do is the absolute dumbest thing possible which is to estimate. So I wanna estimate x of j. I know that i i is x.  This is gonna be the goal.  And lk itself, well I don't care about lk, I care about lj, right? lj is the distance between x and y by definition essentially, right?  So this is gonna be much smaller than l sub j and the amount smaller it'll be is I mean this is equals epsilon times square root of one plus epsilon squared over two, say to the j minus i, right? lj. So in other words, this is something summable is all you care about, right? If k is gonna be, yes, that's less than one. So if k is gonna be less than j, right? This sums just like two, right? As we move on, something a little bit bigger than one. Which means that if we want to prove this, that phi ij of x minus x, this is less than or equals to the sum by just sort of definition of what these things are, right? Of, I'm likely gonna get all my i's and j's messed up here. So I can do the silly thing where I can just sort of add and subtract in all the in-between steps and just do a triangle inequality, plug all these guys in. They're all epsilon times l sub j, but something summable. All right, so what I get is c, which is maybe two even, compute times l sub j, and this is like x minus y. So as claimed below, well, once you start wiggling more than the distance between them, you don't care, things don't move around much. So this map ends up not counting for much. I've got five pictures, I can erase one. So what have we done? We've taken our point on, let's not draw it that way, let's draw it this way, I don't know why I'm intersecting the two, I shouldn't be. Here's y, right? So I've got my two points on si, and I've now projected them up to sj and nothing much has changed, right? They've moved here. So here's my phi ij of x, and this has moved over here somewhere. I mean it's bounced around, but it's gotten up around here, my phi ij of y. So comparing these two distance is now basically equivalent to comparing these two distances, it doesn't matter. So now how do I finish the game and prove this? So let's just look at this for a second. Sorry about the two lines. So by definition, the phi i, which is the composition of all those maps, I can get it by first doing the first i minus j of them and then doing the last j of them, and this is the map we just studied. You get from here to here. Now, what do we know about this mapping here in each of these? Well, this thing guy here is a composition of j of our single projection maps from point to point. If we look at this estimate up here, we know the distance, every time we do this, the distance multiplies by a factor, so in fact, it's close to one, right? And it'll most now multiply by this guy j times. So in fact, all this guy does is multiply distances by at most the square root of one plus epsilon squared to the power j, right? So this is less than or equal to one plus epsilon squared to the j over two times the original distance of these two points, the ij of x minus phi ij. Y, okay. Now, phi ij of x and phi ij of y, what do we know about them? This hasn't moved x by more than some small factor times the distance. This hasn't moved y by more than some small factor times this distance. So this here is less than or equal to, in fact, maybe I'll, let me just do it for you. I'll do it over here so I don't have to mess up my computation. So what happens when we do this twice?  So how far away is phi ij of x minus phi ij of y in comparison to x minus y? We'll just do a triangle inequality. Let's bound this by by, I have a small blackboard, I'm afraid, the sum of that minus that plus that minus that. You can do this by a triangle inequality. So it's just twice this, right? So this is another c, I don't care. I'm an analyst, my c is never the same from line to line. So this is also just bound to by c epsilon times x minus y, which is to say, this distance is up to one plus c epsilon, the same as this distance. Because this didn't move x much and this didn't move y much relative to the distance of x and y. That's why we chose to scale j. Okay, now what? All we actually really have to do at this point is fiddle around with the definition of this and what this is for a minute. So let's just write like four lines and we'll have a bi-holder estimate. We're gonna have to erase another board. I guess that one's gonna go. So this is, so one of the definition of x minus y was that it was roughly equivalent to lj in particular less than or equal to lj. So let me put that back. And I'm basically gonna turn this into lj, manipulate all my j's and turn them back into x minus y just because we can see it cleaner that way. And let me stick straight in the definition of lj. So this is bounded by lj, which is itself precisely equal to what? Square root of one plus epsilon squared. That looks familiar. Over two to the power of j. This is equal to one plus c epsilon. I just wanna collect together my two square root of one plus epsilon squared because why not for j? And now here we're sort of the nifty definition comes from. So let's define, let me try to do this right. Oh, sorry. There's actually a four here, right? When I put in the definition of l sub j, it's four times this. Doesn't matter. It gets it right back in just by definition it's four times that because I start on the interval of like four and then I keep multiplying by this. This is gonna be equal, now equal mind you. So I'm gonna be giving you a definition in one second. I don't like this one plus epsilon squared. I prefer square root of one plus epsilon squared because that gets me back to l j. So what I'm gonna do is simply write this as being the square root of one plus epsilon squared over two to the power of beta. And let's observe something, right? So this is less than one, right? So this is a little bit bigger than this is but this is less than one which means I need to raise to a power it's a little bit less than one. So beta is less than one here. I mean this equality here is my definition of beta, right? It's less than one but clearly as epsilon goes to zero, beta goes to one. What's that? This? The last one, the very last one. Yeah, this line here only holds though when the distance between x minus y is equivalent to l sub j, right? That was the point. So this distance here ended up being so you got this line here, right? Which is always true but that only became x minus y on this level. So I can't let j go to zero. That's the point. I can only do that when the wiggling is more than the distance is less than the distance between x and y. That's the point. Okay, where am I? I'm over here. I just wanna switch beta and j. Well, yes. I don't care, I'm gonna do either. So this is, I wanna do a less than or equal to so I can shove a beta here. And then I'll just, I'll rewrite it because I don't wanna do too much in one line. Two, I'm gonna stick a beta there and a j there. I'm just gonna switch beta and j. And then, so all I did was raise this to a power so I could absorb this into here now. One plus C epsilon times four times the square root of one plus epsilon squared over two to the j to the beta. And this is l sub j by definition, right? So we can get back to that. I might have to make this a little bit bigger. I didn't actually keep track of which one made it less than or equal to. This is a little bit less than one, which actually makes this a little bit smaller but only by an amount that's very, very close to one. Right, so I made this C worse. That's how I made it less than or equal to. L sub j to the beta. And now one more time we turn that x minus y and we're done. And the way I'm doing this, I guess I'm gonna shove a factor that's close to two. You can actually be more careful in that, but I don't care. So, two, whatever. And I may as well just drop the one plus C epsilon and I'm gonna put two there. L sub j x minus y to the beta. The two was because of this guy here. You can be more careful in that, but I didn't. That's it, right? So, starting up here, it's saying we have a holder condition on how these are and the opposite direction is basically exactly the same. You just gotta trace the inequalities the other direction. So, good. So, what's aggravating about this is that it's aggravating. However, it is a comparatively simple situation, right?  It's just a projection from line to line to line to line and you trace through. The clever point you really wanna keep track of here is that when you're doing your estimate, you cannot, you've gotta have both these guys.  If you do this too much, well, actually, you might be able to preserve like room of how or how's worth close, but you'll never get a byholder condition, right? So, you need to preserve the distance between the two. Okay. Good.