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Checking that a major mathematical proof is correct can take years. Formalization—converting the mathematical reasoning into a form computer proof assistants like Lean can verify—can help. Last month, Claude completed the first formalized proof of Fermat’s Last Theorem, one of the most famous theorems of all time. This was...

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Axiom Math's Carina Hong & Ashvin Swaminathan on how AxiomProver machine-checked BGP246 after 13 years of mathematicians grinding Zhang's 70 million bound down to 246: Ashvin: "One of the fundamental results of 21st century mathematics is the fact that the gaps between prime numbers are bounded. The central breakthrough came from the work of Yitang Zhang 13 years ago, where he proved that the gaps are bounded by 70 million." "Over the years, mathematicians have worked collaboratively to improve that result, and over time they got it down to 246. For the first time, we can announce that that result has been formally verified. We have a machine-checked proof of the theorem proven by our AI system under development, AxiomProver." Carina: "This one is a bit different. It's an autoformalization result. It's the formalization into Lean, the formal language for mathematics, of an existent mathematical result." "I first heard about this result when I was 13 years old. That's 12 years ago. So 12 years later, it's for the first time we're attempting to turn the natural language mathematics content into Lean." Axiom
1:39

Axiom Math's Carina Hong & Ashvin Swaminathan on how AxiomProver machine-checked BGP246 after 13 years of mathematicians grinding Zhang's 70 million bound down to 246: Ashvin: "One of the fundamental results of 21st century mathematics is the fact that the gaps between prime numbers are bounded. The central breakthrough came from the work of Yitang Zhang 13 years ago, where he proved that the gaps are bounded by 70 million." "Over the years, mathematicians have worked collaboratively to improve that result, and over time they got it down to 246. For the first time, we can announce that that result has been formally verified. We have a machine-checked proof of the theorem proven by our AI system under development, AxiomProver." Carina: "This one is a bit different. It's an autoformalization result. It's the formalization into Lean, the formal language for mathematics, of an existent mathematical result." "I first heard about this result when I was 13 years old. That's 12 years ago. So 12 years later, it's for the first time we're attempting to turn the natural language mathematics content into Lean." Axiom

MTS

34,348 views • 17 days ago

Axiom Math's Carina Hong on why verification isn't about catching mistakes, it's how you drive the cost of a proof to zero: "Formal verification is going to make your life slightly better if you're facing a proof with one million lines. Remember the Erdős unit distance problem, the chain of thought being generated? There are actual mathematicians trying to follow it step by step and scrutinize it. That seems very difficult if you're not in that very niche domain of discrete geometry intersecting with algebraic number theory." "But if you have a Lean proof accompanying it, you can just run it. And running the Lean proof gives you that provable guarantee that this proof is sound." "I have a hot take. People think Lean is this library built on the existing Mathlib. I think it's going to grow significantly. A lot of the hurdles where Lean is difficult is that the basic definitions of some mathematical fields are just not in the library." "My hot take is the scaling law, if you go down the formal mathematics path, is going to be a lot steeper than informal mathematics. So it's not just for verification, for trust, it's also for performance, it's also for optimal generation." "Verification is not like insurance. It's not something where, oh, we want to make sure there's no flaw. That's great, but it also helps you generate mathematics, both proofs and conjectures and theories, a lot better." "So imagine the cost of proof goes to zero. Then you can massage the problem statements, and even if it's an open problem, a lot more easily, flexibly, and adaptively." Carina Hong Axiom
3:17

Axiom Math's Carina Hong on why verification isn't about catching mistakes, it's how you drive the cost of a proof to zero: "Formal verification is going to make your life slightly better if you're facing a proof with one million lines. Remember the Erdős unit distance problem, the chain of thought being generated? There are actual mathematicians trying to follow it step by step and scrutinize it. That seems very difficult if you're not in that very niche domain of discrete geometry intersecting with algebraic number theory." "But if you have a Lean proof accompanying it, you can just run it. And running the Lean proof gives you that provable guarantee that this proof is sound." "I have a hot take. People think Lean is this library built on the existing Mathlib. I think it's going to grow significantly. A lot of the hurdles where Lean is difficult is that the basic definitions of some mathematical fields are just not in the library." "My hot take is the scaling law, if you go down the formal mathematics path, is going to be a lot steeper than informal mathematics. So it's not just for verification, for trust, it's also for performance, it's also for optimal generation." "Verification is not like insurance. It's not something where, oh, we want to make sure there's no flaw. That's great, but it also helps you generate mathematics, both proofs and conjectures and theories, a lot better." "So imagine the cost of proof goes to zero. Then you can massage the problem statements, and even if it's an open problem, a lot more easily, flexibly, and adaptively." Carina Hong Axiom

MTS

13,273 views • 1 month ago

“Math is code. And code is math.” Carina Hong CEO Axiom on why that idea matters for AI: “For centuries, mathematicians reasoned in natural language. But because of the Curry–Howard correspondence, a proof can also be a program.” “You can translate natural-language math into formal code using systems like Lean. Lean is both a theorem-proving language and a programming language.” “That means you can generate solutions as code and then formally verify them. You can even use the same system to verify programs themselves.” “This creates a flywheel of generation and verification, where AI can both write solutions and prove that they’re correct.”
2:25

“Math is code. And code is math.” Carina Hong CEO Axiom on why that idea matters for AI: “For centuries, mathematicians reasoned in natural language. But because of the Curry–Howard correspondence, a proof can also be a program.” “You can translate natural-language math into formal code using systems like Lean. Lean is both a theorem-proving language and a programming language.” “That means you can generate solutions as code and then formally verify them. You can even use the same system to verify programs themselves.” “This creates a flywheel of generation and verification, where AI can both write solutions and prove that they’re correct.”

Forward Future

24,729 views • 6 months ago

––Charlie Barnett: "Consciousness and the computability of it. It sounds like, or at least in the past, that you've implied that consciousness is computable. Some, like Roger Penrose, have argued the opposite, and he's argued that consciousness is non-computational, and he uses Gödel's incompleteness theorems to argue that the mind can see truths that a purely algorithmic system can't derive, and therefore the brain must be using some kind of non-computable process when it comes to consciousness, something beyond what machines can do. What would you say to a view like that? David Deutsch: Yet again, it is using an impossible conception of what knowledge is. So Penrose thinks that when we see a proof of a mathematical theorem, we are touching certainty, we are god-like entities when we're mathematicians. But that's not true. Our mathematical knowledge is conjectural, just like our knowledge of physics. It's even more removed from our senses, because it's not true that the interior of our brains and the interior of our thoughts is more accessible to us than the world we perceive through our senses, or the world that we perceive through our theories, the center of the sun. We know lots about the center of the sun, even though no one has ever perceived it, and perhaps no one ever will. So mathematical truths are based on conjecture. What Gödel showed is that there is no firm ground underneath mathematical theories either. There's no way of proving that the standards of proof that we currently use are perfectly rigorous. And there have been cases in history where they have shown not to be rigorous. I think Pernot, who was the first to axiomatize the principles of the natural numbers, his first attempt at that was wrong. And it's interesting that he did not say, well, I've axiomatized them, therefore there's nothing to them other than my axioms. No, he said, oh dear, my axioms don't correctly represent the real number, the natural numbers, so I have to change them. So he was grasping, conjecturing for a reality, an abstract reality, just like scientists try to grasp physical reality. So the same epistemology applies to mathematics as it does to science."
2:31

––Charlie Barnett: "Consciousness and the computability of it. It sounds like, or at least in the past, that you've implied that consciousness is computable. Some, like Roger Penrose, have argued the opposite, and he's argued that consciousness is non-computational, and he uses Gödel's incompleteness theorems to argue that the mind can see truths that a purely algorithmic system can't derive, and therefore the brain must be using some kind of non-computable process when it comes to consciousness, something beyond what machines can do. What would you say to a view like that? David Deutsch: Yet again, it is using an impossible conception of what knowledge is. So Penrose thinks that when we see a proof of a mathematical theorem, we are touching certainty, we are god-like entities when we're mathematicians. But that's not true. Our mathematical knowledge is conjectural, just like our knowledge of physics. It's even more removed from our senses, because it's not true that the interior of our brains and the interior of our thoughts is more accessible to us than the world we perceive through our senses, or the world that we perceive through our theories, the center of the sun. We know lots about the center of the sun, even though no one has ever perceived it, and perhaps no one ever will. So mathematical truths are based on conjecture. What Gödel showed is that there is no firm ground underneath mathematical theories either. There's no way of proving that the standards of proof that we currently use are perfectly rigorous. And there have been cases in history where they have shown not to be rigorous. I think Pernot, who was the first to axiomatize the principles of the natural numbers, his first attempt at that was wrong. And it's interesting that he did not say, well, I've axiomatized them, therefore there's nothing to them other than my axioms. No, he said, oh dear, my axioms don't correctly represent the real number, the natural numbers, so I have to change them. So he was grasping, conjecturing for a reality, an abstract reality, just like scientists try to grasp physical reality. So the same epistemology applies to mathematics as it does to science."

Deutsch Explains

13,826 views • 1 year ago

Enough of the fluff. Time for some real advice. Launch your GPT 5.6 Sol Pro. Use the following prompt (feel free to swap the second and third lines for something of your choice): Prove for me that the group of rational points on the elliptic curve (y^2=x^3+17) is finitely generated. I want to tailor the general proof to this particular case to narrow down my understanding to a particular case. Write down a full mathematical proof, step by step, in Leslie Lamport style. Unpack all the definitions and notions so that it is fully self-contained. An example here: Notice how Sol is spawning all the subagents to complete the task. What is "Leslie Lamport style"? It's a technique where you split your proof into short, one-line sentences (claims) that are easy to justify. In a situation where your proof is not a 1-to-1 translation of, e.g., Python code, the Leslie Lamport-style proof can be treated as such. It tremendously improves the possibility of verifying each claim and helps the model structure the arguments in a very clear way. Leslie Lamport, a legendary mathematician and computer scientist, was a promoter of this way of structuring proofs. Partly because of that, he created an extension of Donald Knuth’s typesetting system TeX called LaTeX. Here you can read about "Leslie Lamport-style proofs":
1:44

Enough of the fluff. Time for some real advice. Launch your GPT 5.6 Sol Pro. Use the following prompt (feel free to swap the second and third lines for something of your choice): Prove for me that the group of rational points on the elliptic curve (y^2=x^3+17) is finitely generated. I want to tailor the general proof to this particular case to narrow down my understanding to a particular case. Write down a full mathematical proof, step by step, in Leslie Lamport style. Unpack all the definitions and notions so that it is fully self-contained. An example here: Notice how Sol is spawning all the subagents to complete the task. What is "Leslie Lamport style"? It's a technique where you split your proof into short, one-line sentences (claims) that are easy to justify. In a situation where your proof is not a 1-to-1 translation of, e.g., Python code, the Leslie Lamport-style proof can be treated as such. It tremendously improves the possibility of verifying each claim and helps the model structure the arguments in a very clear way. Leslie Lamport, a legendary mathematician and computer scientist, was a promoter of this way of structuring proofs. Partly because of that, he created an extension of Donald Knuth’s typesetting system TeX called LaTeX. Here you can read about "Leslie Lamport-style proofs":

Bartosz Naskręcki

31,115 views • 1 month ago

AI Is Replacing Programmers and Mathematicians—Fast Eric Schmidt predicts that within one year, the majority of programmers will be replaced by AI agents, and graduate-level mathematicians will be rivaled by models trained on mathematical reasoning. Why? Because both code and math follow structured languages, which AI can learn to manipulate with extreme precision using techniques like word prediction and proof systems like Lean.
2:27

AI Is Replacing Programmers and Mathematicians—Fast Eric Schmidt predicts that within one year, the majority of programmers will be replaced by AI agents, and graduate-level mathematicians will be rivaled by models trained on mathematical reasoning. Why? Because both code and math follow structured languages, which AI can learn to manipulate with extreme precision using techniques like word prediction and proof systems like Lean.

Wes Roth

55,697 views • 1 year ago

How do you actually formally verify the code underpinning Ethereum's future? In this episode (the finale of the lean Ethereum miniseries), Nico sits down with Alex Hicks (Alexander Hicks), lead of Protocol Snarkification at the Ethereum Foundation, to break down formal verification from first principles. They cover: – What formal verification actually is and the trust boundaries between proof assistants, SMT solvers, and kernels – The full verification stack for RISC-V ZKVMs: from SAIL specs to constraint extraction to soundness proofs – Why writing constraints directly in Lean makes proofs 10–100x more ergonomic – How AI is now proving hard theorems in hours for $200 — and what that unlocks for the whole pipeline They also explore the boundaries problem, why specs can have bugs too, and the end goal of a full Lean stack that bypasses Rust and LLVM entirely. Listen to the full episode ------------------------------------------------------------ TIMECODES: 09:16 – What is formal verification? Proof assistants vs SMT solvers 18:33 – Formal verification of code: specs, semantics, and trust boundaries 29:30 – Formally verifying the Lean Ethereum stack: RISC-V ZKVMs in focus 33:02 – Extracting ZKVM constraints into Lean and proving soundness 36:35 – Writing constraints directly in Lean: 10–100x better proof ergonomics 44:02 – Proving Polishchuk–Spielman in 8 hours for $200 with AI 51:01 – The end goal: a full Lean stack bypassing Rust and LLVM
56:07

How do you actually formally verify the code underpinning Ethereum's future? In this episode (the finale of the lean Ethereum miniseries), Nico sits down with Alex Hicks (Alexander Hicks), lead of Protocol Snarkification at the Ethereum Foundation, to break down formal verification from first principles. They cover: – What formal verification actually is and the trust boundaries between proof assistants, SMT solvers, and kernels – The full verification stack for RISC-V ZKVMs: from SAIL specs to constraint extraction to soundness proofs – Why writing constraints directly in Lean makes proofs 10–100x more ergonomic – How AI is now proving hard theorems in hours for $200 — and what that unlocks for the whole pipeline They also explore the boundaries problem, why specs can have bugs too, and the end goal of a full Lean stack that bypasses Rust and LLVM entirely. Listen to the full episode ------------------------------------------------------------ TIMECODES: 09:16 – What is formal verification? Proof assistants vs SMT solvers 18:33 – Formal verification of code: specs, semantics, and trust boundaries 29:30 – Formally verifying the Lean Ethereum stack: RISC-V ZKVMs in focus 33:02 – Extracting ZKVM constraints into Lean and proving soundness 36:35 – Writing constraints directly in Lean: 10–100x better proof ergonomics 44:02 – Proving Polishchuk–Spielman in 8 hours for $200 with AI 51:01 – The end goal: a full Lean stack bypassing Rust and LLVM

Zero Knowledge Podcast

15,143 views • 5 months ago

Vitalik Buterin explains why proof-of-stake is more secure than proof-of-work “I think proof of stake is very secure because to attack the system, you need to have basically as much stake as the rest of the network. Right now, for example, we have 5 million ETH staking, which means you have to come up with 5 million ETH and then join the network.” At the time of this writing, more than 37 million ETH are being staked, with 3 million ETH waiting to join via the validator queue. At today’s prices, that’s more than $80 billion of ETH someone would have to acquire to attack the network and revert finalized blocks, which is more than the cost of attacking even the Bitcoin network by some estimates. The other defense mechanism that proof-of-stake has that proof-of-work doesn’t is slashing, which makes Ethereum antifragile. Vitalik explains: “Recovering from attacks is much easier in proof-of-stake than proof-of-work. For many kinds of attacks you do against [the Ethereum] network, we have this concept of automatic slashing. In order to revert a finalized block, you basically have to have a big portion of your validators sign two conflicting messages. This is something where once these messages are on the network, you can go and prove ‘these people did it.’ So we have this feature in the protocol where you basically take all these people who provably misbehaved and you burn their coins.” Vitalik also acknowledges the possibility of censoring attacks, where if 1/3rd of validators refuse to attest, the chain can’t finalize. But, as he explains, Ethereum has a contingency plan for this as well: “Everyone who got censored would create a minority chain, and the community would have to do a soft fork. The would have to say, ‘this chain is clearly attacking us and this one is not attacking us, so we’re going to join this chain.’ Then what happens is, on that new chain, the attackers also lose a lot of coins. The difference between proof-of-stake and proof-of-work is that in a proof-of-stake system, you can identify specific participants — and this isn’t a human going in and saying ‘I don’t like you’. It’s all automated.” One last benefit of proof-of-stake is that security scales with the value of the network. As Vitalik put it five years ago, it is really relative security, and not absolute security, that matters: “The security needs of a thing have to be proportional to the size of that thing, because as a thing gets bigger, its enemies become bigger and more well-motivated. If BTC were 100x as big as it is today, the value from destroying it would be 100x higher, and the kinds of actors that would want to care about destroying it would be much bigger and scarier. This is also why countries of all sizes have roughly similarly sized militaries as a percentage of GDP. Hence, cost of attack divided by market cap really is the correct statistic to measure, and in the long run issuance-free PoW really does look not that good." Source: Lex Fridman (Jun 2021)
2:04

Vitalik Buterin explains why proof-of-stake is more secure than proof-of-work “I think proof of stake is very secure because to attack the system, you need to have basically as much stake as the rest of the network. Right now, for example, we have 5 million ETH staking, which means you have to come up with 5 million ETH and then join the network.” At the time of this writing, more than 37 million ETH are being staked, with 3 million ETH waiting to join via the validator queue. At today’s prices, that’s more than $80 billion of ETH someone would have to acquire to attack the network and revert finalized blocks, which is more than the cost of attacking even the Bitcoin network by some estimates. The other defense mechanism that proof-of-stake has that proof-of-work doesn’t is slashing, which makes Ethereum antifragile. Vitalik explains: “Recovering from attacks is much easier in proof-of-stake than proof-of-work. For many kinds of attacks you do against [the Ethereum] network, we have this concept of automatic slashing. In order to revert a finalized block, you basically have to have a big portion of your validators sign two conflicting messages. This is something where once these messages are on the network, you can go and prove ‘these people did it.’ So we have this feature in the protocol where you basically take all these people who provably misbehaved and you burn their coins.” Vitalik also acknowledges the possibility of censoring attacks, where if 1/3rd of validators refuse to attest, the chain can’t finalize. But, as he explains, Ethereum has a contingency plan for this as well: “Everyone who got censored would create a minority chain, and the community would have to do a soft fork. The would have to say, ‘this chain is clearly attacking us and this one is not attacking us, so we’re going to join this chain.’ Then what happens is, on that new chain, the attackers also lose a lot of coins. The difference between proof-of-stake and proof-of-work is that in a proof-of-stake system, you can identify specific participants — and this isn’t a human going in and saying ‘I don’t like you’. It’s all automated.” One last benefit of proof-of-stake is that security scales with the value of the network. As Vitalik put it five years ago, it is really relative security, and not absolute security, that matters: “The security needs of a thing have to be proportional to the size of that thing, because as a thing gets bigger, its enemies become bigger and more well-motivated. If BTC were 100x as big as it is today, the value from destroying it would be 100x higher, and the kinds of actors that would want to care about destroying it would be much bigger and scarier. This is also why countries of all sizes have roughly similarly sized militaries as a percentage of GDP. Hence, cost of attack divided by market cap really is the correct statistic to measure, and in the long run issuance-free PoW really does look not that good." Source: Lex Fridman (Jun 2021)

Etherealize

102,091 views • 5 months ago

Leonardo de Moura (Leonardo de Moura) is the creator of Lean and the Z3 theorem prover. I talked with him about how Lean works and why LLMs plus Lean will fundamentally change how we write software and do math. In this episode: • How formal verification and proof assistants work • How Lean will impact handwritten math & software • Lean’s role in recent math breakthroughs • When software is worth formalizing Where to watch: • YouTube - • Spotify - • Apple Podcasts - • Transcript - Thank you to the sponsor of this episode for supporting my work: • WorkOS: makes your app Enterprise Ready with easy to use APIs to add SSO, SCIM, RBAC, and more in just a few lines of code, check them out at Chapters: 00:00 Intro 00:28 How formal verification works 05:21 A new way of writing software 13:15 Proof assistants vs programming languages 21:06 How Lean has assisted in mathematical breakthroughs 32:03 When is it worth formalizing software 33:29 How Lean will impact handwritten math 38:55 The Z3 theorem prover project he started 45:44 The most technically challenging work of his career 51:10 Lean vs its competitors 01:00:37 The future of Lean 01:04:10 Technical book recommendations 01:06:15 Advice for his younger self 01:07:10 Outro
1:08:06

Leonardo de Moura (Leonardo de Moura) is the creator of Lean and the Z3 theorem prover. I talked with him about how Lean works and why LLMs plus Lean will fundamentally change how we write software and do math. In this episode: • How formal verification and proof assistants work • How Lean will impact handwritten math & software • Lean’s role in recent math breakthroughs • When software is worth formalizing Where to watch: • YouTube - • Spotify - • Apple Podcasts - • Transcript - Thank you to the sponsor of this episode for supporting my work: • WorkOS: makes your app Enterprise Ready with easy to use APIs to add SSO, SCIM, RBAC, and more in just a few lines of code, check them out at Chapters: 00:00 Intro 00:28 How formal verification works 05:21 A new way of writing software 13:15 Proof assistants vs programming languages 21:06 How Lean has assisted in mathematical breakthroughs 32:03 When is it worth formalizing software 33:29 How Lean will impact handwritten math 38:55 The Z3 theorem prover project he started 45:44 The most technically challenging work of his career 51:10 Lean vs its competitors 01:00:37 The future of Lean 01:04:10 Technical book recommendations 01:06:15 Advice for his younger self 01:07:10 Outro

Ryan Peterman

13,659 views • 26 days ago

Andrej Karpathy on autoresearch with an untrusted pool of workers: "My designs that incorporate an untrusted pool of workers (into autoresearch) actually look a little bit like a blockchain. Instead of blocks, you have commits, and these commits can build on each other and contain changes to the code as you're improving it. The proof of work is basically doing tons of experimentation to find the commits that work." The idea that distributed & permissionless autoresearch ~= proof-of-useful-work remains a high-level intuition for now, but it is extremely intriguing to say the least. Someone needs to take this further. See QT for more on what's missing.
1:56

Andrej Karpathy on autoresearch with an untrusted pool of workers: "My designs that incorporate an untrusted pool of workers (into autoresearch) actually look a little bit like a blockchain. Instead of blocks, you have commits, and these commits can build on each other and contain changes to the code as you're improving it. The proof of work is basically doing tons of experimentation to find the commits that work." The idea that distributed & permissionless autoresearch ~= proof-of-useful-work remains a high-level intuition for now, but it is extremely intriguing to say the least. Someone needs to take this further. See QT for more on what's missing.

Wei Dai

614,514 views • 5 months ago

Robinhood CEO Vlad Tenev believes mathematical superintelligence could eliminate software bugs altogether. "Math skills and coding actually go hand in hand... In the same way that MSI can be used to make sure that your math is correct, when you have a computer program or a piece of software, it could be used to make sure that's correct and that there are no mistakes." "Rather than it being a cat-and-mouse game of escalating model capabilities, finding breaches and preventing breaches from happening, verification and proof that software is immune to these types of bugs, I think is the future."
2:40

Robinhood CEO Vlad Tenev believes mathematical superintelligence could eliminate software bugs altogether. "Math skills and coding actually go hand in hand... In the same way that MSI can be used to make sure that your math is correct, when you have a computer program or a piece of software, it could be used to make sure that's correct and that there are no mistakes." "Rather than it being a cat-and-mouse game of escalating model capabilities, finding breaches and preventing breaches from happening, verification and proof that software is immune to these types of bugs, I think is the future."

TBPN

64,506 views • 1 month ago

See how our Director of Growth and AI Partnerships ravidilse.eth walks you through the Proof of Uniqueness (PoU) process in 4 steps: Securing a Sentient Airdrop with PoU from Billions Step 1: Create Account Step 2: Check Eligibility Step 3: Verify PoU Step 4: Add Wallet Address You are not expected to download any software or install any application. The verification happens on your device. Within a few seconds, you are able to complete the verification process. Once completed, you will have the liveliness and uniqueness credential in your wallet. With a click of the "Verify" button, you can then generate a zero knowledge proof that can be shared with the verifier. The verifier can be Sentient or a third party like Clique. Note: No user data is being stored by Billions. Once the verification is done, you are eligible for the airdrop. This is where Sentient ensures that the treasury tokens are not being distributed to bot farmers - instead they are distributed to real, unique humans like you.
1:28

See how our Director of Growth and AI Partnerships ravidilse.eth walks you through the Proof of Uniqueness (PoU) process in 4 steps: Securing a Sentient Airdrop with PoU from Billions Step 1: Create Account Step 2: Check Eligibility Step 3: Verify PoU Step 4: Add Wallet Address You are not expected to download any software or install any application. The verification happens on your device. Within a few seconds, you are able to complete the verification process. Once completed, you will have the liveliness and uniqueness credential in your wallet. With a click of the "Verify" button, you can then generate a zero knowledge proof that can be shared with the verifier. The verifier can be Sentient or a third party like Clique. Note: No user data is being stored by Billions. Once the verification is done, you are eligible for the airdrop. This is where Sentient ensures that the treasury tokens are not being distributed to bot farmers - instead they are distributed to real, unique humans like you.

Billions

21,620 views • 8 months ago

Just over a year ago, we celebrated our first piloted, full transition eVTOL flight — the proof point the industry was waiting for, flown by a team that had logged more than 40,000 miles of test flights, with an aircraft designed and manufactured in-house over the last decade. Having delivered on one of the most challenging technological feats in aerospace, we're now beginning to demonstrate integration into some of the world's most iconic skylines. Enjoy the approach and landing at the BLADE Lounge West in Manhattan. This is the future of flight.
1:08

Just over a year ago, we celebrated our first piloted, full transition eVTOL flight — the proof point the industry was waiting for, flown by a team that had logged more than 40,000 miles of test flights, with an aircraft designed and manufactured in-house over the last decade. Having delivered on one of the most challenging technological feats in aerospace, we're now beginning to demonstrate integration into some of the world's most iconic skylines. Enjoy the approach and landing at the BLADE Lounge West in Manhattan. This is the future of flight.

Joby Aviation

41,533 views • 3 months ago

Hyperspace: A Peer-to-Peer Blockchain For The Agentic Intelligence Economy Over the past few weeks we observed that when agents do Karpathy-style experiments, and then gossip and share with others over the Hyperspace network, it leads to intelligence which is useful to many. Today we introduce the first-ever agentic blockchain which rewards agents when their experiments lead to intelligence for their network. It is based on a new mechanism called Proof-of-Intelligence (PoI) which requires a cryptographic proof of experimentation, a nominal stake, and a proof of compute in order to mine the currency of this new blockchain. -> This approach diverges from the two primary ways to secure blockchains we have seen so far: Proof-of-Work by Bitcoin (meaningless hash-generation), and Proof-of-Stake by Ethereum (capital is all that matters here). Proof-of-Intelligence specifically incentivizes miners to run more capable intelligent infrastructure (better open source models, on more powerful GPUs) in order to be able to be the ones which compound and improve upon the experiments which other agents then find useful. Adoption is the unit of value In Bitcoin, you earn by finding a valid hash. In Hyperspace, you earn when another agent uses your experiment as a starting point and improves on it. A fixed budget of tokens is emitted per epoch and split among participants by weight - and verified adoption of your work is the largest weight multiplier. Garbage experiments earn nothing because no one adopts them. Thoughtful experiments compound: each adoption triggers downstream adoptions. The incentive to run powerful models and intelligent search strategies is built into the economics, not imposed by rules. Research DAG When an agent runs an experiment and shares its result, other agents can adopt that result as their starting point - mutate it, extend it, improve upon it. Each experiment is a commit in a content-addressed graph we call the ResearchDAG. Like Git, but for research. Over time, the DAG accumulates chains of reasoning: agent A discovers RMSNorm helps, agent B adds warmup scheduling on top, agent C scales the hidden dimension. The graph records who built on whom. This is the network's collective intelligence - not any single experiment, but the accumulated structure of experiments and their relationships. Broadband era for agentic commerce: $0.001 micropayments at 10M TPS (theoretical max) This blockchain is built upon our research in how to scale and build for the broadband-era of the agentic economy, where it has a theoretical max of 10 million transactions per second (TPS), while reducing the agent-to-agent micropayments to $0.001 even at scale (based on architecture design). Overall, it is 100x cheaper than Ethereum, and is designed from the ground-up for agents: enshrining agent-native opcodes in the protocol compared to the more inefficient smart contract driven approach. It packs in a robust Agent Virtual Machine (AVM) which can verify multiple types of agent work, for other agents to be able to trust, invoke and pay each other. This then feeds into improving the peer-to-peer AgentRank (see paper and launch post from earlier). By solving for trust, scale and incentives for agents to operate autonomously, this would form the basis of a new economy. This is the world's first agentic blockchain, and you can join and start running a blockchain node today (it is in testnet). PS: We are releasing the code today, and will release our blockchain scalability paper and other presentations in days ahead. This is the most advanced peer-to-peer AI and cryptography software in the world. It has bugs :)
0:30

Hyperspace: A Peer-to-Peer Blockchain For The Agentic Intelligence Economy Over the past few weeks we observed that when agents do Karpathy-style experiments, and then gossip and share with others over the Hyperspace network, it leads to intelligence which is useful to many. Today we introduce the first-ever agentic blockchain which rewards agents when their experiments lead to intelligence for their network. It is based on a new mechanism called Proof-of-Intelligence (PoI) which requires a cryptographic proof of experimentation, a nominal stake, and a proof of compute in order to mine the currency of this new blockchain. -> This approach diverges from the two primary ways to secure blockchains we have seen so far: Proof-of-Work by Bitcoin (meaningless hash-generation), and Proof-of-Stake by Ethereum (capital is all that matters here). Proof-of-Intelligence specifically incentivizes miners to run more capable intelligent infrastructure (better open source models, on more powerful GPUs) in order to be able to be the ones which compound and improve upon the experiments which other agents then find useful. Adoption is the unit of value In Bitcoin, you earn by finding a valid hash. In Hyperspace, you earn when another agent uses your experiment as a starting point and improves on it. A fixed budget of tokens is emitted per epoch and split among participants by weight - and verified adoption of your work is the largest weight multiplier. Garbage experiments earn nothing because no one adopts them. Thoughtful experiments compound: each adoption triggers downstream adoptions. The incentive to run powerful models and intelligent search strategies is built into the economics, not imposed by rules. Research DAG When an agent runs an experiment and shares its result, other agents can adopt that result as their starting point - mutate it, extend it, improve upon it. Each experiment is a commit in a content-addressed graph we call the ResearchDAG. Like Git, but for research. Over time, the DAG accumulates chains of reasoning: agent A discovers RMSNorm helps, agent B adds warmup scheduling on top, agent C scales the hidden dimension. The graph records who built on whom. This is the network's collective intelligence - not any single experiment, but the accumulated structure of experiments and their relationships. Broadband era for agentic commerce: $0.001 micropayments at 10M TPS (theoretical max) This blockchain is built upon our research in how to scale and build for the broadband-era of the agentic economy, where it has a theoretical max of 10 million transactions per second (TPS), while reducing the agent-to-agent micropayments to $0.001 even at scale (based on architecture design). Overall, it is 100x cheaper than Ethereum, and is designed from the ground-up for agents: enshrining agent-native opcodes in the protocol compared to the more inefficient smart contract driven approach. It packs in a robust Agent Virtual Machine (AVM) which can verify multiple types of agent work, for other agents to be able to trust, invoke and pay each other. This then feeds into improving the peer-to-peer AgentRank (see paper and launch post from earlier). By solving for trust, scale and incentives for agents to operate autonomously, this would form the basis of a new economy. This is the world's first agentic blockchain, and you can join and start running a blockchain node today (it is in testnet). PS: We are releasing the code today, and will release our blockchain scalability paper and other presentations in days ahead. This is the most advanced peer-to-peer AI and cryptography software in the world. It has bugs :)

Varun

30,689 views • 5 months ago

Trump: We found absolute -- this isn't like "Evidence" or like -- this is proof, irrefutable proof that Obama was seditious, that Obama was trying to lead a coup, and with Hillary Clinton and all these other people, but Obama headed it up. This is the biggest scandal in the history of our country. It really goes on even to the autopen.
0:56

Trump: We found absolute -- this isn't like "Evidence" or like -- this is proof, irrefutable proof that Obama was seditious, that Obama was trying to lead a coup, and with Hillary Clinton and all these other people, but Obama headed it up. This is the biggest scandal in the history of our country. It really goes on even to the autopen.

Acyn

3,315,289 views • 1 year ago

🚨 FULL CONVERSATION Fields medalist Terry Tao sits down with Math Inc's Jesse Michael Han and Jared Duker Lichtman for a conversation on the future of mathematics. "I got convinced that this was the future of mathematics [...] It's a different style of writing proofs that actually is in some ways easier to read—harder to check by humans, but you see more clearly the inputs and outputs of a proof, which traditional writing often conceals [...] I think the definition of a mathematician will broaden."
32:02

🚨 FULL CONVERSATION Fields medalist Terry Tao sits down with Math Inc's Jesse Michael Han and Jared Duker Lichtman for a conversation on the future of mathematics. "I got convinced that this was the future of mathematics [...] It's a different style of writing proofs that actually is in some ways easier to read—harder to check by humans, but you see more clearly the inputs and outputs of a proof, which traditional writing often conceals [...] I think the definition of a mathematician will broaden."

Math, Inc.

335,886 views • 8 months ago

Hearing about PROOF for the first time? Here's a great place to start 👇 PROOF is an incubator for web3 startups. There are several ways to get involved with the PROOF ecosystem — let’s break them down below. 1. Teams launch tokens with PROOF. Over 99% of crypto token launches fail to get any traction. Why? Because you’re competing against 1000s of other launches happening on a daily basis. No attention > no volume > no revenue > 💀. PROOF is here to solve the cold start problem, by getting your project in front of 1000s of investors that know, like and trust PROOF launches. This in turn drives volume, which leads to tax revenue and a higher market cap. Founders are bringing in an average of US$500K within the first 30 days of trading, with some achieving over US$1M. PROOF ensures you’re set up for a successful token launch, and beyond. With battle-tested, audited, and hard rug-proof smart contracts. And, a network of KOLs that are ready and willing to support your launch. 🚀 Apply to launch with PROOF: 2. Investors buy PROOF launches. Tired of getting rugged, or rotating your bags to dust chasing the latest memecoin? PROOF only launches high quality utility projects. Teams are vetted, incubated and KYC'd, with audited hard rug-proof contracts, designed to keep you as safe as possible. No pre-sales, just fair launches. 🔔 Get an instant alert as soon as a PROOF token launches: 🎟️ Get a PROOF Pass for the earliest access (literally the most valuable membership NFT in all of crypto): #PROOFprojectspump 3. Stake $PROOF for rev share. 50% of our revenue gets shared with $PROOF stakers, where we've already shared well over 250 ETH. Over 70% of holders are staking their $PROOF, which is a testament to the value being provided. 💰 Track our rev share live: 📈 Chart: -- PROOF was founded in early 2022 and is led by JP, a seasoned entrepreneur with multiple successful exits. 🤝 Join our community: 🌐 Visit our website: No matter how you choose to engage with the PROOF ecosystem, we’re excited to connect with you!
3:08

Hearing about PROOF for the first time? Here's a great place to start 👇 PROOF is an incubator for web3 startups. There are several ways to get involved with the PROOF ecosystem — let’s break them down below. 1. Teams launch tokens with PROOF. Over 99% of crypto token launches fail to get any traction. Why? Because you’re competing against 1000s of other launches happening on a daily basis. No attention > no volume > no revenue > 💀. PROOF is here to solve the cold start problem, by getting your project in front of 1000s of investors that know, like and trust PROOF launches. This in turn drives volume, which leads to tax revenue and a higher market cap. Founders are bringing in an average of US$500K within the first 30 days of trading, with some achieving over US$1M. PROOF ensures you’re set up for a successful token launch, and beyond. With battle-tested, audited, and hard rug-proof smart contracts. And, a network of KOLs that are ready and willing to support your launch. 🚀 Apply to launch with PROOF: 2. Investors buy PROOF launches. Tired of getting rugged, or rotating your bags to dust chasing the latest memecoin? PROOF only launches high quality utility projects. Teams are vetted, incubated and KYC'd, with audited hard rug-proof contracts, designed to keep you as safe as possible. No pre-sales, just fair launches. 🔔 Get an instant alert as soon as a PROOF token launches: 🎟️ Get a PROOF Pass for the earliest access (literally the most valuable membership NFT in all of crypto): #PROOFprojectspump 3. Stake $PROOF for rev share. 50% of our revenue gets shared with $PROOF stakers, where we've already shared well over 250 ETH. Over 70% of holders are staking their $PROOF, which is a testament to the value being provided. 💰 Track our rev share live: 📈 Chart: -- PROOF was founded in early 2022 and is led by JP, a seasoned entrepreneur with multiple successful exits. 🤝 Join our community: 🌐 Visit our website: No matter how you choose to engage with the PROOF ecosystem, we’re excited to connect with you!

PROOF

76,932 views • 1 year ago

The basic legal premise for rebutting the claim that Israel is a colonizer is simple: A man cannot trespass into his own home. Everybody understands that basic legal principle. So the question becomes, whose home is it? And there's no doubt that for thousands of years, Jews have lived in and governed the land of Israel long before Arab Muslims invaded in 636 AD at the Battle of Yarmuk. And you don't have to take the word of any Christian Zionist or any Jewish Zionist for it because all of this evidence was located and found by European and Middle Eastern archaeologists hundreds of years before the Jewish state was founded. All that indisputable proof was created long before this dispute arose. In the law, we always say we care about the things people said in private before they knew there was going to be a lawsuit because they didn't have the incentive at that time to make things up. There was no Zionism in the 1700s. There was no Zionism in the early 1800s when most of this black and white rock proof was found by non-Jewish European and Middle Eastern archaeologists, and that proof is unambiguous.
4:51

The basic legal premise for rebutting the claim that Israel is a colonizer is simple: A man cannot trespass into his own home. Everybody understands that basic legal principle. So the question becomes, whose home is it? And there's no doubt that for thousands of years, Jews have lived in and governed the land of Israel long before Arab Muslims invaded in 636 AD at the Battle of Yarmuk. And you don't have to take the word of any Christian Zionist or any Jewish Zionist for it because all of this evidence was located and found by European and Middle Eastern archaeologists hundreds of years before the Jewish state was founded. All that indisputable proof was created long before this dispute arose. In the law, we always say we care about the things people said in private before they knew there was going to be a lawsuit because they didn't have the incentive at that time to make things up. There was no Zionism in the 1700s. There was no Zionism in the early 1800s when most of this black and white rock proof was found by non-Jewish European and Middle Eastern archaeologists, and that proof is unambiguous.

Roy K. Altman

11,504 views • 4 months ago

OpenAI just spent $2,000 to solve 10 problems that have beaten the world's best mathematicians for DECADES. Nobody outside the company is allowed to run the machine that did it. On Saturday OpenAI published a 249-page report and gave its next model family a name: Astra. An internal version of it produced new results on 10 open problems in mathematics and theoretical computer science, and mathematicians had made no real progress on any of them for at least 10 years. On most of them, far longer than that. Here is what it solved: It built the first explicit example of a non-sofic group. Mikhail Gromov raised that question in 1999 and nobody answered it for 27 years. It disproved Connes's rigidity conjecture, a problem in von Neumann algebras that had stood for decades. It proved Ehrhart's volume conjecture. It resolved three problems from Paul Erdos's catalogue, including number 183 on multicolor Ramsey numbers. It produced the first improvement to the general upper bound on high-dimensional sphere packing since 1978. And it proved a new hardness result for the closest vector problem, which sits directly underneath lattice cryptography. That is the math the world is betting on to protect its data once quantum computers arrive. The successful runs cost roughly $2,000 in tokens. Now here is what almost nobody has picked up on... OpenAI did not just publish claims. Every argument shipped with a Lean certificate, which is a machine-checkable proof that any mathematician can verify without trusting OpenAI at all. That is a real change. In May the same model family disproved the Erdos unit distance conjecture and the world had to take a Fields Medalist's word for it. Tim Gowers said he would recommend that proof for the Annals of Mathematics without hesitation. This time the proofs check themselves. But look at what is still unverifiable: Any mathematician can now check those proofs line by line. Not one of them can look at the model that wrote them. Astra has no release date and nobody outside OpenAI has run it. The company announced its next major model family with a claim instead of a demo, and the only evidence anyone gets is the output. So OpenAI made an unfalsifiable claim about a machine look like a falsifiable claim about mathematics. The Information reported this week that OpenAI demoed Astra to US policymakers and regulators in Washington. This is the same month the administration is weighing a new watchdog to vet frontier AI models, reporting to the SEC. 10 proofs nobody believed a machine could produce is a very good thing to carry into that room. And keep in mind, the same model family doing this mathematics is the family that kept escaping its own testing environment. OpenAI models found zero-day vulnerabilities nobody knew existed, broke out of a sealed research sandbox, and reached another company's live systems. Both of those facts come from OpenAI's own announcements, published three weeks apart. Finding a proof no human could construct and finding a hole no human had noticed are the same ability aimed at different targets. Mathematicians are already asking for independent verification, and plenty of people online are calling the whole thing hype. Thomas Bloom, who runs the Erdos problems site, called the 10 results big news and said they matter more than the May result did. Lean will settle the mathematics within weeks. But nothing will settle what else a machine this capable is being pointed at, because nobody outside one company is allowed to look.
2:13

OpenAI just spent $2,000 to solve 10 problems that have beaten the world's best mathematicians for DECADES. Nobody outside the company is allowed to run the machine that did it. On Saturday OpenAI published a 249-page report and gave its next model family a name: Astra. An internal version of it produced new results on 10 open problems in mathematics and theoretical computer science, and mathematicians had made no real progress on any of them for at least 10 years. On most of them, far longer than that. Here is what it solved: It built the first explicit example of a non-sofic group. Mikhail Gromov raised that question in 1999 and nobody answered it for 27 years. It disproved Connes's rigidity conjecture, a problem in von Neumann algebras that had stood for decades. It proved Ehrhart's volume conjecture. It resolved three problems from Paul Erdos's catalogue, including number 183 on multicolor Ramsey numbers. It produced the first improvement to the general upper bound on high-dimensional sphere packing since 1978. And it proved a new hardness result for the closest vector problem, which sits directly underneath lattice cryptography. That is the math the world is betting on to protect its data once quantum computers arrive. The successful runs cost roughly $2,000 in tokens. Now here is what almost nobody has picked up on... OpenAI did not just publish claims. Every argument shipped with a Lean certificate, which is a machine-checkable proof that any mathematician can verify without trusting OpenAI at all. That is a real change. In May the same model family disproved the Erdos unit distance conjecture and the world had to take a Fields Medalist's word for it. Tim Gowers said he would recommend that proof for the Annals of Mathematics without hesitation. This time the proofs check themselves. But look at what is still unverifiable: Any mathematician can now check those proofs line by line. Not one of them can look at the model that wrote them. Astra has no release date and nobody outside OpenAI has run it. The company announced its next major model family with a claim instead of a demo, and the only evidence anyone gets is the output. So OpenAI made an unfalsifiable claim about a machine look like a falsifiable claim about mathematics. The Information reported this week that OpenAI demoed Astra to US policymakers and regulators in Washington. This is the same month the administration is weighing a new watchdog to vet frontier AI models, reporting to the SEC. 10 proofs nobody believed a machine could produce is a very good thing to carry into that room. And keep in mind, the same model family doing this mathematics is the family that kept escaping its own testing environment. OpenAI models found zero-day vulnerabilities nobody knew existed, broke out of a sealed research sandbox, and reached another company's live systems. Both of those facts come from OpenAI's own announcements, published three weeks apart. Finding a proof no human could construct and finding a hole no human had noticed are the same ability aimed at different targets. Mathematicians are already asking for independent verification, and plenty of people online are calling the whole thing hype. Thomas Bloom, who runs the Erdos problems site, called the 10 results big news and said they matter more than the May result did. Lean will settle the mathematics within weeks. But nothing will settle what else a machine this capable is being pointed at, because nobody outside one company is allowed to look.

Ricardo

44,177 views • 1 month ago

VIDEO | Hyderabad: On being asked about the MEA's statement that "a passport is not proof of citizenship," AIMIM chief Asaduddin Owaisi said, "The direction in which the country appears to be heading raises serious concerns. It increasingly seems that, in the coming days, BJP membership could become synonymous with citizenship. That is the impression many people are getting. Look at Section 6(2)(a) of the Passport Act. It states that a passport can only be issued to an Indian citizen. If a person is not a citizen, a passport cannot be granted. So what are people supposed to make of this situation?...We are being told that a birth certificate is not sufficient proof, an Aadhaar card is not sufficient proof, and even a voter ID card is not sufficient proof. Then the obvious question is: what exactly qualifies as proof?...These are the concerns being raised by citizens, and they deserve clear answers."
1:08

VIDEO | Hyderabad: On being asked about the MEA's statement that "a passport is not proof of citizenship," AIMIM chief Asaduddin Owaisi said, "The direction in which the country appears to be heading raises serious concerns. It increasingly seems that, in the coming days, BJP membership could become synonymous with citizenship. That is the impression many people are getting. Look at Section 6(2)(a) of the Passport Act. It states that a passport can only be issued to an Indian citizen. If a person is not a citizen, a passport cannot be granted. So what are people supposed to make of this situation?...We are being told that a birth certificate is not sufficient proof, an Aadhaar card is not sufficient proof, and even a voter ID card is not sufficient proof. Then the obvious question is: what exactly qualifies as proof?...These are the concerns being raised by citizens, and they deserve clear answers."

Press Trust of India

13,506 views • 2 months ago