1 00:00:00,040 --> 00:00:02,160 Welcome back to the Deep Dive. Good to be here. 2 00:00:02,560 --> 00:00:05,800 You know, this is where we try to tackle those really dense, 3 00:00:05,920 --> 00:00:08,720 complicated ideas, right? And hopefully boil them down 4 00:00:08,720 --> 00:00:11,800 into something well usable knowledge. 5 00:00:11,800 --> 00:00:13,400 Exactly. Try and distill it. 6 00:00:13,840 --> 00:00:17,800 And today, wow, we're definitely not just talking about abstract 7 00:00:17,800 --> 00:00:20,640 math and a vacuum. We're stepping into the ring 8 00:00:20,920 --> 00:00:23,760 with what might just be the ultimate intellectual challenge. 9 00:00:24,120 --> 00:00:25,640 You must be the Millennium Prize problem. 10 00:00:25,640 --> 00:00:28,800 That's exactly it. The seven huge unsolved 11 00:00:28,800 --> 00:00:31,960 questions posed by the Clay Institute way back in 2000. 12 00:00:32,159 --> 00:00:36,760 And each one comes with a $1 million bounty attached. 13 00:00:36,800 --> 00:00:38,760 It's high stakes math for sure. It really is. 14 00:00:38,760 --> 00:00:42,320 And what's kind of staggering is, since 2000, only one has 15 00:00:42,320 --> 00:00:44,320 actually been solved. That's right, the Poincare 16 00:00:44,400 --> 00:00:48,080 conjecture, Grigori Pearlman back in 2006. 17 00:00:48,280 --> 00:00:51,040 An incredible achievement. Which leaves six giants still 18 00:00:51,040 --> 00:00:53,240 standing decades later. They're resisting all the 19 00:00:53,240 --> 00:00:56,440 traditional proof methods. So you have to ask if the best 20 00:00:56,440 --> 00:00:59,480 minds haven't cracked these with standard approaches in what, 21 00:00:59,480 --> 00:01:02,840 nearly 25 years? Maybe we need a different angle, 22 00:01:02,840 --> 00:01:04,360 a shortcut? Precisely. 23 00:01:04,760 --> 00:01:07,400 And that shortcut is what we're diving into today. 24 00:01:07,400 --> 00:01:12,440 We're shifting focus from that image of the lone genius, you 25 00:01:12,440 --> 00:01:14,760 know, staring at a blackboard for years towards what? 26 00:01:14,920 --> 00:01:16,320 Towards a computational approach. 27 00:01:16,320 --> 00:01:19,240 A strategy that Marco Saba introduced, and it's been 28 00:01:19,240 --> 00:01:23,120 formalized more recently by XA, is Grok 4. 29 00:01:23,400 --> 00:01:26,360 They called the Assume and Verify methodology, or A and V 30 00:01:26,360 --> 00:01:27,880 for short. I love this idea. 31 00:01:27,880 --> 00:01:31,520 It feels almost subversive instead of demanding this 32 00:01:31,520 --> 00:01:35,040 perfect, timeless formal proof that covers everything. 33 00:01:35,160 --> 00:01:37,600 Which might be impossible, or at least incredibly hard. 34 00:01:37,600 --> 00:01:39,520 Right. Instead, you use the massive 35 00:01:39,520 --> 00:01:43,080 computing power we have now, all the data, the fast algorithms, 36 00:01:43,320 --> 00:01:45,760 the specialized software. And you turn these problems, 37 00:01:45,760 --> 00:01:49,400 these huge conjectures, into, well, numerical laboratories. 38 00:01:49,400 --> 00:01:51,640 You run experiments. It feels like it makes them more 39 00:01:51,640 --> 00:01:53,680 accessible somehow. That's a big part of it. 40 00:01:53,680 --> 00:01:56,440 It democratizes the inquiry. Now, we have to be super clear. 41 00:01:56,560 --> 00:01:58,080 This doesn't replace formal proof. 42 00:01:58,080 --> 00:02:00,840 OK, important distinction. Finite checks can't give 43 00:02:00,840 --> 00:02:02,440 infinite certainty. Exactly. 44 00:02:02,680 --> 00:02:07,120 But what AMV can do is build incredible empirical confidence. 45 00:02:07,400 --> 00:02:10,280 It can sniff out counter examples much faster than theory 46 00:02:10,280 --> 00:02:12,280 might. And it gives hints like 47 00:02:12,280 --> 00:02:15,080 guideposts. Yes, critical heuristics. 48 00:02:15,240 --> 00:02:17,720 It tells the theorists where the tricky parts are, where the 49 00:02:17,720 --> 00:02:19,360 breakthroughs really need to focus. 50 00:02:19,600 --> 00:02:22,960 So for you, the listener, the goal today is to understand this 51 00:02:22,960 --> 00:02:27,520 empirical coherence, how computation builds a case for, 52 00:02:27,720 --> 00:02:30,680 or maybe even against these massive conjectures. 53 00:02:30,680 --> 00:02:34,040 OK, let's really unpack this assume and verify strategy. 54 00:02:34,600 --> 00:02:36,560 The need for something new seems clear. 55 00:02:36,560 --> 00:02:39,240 Progress on these problems has been, well, slow. 56 00:02:39,280 --> 00:02:42,280 Glacial in some cases. But is this shift to computation 57 00:02:42,280 --> 00:02:46,040 just about having faster computers than we did 30 years 58 00:02:46,040 --> 00:02:48,720 ago, or is there more to it? Oh, it's definitely more than 59 00:02:48,720 --> 00:02:51,360 just faster processors. I mean, that helps obviously, 60 00:02:51,720 --> 00:02:55,080 but 30 years ago we really lacked the the integration and 61 00:02:55,080 --> 00:02:56,920 the specialized tools. What kind of cools are we 62 00:02:56,920 --> 00:02:58,600 talking about? Well, these problems are deep, 63 00:02:58,600 --> 00:03:00,200 right? They touch on really complex 64 00:03:00,200 --> 00:03:02,880 math, so you need sophisticated algorithms for one. 65 00:03:03,280 --> 00:03:06,400 And crucially, you need high performance software libraries 66 00:03:06,400 --> 00:03:08,960 that actually implement these algorithms efficiently. 67 00:03:09,240 --> 00:03:12,400 So not just writing code from scratch in a basic language, you 68 00:03:12,400 --> 00:03:14,320 need toolkits. Exactly. 69 00:03:14,400 --> 00:03:18,240 Highly specialized packages, things like SIM, P and Python 70 00:03:18,240 --> 00:03:21,480 are great for symbolic math, doing precise calculations on 71 00:03:21,480 --> 00:03:24,200 smaller scales. Then for problems involving 72 00:03:24,200 --> 00:03:28,200 logic like PBSNP you use things called SAT solvers. 73 00:03:28,320 --> 00:03:31,480 There are libraries like Pisat that give you access to 74 00:03:31,480 --> 00:03:34,080 state-of-the-art solvers. And for something like fluid 75 00:03:34,080 --> 00:03:38,240 dynamics, Navier Stokes. There you need heavy machinery. 76 00:03:38,520 --> 00:03:40,600 Packages like Dedalis are popular. 77 00:03:40,600 --> 00:03:43,480 They use what are called spectral methods, which are very 78 00:03:43,480 --> 00:03:45,360 accurate for those kinds of simulations. 79 00:03:45,720 --> 00:03:49,160 These tools are what let us turn the abstract math into something 80 00:03:49,160 --> 00:03:50,880 you can actually run. Got it. 81 00:03:51,080 --> 00:03:54,760 So these tools enable this A&B strategy, which you said is like 82 00:03:55,160 --> 00:03:57,560 proof by refutation. Yeah, trying to break it. 83 00:03:57,560 --> 00:03:59,760 That's the core idea, yeah. It shifts the whole mindset. 84 00:03:59,760 --> 00:04:03,560 Instead of asking, can I prove this is true everywhere forever, 85 00:04:03,840 --> 00:04:06,160 which is daunting. You ask, can I find even one 86 00:04:06,160 --> 00:04:09,160 instance where this breaks using all the computational power I 87 00:04:09,160 --> 00:04:10,800 have? Exactly, and there's a clear 88 00:04:10,800 --> 00:04:13,480 four step process. Step one is foundational. 89 00:04:13,480 --> 00:04:17,079 OK, what's step one? Step one, assume the conjecture. 90 00:04:17,560 --> 00:04:19,440 You start by taking the statement as given. 91 00:04:19,959 --> 00:04:23,800 Or sometimes, like we'll see with PBSNP, you assume the 92 00:04:23,800 --> 00:04:25,000 opposite is true. OK. 93 00:04:25,000 --> 00:04:27,920 Assume it, then what? The really crucial part here, 94 00:04:27,920 --> 00:04:31,040 the intellectual work, is translating that abstract 95 00:04:31,040 --> 00:04:34,400 assumption into concrete computable consequences. 96 00:04:35,440 --> 00:04:40,120 So not just assuming it, but figuring out if this is true, 97 00:04:40,440 --> 00:04:42,840 what must we be able to measure or calculate? 98 00:04:42,840 --> 00:04:46,040 Precisely what specific numerical bounds have to hold? 99 00:04:46,040 --> 00:04:49,080 What scaling laws should we see? Are there mathematical 100 00:04:49,080 --> 00:04:51,000 invariants that must be constant? 101 00:04:51,320 --> 00:04:54,280 This is the bridge from the idea to the experiment. 102 00:04:54,280 --> 00:04:55,840 OK, so take the Riemann Hypothesis. 103 00:04:55,840 --> 00:04:57,880 That's about where the zeros of a function are, right? 104 00:04:57,960 --> 00:04:59,400 Very abstract. Very abstract. 105 00:04:59,400 --> 00:05:02,480 It's about the non trivial zeros of the Riemann Zeta function 106 00:05:02,480 --> 00:05:05,000 lying on a specific line in the complex plane. 107 00:05:05,160 --> 00:05:07,600 So if that's true, the computable consequence is 108 00:05:08,040 --> 00:05:10,080 something about prime numbers. Exactly. 109 00:05:10,360 --> 00:05:14,320 It implies a very specific limit on how much the actual count of 110 00:05:14,320 --> 00:05:16,880 prime numbers can deviate from the best mathematical 111 00:05:16,880 --> 00:05:19,600 approximation that deviation is measurable. 112 00:05:19,760 --> 00:05:22,000 And that measurable thing is what you test in Step 2. 113 00:05:22,240 --> 00:05:23,560 Absolutely. Step 2. 114 00:05:23,840 --> 00:05:27,320 Generate testable predictions. You take those consequences, the 115 00:05:27,320 --> 00:05:31,120 bounds, the scalings, and you formulate them as specific 116 00:05:31,120 --> 00:05:33,760 hypotheses you can actually check with finite data. 117 00:05:33,920 --> 00:05:37,600 Because we can't check infinite primes, but we can count primes 118 00:05:37,600 --> 00:05:40,760 up to, say 10 to the power of 20. 119 00:05:40,920 --> 00:05:42,560 Right. We can count primes up to 120 00:05:42,560 --> 00:05:46,320 enormous numbers now, so we generate predictions verifiable 121 00:05:46,320 --> 00:05:48,280 within that reachable finite domain. 122 00:05:48,600 --> 00:05:51,800 And then Step 3 is where the computers earn their keep. 123 00:05:52,160 --> 00:05:56,360 Step three, verify empirically. Now you use those specialized 124 00:05:56,360 --> 00:05:59,280 software tools. Simpi Π SAT did list whatever 125 00:05:59,360 --> 00:06:01,600 fits the problem, and you run the tests. 126 00:06:01,760 --> 00:06:04,280 You check if the predictions hold up against actual 127 00:06:04,280 --> 00:06:05,920 computation. What happens then? 128 00:06:06,120 --> 00:06:08,800 Two possibilities. 2 main outcomes if the predictions hold 129 00:06:08,800 --> 00:06:10,880 true. If the numbers match the theory 130 00:06:11,240 --> 00:06:13,520 well, you gain empirical coherence. 131 00:06:13,520 --> 00:06:15,960 Coherence, not proof. Not proof, but it's 132 00:06:15,960 --> 00:06:18,560 significantly strengthens your confidence in that initial 133 00:06:18,560 --> 00:06:21,040 assumption. It shows the conjecture behaves 134 00:06:21,040 --> 00:06:23,520 as expected in the real world, as far as we can test. 135 00:06:23,520 --> 00:06:26,360 But what if it doesn't hold? What if the computation breaks 136 00:06:26,360 --> 00:06:28,880 the prediction? That's where it gets really 137 00:06:28,880 --> 00:06:32,080 exciting. A failure isn't just a failure, 138 00:06:32,200 --> 00:06:35,320 it's a refutation. It refutes the consequence you 139 00:06:35,320 --> 00:06:38,280 tested. Which means it must also refute 140 00:06:38,280 --> 00:06:39,880 the original assumption you started with. 141 00:06:39,880 --> 00:06:42,720 Exactly. That failure is potentially a 142 00:06:42,720 --> 00:06:45,200 massive discovery. It could be the counter example 143 00:06:45,200 --> 00:06:48,000 everyone's been looking for. Which leads naturally to Step 4. 144 00:06:48,080 --> 00:06:51,120 Iterate on falsity, right? This is basically the scientific 145 00:06:51,120 --> 00:06:54,800 method in pure math if your initial assumption gets shot 146 00:06:54,800 --> 00:06:57,000 down by the data. Say you find a prime 147 00:06:57,000 --> 00:07:00,000 distribution error that violates the Riemann Hypothesis bound. 148 00:07:00,240 --> 00:07:02,160 Then you immediately flip your assumption. 149 00:07:02,240 --> 00:07:05,720 You start assuming the negation is true, and then you generate 150 00:07:05,720 --> 00:07:08,560 new predictions based on that. You start actively hunting for 151 00:07:08,560 --> 00:07:10,720 the properties of this counterexample you've just found 152 00:07:10,720 --> 00:07:13,040 evidence for. OK, this framework makes sense, 153 00:07:13,440 --> 00:07:17,000 but I have to push back on this coherence versus proof point. 154 00:07:17,800 --> 00:07:19,400 If it's not proof, isn't there a risk? 155 00:07:19,400 --> 00:07:22,120 What kind of risk? The risk that we spend, you 156 00:07:22,120 --> 00:07:25,280 know, millions of CPU hours building up all this coherence, 157 00:07:25,480 --> 00:07:30,880 but it's just, well, finding patterns in finite data that 158 00:07:30,880 --> 00:07:34,040 don't actually hold up in the infinite case, a distraction 159 00:07:34,040 --> 00:07:35,920 from the real theoretical work needed. 160 00:07:36,120 --> 00:07:39,120 That's a very valid concern, and it's why we constantly stress 161 00:07:39,120 --> 00:07:40,880 the distinction. This is not a replacement for 162 00:07:40,880 --> 00:07:42,600 proof. Think of it like simulation in 163 00:07:42,600 --> 00:07:46,040 physics or engineering. We simulate a new airplane wing 164 00:07:46,040 --> 00:07:48,320 design under millions of conditions. 165 00:07:48,760 --> 00:07:51,760 If it passes all tests, we have extremely high confidence it's 166 00:07:51,760 --> 00:07:53,600 safe. Does it prove it will never fail 167 00:07:53,600 --> 00:07:55,880 under some bizarre, untested circumstance? 168 00:07:55,880 --> 00:07:57,720 No. But it builds enormous 169 00:07:57,720 --> 00:08:00,600 confidence and tells us where the likely weak points aren't. 170 00:08:01,040 --> 00:08:04,480 So A&V helps narrow the search base for the theorists. 171 00:08:04,480 --> 00:08:07,880 Massively, if you check the Riemann hypothesis up to N 13 172 00:08:07,880 --> 00:08:12,040 zeros, or verify its consequence for primes up to 1020 and find 173 00:08:12,040 --> 00:08:13,720 nothing wrong. Which people have done. 174 00:08:13,720 --> 00:08:16,360 Which people have done in just seconds for the prime check, by 175 00:08:16,360 --> 00:08:18,360 the way. Then the burden shifts. 176 00:08:18,560 --> 00:08:20,800 It tells the pure mathematician, look if there's a 177 00:08:20,800 --> 00:08:23,120 counterexample. It's hiding way out there in the 178 00:08:23,120 --> 00:08:25,640 asymptotic weeds. Don't waste time looking for 179 00:08:25,640 --> 00:08:28,440 simple ones. And conversely, if a simulation 180 00:08:28,440 --> 00:08:32,120 did find a glitch, then. Bam, you have a concrete 181 00:08:32,159 --> 00:08:34,480 numerical counterexample, something tangible. 182 00:08:34,760 --> 00:08:38,039 A theorist can grab that specific data point and analyze 183 00:08:38,039 --> 00:08:41,120 why it breaks the pattern. It turns a philosophical maze 184 00:08:41,120 --> 00:08:43,880 into analyzing a specific observable defect. 185 00:08:44,039 --> 00:08:46,640 It's like giving the theorists a high-powered computational 186 00:08:46,640 --> 00:08:48,800 compass, even if it's not the final map. 187 00:08:48,960 --> 00:08:51,160 That's a great analogy. It guides the search 188 00:08:51,160 --> 00:08:52,280 efficiently. All right. 189 00:08:52,280 --> 00:08:54,040 I'm sold on the strategies potential. 190 00:08:54,440 --> 00:08:57,800 Let's get into the specifics. Let's see A&V in action on the 191 00:08:57,800 --> 00:09:03,960 actual Millennium problems. OK, first up, the big one, maybe 192 00:09:03,960 --> 00:09:07,960 the most famous unsolved problem in math, the Riemann Hypothesis. 193 00:09:08,040 --> 00:09:11,120 The king of number theory. Yeah, as we mentioned it's about 194 00:09:11,120 --> 00:09:13,840 the location of the non trivial zeros of the Riemann Zeta 195 00:09:13,840 --> 00:09:16,000 function. The conjecture is they all lie 196 00:09:16,000 --> 00:09:19,200 on a specific line, the critical line, where the real part is 12. 197 00:09:19,520 --> 00:09:22,640 And this seemingly obscure statement about complex numbers 198 00:09:22,640 --> 00:09:26,400 has deep ties to how prime numbers are spread out. 199 00:09:26,520 --> 00:09:28,880 Incredibly deep ties. It's fundamentally about the 200 00:09:28,880 --> 00:09:32,040 regularity, or the lack thereof, in the distribution of primes. 201 00:09:32,320 --> 00:09:35,440 If RH is true, the primes are distributed as regularly as 202 00:09:35,440 --> 00:09:37,680 possible. And the envy strategy takes this 203 00:09:37,680 --> 00:09:39,360 abstract idea and makes it concrete. 204 00:09:39,360 --> 00:09:41,040 How? Through that bound you 205 00:09:41,040 --> 00:09:43,040 mentioned. Exactly through the von Koch 206 00:09:43,040 --> 00:09:45,640 Schoenfeld bound. This is the computable 207 00:09:45,640 --> 00:09:48,360 consequence derived from assuming RH is true. 208 00:09:48,760 --> 00:09:51,560 So remind us what that bound actually limits? 209 00:09:51,560 --> 00:09:54,440 It limits the error you make when you estimate the number of 210 00:09:54,440 --> 00:09:58,320 primes up to a certain number X. The best estimate we have is a 211 00:09:58,320 --> 00:10:01,280 function called the logarithmic integral lead X. 212 00:10:01,600 --> 00:10:05,120 The actual count is X. OK, lead X is the guess, Lex is 213 00:10:05,120 --> 00:10:08,720 the reality, right? The bound says if RH is true, 214 00:10:08,880 --> 00:10:11,160 then the absolute difference between the guess and the 215 00:10:11,160 --> 00:10:16,080 reality X X has to be smaller than a specific quantity, The 216 00:10:16,080 --> 00:10:19,400 square root of X times the logarithm of X all divided by 8. 217 00:10:19,560 --> 00:10:21,960 So air X log X. Precisely. 218 00:10:22,440 --> 00:10:24,000 But there is a catch in the formal statement. 219 00:10:24,480 --> 00:10:27,800 Technically, that bound is only proven to hold if RH is true and 220 00:10:27,800 --> 00:10:31,240 for very very large X. The paper mentions X greater 221 00:10:31,240 --> 00:10:34,480 than or equal to 10 to the 28th power. 10 to the 28th, that's 222 00:10:34,560 --> 00:10:36,920 astronomically large. How can we possibly test that 223 00:10:36,920 --> 00:10:38,640 with computers? We can't reach that high, can 224 00:10:38,640 --> 00:10:39,960 we? Not directly, no. 225 00:10:40,400 --> 00:10:42,960 So the A&B experimental design adapts. 226 00:10:42,960 --> 00:10:46,240 We don't test the bound at 10:28, instead we test it at the 227 00:10:46,240 --> 00:10:49,160 largest values of X we can reach computationally. 228 00:10:49,280 --> 00:10:52,640 And we check for coherence. We see if the error behaves as 229 00:10:52,640 --> 00:10:54,520 if it's respecting that kind of limit. 230 00:10:54,560 --> 00:10:57,040 Exactly. And we lean on the mountain of 231 00:10:57,040 --> 00:10:59,840 existing evidence. Decades of computation have 232 00:10:59,840 --> 00:11:01,640 already checked the zeros themselves. 233 00:11:01,640 --> 00:11:04,320 Billions and billions of them. The latest checks are past 234 00:11:04,320 --> 00:11:08,320 10,013 zeros, and none have been found off the critical line. 235 00:11:08,320 --> 00:11:11,040 OK, so our A&B task here is twofold. 236 00:11:11,040 --> 00:11:14,560 One verify that this consequence about prime counting holds for 237 00:11:14,560 --> 00:11:17,640 the largest accessible numbers to keep pushing that 238 00:11:17,640 --> 00:11:21,440 computational frontier, always hoping, maybe perversely, to be 239 00:11:21,440 --> 00:11:24,120 the first to find a violation. We can see this really clearly 240 00:11:24,120 --> 00:11:26,120 with a small example, right? Something runnable? 241 00:11:26,280 --> 00:11:29,880 Let's say test X = 1,000,000. Yeah, X well, is one O 6. 242 00:11:29,880 --> 00:11:32,120 That's easily doable. We can use simply for this, 243 00:11:32,120 --> 00:11:34,520 because it handles the precision needed for the logarithmic 244 00:11:34,520 --> 00:11:38,000 integral correctly, even if it's slower for huge numbers. 245 00:11:38,000 --> 00:11:39,600 And running that, how long does it take? 246 00:11:39,600 --> 00:11:42,200 What's the result? On a standard machine, maybe a 247 00:11:42,200 --> 00:11:46,680 second or two, you calculate 1,000,000, the exact number of 248 00:11:46,680 --> 00:11:50,080 primes up to 1,000,000, and you calculate Lil 1,000,000. 249 00:11:50,200 --> 00:11:52,240 And the difference the actual error. 250 00:11:52,280 --> 00:11:55,320 The output shows the error. The lex lex part comes out to be 251 00:11:55,320 --> 00:11:58,840 around 23 point. Five OK 23.5 Now what does the 252 00:11:58,840 --> 00:12:00,800 von Koch Schoenfeld formula predict? 253 00:12:00,800 --> 00:12:05,080 The maximum allowable error should be at X = 1,000,000. 254 00:12:05,080 --> 00:12:08,480 You plug XO is one O 6 into X log X. 255 00:12:08,920 --> 00:12:12,360 That calculation gives you a bound of approximately 37.2. 256 00:12:12,640 --> 00:12:16,440 So the actual error 23.5 is way less than the maximum allowed 257 00:12:16,440 --> 00:12:19,120 error 37.2. Comfortably less so. 258 00:12:19,120 --> 00:12:22,160 The coherence holds beautifully. The error is well within the 259 00:12:22,160 --> 00:12:25,240 safety margin predicted by. Assuming RH is true, It behaves 260 00:12:25,240 --> 00:12:27,320 exactly as expected. And this is where the tools 261 00:12:27,320 --> 00:12:29,080 matter. That sippy check is great for 262 00:12:29,080 --> 00:12:31,160 understanding the principle takes a second, but it won't 263 00:12:31,160 --> 00:12:33,200 scale to checking X equal 10/20, right? 264 00:12:33,280 --> 00:12:35,640 Absolutely not. Sippy is doing exact symbolic 265 00:12:35,640 --> 00:12:38,480 math, which gets very slow to go to really large numbers like 266 00:12:38,480 --> 00:12:40,240 1020. You need to switch gears to what 267 00:12:40,440 --> 00:12:44,840 you switch to highly optimized C libraries that are specifically 268 00:12:44,840 --> 00:12:47,920 designed for prime counting. The source mentioned prime count 269 00:12:47,920 --> 00:12:50,200 as an example. These trade the symbolic 270 00:12:50,200 --> 00:12:53,480 perfection for raw speed using clever algorithms. 271 00:12:53,520 --> 00:12:55,720 And how fast are those checking up to 1020? 272 00:12:56,280 --> 00:12:57,560 That sounds like it should take ages. 273 00:12:57,600 --> 00:12:59,880 That's the amazing part. The source notes that checking 274 00:12:59,880 --> 00:13:04,720 the prime count error relation up to X = 1020 takes only about 275 00:13:04,720 --> 00:13:08,040 30 seconds using prime count. 30 seconds to check something 276 00:13:08,040 --> 00:13:10,560 related to the Ryman hypothesis. Up to 10 to the 20. 277 00:13:10,600 --> 00:13:12,960 Yep. Because the relationship itself, 278 00:13:12,960 --> 00:13:15,920 the comparison between CLEX and legs and the bound, is 279 00:13:15,920 --> 00:13:18,600 computationally simple once you have the prime count, and the 280 00:13:18,600 --> 00:13:21,920 prime counting algorithms are incredibly refined, it's a 281 00:13:21,920 --> 00:13:25,120 fantastic example of how computation provides extremely 282 00:13:25,120 --> 00:13:27,960 strong coherence very quickly. OK, let's switch gears 283 00:13:27,960 --> 00:13:31,200 completely from the smooth world of prime numbers and complex 284 00:13:31,200 --> 00:13:34,600 functions to the jagged, discrete world of computational 285 00:13:34,600 --> 00:13:38,680 complexity, P versus NP. Ah, yes, the question of 286 00:13:38,680 --> 00:13:41,360 efficient computation. Can every problem whose solution 287 00:13:41,360 --> 00:13:44,960 is easy to check, that's NP nondeterministic polynomial 288 00:13:44,960 --> 00:13:48,640 time, also be easy to solve? That's P polynomial time. 289 00:13:48,800 --> 00:13:53,440 If P did equal NP, the consequences would be, well, 290 00:13:53,480 --> 00:13:55,640 world changing. We're absolutely transformative. 291 00:13:55,840 --> 00:13:58,280 It would mean things we currently think are impossibly 292 00:13:58,280 --> 00:14:01,760 hard, like breaking most modern encryption, finding optimal 293 00:14:01,760 --> 00:14:04,120 solutions for logistics nightmares like the traveling 294 00:14:04,120 --> 00:14:07,840 salesman problem, or solving complex protein folding. 295 00:14:08,320 --> 00:14:11,960 All could potentially be solved efficiently in polynomial time. 296 00:14:12,480 --> 00:14:15,880 But the gut feeling, the intuition in the field, leans 297 00:14:15,880 --> 00:14:18,320 heavily the other way. The overwhelming consensus, 298 00:14:18,320 --> 00:14:21,720 yeah, is that P does not equal NP, that there really are 299 00:14:21,720 --> 00:14:24,560 problems that are easy to check but fundamentally hard to solve. 300 00:14:24,760 --> 00:14:27,600 So for the A&V strategy, we actually start by assuming the 301 00:14:27,600 --> 00:14:31,040 opposite of PNP. We assume P is not equal to NP. 302 00:14:31,040 --> 00:14:32,640 Exactly. That's our starting assumption 303 00:14:32,640 --> 00:14:37,000 for the test PNP. OK, if PNP is true, what's the 304 00:14:37,000 --> 00:14:38,880 computable consequence? What should we be able to 305 00:14:38,880 --> 00:14:41,920 measure? If PNP, then these hard NP 306 00:14:41,920 --> 00:14:45,080 complete problems like the classic 3 SAT problem must 307 00:14:45,080 --> 00:14:48,360 exhibit runtimes that scale exponentially or at least super 308 00:14:48,360 --> 00:14:50,960 polynomially as the problem size increases. 309 00:14:51,240 --> 00:14:53,920 They can't scale nicely like polynomial problems. 310 00:14:54,000 --> 00:14:56,360 So we're looking for that exponential blow up in 311 00:14:56,360 --> 00:14:58,120 computation time. That's the signal. 312 00:14:58,120 --> 00:14:59,520 That's the signal we're hunting for. 313 00:14:59,520 --> 00:15:02,480 The experimental design is quite clever here. 314 00:15:02,520 --> 00:15:04,840 How does it work? First you need pest cases. 315 00:15:05,040 --> 00:15:08,720 You generate random instances of a known NP complete problem, 316 00:15:08,760 --> 00:15:11,960 typically three SAT. That's a Boolean satisfiability 317 00:15:11,960 --> 00:15:13,280 problem. Trying to find if there's a 318 00:15:13,280 --> 00:15:16,360 combination of true, false inputs that makes a logical 319 00:15:16,360 --> 00:15:17,920 formula true. Exactly. 320 00:15:17,920 --> 00:15:21,360 But you don't just generate any random three SAT instance, you 321 00:15:21,360 --> 00:15:24,520 generate them very carefully, right at a specific density. 322 00:15:24,520 --> 00:15:25,600 Density. What does that mean? 323 00:15:25,640 --> 00:15:28,360 You mentioned a number 4.26. Right. 324 00:15:28,760 --> 00:15:32,160 It turns out that for random three SAT, the hardness depends 325 00:15:32,160 --> 00:15:35,400 critically on the ratio of clauses constraints to 326 00:15:35,400 --> 00:15:38,280 variables. If you have too few clauses, the 327 00:15:38,280 --> 00:15:40,920 problem is easy. Too many, and it's usually 328 00:15:40,920 --> 00:15:44,360 impossible, unsatisfiable, and often easy to prove so. 329 00:15:44,600 --> 00:15:48,560 But right around that magic number, about 4.26 clauses per 330 00:15:48,560 --> 00:15:50,400 variable. That's the sweet spot, the 331 00:15:50,400 --> 00:15:52,600 so-called phase transition region. 332 00:15:53,080 --> 00:15:55,760 These instances are computationally the hardest. 333 00:15:55,960 --> 00:15:59,120 They're often satisfiable, but finding the solution takes the 334 00:15:59,120 --> 00:16:01,640 longest time. They're the most likely to show 335 00:16:01,640 --> 00:16:04,200 that exponential scaling if PNP is true. 336 00:16:04,480 --> 00:16:08,640 So you generate these maximally difficult instances, then what? 337 00:16:08,880 --> 00:16:11,560 Then you throw them at a powerful SAT solver. 338 00:16:12,000 --> 00:16:15,520 The source mentions Kisat as a modern high performance solver, 339 00:16:15,520 --> 00:16:18,040 often used via a Python wrapper like Pisat and. 340 00:16:18,040 --> 00:16:19,800 You measure how long it takes to solve them. 341 00:16:20,080 --> 00:16:22,400 Different sizes, different N. Precisely, you run it for 342 00:16:22,400 --> 00:16:27,040 instances with say N50 variables N 100, N 150, N 200 and so on, 343 00:16:27,200 --> 00:16:30,400 carefully logging the CPU time for each successful solve. 344 00:16:30,560 --> 00:16:34,680 OK, now you have data problem size N versus time. 345 00:16:35,000 --> 00:16:36,880 How do you test the PNB assumption? 346 00:16:36,880 --> 00:16:39,680 You do a statistical analysis. You try to fit that time data to 347 00:16:39,680 --> 00:16:41,280 two different mathematical models. 348 00:16:41,280 --> 00:16:42,960 Model 1. Polynomial scaling. 349 00:16:43,000 --> 00:16:46,280 Right, that would look like log time is proportional to log NA 350 00:16:46,320 --> 00:16:49,120 straight line on a log log plot. This would support PNB. 351 00:16:49,160 --> 00:16:50,800 Model 2. Exponential scaling. 352 00:16:51,040 --> 00:16:53,760 Yeah, that would look like log time is proportional to N 353 00:16:53,760 --> 00:16:56,000 itself. A straight line on a semi log 354 00:16:56,000 --> 00:17:01,640 plot log time versus linear N This supports PANP. 355 00:17:01,640 --> 00:17:04,720 And you see which model fits the actual runtime data better. 356 00:17:04,839 --> 00:17:07,160 Exactly. You look at measures like the 357 00:17:07,160 --> 00:17:09,880 R-squared value, how well the curve fits the points. 358 00:17:09,920 --> 00:17:14,720 So what do the results show for these tests run up to say and 359 00:17:14,720 --> 00:17:16,760 200 variables? The results are pretty clear 360 00:17:16,760 --> 00:17:17,800 cut. According to the source 361 00:17:17,800 --> 00:17:21,280 material, the exponential model provides a much better fit to 362 00:17:21,280 --> 00:17:23,119 the data than the polynomial model. 363 00:17:23,160 --> 00:17:25,079 Really strong support for PNP then. 364 00:17:25,200 --> 00:17:28,480 Very strong empirical support. The analysis even estimates the 365 00:17:28,480 --> 00:17:31,240 constant in the exponent. It found that log time is 366 00:17:31,240 --> 00:17:34,520 roughly proportional to .03 * n. What? 367 00:17:34,520 --> 00:17:37,640 Does that .03 mean practically? It means for every single 368 00:17:37,640 --> 00:17:41,120 variable you add to the problem, the runtime increases by a 369 00:17:41,120 --> 00:17:45,960 multiplicative factor of E 0.03, which is a about 3% increase. 370 00:17:46,320 --> 00:17:49,440 Doesn't sound like much, but it compounds exponentially at 100 371 00:17:49,440 --> 00:17:52,120 variables and the time multiplies by E3, which is about 372 00:17:52,120 --> 00:17:55,120 20 times harder. Wow, OK, that's the numerical 373 00:17:55,120 --> 00:17:58,560 signature of that exponential barrier we expect if PANP. 374 00:17:58,720 --> 00:18:01,520 It absolutely is. It's strongly reinforces the 375 00:18:01,520 --> 00:18:04,640 consensus view. In this check up to any 200. 376 00:18:04,880 --> 00:18:07,680 How long does that take? The source suggests this kind of 377 00:18:07,680 --> 00:18:10,920 test, generating the instances and running the solver across 378 00:18:10,920 --> 00:18:14,880 these sizes might take around 10 seconds on a modern machine. 10 379 00:18:14,880 --> 00:18:18,040 seconds to get strong numerical evidence supporting one of the 380 00:18:18,040 --> 00:18:20,480 biggest questions in computer science and mathematics. 381 00:18:20,480 --> 00:18:23,960 It's pretty efficient, yeah. Now if the polynomial model had 382 00:18:23,960 --> 00:18:26,520 fit better, that would have been earth shattering. 383 00:18:26,520 --> 00:18:28,280 It would have immediately told researchers. 384 00:18:28,280 --> 00:18:31,760 Hold on, maybe our intuition about PVSNP is wrong. 385 00:18:32,000 --> 00:18:34,080 Start looking for polynomial algorithms. 386 00:18:34,360 --> 00:18:36,680 But it didn't. And does this exponential 387 00:18:36,680 --> 00:18:39,680 scaling hold up as you go to a much larger in? 388 00:18:39,680 --> 00:18:42,480 That's the thing, the exponential nature becomes even 389 00:18:42,480 --> 00:18:46,520 more apparent while the N 2 100 check takes seconds, stealing up 390 00:18:46,520 --> 00:18:50,160 to say and 10,000 variables, which is common in serious 391 00:18:50,160 --> 00:18:52,480 research. That 3% per variable really 392 00:18:52,480 --> 00:18:54,680 starts to add up. Oh yeah, the runtime balloons 393 00:18:54,680 --> 00:18:57,720 into minutes or even hours. Even using parallel computing, 394 00:18:58,160 --> 00:19:01,200 the difficulty scales exactly as the PMP assumption predicts. 395 00:19:01,200 --> 00:19:04,280 For these hard instances, the coherence is very strong. 396 00:19:04,520 --> 00:19:06,600 OK, let's shift again. We've done numbers, we've done 397 00:19:06,600 --> 00:19:10,240 computation theory. Now let's talk about fluid 398 00:19:10,240 --> 00:19:16,000 dynamics, water, air turbulence, the Navier Stokes equations. 399 00:19:16,160 --> 00:19:18,520 Right. These equations are fundamental 400 00:19:18,520 --> 00:19:22,160 to describing how fluids move everything from, you know, water 401 00:19:22,160 --> 00:19:25,480 flowing in a pipe to weather patterns to the air flowing over 402 00:19:25,480 --> 00:19:28,120 an airplane wing. They're incredibly important in 403 00:19:28,120 --> 00:19:31,160 science and engineering. But there's a catch a Millennium 404 00:19:31,160 --> 00:19:32,880 problem hidden in there. There is. 405 00:19:32,880 --> 00:19:35,240 It's about the smoothness of the solutions. 406 00:19:35,600 --> 00:19:40,400 The question is if you start with a perfectly smooth, well 407 00:19:40,400 --> 00:19:44,240 behaved initial state for the fluid, say perfectly defined 408 00:19:44,240 --> 00:19:46,760 velocities everywhere. With perfectly Stillwater or a 409 00:19:46,760 --> 00:19:49,080 very general predictable flow. Exactly. 410 00:19:49,200 --> 00:19:51,840 Will the solution, the way the fluid evolves according to the 411 00:19:51,840 --> 00:19:54,640 equations always remain smooth and well behaved for all future 412 00:19:54,640 --> 00:19:57,200 times? Or could something catastrophic 413 00:19:57,200 --> 00:19:59,360 happen? Could a singularity form a point 414 00:19:59,360 --> 00:20:01,880 in space and time where the fluid velocity or pressure 415 00:20:01,880 --> 00:20:04,960 suddenly becomes infinite? Or maybe the derivatives become 416 00:20:04,960 --> 00:20:06,920 undefined? This is called a blow up. 417 00:20:07,200 --> 00:20:08,400 Blow up? That sounds bad. 418 00:20:08,520 --> 00:20:12,200 It would be mathematically profound if blow up can occur 419 00:20:12,200 --> 00:20:15,200 from smooth initial data. It means the Naver Stokes 420 00:20:15,200 --> 00:20:19,800 equations might be well incomplete or breakdown under 421 00:20:19,800 --> 00:20:22,560 certain conditions. It would mean our fundamental 422 00:20:22,560 --> 00:20:25,120 model of fluid flow has a hidden flaw. 423 00:20:25,680 --> 00:20:28,680 So the Millennium conjecture is that blow up doesn't happen, 424 00:20:29,040 --> 00:20:32,680 that smooth initial conditions always lead to smooth solutions 425 00:20:32,680 --> 00:20:34,240 globally. That's the conjecture. 426 00:20:34,240 --> 00:20:37,320 Basically global existence of smooth solutions. 427 00:20:37,320 --> 00:20:40,560 OK, so how does assume and verify tackle this? 428 00:20:40,920 --> 00:20:44,200 We assume smoothness holds. We assume the conjecture is true 429 00:20:44,240 --> 00:20:47,560 that solutions stay smooth. And the test is to actively hunt 430 00:20:47,560 --> 00:20:50,640 for a situation where that assumption breaks to try and 431 00:20:50,640 --> 00:20:52,320 provoke a blow up in a simulation. 432 00:20:52,360 --> 00:20:55,000 Precisely, you're using computation to search for the 433 00:20:55,000 --> 00:20:57,280 spontaneous formation of a singularity. 434 00:20:57,520 --> 00:21:00,080 How do you even set that up? Simulating fluid sounds 435 00:21:00,080 --> 00:21:01,960 complicated. It is very complicated. 436 00:21:02,040 --> 00:21:05,280 The experimental design usually involves simulating the fluid in 437 00:21:05,280 --> 00:21:07,640 a simplified domain, often a 3D Taurus A. 438 00:21:07,640 --> 00:21:10,920 Taurus like a doughnut shape. It's mathematically convenient. 439 00:21:11,480 --> 00:21:16,400 A Taurus T333 is a box where the opposite faces are connected, so 440 00:21:16,400 --> 00:21:19,280 if fluid flows out one side, it comes back in the opposite side. 441 00:21:19,640 --> 00:21:22,840 It avoids having to deal with complex boundary conditions at 442 00:21:22,840 --> 00:21:26,400 walls, letting you focus purely on the internal dynamics of the 443 00:21:26,400 --> 00:21:28,680 fluid itself. OK, so simulate on a Taurus, 444 00:21:28,680 --> 00:21:30,960 what do you start with? You start with random but 445 00:21:30,960 --> 00:21:34,080 mathematically smooth initial velocity fields. 446 00:21:34,520 --> 00:21:37,640 Then you let the Naver Stokes equations evolve the state 447 00:21:37,640 --> 00:21:40,000 forward in time using numerical methods. 448 00:21:40,000 --> 00:21:42,120 And you need good numerical methods, presumably. 449 00:21:42,120 --> 00:21:45,240 Extremely good ones. The source specifically mentions 450 00:21:45,240 --> 00:21:47,880 using high order spectral methods as implemented in 451 00:21:47,880 --> 00:21:50,840 libraries like Daedalus. Spectral methods are known for 452 00:21:50,840 --> 00:21:53,920 being very accurate for smooth flows, which is crucial if 453 00:21:53,920 --> 00:21:55,600 you're trying to see if smoothness breaks. 454 00:21:55,640 --> 00:21:59,120 Right, so the simulation rent, what are you watching for? 455 00:21:59,120 --> 00:22:02,080 What's the red flag that screams potential blow up? 456 00:22:02,280 --> 00:22:05,560 The key metric to monitor is often the maximum absolute 457 00:22:05,560 --> 00:22:07,320 vorticity. Vorticity. 458 00:22:07,520 --> 00:22:09,760 What's that? Vorticity measures the local 459 00:22:09,760 --> 00:22:13,200 spinning motion of the fluid. Think of tiny whirlpools or 460 00:22:13,200 --> 00:22:16,160 eddies. If energy starts to concentrate 461 00:22:16,160 --> 00:22:19,320 intensely in a very small region, which is what you might 462 00:22:19,320 --> 00:22:22,640 expect near a singularity, the vorticity in that region would 463 00:22:22,640 --> 00:22:25,280 spike dramatically. So if vorticity goes through the 464 00:22:25,280 --> 00:22:28,680 roof, that's the warning sign. That's the main indicator. 465 00:22:28,880 --> 00:22:33,360 A common criterion used in these AMV test is if the maximum 466 00:22:33,360 --> 00:22:36,600 corticity anywhere in the simulation suddenly double s or 467 00:22:36,600 --> 00:22:40,320 triples or increases by some large factor over a very short 468 00:22:40,320 --> 00:22:44,320 time, that signals you might be witnessing the onset of a blow 469 00:22:44,360 --> 00:22:45,480 up. OK, makes sense. 470 00:22:45,480 --> 00:22:47,680 So people run these sophisticated simulations. 471 00:22:48,760 --> 00:22:50,760 What do they find? Does blow up happen often? 472 00:22:50,760 --> 00:22:52,080 This is where it gets interesting. 473 00:22:52,120 --> 00:22:55,680 The overwhelming result from countless simulations is no 474 00:22:55,920 --> 00:22:58,280 intrinsic blow up from smooth initial data. 475 00:22:58,280 --> 00:23:00,640 Seems to be very rare, if it happens at all in these 476 00:23:00,640 --> 00:23:02,640 settings. So the simulations support the 477 00:23:02,640 --> 00:23:04,400 conjecture. They support smoothness. 478 00:23:04,400 --> 00:23:07,160 They provide strong empirical coherence for the smoothness 479 00:23:07,160 --> 00:23:09,400 conjecture. Yes, the solutions tend to 480 00:23:09,400 --> 00:23:11,960 remain well behaved. I sense a but coming. 481 00:23:12,040 --> 00:23:15,200 There's a huge but and it's computational cost. 482 00:23:15,800 --> 00:23:19,240 This problem highlights the limitations of AMV perhaps more 483 00:23:19,240 --> 00:23:22,000 brutally than any other. The scalability issue again. 484 00:23:22,000 --> 00:23:25,240 Massively so. You can run a course simulation, 485 00:23:25,240 --> 00:23:30,480 maybe a 64 by 64 by 64 grid on a decent workstation in minutes or 486 00:23:30,480 --> 00:23:32,520 hours and it will likely look smooth. 487 00:23:32,840 --> 00:23:34,600 But that's not enough to be sure. 488 00:23:34,600 --> 00:23:37,800 Not nearly enough. Turbulence involves interactions 489 00:23:37,800 --> 00:23:42,720 across a vast range of scales, from large eddies down to tiny, 490 00:23:42,720 --> 00:23:45,640 tiny dissipative structures. To be confident you're not 491 00:23:45,800 --> 00:23:49,080 seeing a singularity that forms only at very small scales, you 492 00:23:49,080 --> 00:23:52,200 need incredibly high resolution. Like what kind of resolution? 493 00:23:52,200 --> 00:23:54,920 We're talking grids like 5-2 by 5 by two or five by 5 two or 494 00:23:54,920 --> 00:23:58,400 even 10243 or higher resolving those tiny scales. 495 00:23:58,400 --> 00:24:00,840 And simulating that. Takes enormous resources. 496 00:24:00,840 --> 00:24:04,000 We're talking hours or even days running on large clusters with 497 00:24:04,000 --> 00:24:07,200 many GPU's working in parallel. It's pushing the limits of even 498 00:24:07,200 --> 00:24:10,360 supercomputers. So AMV tells us smoothness is 499 00:24:10,360 --> 00:24:12,680 the norm and simulations we can afford to run. 500 00:24:12,800 --> 00:24:13,920 Right. It builds confidence. 501 00:24:14,080 --> 00:24:17,400 But it hasn't definitively ruled out that a blow up couldn't 502 00:24:17,400 --> 00:24:20,960 happen in a scenario requiring resolution or simulation time 503 00:24:21,200 --> 00:24:22,800 that's still beyond our current reach. 504 00:24:22,840 --> 00:24:24,840 Exactly. We're compute bound. 505 00:24:25,040 --> 00:24:28,120 The empirical evidence leans towards smoothness, but the 506 00:24:28,120 --> 00:24:31,360 final word requires either a theoretical breakthrough or 507 00:24:31,360 --> 00:24:34,320 vastly more powerful computation than we have today. 508 00:24:34,560 --> 00:24:37,680 OK, so we've seen Ryman Hypothesis and P Visas MP where 509 00:24:37,680 --> 00:24:42,080 AMV gets pretty quick strong coherence, then Navier Stokes 510 00:24:42,080 --> 00:24:45,720 where it supports the conjecture but hits a massive computational 511 00:24:45,720 --> 00:24:47,560 wall. A very different feel for each, 512 00:24:47,600 --> 00:24:49,200 yeah? Let's apply that same detailed 513 00:24:49,200 --> 00:24:51,680 A&V lens to the other three Millennium problems. 514 00:24:51,680 --> 00:24:53,680 We need to make sure we unpack the jargon here, too. 515 00:24:53,920 --> 00:24:56,720 There's Birch and Swinnerton, Dyer, Hodge and Yang Mills. 516 00:24:56,880 --> 00:24:59,720 Right, covering number theory, algebraic geometry, and 517 00:24:59,720 --> 00:25:02,920 fundamental physics. The AMD strategy translating the 518 00:25:02,920 --> 00:25:06,320 abstract idea into a computable test applies to all of them, but 519 00:25:06,320 --> 00:25:09,880 the specifics differ greatly. Hashtag tag, tag tag tag, The 520 00:25:09,880 --> 00:25:11,560 Birch and Swinnerton Dyer conjecture. 521 00:25:11,720 --> 00:25:13,960 BSD. Let's start with BSD, Birch and 522 00:25:13,960 --> 00:25:15,640 Swinnerton Dyer. This one's back in number 523 00:25:15,640 --> 00:25:18,280 theory, right? And it's about elliptic curves. 524 00:25:18,280 --> 00:25:21,120 What are those in simple terms? Yeah, number theory again. 525 00:25:21,800 --> 00:25:24,400 An elliptic curve isn't actually an ellipse. 526 00:25:24,400 --> 00:25:26,720 It's typically defined by a cubic equation. 527 00:25:26,800 --> 00:25:30,240 Something like Y2 cools by 3 + X + b. 528 00:25:31,120 --> 00:25:34,280 They are smooth curves, and mathematicians are fascinated by 529 00:25:34,280 --> 00:25:37,960 their points, specifically points where both X&Y are 530 00:25:37,960 --> 00:25:41,440 rational numbers fractions. Finding fractional solutions to 531 00:25:41,440 --> 00:25:43,200 this cubic equation. Exactly. 532 00:25:43,480 --> 00:25:46,960 The BSD conjecture tries to connect two very different ways 533 00:25:46,960 --> 00:25:50,000 of measuring the complexity or richness of these rational 534 00:25:50,000 --> 00:25:51,920 points on the curve. Two different measures. 535 00:25:51,920 --> 00:25:53,880 What are they? On one side you have the 536 00:25:53,880 --> 00:25:56,440 algebraic rank. This is, roughly speaking, a 537 00:25:56,440 --> 00:25:59,080 measure of how many independent rational points there are. 538 00:25:59,240 --> 00:26:03,680 It's about counting 0 12A whole number describing the structure 539 00:26:03,680 --> 00:26:06,120 of the rational points. OK, algebraic rank accounting 540 00:26:06,120 --> 00:26:07,080 points. What's the other side? 541 00:26:07,120 --> 00:26:08,880 The other side is the analytic rank. 542 00:26:09,120 --> 00:26:11,880 This comes from calculus, specifically complex analysis. 543 00:26:12,120 --> 00:26:15,120 You associate a special function called an L series to the 544 00:26:15,120 --> 00:26:17,240 elliptic curve. It's a bit like the Raymond Zeta 545 00:26:17,240 --> 00:26:20,000 function. The analytic rank is defined by 546 00:26:20,000 --> 00:26:23,600 how this L series behaves right at a specific point, S1. 547 00:26:23,920 --> 00:26:27,360 Specifically, it's the order of the zero of the L series at S1. 548 00:26:27,520 --> 00:26:29,880 Whoa. OK, SO1 rank is from counting 549 00:26:29,880 --> 00:26:32,520 fractional points. The other is from the behavior 550 00:26:32,520 --> 00:26:36,320 of a fancy calculus function and BSD claims. 551 00:26:36,680 --> 00:26:39,800 BSD claims these two ranks derived in completely different 552 00:26:39,800 --> 00:26:42,360 mathematical universes are always equal. 553 00:26:42,680 --> 00:26:45,320 The analytic rank equals the algebraic rank. 554 00:26:45,520 --> 00:26:47,080 That sounds like a really deep connection. 555 00:26:47,360 --> 00:26:49,400 How on earth do you test that with A&V? 556 00:26:49,400 --> 00:26:52,680 This is a perfect case for A&V via massive data checking. 557 00:26:52,680 --> 00:26:56,520 We have large databases of elliptic curves like the Cremona 558 00:26:56,520 --> 00:26:58,120 database but. People have already cataloged 559 00:26:58,120 --> 00:27:00,320 millions of these curves. Oh yes, their properties 560 00:27:00,320 --> 00:27:02,600 computed. We use computational algebra 561 00:27:02,600 --> 00:27:04,760 systems. Sage Math is a very powerful the 562 00:27:04,760 --> 00:27:07,760 open source one mentioned that can access these databases. 563 00:27:07,840 --> 00:27:09,840 And the test is. For a given curve in the 564 00:27:09,840 --> 00:27:13,040 database, you compute its algebraic rank, which involves 565 00:27:13,040 --> 00:27:16,000 sophisticated algorithms for finding points, and you compute 566 00:27:16,000 --> 00:27:19,360 its analytic rank, which involves numerically evaluating 567 00:27:19,360 --> 00:27:21,880 the L series and its derivatives near S1. 568 00:27:22,000 --> 00:27:25,680 And you just check does integer A equal integer B. 569 00:27:25,840 --> 00:27:28,880 That's the core empirical check. You do this for as many curves 570 00:27:28,880 --> 00:27:32,040 as possible, especially those with low conductor, the measure 571 00:27:32,040 --> 00:27:34,120 of complexity. What's the verdict from these 572 00:27:34,120 --> 00:27:37,280 massive checks? The coherence is overwhelming. 573 00:27:37,680 --> 00:27:40,240 The source material confirms that millions of curves have 574 00:27:40,240 --> 00:27:42,600 been tested over the decades since the conjecture was 575 00:27:42,600 --> 00:27:45,920 proposed, and for all the curves where both ranks could be 576 00:27:45,920 --> 00:27:50,200 reliably computed, they match. No counterexamles found. 577 00:27:50,280 --> 00:27:52,960 None have been confirmed. The computational evidence is 578 00:27:52,960 --> 00:27:55,600 incredibly strong in favor of BSD. 579 00:27:56,160 --> 00:27:59,520 A single check on one curve might take seconds or minutes 580 00:27:59,520 --> 00:28:02,880 using Sage math, but the cumulative result built over 581 00:28:02,880 --> 00:28:05,160 years provides enormous confidence. 582 00:28:05,360 --> 00:28:07,640 Hashtag, tag tag hashtag The Hodge conjecture. 583 00:28:07,840 --> 00:28:10,000 OK, Next up, the Hodge conjecture. 584 00:28:10,320 --> 00:28:13,080 This one sounds intimidating. Algebraic geometry. 585 00:28:13,280 --> 00:28:16,440 Yeah, this one dives deep into the geometry of shapes defined 586 00:28:16,440 --> 00:28:19,240 by polynomial equations. These shapes are called 587 00:28:19,240 --> 00:28:21,960 algebraic varieties. The Hodge conjecture is often 588 00:28:21,960 --> 00:28:24,400 considered one of the most abstract and difficult of the 589 00:28:24,400 --> 00:28:26,440 seven problems. What's the fundamental idea it's 590 00:28:26,440 --> 00:28:28,720 trying to capture? OK, let's try to simplify. 591 00:28:29,480 --> 00:28:32,880 When you study the topology of these geometric shapes, you can 592 00:28:32,880 --> 00:28:36,280 break down certain mathematical structures associated with them. 593 00:28:36,280 --> 00:28:40,160 They're cohomology groups into pieces called Hodge glasses. 594 00:28:40,600 --> 00:28:43,480 Some pieces depend on the shapes complex structure. 595 00:28:43,680 --> 00:28:47,000 OK, complex structure pieces. Then, separately, you can study 596 00:28:47,000 --> 00:28:49,600 geometric substructures within the main shape that are 597 00:28:49,600 --> 00:28:52,320 themselves defined by polynomial equations. 598 00:28:52,760 --> 00:28:54,400 These are called algebraic cycles. 599 00:28:54,400 --> 00:28:56,880 Think of lines or curves lying on a surface. 600 00:28:57,320 --> 00:29:00,400 So algebraic cycles are polynomial substructures. 601 00:29:00,480 --> 00:29:03,880 Right, the Hodge conjecture focuses on a specific type of 602 00:29:03,880 --> 00:29:06,640 Hodge class, the ones called Hodge PP classes. 603 00:29:07,320 --> 00:29:10,720 It claims that these specific topological pieces which seem 604 00:29:10,720 --> 00:29:14,400 tied to the complex structure are actually always combinations 605 00:29:14,400 --> 00:29:17,160 of these simpler, purely algebraic cycles. 606 00:29:17,280 --> 00:29:20,480 It's saying something seemingly complex and analytic is actually 607 00:29:20,480 --> 00:29:23,040 built from simpler algebraic blocks. 608 00:29:23,240 --> 00:29:24,440 That's a good way to think about it. 609 00:29:24,520 --> 00:29:27,520 It connects the deep analytic topology of the space back to 610 00:29:27,520 --> 00:29:30,000 the polynomial equations that defined it in the first place. 611 00:29:30,320 --> 00:29:32,640 How do you even begin to test something like that 612 00:29:32,640 --> 00:29:35,520 computationally with A&V? You can't test it for all 613 00:29:35,520 --> 00:29:38,840 varieties of course. The A&V approach here is to test 614 00:29:38,840 --> 00:29:41,440 it on specific manageable examples. 615 00:29:41,720 --> 00:29:44,680 You look at relatively low dimensional or low degree of 616 00:29:44,680 --> 00:29:46,920 varieties of what the source mentions. 617 00:29:46,920 --> 00:29:49,960 Things like cubic hyper surfaces, shapes defined by 618 00:29:49,960 --> 00:29:54,000 degree 3, polynomials in five dimensional projective space 619 00:29:54,760 --> 00:29:56,680 five. You need specialized software 620 00:29:56,680 --> 00:29:59,240 for this kind of thing. Software Macaulay 2 is a 621 00:29:59,240 --> 00:30:02,080 standard powerhouse in computational algebraic 622 00:30:02,080 --> 00:30:03,960 geometry. You might use it through a 623 00:30:03,960 --> 00:30:06,800 Python interface to make it easier to script the tests. 624 00:30:07,160 --> 00:30:08,680 And what's the actual computation? 625 00:30:08,680 --> 00:30:11,760 What numbers are you comparing? You compute properties related 626 00:30:11,760 --> 00:30:15,040 to both sides of the conjecture. For your chosen variety, you'd 627 00:30:15,040 --> 00:30:18,360 compute the structure of its Hodge classes, often summarized 628 00:30:18,360 --> 00:30:19,920 in something called the Hodge diamond. 629 00:30:20,160 --> 00:30:21,760 I get the Hodge. Diamond, and then you compute 630 00:30:21,760 --> 00:30:24,160 the structure of its algebraic cycles represented by 631 00:30:24,160 --> 00:30:26,160 mathematical objects called Chow groups. 632 00:30:26,760 --> 00:30:30,120 The A&V test then essentially checks if the part of the Hodge 633 00:30:30,120 --> 00:30:33,680 diamond corresponding to the P classes matches the structure 634 00:30:33,680 --> 00:30:36,680 predicted by the Chow groups. Does the dimension match? 635 00:30:36,920 --> 00:30:40,040 Do the generators correspond? It's a structural comparison. 636 00:30:40,160 --> 00:30:43,760 Exactly, and the source notes that the conjecture is known to 637 00:30:43,760 --> 00:30:47,200 hold true for many low dimensional and low degree cases 638 00:30:47,200 --> 00:30:49,120 where these computations are feasible. 639 00:30:49,280 --> 00:30:51,080 How long does a check like that take? 640 00:30:51,080 --> 00:30:54,240 For a single, well chosen small example that's computable, 641 00:30:54,440 --> 00:30:57,920 running the Macaulay 2 scripts might take seconds or minutes. 642 00:30:58,240 --> 00:31:01,240 It verifies the principle holds in concrete cases. 643 00:31:01,760 --> 00:31:05,200 The difficulty, like with Navier Stokes, is scaling this up to 644 00:31:05,200 --> 00:31:08,720 more complex varieties where the computations become intractable 645 00:31:08,720 --> 00:31:11,280 very quickly. But for the testable cases, the 646 00:31:11,280 --> 00:31:15,440 coherence holds #tach Hashtag, the Yang Mills existence, and 647 00:31:15,440 --> 00:31:17,760 Mascat. OK, our last problem, Yang Mills 648 00:31:17,760 --> 00:31:21,160 existence and Mascat. It sounds like physics, Quantum 649 00:31:21,160 --> 00:31:23,840 physics. Exactly this one is fundamental 650 00:31:23,840 --> 00:31:26,000 to the standard Model of particle physics. 651 00:31:26,320 --> 00:31:28,800 Yang Mills theories are the mathematical framework 652 00:31:28,800 --> 00:31:32,360 describing fundamental forces, particularly the strong nuclear 653 00:31:32,360 --> 00:31:35,480 force, which holds atomic nuclei together, and the weak nuclear 654 00:31:35,480 --> 00:31:36,920 force. And the Millennium problem has 655 00:31:36,920 --> 00:31:39,520 two parts, existence and mass gap. 656 00:31:39,760 --> 00:31:41,680 That's right. The existence part is about 657 00:31:41,680 --> 00:31:44,560 proving that the quantum version of Yang Mills theory is 658 00:31:44,560 --> 00:31:46,800 mathematically well defined and consistent. 659 00:31:47,160 --> 00:31:50,920 The mass gap part is a specific prediction of such a consistent 660 00:31:50,920 --> 00:31:52,480 theory. What is the mass gap? 661 00:31:52,480 --> 00:31:54,200 Predict. It predicts that the lightest 662 00:31:54,200 --> 00:31:56,800 particles predicted by the theory excitations of the 663 00:31:56,800 --> 00:32:00,200 quantum field, sometimes called glue balls in the context of the 664 00:32:00,200 --> 00:32:04,560 strong force, must have a strictly positive minimum mass. 665 00:32:05,000 --> 00:32:09,000 There must be a gap between the vacuum state 0 energy and the 666 00:32:09,000 --> 00:32:11,120 energy of the lightest possible particle. 667 00:32:11,600 --> 00:32:13,400 Let's call this minimum mass delta. 668 00:32:13,400 --> 00:32:16,760 The conjecture is 0. Why is that important? 669 00:32:16,760 --> 00:32:20,560 What if the mass gap was 0? If zero were zero, the theory 670 00:32:20,560 --> 00:32:23,600 would predict massless force carriers interacting over long 671 00:32:23,600 --> 00:32:25,920 distances. But we know the strong nuclear 672 00:32:25,920 --> 00:32:27,640 force is short range. It doesn't leak out of the 673 00:32:27,640 --> 00:32:31,000 nucleus. A positive mass gap is essential 674 00:32:31,000 --> 00:32:33,160 for explaining this short range nature. 675 00:32:33,200 --> 00:32:36,680 OK so the AMV test here assumes the theory exists and has a mass 676 00:32:36,680 --> 00:32:38,880 gap yes? And the computer to consequence 677 00:32:39,000 --> 00:32:42,920 is that simulations should actually find a non 0 mass for 678 00:32:42,920 --> 00:32:45,320 the lightest particle. That's the core idea. 679 00:32:45,600 --> 00:32:48,880 You need to simulate the quantum field theory and extract the 680 00:32:48,880 --> 00:32:50,800 particle masses from the simulation. 681 00:32:51,160 --> 00:32:53,600 How on earth do you simulate quantum field theory? 682 00:32:53,840 --> 00:32:55,520 That sounds even harder than fluids. 683 00:32:55,520 --> 00:32:57,960 It requires the most computationally intensive 684 00:32:57,960 --> 00:33:02,040 technique we've discussed, lattice QCD or lattice quantum 685 00:33:02,040 --> 00:33:04,400 chromo dynamics for the strong force case. 686 00:33:04,400 --> 00:33:06,800 Lattice like a grid. Exactly. 687 00:33:06,800 --> 00:33:09,880 You can't simulate continuous space-time directly, so you 688 00:33:09,880 --> 00:33:14,360 approximate it with a discrete 4 dimensional grid, the lattice. 689 00:33:14,800 --> 00:33:17,200 You then simulate the interactions of the fundamental 690 00:33:17,200 --> 00:33:20,720 fields like gluons. For the strong force on the 691 00:33:20,720 --> 00:33:22,320 points and links of this lattice. 692 00:33:22,440 --> 00:33:24,600 It requires massive parallel computing. 693 00:33:24,920 --> 00:33:28,200 So the experimental design is this huge lattice simulation. 694 00:33:28,680 --> 00:33:31,640 What are you measuring in the simulation to find the mass gap? 695 00:33:31,840 --> 00:33:34,040 You calculate things called glue ball correlators. 696 00:33:34,040 --> 00:33:36,720 A correlator measures how the properties of a particle created 697 00:33:36,720 --> 00:33:39,440 at one point in space-time are related to its properties at 698 00:33:39,440 --> 00:33:41,880 another point. OK, for a massive particle this 699 00:33:41,880 --> 00:33:44,640 correlation decays exponentially with distance. 700 00:33:45,000 --> 00:33:47,800 The rate of that exponential decay is directly related to the 701 00:33:47,800 --> 00:33:50,880 particles mass. So you simulate, calculate these 702 00:33:50,880 --> 00:33:53,640 correlators, see how fast they decay, and that tells you the 703 00:33:53,640 --> 00:33:55,440 mass. Precisely, you compute the 704 00:33:55,440 --> 00:33:58,480 correlators for the lightest expected glue Ball State and you 705 00:33:58,480 --> 00:34:00,760 fit their decay to an exponential function. 706 00:34:01,080 --> 00:34:03,520 The mass comes from the exponent in that fit. 707 00:34:03,920 --> 00:34:06,960 The A&V check is. Does this fitting procedure 708 00:34:06,960 --> 00:34:10,639 consistently yield a positive non 0 mass? 709 00:34:10,760 --> 00:34:14,520 And what is the data from these huge lattice QCD simulations 710 00:34:14,520 --> 00:34:16,880 show? The empirical data is strongly 711 00:34:16,880 --> 00:34:18,840 supportive of the mass gap conjecture. 712 00:34:19,199 --> 00:34:20,960 Really, they find a positive mass. 713 00:34:20,960 --> 00:34:25,000 Yes, large scale simulations, often using approximations like 714 00:34:25,000 --> 00:34:28,600 quenched SU-3, Yang Mills theory, consistently produce 715 00:34:28,600 --> 00:34:31,800 finite positive masses for the lightest glue ball states. 716 00:34:32,280 --> 00:34:34,719 The source mentions estimates for the lightest scalar glue 717 00:34:34,719 --> 00:34:39,400 ball lane plus Cree dollars are around 1.6 GV giga electron 718 00:34:39,400 --> 00:34:41,520 volts. So the simulations confirm the 719 00:34:41,520 --> 00:34:44,239 physical prediction needed for the theory to match reality. 720 00:34:44,440 --> 00:34:46,639 That's the coherence. That's the A&V coherence 721 00:34:46,639 --> 00:34:48,480 exactly, but again, we have to mention the cost. 722 00:34:48,480 --> 00:34:51,320 Let me guess, GPU farms and lots of time? 723 00:34:51,480 --> 00:34:54,120 You got it. While simplified toy models 724 00:34:54,120 --> 00:34:57,240 might run quickly, getting reliable results from full scale 725 00:34:57,240 --> 00:35:02,000 lattice QCD requires hours, days, or even weeks on dedicated 726 00:35:02,000 --> 00:35:05,200 supercomputing resources, usually massive GPU farms. 727 00:35:05,760 --> 00:35:07,960 It's another area where computation provides strong 728 00:35:07,960 --> 00:35:09,680 evidence, but is incredibly demanding. 729 00:35:09,840 --> 00:35:11,880 Wow. OK, that's an incredible tour 730 00:35:11,880 --> 00:35:17,080 across 6 monumental problems. Ryman hypothesis, PBSNP, Navier, 731 00:35:17,080 --> 00:35:21,840 Stokes, BSD, Hodge, Yang Mills all probe using this assume and 732 00:35:21,840 --> 00:35:25,280 verify computational lens. It really shows the versatility 733 00:35:25,280 --> 00:35:28,120 of the approach, doesn't it? The same core logic applies even 734 00:35:28,120 --> 00:35:30,120 if the tools and details vary wildly. 735 00:35:30,240 --> 00:35:31,920 Absolutely. So for someone listening, maybe 736 00:35:31,920 --> 00:35:34,160 trying to get a handle on these fields efficiently, let's 737 00:35:34,280 --> 00:35:37,480 quickly synthesize the big advantages of this A&V approach. 738 00:35:37,760 --> 00:35:40,120 Why is it valuable? OK, advantage 1 speed. 739 00:35:40,360 --> 00:35:44,160 We saw this clearly with RH and PVS and P tests that give strong 740 00:35:44,160 --> 00:35:46,480 initial coherence can run in seconds or minutes. 741 00:35:46,480 --> 00:35:49,400 This is orders of magnitude faster than waiting decades for 742 00:35:49,400 --> 00:35:51,640 a potential theoretical proof of refutation. 743 00:35:51,920 --> 00:35:54,480 It massively accelerates the learning and discovery cycle. 744 00:35:54,480 --> 00:35:56,280 Right. Get feedback quickly advantage 745 00:35:56,280 --> 00:35:59,040 too. It generates valuable resources. 746 00:35:59,520 --> 00:36:02,560 The process itself creates things like benchmark data sets, 747 00:36:02,560 --> 00:36:06,320 hard three SAT instances, databases of computed 748 00:36:06,320 --> 00:36:09,840 properties, elliptic curve ranks, or well tested simulation 749 00:36:09,840 --> 00:36:12,200 codes. These become tools for the 750 00:36:12,200 --> 00:36:15,880 entire research community, even if the main conjecture remains 751 00:36:15,880 --> 00:36:17,720 unsolved. That's a great side effect. 752 00:36:17,800 --> 00:36:20,440 What else? Advantage 3 and maybe the most 753 00:36:20,440 --> 00:36:23,800 powerful theoretically direct refutation potential. 754 00:36:24,360 --> 00:36:27,600 If an AMV check fails. If the computation spits out a 755 00:36:27,600 --> 00:36:30,880 number that violates the predicted consequence, that's 756 00:36:30,880 --> 00:36:33,440 potentially hewed. It's not just inconclusive, it's 757 00:36:33,640 --> 00:36:35,520 wrong assumption found. Exactly. 758 00:36:35,600 --> 00:36:38,000 It provides a concrete counterexample, or at least 759 00:36:38,000 --> 00:36:40,960 points precisely to where the theory breaks down numerically. 760 00:36:41,240 --> 00:36:43,520 That's incredibly valuable information for theorists. 761 00:36:43,520 --> 00:36:45,680 It tells them exactly where to focus their efforts. 762 00:36:45,680 --> 00:36:48,320 And the last advantage may be more personal for the learner. 763 00:36:48,320 --> 00:36:51,120 It builds critical intuition actually seeing the prime 764 00:36:51,120 --> 00:36:54,680 counting error stay within the RH bound, or watching the SAT 765 00:36:54,760 --> 00:36:56,720 solver time explode exponentially. 766 00:36:56,960 --> 00:37:00,400 It gives you a gut feeling and experiential understanding of 767 00:37:00,400 --> 00:37:02,840 the problems behavior that you just can't get from reading 768 00:37:02,840 --> 00:37:06,240 abstract theorems. OK, speed, resources, 769 00:37:06,240 --> 00:37:11,120 refutation, intuition. Powerful stuff, but we have to 770 00:37:11,120 --> 00:37:13,440 be critical. We touched on limitations 771 00:37:13,440 --> 00:37:15,080 throughout. Let's bring them together. 772 00:37:15,080 --> 00:37:19,320 What are the big caveats? #1 Underline bolded flashing 773 00:37:19,320 --> 00:37:21,680 lights. Finite checks are not infinite 774 00:37:21,680 --> 00:37:23,760 proof. This is the absolute bedrock 775 00:37:23,760 --> 00:37:27,040 limitation. We can check 10-13 zeros of the 776 00:37:27,040 --> 00:37:30,480 Zeta function or simulate a fluid for a billion time steps, 777 00:37:30,800 --> 00:37:33,640 but that doesn't prove the property holds for the 1013 plus 778 00:37:33,640 --> 00:37:37,600 for zero or for all time. The asymptotic beast might still 779 00:37:37,600 --> 00:37:40,120 be lurking just beyond our computational reach. 780 00:37:40,160 --> 00:37:43,520 Always limitation 2. Algorithmic and implementation 781 00:37:43,520 --> 00:37:45,240 errors. We're relying on complex 782 00:37:45,240 --> 00:37:47,880 software. Prime count, Dedalus, lattice, 783 00:37:47,880 --> 00:37:49,800 QCD codes. These are written by humans. 784 00:37:49,800 --> 00:37:52,320 They can have bugs. So a seeming refutation could 785 00:37:52,320 --> 00:37:54,240 actually just be a bug in the simulation code. 786 00:37:54,280 --> 00:37:56,600 It's a real risk. Verifying the correctness of the 787 00:37:56,600 --> 00:37:58,400 tools themselves is a major challenge. 788 00:37:58,640 --> 00:38:01,240 A subtle bug could lead you entirely astray, giving false 789 00:38:01,240 --> 00:38:04,560 coherence or false refutation. OK. 3rd limitation. 790 00:38:04,680 --> 00:38:07,520 The scalability wall. We saw this starkly with the 791 00:38:07,520 --> 00:38:11,200 Naver, Stokes and Yang Mills. To truly test the conjectures in 792 00:38:11,200 --> 00:38:14,360 the regimes where interesting things might happen, very small 793 00:38:14,360 --> 00:38:17,640 scales, very long times, very strong couplings, you need 794 00:38:17,640 --> 00:38:21,000 computational resources that are often prohibitively expensive, 795 00:38:21,280 --> 00:38:24,360 requiring dedicated supercomputers or GPU farms. 796 00:38:24,360 --> 00:38:27,880 So for some problems, A&V can only explore a limited part of 797 00:38:27,880 --> 00:38:29,720 the landscape effectively. Exactly. 798 00:38:29,720 --> 00:38:32,640 The cost prevents us from doing the definitive computational 799 00:38:32,640 --> 00:38:35,880 experiment in those cases. And the final, maybe most subtle 800 00:38:35,880 --> 00:38:38,080 limitation. It's that empirical scaling 801 00:38:38,080 --> 00:38:40,800 might be misleading. The behavior we observe in the 802 00:38:40,800 --> 00:38:44,320 computable range might not be the true asymptotic behavior. 803 00:38:44,360 --> 00:38:46,440 How so? Remember PVSNP? 804 00:38:47,160 --> 00:38:50,960 We see clear exponential scaling for three SAT up to thousands of 805 00:38:50,960 --> 00:38:53,680 variables. It looks convincingly like PAMP, 806 00:38:54,600 --> 00:38:57,480 but is it theoretically impossible that at N10100 807 00:38:57,480 --> 00:39:00,480 variables some completely different polynomial behavior 808 00:39:00,480 --> 00:39:02,400 takes over? We can't rule it out just from 809 00:39:02,400 --> 00:39:04,680 the simulations at smaller N. We can't. 810 00:39:04,800 --> 00:39:08,000 What looks exponential in our lab might, in the infinite 811 00:39:08,000 --> 00:39:11,480 limit, turn out to be just a very, very steep polynomial, or 812 00:39:11,480 --> 00:39:14,640 something else entirely. We're extrapolating from finite 813 00:39:14,640 --> 00:39:16,760 data, and that's always inherently risky. 814 00:39:16,840 --> 00:39:19,320 These limitations really frame the results. 815 00:39:19,480 --> 00:39:21,560 Let's make it concrete with those runtime examples. 816 00:39:21,560 --> 00:39:24,160 Again, show the difference between what you, the listener, 817 00:39:24,160 --> 00:39:26,680 might run on a laptop versus professional research. 818 00:39:26,680 --> 00:39:30,440 OK Raymond hypothesis checking the prime count error. 819 00:39:30,760 --> 00:39:33,640 Your laptop test may be up to X1 on 9 or 1010. 820 00:39:33,640 --> 00:39:35,960 Using simpi might take a few seconds to confirm the 821 00:39:35,960 --> 00:39:36,800 principle. OK. 822 00:39:36,920 --> 00:39:40,440 The pro level. Using Optimize C++ a checking up 823 00:39:40,440 --> 00:39:45,240 to X120 takes about 30 seconds. Here the scaling is fantastic, 824 00:39:45,320 --> 00:39:48,480 huge increase in verification range for minimal extra time. 825 00:39:48,520 --> 00:39:53,640 Right now PVSNPSAT solving. Your laptop test may be 826 00:39:53,640 --> 00:39:56,480 generating and solving instances up to end 200 at the phase 827 00:39:56,480 --> 00:39:58,600 transition. That takes around 10 seconds. 828 00:39:58,720 --> 00:40:01,600 Strong evidence quickly pro level pushing to end tool 10,000 829 00:40:01,600 --> 00:40:04,280 or more, which requires more sophisticated instance 830 00:40:04,280 --> 00:40:07,960 generation and longer runs. That pushes into minutes, maybe 831 00:40:07,960 --> 00:40:10,480 10s of minutes depending on the instances and the solver. 832 00:40:11,040 --> 00:40:13,280 The exponential cost starts to bite. 833 00:40:13,280 --> 00:40:16,600 And the big ones, Yang Mills or Navier Stokes. 834 00:40:17,120 --> 00:40:21,640 Laptop level you can run a tiny toy lattice simulation, maybe 84 835 00:40:21,640 --> 00:40:27,520 grid or a very low res fluid SIM 323 that might take seconds or 836 00:40:27,520 --> 00:40:29,160 minutes. You can see the code works. 837 00:40:29,160 --> 00:40:31,800 Maybe verify basic properties. It tells you almost nothing 838 00:40:31,800 --> 00:40:33,280 about the actual Millennium problem. 839 00:40:33,400 --> 00:40:34,920 Pretty much nothing definitive, no. 840 00:40:35,200 --> 00:40:39,640 The professional level requires those huge grids, 523 fluids, 841 00:40:39,640 --> 00:40:44,400 maybe 644 or larger lattices run for long simulation times. 842 00:40:44,640 --> 00:40:48,080 That's hours, days, even weeks on dead indicated GP clusters 843 00:40:48,080 --> 00:40:51,280 costing millions. The gap between accessible and 844 00:40:51,280 --> 00:40:52,800 research grade is enormous there. 845 00:40:52,840 --> 00:40:54,680 That comparison really highlights the different 846 00:40:54,680 --> 00:40:56,520 characters of the problems computationally. 847 00:40:56,520 --> 00:40:58,600 It does. Some yield strong coherence 848 00:40:58,600 --> 00:41:01,080 quickly and cheaply. Others require herculean 849 00:41:01,080 --> 00:41:03,400 computational effort to get meaningful results. 850 00:41:03,600 --> 00:41:05,760 So let's synthesize what's the big take away here? 851 00:41:05,960 --> 00:41:08,680 What does this AMV approach really mean for how we tackle 852 00:41:08,680 --> 00:41:11,040 these problems? I think it fundamentally changes 853 00:41:11,040 --> 00:41:13,520 the landscape. It transforms these abstract 854 00:41:13,520 --> 00:41:16,520 mathematical mountains into essentially executable 855 00:41:16,520 --> 00:41:19,120 experiments. It provides a powerful, 856 00:41:19,120 --> 00:41:22,760 systematic way to build evidence, find flaws, and guide 857 00:41:22,760 --> 00:41:25,360 theoretical work. It accelerates discovery. 858 00:41:25,480 --> 00:41:28,440 And the evidence we've reviewed today, the empirical coherence, 859 00:41:28,800 --> 00:41:31,720 it seems to strongly support some long held beliefs. 860 00:41:31,920 --> 00:41:34,680 Very much so. The data strongly favors the 861 00:41:34,680 --> 00:41:38,440 Ryman hypothesis being true. It strongly favors PNP. 862 00:41:38,520 --> 00:41:41,720 It supports the Birch and Swinnerton Dyer conjecture and 863 00:41:41,720 --> 00:41:44,360 the existence of a mass gap in Yang Mills theory. 864 00:41:44,920 --> 00:41:48,320 The computational results align with the prevailing mathematical 865 00:41:48,320 --> 00:41:50,400 and physical intuition in these cases. 866 00:41:50,480 --> 00:41:53,320 But for others like Navier, Stokes, smoothness or maybe 867 00:41:53,320 --> 00:41:56,920 Hodge, the computation supports the conjecture but also screams 868 00:41:57,120 --> 00:42:00,280 we need more power or we need a new theoretical idea. 869 00:42:00,280 --> 00:42:03,480 Precisely the limits of the A&V approach in those areas 870 00:42:03,480 --> 00:42:06,160 highlight exactly where the next breakthroughs are most needed. 871 00:42:06,280 --> 00:42:09,000 Whether they come from better algorithms, faster hardware, or 872 00:42:09,000 --> 00:42:12,440 fundamentally new mathematical insights, A&V acts as the best 873 00:42:12,440 --> 00:42:13,920 heuristic guide we currently have. 874 00:42:14,080 --> 00:42:17,280 Now, looking forward, you mentioned the sort material 875 00:42:17,280 --> 00:42:19,600 hints at the next frontier. What is it? 876 00:42:19,800 --> 00:42:23,440 It's about bringing in another computational paradigm, machine 877 00:42:23,440 --> 00:42:26,800 learning. AI how would that fit into 878 00:42:27,120 --> 00:42:30,080 assume and verify? Think about those problems where 879 00:42:30,080 --> 00:42:33,040 we hit the scalability wall, like Navier Stokes. 880 00:42:33,240 --> 00:42:36,200 We generate petabytes of simulation data, but we're often 881 00:42:36,200 --> 00:42:39,200 just looking for simple signals like did vorticity double? 882 00:42:39,200 --> 00:42:42,080 Right, a fairly crude check. Machine learning offers a way to 883 00:42:42,080 --> 00:42:44,440 analyze that simulation data much more deeply. 884 00:42:44,720 --> 00:42:47,440 Instead of setting simple thresholds, you could train ML 885 00:42:47,440 --> 00:42:50,920 models, maybe deep neural networks, to detect subtle 886 00:42:50,920 --> 00:42:54,680 complex emergent patterns in the flow data that might be the 887 00:42:54,680 --> 00:42:58,200 precursors to a singularity. So the AI could spot the warning 888 00:42:58,200 --> 00:43:01,760 signs much earlier, or patterns that a human analyst or a simple 889 00:43:01,760 --> 00:43:03,960 rule would completely miss. That's the hope. 890 00:43:04,120 --> 00:43:06,840 It could act as a highly sophisticated anomaly detector 891 00:43:06,840 --> 00:43:09,400 within the ANV framework. It could sift through the 892 00:43:09,400 --> 00:43:12,680 massive data sets generated by the simulations and flag the 893 00:43:12,680 --> 00:43:16,200 truly interesting, potentially dangerous events that warrant 894 00:43:16,200 --> 00:43:19,120 closer inspection or even higher resolution follow up 895 00:43:19,120 --> 00:43:21,360 simulations. That could potentially get us 896 00:43:21,360 --> 00:43:25,120 around the raw computational power bottleneck to some extent 897 00:43:25,520 --> 00:43:28,280 by making the search smarter. It could make the search vastly 898 00:43:28,280 --> 00:43:30,960 more efficient. Instead of simulating everything 899 00:43:30,960 --> 00:43:33,520 at maximum resolution, you simulate broadly. 900 00:43:33,600 --> 00:43:36,640 Let the ML find the suspicious regions and then zoom in 901 00:43:36,640 --> 00:43:40,200 computationally just on those. It could be the key to probing 902 00:43:40,200 --> 00:43:43,200 those hard to reach regimes. Turning the numerical lab into 903 00:43:43,200 --> 00:43:45,720 an intelligent hunting ground for counter examples. 904 00:43:46,160 --> 00:43:49,160 That's a fascinating prospect. It really feels like the next 905 00:43:49,160 --> 00:43:52,800 logical step, integrating data-driven AI insights with the 906 00:43:52,800 --> 00:43:56,600 physics based A&V simulations. So our final thought for you, 907 00:43:56,640 --> 00:44:00,320 the listener really comes back. This idea of accessibility, this 908 00:44:00,320 --> 00:44:03,040 whole assume and verify approach, from the quick RH 909 00:44:03,040 --> 00:44:06,400 checks to the complex physics simulations, it's all built on 910 00:44:06,400 --> 00:44:08,480 code. Code that in many cases, 911 00:44:08,480 --> 00:44:11,960 particularly for the basic setups, is becoming increasingly 912 00:44:11,960 --> 00:44:14,680 open and available. Which means these monumental 913 00:44:14,680 --> 00:44:17,440 challenges, the deepest questions in mathematics and 914 00:44:17,440 --> 00:44:20,600 physics, they aren't solely confined to the ivory tower 915 00:44:20,600 --> 00:44:22,400 anymore. They're not just philosophical 916 00:44:22,400 --> 00:44:24,680 debates. They've become, in a very real 917 00:44:24,680 --> 00:44:28,480 sense, executable, verifiable computational problems. 918 00:44:28,480 --> 00:44:31,480 And the amazing implication is that the path towards 919 00:44:31,720 --> 00:44:34,400 understanding, contributing, maybe even finding that 920 00:44:34,400 --> 00:44:37,200 $1,000,000 insight, it might just start with downloading some 921 00:44:37,200 --> 00:44:39,760 code, understanding it, and running an experiment on the 922 00:44:39,760 --> 00:44:41,080 computer you have right in front of you. 923 00:44:41,400 --> 00:44:42,880 The journey begins with computation.