Shan WAN’s Post

The Search Space Illusion Maxwell, Shannon, Landauer: three physicists explained the true cost of computation. None of them made it into Computer Science 101. We are attempting extropic systems on an entropic substrate — and wondering why it’s so expensive Maxwell showed that knowing has a cost. Shannon showed that transmission has limits. Landauer showed that forgetting burns energy. The remaining question is not whether computation obeys physics. It is whether we continue to build systems that fight it inefficiently — or design systems that make error unreachable from the start.Not better search. No search. Intelligence is not expansion. It is collapse. https://lnkd.in/e8XTVhBd

This model is the fundament of #complexidealism. This article presents a fundamental, discrete mathematical framework for computational models in physics and artificial intelligence, circumventing the reliance on transcen-dental constants (e, π) and the imaginary unit (i). We hypothesize that computational reality is grounded in the operational counting of discrete values. By utilizing three fundamental counters-the placeholder (N), the drive (k), and the universal flow (Υ)-we define the Generative Universal Transformer. We demonstrate that this 2 × 2 scaled rotation matrix natively generates exact rotational geometries, Pythagorean triples, and continuous-like phenomena through strictly rational arithmetic. Furthermore, we prove that this discrete matrix operator perfectly replicates the Continuous Fourier Transform and the Matrix Exponential, while seamlessly generating natural growth structures such as the Golden Spiral and Lucas sequences. This framework proves that the operational engine of physical and computational systems can be modelled entirely through discrete, area-preserving matrix mechanics. The universe does not need to know all digits of π to rotate. The only relevant question for intelligence is what we count. https://lnkd.in/e4sj-Q-2

"Ordinary Least Squares is a Special Case of Transformer" by Xiaojun Tan, Yuchen Zhao "The statistical essence of the Transformer architecture has long remained elusive: Is it a universal approximator, or a neural network version of known computational algorithms? Through rigorous algebraic proof, we show that the latter better describes Transformer's basic nature: Ordinary Least Squares (OLS) is a special case of the single-layer Linear Transformer. Using the spectral decomposition of the empirical covariance matrix, we construct a specific parameter setting where the attention mechanism's forward pass becomes mathematically equivalent to the OLS closed-form projection. This means attention can solve the problem in one forward pass, not by iterating. Building upon this prototypical case, we further uncover a decoupled slow and fast memory mechanism within Transformers. Finally, the evolution from our established linear prototype to standard Transformers is discussed. This progression facilitates the transition of the Hopfield energy function from linear to exponential memory capacity, thereby establishing a clear continuity between modern deep architectures and classical statistical inference." Paper: https://lnkd.in/dD_ZEuVa #machinelearning

Criticality, Phase Transitions, and the Emergence of Meaning: A Dynamical Perspective on Value in Hyper-Aether Systems Abstract We propose a unified dynamical framework for understanding meaning, concept formation, and creativity in AI-native computational systems. Building on the Hyper-Aether architecture, we interpret the value function V (G) over graph-structured computational states as an energy landscape governing system dynamics. We show that (1) meaning corresponds to stable attractors, (2) concept formation corresponds to phase transitions, and (3) creativity emerges from critical fluctuations near phase boundaries. This framework connects reinforcement learning, statistical physics, and semantic computation into a single theoretical model. https://lnkd.in/gR_g4-5b

Between the Discrete and the Continuous: Two Languages, One Reality We often speak as if the world is forced to choose: discrete or continuous, digital or analog, particles or fields. But nature does not sign contracts with our categories. It simply is—and we approximate. At one scale, reality looks discrete: quanta of energy, particles, bits, decisions. At another, it dissolves into continuity: fields, waves, smooth geometries, differential equations that flow like rivers of logic. And yet, neither description is complete on its own. In physics, this duality is not philosophical—it is operational. Quantum systems show discrete spectra, while their evolution is governed by continuous wavefunctions. Classical mechanics assumes smooth spacetime, yet numerical simulation forces us into discretization. We live in the tension between what is and what we can compute. Mathematics formalizes this bridge. From discrete to continuous: • factorials extend into the Gamma function • finite differences converge to derivatives We pass from: f′(x) = lim_{h→0} [f(x+h) − f(x)] / h A discrete increment becomes a continuous slope in the limit of vanishing scale. From continuous to discrete: we reverse the lens. We approximate smooth fields by finite representations: u(x) ≈ Σ Nᵢ(x) uᵢ A continuous reality becomes a weighted sum of local anchors—nodes in a mesh, samples in a signal, agents in a network. This is not a loss. It is a translation. And perhaps that is the key idea: we are not moving between truth and approximation, but between languages of truth. Continuous models give us elegance, symmetry, and analytical power. Discrete models give us computability, structure, and control. The most powerful scientific frameworks today are not pure—they are hybrid. Finite element methods, spectral methods, multiscale models: all of them live in this in-between space where continuity is simulated by discreteness, and discreteness is interpreted through continuity. But there is a deeper question emerging beneath engineering and mathematics: Is reality itself one of these, or something prior to both? Some modern directions in theoretical physics suggest that spacetime may be fundamentally discrete, with continuity emerging as an approximation. Others suggest the opposite: that discreteness is an artifact of observation, and the universe is fundamentally a smooth structure we sample imperfectly. There is a third possibility, more radical and perhaps more honest: Reality may not be either. It may be a structure from which both continuous fields and discrete quanta emerge as different projections—like shadows cast by the same underlying geometry. If so, then the real task of science is not to decide between discrete and continuous descriptions, but to build the transformations between them with increasing fidelity. Because every time we move from a mesh to a field, from samples to waves, from atoms to continua—or back—we are not simplifying reality.

Black holes. Artificial General Intelligence. Human consciousness. What do they have in common? ⚛️ They are all bound by the concept of a "Singularity"—a mathematical, physical, and computational point where the standard architecture of a system completely breaks down. As engineers, we are used to predictable systems. You put data in, you get an expected output. But what happens when the architecture scales to infinity? 🔹 In Physics: When mass scales infinitely into a localized point, gravity breaks the standard model. We call it a Black Hole. 🔹 In Code: When an intelligence system scales its own capacity to optimize algorithms exponentially, human prediction models fail. We call it the AI Singularity. 🔹 In Human Biology: When the human mind attempts to process reality beyond biological constraints—either through hyper-connectivity or eventual merging with digital interfaces—the boundary of the ego dissolves. The universe seems to have a built-in mechanism for resetting the rules when a system reaches maximum capacity. Matter compresses into a new physical reality. Computation compresses into superintelligence. We are currently engineering the digital version of a black hole. Once we cross the threshold of recursive self-improvement, we pass the event horizon. What happens on the other side? #SystemArchitecture #AI #AGI #Physics #SoftwareEngineering #TechInnovation

🚀 A new tool for studying complex dynamics: 🧨 DYNAMITE 🧨 We’re excited to share DYNAMITE (DYNAmical Mean-fIeld Time Evolution solver) — a high-performance framework designed to solve Dynamical Mean-Field Equations (DMFE) at unprecedented time scales. 🤔  Why does it matter? DMFEs are essential for understanding systems with slow dynamics and long memory, with applications ranging from statistical physics to machine learning. However, solving them numerically has long been limited by prohibitive computational costs. ⚡ What does DYNAMITE bring? - Access to extremely long time scales (up to ~10⁷) - Efficient scaling via non-uniform interpolation and numerical “renormalization” - GPU acceleration with orders-of-magnitude speedups - Open-source, reproducible, and extensible codebase 💡 This unlocks new possibilities in studying: - Aging dynamics in disordered systems - Spin glasses and complex energy landscapes - Neural network models and inference problems 📎 Learn more and explore the code: https://lnkd.in/gXD96RmN A significant step forward for researchers working on complex systems and out-of-equilibrium dynamics. #Physics #HPC #ComputationalPhysics #MachineLearning #OpenScience #Research

Quantum Searches Become Vastly Faster with Reinforcement Learning Reducing the time to find a specific item within a quantum system from scaling with the square root to the natural logarithm of its size represents a fundamental shift in computational efficiency. This advance means a search in a system of ten million parts could, in theory, be completed in just twenty-three steps, rather than three thousand one hundred and sixty-two. Numerical simulations suggest this reinforcement strategy also tolerates exponentially more noise than standard approaches, paving the way for more reliable quantum algorithms. #quantum #quantumcomputing #technology https://lnkd.in/emmy3SgT

New mathematical study argues the universe may not be a simulation, challenging one of the internet’s biggest ideas. The simulation hypothesis asks whether reality could be like advanced software created by a higher intelligence. It became popular because computers keep improving rapidly. If civilizations can build virtual worlds, some thinkers wondered whether our world might also be one. The new argument focuses on mathematics and limits. Simulating an entire universe in perfect detail could require unimaginable computing power, storage, and energy. Every particle, force, and interaction would need updating continuously. Even if shortcuts were used, the complexity may become overwhelming. The study suggests the costs of running such a complete reality could be far beyond practicality. This does not automatically prove the universe is definitely not simulated. It shows one possible route may be less plausible than many people assume. Philosophy often works this way. A claim is tested through logic, assumptions, and consequences. If the assumptions fail, confidence drops. But final answers remain difficult when the subject reaches beyond direct experimental access today. There are also deeper questions. Would a simulation need to model every atom, or only what observers notice? Could reality use compressed rules instead of brute force detail? Might strange physics be signs of code, or simply nature being nature? Different versions of the hypothesis lead to different conclusions, which is why debate continues across science and philosophy. The most useful lesson may not be whether we live inside software. It is that humans keep asking bold questions about existence itself. We test ideas, challenge assumptions, and search for deeper patterns. Whether reality is fundamental or simulated, the stars still shine, choices still matter, and discovery remains real to us. Wonder survives either answer beautifully today always. #universe #physics

Everyone calls it 'physics-informed.' But informed by what, exactly? If you embed Navier-Stokes into your loss function, that's physics. Conservation of mass, momentum, energy. No argument there. But what about a turbulence closure model? Say SST k-ω. That's not physics. That's a set of calibrated assumptions Menter put together in 1994. The constants were tuned to match specific experiments. The Boussinesq hypothesis underneath it isn't a law, it's an analogy borrowed from how molecular viscosity works, applied to turbulence where it doesn't necessarily hold. So when your PINN 'satisfies the governing equations,' which governing equations? The real ones, or the ones we agreed to use because the real ones are too expensive to solve? This distinction doesn't get enough attention. Then there's the loss curve problem. Training loss is decreasing, great. But neural networks have spectral bias. They pick up smooth, low-frequency features first. The sharp stuff, shocks, boundary layers, the thin regions where engineering failures actually happen, those get learned last, if at all. This isn't speculation, it's shown through NTK theory. You can have a beautiful loss curve and a model that's completely blind to the most critical features in your domain. Classical numerical methods took decades to build convergence guarantees, error estimators, mesh independence checks. We don't have any of that for PINNs yet. From my perspective the validation culture simply isn't there. None of this means SciML is useless. It means it's too early to trust it blindly. The people who will push it forward are the ones asking where it breaks, not the ones celebrating where it works. What's your experience been? #SciML #PhysicsInformedML #CFD #NumericalMethods #DeepLearning