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Eigenvalues are why Google's PageRank converges, why PCA compresses 1,000-dimensional embeddings into 2D without losing structure, and why quantum mechanics gives discrete energy levels instead of a continuous blur. The same matrix operation that finds the most important directions in a dataset is what transformer attention heads use to... show more

Eron

5,221 subscribers

21,712 views • 2 months ago •via X (Twitter)

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9 Comments

Kravn2 months ago

one inaccuracy here: attention computes Q, K, V through plain matrix multiplication, not eigendecomposition, eigenvalues don't actually show up on every forward pass

AdvancedLivingSystems2 months ago

However Define an electron This is its precessional envelope in an anisotropic metal as a chip regulates flow in 2 states: ¹ Conduct ² Insulation These chips-to-memory cannot be hacked in mfg because no voltage is "null", secure vs digital probabilistic wideband processors By using black_photons instead of using fiber-optics decodes to hex, eliminates spicing a big deal, yes ? In any case this allows near-deterministic results to 9-decimals as a computational framework no probabilistic algo needed for most transactions 🛌🏻💤 This is #DiscoveryPhysics no_papers no_consensus, @supergrokheavy & @imagine astute services 🦆 @tmallard, @AdvLivingSys sessions, Thomas I. Mallard, ORCID 0009-0008-7386-5087, on this & related topics, factors as sole author ~~~ For most sim/models: picometer³/pm² sample, g/cm³ density, quectosecond to Planck ticks & use quarternion-algebra & 299792458 m/s or no_go For neutrinos quectometer³/qm², 1e-30m sample, g/mm³ density & hundredths of Planck ticks Bola-math frequency: 1-Hz = exactly 0.42 = (1-sidereal rotation of a mass) ÷ (1-revolution of the pair) or no_go Minimum masses: 3.271191e-33g leptons quarks half that are the minimum mass such that: 1.4699999259436345e-19 J = 1.6355955e-33g • c², a quark static_mass And, at c = no-mass, no-division by zero mass in calcs, an emf vortex has axial-inertia thus effective_mass > 0, integrals with feasible limits to variables Radius of p+ assume e-: ≥ 0.70710678118764752440 pm This is deterministic lab work vs probabilistic by fly-off frequency 🔧

Dr Abhishek2 months ago

Strong point. Real innovation is not only about building faster tools, but building technology that is useful, reliable, and meaningful for people. Exciting to see this direction. 🤖⚡ #AI #Technology #Innovation

Scythe2 months ago

Welcome to math 202

TheoryOfNearlyEverything2 months ago

ersma zählen lernen du idiot

Orbiter992 months ago

Tony Stark here

45 902 months ago

Yes

Tesseract2 months ago

Were rolling out a much better algorithm very soon, you could say inspired by Google or I suppose Matlab more. Look out for GENIE, where searching pizza will get you to pizza.

Wyoming Iliad2 months ago

is it true that "Eigenvalues are why Google's PageRank converges, why PCA compresses 1,000-dimensional embeddings into 2D without losing structure, and why quantum mechanics gives discrete energy levels instead of a continuous blur." or are there other aspects of hardware and software involved?

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Every quantum state in the universe is a vector - not in 3D space, but in a complex space with potentially infinite dimensions where the sum of squared coefficients always equals exactly 1. That constraint is the same math LLMs use to normalize probability distributions over tokens. A wavefunction collapsing to a measurement and a softmax choosing the next word are the same operation - projecting an infinite-dimensional vector onto a basis. Claude and GPT do quantum math on every forward pass. 14 minutes. Bookmark & watch today. Quantum mechanics and LLMs speak the same language.

Every quantum state in the universe is a vector - not in 3D space, but in a complex space with potentially infinite dimensions where the sum of squared coefficients always equals exactly 1. That constraint is the same math LLMs use to normalize probability distributions over tokens. A wavefunction collapsing to a measurement and a softmax choosing the next word are the same operation - projecting an infinite-dimensional vector onto a basis. Claude and GPT do quantum math on every forward pass. 14 minutes. Bookmark & watch today. Quantum mechanics and LLMs speak the same language.

Eron

18,166 views • 2 months ago

For years, AdamW has been the default optimizer behind every LLM training run on the planet. A new optimizer called Muon is twice as computationally efficient - and the core idea is embarrassingly elegant: orthogonalize the momentum matrix. The same singular value decomposition that powers every matrix operation inside transformer attention layers, turned into the optimizer itself. Adam treats all parameters as one flat vector, ignoring the matrix structure entirely. Muon sees the 2D shape and amplifies rare gradient directions that Adam misses - the directions that capture nuanced patterns in data. Full SVD is too expensive to run every step, so Muon approximates it with Newton-Schulz iterations: five rounds of matrix multiplications that push all singular values toward one without ever computing the decomposition. Pure GPU-native operations, no overhead. For stable training at scale they added QK-clip - if attention logits exceed a threshold, rescale the query and key projection weights per head. DeepSeek's multi-head latent attention gets its own variant called MuonClip that handles shared rotary key matrices without double-rescaling. 18 minutes. Bookmark & watch today. The optimizer that might replace AdamW inside every LLM training pipeline.

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Deliriumm

14,292 views • 12 days ago

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Andrew Ng

132,792 views • 1 year ago

In 1747, Euler spent 133 pages proving that logarithms have infinitely many solutions - and in doing so accidentally revealed why every LLM, every signal processor, and every Fourier transform works the way it does. The imaginary unit I doesn't break exponential functions. It rotates them, turning exponential growth into the sine and cosine waves that live inside every transformer attention layer. The proof runs through Briggs' 1617 zooming algorithm, complex polar coordinates, and a thin-triangle argument that connects imaginary exponents to angles in radians. When you plug I into the exponential function, you don't get growth - you get rotation confined to the unit circle. The same unit circle that softmax maps every attention score onto before the GPU multiplies them. 27 minutes. Bookmark & watch today. The most beautiful equation in mathematics and why it powers every model you have ever used.

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Deliriumm

128,190 views • 1 month ago

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DeepLearning.AI

36,832 views • 1 year ago

In 1843, Hamilton realized complex numbers needed not one more dimension but two. One real part, three imaginary - i, j, k. He carved the equation into a Dublin bridge on October 16th. That number system now runs inside every 3D graphics engine, every robotics controller, and every LLM embedding space that maps language into high-dimensional geometry on GPU clusters today. Quaternion multiplication spins two perpendicular circles simultaneously - something that can only happen in four dimensions. i times j is k, but j times i is -k. Rotations don't commute, like a Rubik's cube. That's exactly why game engines and spacecraft use quaternions instead of matrices - no gimbal lock, no numerical drift. The same rotation math that every transformer uses when computing rotary position embeddings to encode token order in the complex plane. 2 minutes. Bookmark & watch today. The number system that powers every 3D engine and nobody taught you about.

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Eron

23,288 views • 1 day ago

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Morlex

34,790 views • 2 months ago

He broke the Bank of England, made $1 billion in a single day, ran a $12 billion fund for 30 years, and never had one losing year this is him sitting down with Bloomberg to explain why he just put most of his portfolio into one AI stock and why a guy who shorted entire currencies just put most of his portfolio into one AI stock bookmark & watch it today ↓

He broke the Bank of England, made $1 billion in a single day, ran a $12 billion fund for 30 years, and never had one losing year this is him sitting down with Bloomberg to explain why he just put most of his portfolio into one AI stock and why a guy who shorted entire currencies just put most of his portfolio into one AI stock bookmark & watch it today ↓

slash1s

225,314 views • 5 months ago

$In 2015, LIGO measured spacetime stretching by less than a thousandth of a proton's diameter - and from that single measurement physicists calculated that spacetime is 10²⁰ times stiffer than steel. A recent paper asked the next question: if spacetime has stiffness, does it also have a breaking point? The same elastic energy framework that governs every neural network weight update and every tensor operation on GPU clusters - applied to the fabric of reality itself. The math is brutal. Apply Griffith's 1921 fracture mechanics to a Planck-sized cube of space and the Young's modulus cancels entirely - the energy at fracture depends only on surface energy times the Planck length squared. The maximum tearing force converges on 10⁴⁴ newtons, the Planck force. The frequency where gravitational wave stiffness meets fracture energy is 10⁴³ hertz - the Planck frequency. Every part of the analogy independently converges on the same scale where Einstein's smooth spacetime breaks down and quantum gravity begins. 43 minutes. Bookmark & watch today. The physics of whether reality itself has a breaking point.$

In 2015, LIGO measured spacetime stretching by less than a thousandth of a proton's diameter - and from that single measurement physicists calculated that spacetime is 10²⁰ times stiffer than steel. A recent paper asked the next question: if spacetime has stiffness, does it also have a breaking point? The same elastic energy framework that governs every neural network weight update and every tensor operation on GPU clusters - applied to the fabric of reality itself. The math is brutal. Apply Griffith's 1921 fracture mechanics to a Planck-sized cube of space and the Young's modulus cancels entirely - the energy at fracture depends only on surface energy times the Planck length squared. The maximum tearing force converges on 10⁴⁴ newtons, the Planck force. The frequency where gravitational wave stiffness meets fracture energy is 10⁴³ hertz - the Planck frequency. Every part of the analogy independently converges on the same scale where Einstein's smooth spacetime breaks down and quantum gravity begins. 43 minutes. Bookmark & watch today. The physics of whether reality itself has a breaking point.

Eron

16,766 views • 12 days ago

The fourth dimension is not impossible to visualize - that claim is at best severely limiting and at worst completely wrong. Every LLM operates in embedding spaces of 4,096 dimensions or more, and the geometric intuitions that make transformer attention work are direct extensions of the same logic this video builds from scratch. The key insight is extrusion. A square is an infinite stack of line segments. A cube is an infinite stack of squares. A hypercube is an infinite stack of cubes - each cross-section a full 3D universe, stacked along a fourth axis perpendicular to all three. The deck of cards model makes this concrete: project 3D space onto a flat surface, free up the perpendicular dimension, and suddenly four dimensions have a home. 18 minutes. Bookmark & watch today. The clearest path into four-dimensional thinking anyone has made.

The fourth dimension is not impossible to visualize - that claim is at best severely limiting and at worst completely wrong. Every LLM operates in embedding spaces of 4,096 dimensions or more, and the geometric intuitions that make transformer attention work are direct extensions of the same logic this video builds from scratch. The key insight is extrusion. A square is an infinite stack of line segments. A cube is an infinite stack of squares. A hypercube is an infinite stack of cubes - each cross-section a full 3D universe, stacked along a fourth axis perpendicular to all three. The deck of cards model makes this concrete: project 3D space onto a flat surface, free up the perpendicular dimension, and suddenly four dimensions have a home. 18 minutes. Bookmark & watch today. The clearest path into four-dimensional thinking anyone has made.

Eron

12,567 views • 1 month ago

In 1957, Stephen Smale proved you can turn a sphere inside out without cutting, tearing, or creasing it - even though you can't do it with a circle. The proof was so abstract it took 7 years before anyone found a practical method. The key is a topological invariant called the turning number - the same class of invariants that every LLM uses when its embedding layers preserve structure across high-dimensional transformations on GPU clusters. The trick is corrugation. Divide the sphere into guide strips connected by wavy segments - the waviness gives enough flexibility for strips to twist past each other without creasing. Push the polar caps through, twist them in opposite directions to convert loops into twisting, then push the equator back through the center. 16 rotated copies of one building block construct the entire eversion. A circle can't do this because its turning number changes from +1 to -1 when reversed. Spheres have an extra dimension where saddles, bowls, and domes interact - their invariant stays the same regardless of which face is out. The same dimensional escape that lets attention heads resolve contradictions in higher-dimensional latent space that look impossible in any lower projection. 19 minutes. Bookmark & watch today. The proof that spheres break a rule circles never can.

In 1957, Stephen Smale proved you can turn a sphere inside out without cutting, tearing, or creasing it - even though you can't do it with a circle. The proof was so abstract it took 7 years before anyone found a practical method. The key is a topological invariant called the turning number - the same class of invariants that every LLM uses when its embedding layers preserve structure across high-dimensional transformations on GPU clusters. The trick is corrugation. Divide the sphere into guide strips connected by wavy segments - the waviness gives enough flexibility for strips to twist past each other without creasing. Push the polar caps through, twist them in opposite directions to convert loops into twisting, then push the equator back through the center. 16 rotated copies of one building block construct the entire eversion. A circle can't do this because its turning number changes from +1 to -1 when reversed. Spheres have an extra dimension where saddles, bowls, and domes interact - their invariant stays the same regardless of which face is out. The same dimensional escape that lets attention heads resolve contradictions in higher-dimensional latent space that look impossible in any lower projection. 19 minutes. Bookmark & watch today. The proof that spheres break a rule circles never can.

Eron

28,367 views • 8 days ago

Schrodinger's equation, Neural ODEs, and every LLM built on Mamba use one operation: raising a matrix to a power. Sounds like nonsense until you see it. A Stanford math grad just released a 27-minute walkthrough starting from a mass on a spring and building up to quantum mechanics. Publicly free. Bookmark & watch this weekend. Same operation, from 1926 to 2025.

Schrodinger's equation, Neural ODEs, and every LLM built on Mamba use one operation: raising a matrix to a power. Sounds like nonsense until you see it. A Stanford math grad just released a 27-minute walkthrough starting from a mass on a spring and building up to quantum mechanics. Publicly free. Bookmark & watch this weekend. Same operation, from 1926 to 2025.

fintex

44,082 views • 3 months ago

The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never touch but link through each other. It's the same structure that describes qubit states on the Bloch sphere - the geometry every quantum computer runs on. Most people think four dimensions are impossible to visualize. Stereographic projection collapses them onto a screen without losing the topology - the same trick GPS satellites use to map curved space onto flat coordinates. 12 minutes. Bookmark & watch today. The math is stranger than the universe.

The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never touch but link through each other. It's the same structure that describes qubit states on the Bloch sphere - the geometry every quantum computer runs on. Most people think four dimensions are impossible to visualize. Stereographic projection collapses them onto a screen without losing the topology - the same trick GPS satellites use to map curved space onto flat coordinates. 12 minutes. Bookmark & watch today. The math is stranger than the universe.

Eron

14,881 views • 2 months ago

CEO of Citadel: "no one is more wrong than I am today", he built the most profitable hedge fund in history in this interview he explains why he hired a Russian rocket scientist, why being the smartest in the room is a mistake, and why being right 54% of the time made $90 billion Bookmark & watch it. Then read the article below - The 77-year-old formula that explains why a small edge is all you need ↓

CEO of Citadel: "no one is more wrong than I am today", he built the most profitable hedge fund in history in this interview he explains why he hired a Russian rocket scientist, why being the smartest in the room is a mistake, and why being right 54% of the time made $90 billion Bookmark & watch it. Then read the article below - The 77-year-old formula that explains why a small edge is all you need ↓

Сarm1ne

4,260,634 views • 6 months ago

Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

Complex numbers aren't imaginary. They're the only way to describe a wave that has both amplitude and phase at the same time - and every LLM processes language as exactly that kind of wave through its attention layers. The same math that describes quantum wave interference is what makes Fourier transforms work, which is what lets GPT process tokens in parallel instead of one at a time. Descartes named them "imaginary" before anyone understood what they were. 20 minutes. Bookmark & watch today. The confusion clears up faster than any textbook manages.

Eron

43,247 views • 2 months ago

$Andrew Ng just revealed why the AI companies throwing the most compute at the problem are going to lose. The winner of the intelligence race won’t use the most compute. They’ll waste the least. Ng: “Most of your high-dimensional data lies on a lower-dimensional subspace. It’s just a fact of life.” Here’s what that means in practice. You have a 10,000-dimensional dataset. Every dimension dragged through every calculation. Every training cycle hauling dead weight the model will never use. Ng: “You’re carrying around these 10,000-dimensional examples throughout your whole training process.” That bloat isn’t just inefficient. It’s a tax on every computation you run. Memory bandwidth. Network bandwidth. Computational speed. All of it eaten by dimensions that contribute nothing to intelligence. They contribute noise. The insight that separates the architects from the arms race: that 10,000-dimensional dataset is almost entirely captured by a much smaller subspace. The signal lives in a fraction of the space you’re paying to process. Compress it. 10,000 dimensions down to 1,000. Ng: “You can run your learning algorithm on a much lower-dimensional set of data and it may be much more efficient.” Same hardware. Same budget. A fraction of the friction. Brute force is the strategy of whoever has the deepest pockets. Compression is the strategy of whoever actually understands the problem. The companies that master this don’t just build faster models. They build models that find more truth in less data than anything scaling blindly ever will. Intelligence was never about processing everything. It’s about knowing what to cut.$

Andrew Ng just revealed why the AI companies throwing the most compute at the problem are going to lose. The winner of the intelligence race won’t use the most compute. They’ll waste the least. Ng: “Most of your high-dimensional data lies on a lower-dimensional subspace. It’s just a fact of life.” Here’s what that means in practice. You have a 10,000-dimensional dataset. Every dimension dragged through every calculation. Every training cycle hauling dead weight the model will never use. Ng: “You’re carrying around these 10,000-dimensional examples throughout your whole training process.” That bloat isn’t just inefficient. It’s a tax on every computation you run. Memory bandwidth. Network bandwidth. Computational speed. All of it eaten by dimensions that contribute nothing to intelligence. They contribute noise. The insight that separates the architects from the arms race: that 10,000-dimensional dataset is almost entirely captured by a much smaller subspace. The signal lives in a fraction of the space you’re paying to process. Compress it. 10,000 dimensions down to 1,000. Ng: “You can run your learning algorithm on a much lower-dimensional set of data and it may be much more efficient.” Same hardware. Same budget. A fraction of the friction. Brute force is the strategy of whoever has the deepest pockets. Compression is the strategy of whoever actually understands the problem. The companies that master this don’t just build faster models. They build models that find more truth in less data than anything scaling blindly ever will. Intelligence was never about processing everything. It’s about knowing what to cut.

Dustin

215,857 views • 8 months ago

YOU ARE IN THE MOST EPIC BATTLE IN ALL OF HUMAN EXISTENCE. What an absolutely incredible honor it is to be participating in the most powerful soul-strengthening experience that ever was, and ever will be created with you all 👊🔥 If you aren't thinking about life this way, then you SHOULD be. Because this is exactly what it is, and precisely why YOU WERE CHOSEN TO BE HERE IN THIS EXACT MOMENT! Cut that self-pitying shit out immediately, and open the throttle full blast as you unleash every ounce of your God-given power into this game. There has NEVER been a better, or more important time to throw all hesitance out the window, and project 1,000% OF YOURSELF into every aspect of your reality. [🎥IG: andrethealchemist]

YOU ARE IN THE MOST EPIC BATTLE IN ALL OF HUMAN EXISTENCE. What an absolutely incredible honor it is to be participating in the most powerful soul-strengthening experience that ever was, and ever will be created with you all 👊🔥 If you aren't thinking about life this way, then you SHOULD be. Because this is exactly what it is, and precisely why YOU WERE CHOSEN TO BE HERE IN THIS EXACT MOMENT! Cut that self-pitying shit out immediately, and open the throttle full blast as you unleash every ounce of your God-given power into this game. There has NEVER been a better, or more important time to throw all hesitance out the window, and project 1,000% OF YOURSELF into every aspect of your reality. [🎥IG: andrethealchemist]

Ben Wehrman

21,656 views • 26 days ago

Heinz Hopf discovered this in 1931. Roger Penrose called it "an element of the architecture of our world." It shows up in at least 8 areas of physics, yet most mathematicians spend entire careers never touching it. The structure that defines every qubit in quantum computing, every rotation a robotics controller computes, and the geometry inside every LLM embedding space operating on GPU clusters today - and nobody talks about it. The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never intersect but each links through every other exactly once. The Bloch sphere every quantum computer operates on, the fiber bundles underlying gauge theory, and the quaternion math inside every 3D graphics engine and every embedding rotation on the planet. Bookmark & watch today. The most important shape nobody talks about.

Heinz Hopf discovered this in 1931. Roger Penrose called it "an element of the architecture of our world." It shows up in at least 8 areas of physics, yet most mathematicians spend entire careers never touching it. The structure that defines every qubit in quantum computing, every rotation a robotics controller computes, and the geometry inside every LLM embedding space operating on GPU clusters today - and nobody talks about it. The Hopf fibration maps a 4D hypersphere onto a regular sphere using circles that never intersect but each links through every other exactly once. The Bloch sphere every quantum computer operates on, the fiber bundles underlying gauge theory, and the quaternion math inside every 3D graphics engine and every embedding rotation on the planet. Bookmark & watch today. The most important shape nobody talks about.

Eron

27,256 views • 3 days ago

$Someone converted the Mandelbrot set into sound by mapping fractal orbits to audio frequencies in real time on a GPU - and accidentally built an infinite synthesizer. Every point in the fractal converges to a repeating orbit with its own period and shape. Map those coordinates to speaker amplitude at 8kHz and you get an instrument where zooming in adds harmonics on top of harmonics - the same principle behind every DSP audio engine ever built. 8 minutes. Bookmark & watch today. The most unexpected use of a GPU you'll see this week.$

Someone converted the Mandelbrot set into sound by mapping fractal orbits to audio frequencies in real time on a GPU - and accidentally built an infinite synthesizer. Every point in the fractal converges to a repeating orbit with its own period and shape. Map those coordinates to speaker amplitude at 8kHz and you get an instrument where zooming in adds harmonics on top of harmonics - the same principle behind every DSP audio engine ever built. 8 minutes. Bookmark & watch today. The most unexpected use of a GPU you'll see this week.

Eron

27,390 views • 2 months ago