Update on Complex Neural Networks: The Phase-Coherent Transformer (PCT) has solved PathX. PathX is regarded as an extremely difficult long-range context problem even among machine learning tasks; fewer than one in a thousand models/LLMs—including practical ones—can solve it. Even state-of-the-art frontier models cannot solve it directly. This time, a variant of PCT—complexified Screening (titled “Screening is Enough”) augmented with phase-preserving complex diagonal matrix operations—solved PathX with 92% accuracy, once again confirming that preserving phase information forms an effective inductive bias. TBH, the fact that models which do not preserve phase information dropped out of PathX as complex NNs may be more important than the fact that PathX was solved itself. PCT (complex Screening + PCR) is the first purely complex neural network to solve PathX. The original model from which the diagonal recurrence was borrowed was only about 60% complex-valued. Confidence in the claim that “a Transformer for complex NNs, which had not existed until now, has been found” has been strengthening through various experiments. Looking ahead, I expect this family of models to become an entirely new class of models that can simultaneously achieve deep understanding of natural language text and time-series/signal analysis. paper： (note: the latest version is in July) model:

Your AI agent's payment log is its strategy log. SubEtha is a private x402 prototype on zERC20 : payments go through, but the payer–payee link never hits the chain. It hides who paid whom. Write-up:

You can boost generative AIs like Claude with Lean — not just during the spec phase, but also for security hardening and bug hunting after implementation. Here’s a cool result from recently. There was an attack vector in Plonky2-whir that even the strongest AI models kept missing no matter how hard they searched. We found it with Lean + Claude, so I turned it into a reusable skill.This skill is pretty beginner-friendly. You can just paste it into Claude or Codex and try it out even if you don’t know anything about Lean. I think the difference between using Lean to decide on specs or database design and then verify them, versus regular vibe coding, is kind of like building with stones or bricks versus using reinforced concrete. At first the speed feels similar, but eventually you reach a height where stacking stones just won’t cut it anymore, and the gap in scale and safety becomes really obvious. Lean is basically the reinforced concrete of software development.Until about 10 years ago, I honestly saw formal verification and logic-based stuff as something that got pushed to the sidelines by machine learning — like “not practical,” “doesn’t make money,” or “another failed expert systems thing.” Fuzzy logic was used in rice cookers and trains, and even in fighter jets later on, but pure logic systems like Prolog felt like they were no longer the main character after deep learning took over. So seeing Lean now working really well with machine learning and delivering these surprisingly strong results feels pretty special to me, since I’ve always liked the logic side of things.

Diamond iO Update We are excited to share that we have introduced several new optimization techniques for Diamond iO, implemented the full end-to-end pipeline with realistic lattice parameters, and produced concrete performance estimates. Latest paper: Latest implementation: Last year’s initial Diamond iO implementation deliberately left out one crucial component: FHE evaluation over BGG+ encodings. The reason was simple: the noise growth was too large to support realistic lattice parameters. This component is not specific to Diamond iO. FHE evaluation over BGG+ encodings has been used as a theoretical building block in advanced lattice-based cryptography for more than a decade, but there has been very little work on making it actually implementable. In the latest update, we make this component concrete. We introduce several techniques that substantially reduce noise growth, including native lookup-table evaluation, constant-depth arithmetic circuits, and specialized noise refreshing. The new implementation also runs all major lattice operations on GPUs. Together, these optimizations allow us to instantiate the full Diamond iO pipeline with realistic lattice parameters: a modulus size of at most 1540 bits and lattice dimension 2^{16}, comparable to large parameter sets used in CKKS-style FHE implementations. The total running time is still far beyond practical use, but we can now estimate the end-to-end cost rather than leaving a major component abstract. What improves? Our estimates suggest that Diamond iO reduces the overhead of the main bottleneck in most modern iO schemes: the transformation from functional encryption (FE) to iO. Compared with prior FE-to-iO transformations, Diamond iO reduces this overhead by at least a factor of 2^118, making the transformation almost as efficient as the underlying FE scheme itself. In other words, Diamond iO appears to largely remove the FE-to-iO transformation as the dominant cost. What remains? Diamond iO is still very far from practical. Once the FE-to-iO overhead is reduced, the bottleneck shifts to the final FE decryption step, which requires FHE evaluation over BGG+ encodings. Our current estimate is that this step would still take about 10^36 hours. As explained in Vitalik’s recent post, much of the inefficiency in modern iO comes from the need to stack a complicated tower of cryptographic primitives: FHE × ABE × GC × XiO × ... Diamond iO significantly simplifies this picture: the construction is essentially simple matrix operations plus an FHE × ABE layer, where the ABE component is essentially BGG+ encodings. In our view, this gives a much clearer roadmap toward practical iO: further simplify the remaining FHE × ABE layer, and the gap to practicality could shrink dramatically.

A ten-thousand word monster post trying to cover the entire tech tree behind the main lineage of obfuscation (iO) protocols: Special thanks to all who helped!

Loopring, the oldest zkRollup from 2019, will end. I'm a little shocked because I still use it occasionally, but I feel like I can't help it with this congestion and this market. As a DEX, I've seen it since 2017, and it's a project that I have a lot of memories of. Thanks to Loopring, I stepped into zkRollup. While I was developing Plasma and giving up once, I was skeptical about zkRollup, but they proved it at the very beginning. When I saw it, I was really moved. Thank you so much!!

Tomorrow in Zurich 🇨🇭, our Co-Founder, Mai Fujimoto, will be participating in a roundtable at the Point Zero Forum on the theme of "Identity Wallets, Trust Infrastructure, and Preparing for the Agentic Economy." We are truly honored to join such an outstanding group of participants representing the fields of finance and digital identity, including representatives from Visa, Circle, TRM Labs, the World Bank, Cardano Foundation, Banking Circle, the Bangko Sentral ng Pilipinas, the United Nations University, and GLEIF. As the co-founder of zERC20, Mai will be speaking about the importance of privacy in digital finance. We look forward to discussing how trusted digital identity systems and privacy-preserving technologies can help build a safer and more trustworthy financial infrastructure as we move toward a future where AI agents actively participate in economic activity.

I believe that Ethereum is only one community that can do deep tech of Blockchain and AI. I insist that Ethereum community should do more than the x402 adoption for the AI space. ofc, x402 is great. Ethereans made the best blockchain, and should make the best AI as well. Alibaba did that. Why can't we?

I was really lucky to come across a very interesting phenomenon in complex neural networks, so I wanted to share it with you. It’s called the Phase Coherent Transformer. Under specific conditions, complex-valued neural networks (CvNNs) exhibit significantly better generalization and superior performance across multiple tasks. I’ve written a detailed experimental report in the paper.Until now, CvNNs were thought to be good mainly in a narrow range of tasks like signal analysis and phase information processing. After Transformers and attention mechanisms came along, the gap with real-valued networks only kept widening. In fact, in this experiment as well, turning a standard Transformer into a complex version mostly just led to performance degradation (except for phase-related tasks). But now, This transformer seems really suited for complex neural networks, unlike before, and it’s showing completely unexpected behaviors and impressive benchmark results. I suspected that “the softmax in Transformers might be causing the loss of phase information in complex weights across multiple layers.” So I looked for attention structures that can preserve phase information, and ran experiments. Approaches like Sigmoid attention and the recently announced Multiscreen have this kind of structure. After complexifying them and tuning for better multi-layer phase control, I named the resulting model the Phase Coherent Transformer (PCT).The experimental results showed that it greatly outperforms many real-valued NNs on tasks where complex NNs had traditionally struggled, while still retaining its advantage on phase-related tasks. Paper preprint: It’s not yet clear how this will impact AI overall, but I think it’s a pretty significant development for the field of complex neural networks at least.This naturally brings up one important question: Q: Should today’s AI stay based on real numbers? In my field, circuit-style cryptography (ZKP, WE), the algebraic structures used as function inputs have naturally expanded into extension fields when necessary. It wouldn’t be surprising if AI follows a similar natural path. Moreover, CvNNs can likely associate and infer from data (=responses and signals from circuits or functions) under the assumption that phase information is hidden inside them. This feels incredibly powerful, because it could allow much more sophisticated handling not only of circuit outputs but also of intermediate state traces.