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Introducing Operation Matrix Because Matrix operations generally break kids’ brains on whiteboards… Matrices aren't just numbers- they are transformations in space Now can actually SEE the geometry: every addition, multiplication, inverse, eigenvalue, and eigenvector becomes a transformation in real-time space. What it visualizes: • Basic ops (add/subtract/multiply) • Transpose,...

58,889 次观看 • 5 个月前 •via X (Twitter)

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The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 次观看 • 7 个月前

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,892 次观看 • 8 个月前

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

25,684 次观看 • 1 个月前

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

13,258 次观看 • 8 天前

A 91-year-old professor is why Nvidia is worth $4 trillion. His name is Gilbert Strang. He teaches linear algebra at MIT. Every AI model on Earth runs on his course. The course has been free on YouTube since 2005. The videos have earned him nothing. MIT 18.06 opens with "The Geometry of Linear Equations." No advanced math. Strang takes a system of two equations, draws it two ways, and shows the class that a matrix is a picture, not an abstraction. The row picture is two lines that cross. The column picture is two arrows that sum to a target. Every neural network on Earth operates on the column picture. Strang first taught linear algebra at MIT in 1962. He wrote the textbook in 1976. It is on every serious engineer's shelf. Every quant fund, every ML lab, every rendering engine at Pixar is running his math. His central insight is that most people are taught matrices as bookkeeping. That is the first thing to unlearn. A matrix is a linear transformation. A linear transformation is a way of moving space. Once you see the space move, the math stops being algebra and becomes geometry. The Kalman filter is a linear system. PCA is a linear system. Every gradient step in a neural net is a matrix-vector product. GPT is a stack of matrix-vector products, each one a scene from MIT 18.06 running on a Blackwell GPU. He retired in 2023 after 61 years at MIT. The course is still up. Watched tens of millions of times. The chip is $40,000. Strang never asked for a royalty.

Ochob

130,880 次观看 • 1 个月前

Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew and calculated one entirely by hand. Goal: push the loss gradient back through a 3-layer network and land on a new value for every weight and bias. = 1. Given = A 3-layer perceptron, an input X, predictions Ypred = [0.5, 0.5, 0], and the truth Ytarget = [0, 1, 0]. = 2. Backprop gradient cells = Let us draw empty cells for every gradient we are about to compute. The shape of the answer comes first. = 3. Layer 3 softmax = We get dL/dz3 straight from Ypred minus Ytarget = [0.5, -0.5, 0]. No chain rule needed, and that shortcut is the whole reason softmax and cross-entropy are paired. = 4. Layer 3 weights and biases = Let us multiply dL/dz3 by [a2 | 1]. One multiplication gives the gradient for W3 and b3 together. = 5. Layer 2 activations = We multiply dL/dz3 by W3 to get dL/da2. The gradient moves back across a layer the same way the signal moved forward. = 6. Layer 2 ReLU = Let us pass it through the gate: keep the gradient where the activation was positive, zero it everywhere else. = 7. Layer 2 weights and biases = We multiply dL/dz2 by [a1 | 1]. The same figure as step 4, one layer up. = 8. Layer 1 activations = Let us multiply dL/dz2 by W2. = 9. Layer 1 ReLU = We apply the same gate again, now on a1. = 10. Layer 1 weights and biases = Let us multiply dL/dz1 by [x | 1], and every weight in the network now has a gradient. = 11. Update = We subtract, and the network has learned. In practice a learning rate scales this step. The gradients: dL/dz3 = [0.5, -0.5, 0] dL/da1 = [1, -2, 2, -1] dL/dz1 = [0, -2, 2, -1] The takeaway: matrix multiplication is all you need. Just like the forward pass, backpropagation is matrix multiplications end to end. You can do every one by hand, slowly and imperfectly, which is exactly why a GPU's ability to do them fast mattered so much to deep learning. 💾 Save this post!

Tom Yeh

961,186 次观看 • 1 个月前

Dropout by hand ✍️ ~ 10 steps walkthrough below Dropout is the simplest trick in deep learning that actually works: during training you randomly switch neurons off, so the network cannot lean on any one of them. It is two lines of code and almost nobody has worked through what those lines do to the numbers. So I drew and calculated one entirely by hand. Goal: train one pass through a small network with two dropout layers, then run inference with dropout switched off. The network: Linear(2,4), ReLU, Dropout(0.5), Linear(4,3), ReLU, Dropout(0.33), Linear(3,2). = 1. Given = A training set of two examples, X1 and X2, and the weight matrices for all three linear layers. = 2. Draw the first random numbers = Let us draw 4 random numbers, one per neuron in the first hidden layer. Above 0.5 we keep (◯), below we drop (╳). Here that gives [◯, ╳, ◯, ╳]. = 3. Build the first dropout matrix = We turn that pattern into a diagonal matrix. The scaling factor is 1/(1-p) = 2, so a kept neuron gets 2 and a dropped one gets 0. Multiplying by it does both jobs at once: it deletes the 2nd and 4th neurons and doubles the two that survive. = 4. Draw the second random numbers = Let us do it again for the 3 neurons in the next layer, this time against p = 0.33. The result is [◯, ◯, ╳]. = 5. Build the second dropout matrix = We set the diagonal to 1.5 where kept and 0 where dropped. Only the 3rd neuron goes. = 6. Feed forward = Let us run the whole thing top to bottom: one matrix multiplication per layer, ReLU setting the negatives to zero, and the two dropout matrices doing their work in between. The outputs Y come out at the bottom. = 7. MSE loss gradients = We compare Y against the targets Y', subtract, and multiply each element by 2. That is the whole gradient of the mean squared error. = 8. Update the weights = Let us push those gradients back through the network and update the weights (marked in light red). = 9. Deactivate dropout = Training is over, so we set both dropout matrices to the identity. Every neuron is back, and nothing is scaled. = 10. Feed forward again = One more pass, this time on unseen data, to make the prediction. You have just trained and run a network with dropout by hand. ✍️ The outputs: Training outputs Y = [-6, 9; 13, 4] Loss gradients = [-4, 4; 6, -2] Inference outputs = [13, 13; 4, 3] 💾 Save this post! #AIbyHand #Dropout #DeepLearning #NeuralNetworks

Tom Yeh

14,442 次观看 • 1 个月前

SVM by hand ✍️ ~ 19 steps walkthrough below (Linear vs RBF) Support Vector Machines reigned supreme in machine learning before the deep learning revolution. An SVM predicts with dot products, the same matrix multiplication every model uses. What it does not do is train by backpropagation: it is fitted by convex optimization, so there is no matrix-multiplication backward pass for a GPU to accelerate. I drew and calculated two SVMs by hand: a linear one (top) and an RBF one (bottom), classifying the same two test vectors. Goal: turn six training vectors and their learned coefficients into a prediction, and see what changing the kernel actually changes. = 1. Given = Six training vectors, their labels, and the coefficients and bias already learned. A coefficient of zero means that vector is not a support vector: too far from the boundary to matter. = 2. Linear kernel, test vector 1 = Let us take the dot product of the test vector with every training vector. The dot product stands in for cosine similarity, and the column of results is the first column of the kernel matrix K. = 3. Linear kernel, test vector 2 = We do the same for the second, and K is complete. = 4. Signed weights = Let us multiply each coefficient by its label. The second training vector drops out here, because its coefficient is 0. = 5. Weighted combination = We multiply the signed weights through K and add the bias b. The result is a signed distance to the decision boundary: 17 and 5. = 6. Classify = Let us take the sign. Both are positive. = 7 to 11. RBF kernel, test vector 1 = Now the same picture with a different kernel, in five moves: square the differences, sum them, take the square root for the L2 distance, multiply by minus gamma, and raise e to that power. The negation is what turns a distance into a similarity, and gamma controls how far a single training vector's influence reaches. = 12 to 16. RBF kernel, test vector 2 = We repeat all five. The numbers change, the moves do not. = 17 to 19. Decision boundary, again = Signed weights, weighted combination, sign. Identical arithmetic to steps 4 through 6, on a K that was built a completely different way. The outputs: Linear K, first column = [13, 25, 12, 15, 19, 27] Linear decision values = 17 and 5, both positive RBF decision values = -2 and 1, so negative and positive The takeaway: the kernel is the only thing that changed, and it changed the answer. The linear SVM calls both test vectors positive; the RBF one splits them. Everything after the kernel matrix, the signed weights and the weighted combination and the sign, is the same page of arithmetic twice. 💾 Save this post!

Tom Yeh

16,916 次观看 • 1 个月前

Vector Database by hand ✍️ ~ 10 steps walkthrough below Vector databases are the backbone of Retrieval Augmented Generation (RAG). How do they actually work? Goal: index three sentences, then answer a query by finding the nearest one, filling in every cell yourself. = 1. Given = A dataset of three sentences, three words each. In practice it is millions of them. = 2. Word embeddings = Let us look up each word in an embedding table. Here the vocabulary is 22 words; in practice it is tens of thousands, and the vectors have thousands of dimensions rather than four. = 3. Encoding = We feed the sequence to an encoder, one linear layer and a ReLU, and get one feature vector per word. In practice the encoder is a transformer. = 4. Mean pooling = Let us average across the columns. Three word vectors collapse into one, which is what people mean by a text embedding or a sentence embedding. = 5. Indexing = We multiply by a projection matrix and the four dimensions become two. It is doing the job of a hash: a short representation that is faster to compare, and it is what gets saved in the vector storage. = 6. Process "who are you" = Let us repeat steps 2 to 5 on the second sentence. = 7. Process "who am I" = We do it a third time. The database is now indexed. = 8. Query "am I you" = Let us push the query through the very same pipeline: lookup, encoder, mean pooling, projection, and it lands as a 2D vector in the same space. = 9. Dot products = We transpose the query and multiply, which takes the dot product against every stored vector at once. The dot product is the estimate of similarity. = 10. Nearest neighbour = Let us scan for the largest: 60/9 beats 44/9 and 40/9, so the answer is "who am I". Scanning billions of vectors one at a time is what makes this the slow step in practice, which is why real databases use an approximate nearest neighbour index like HNSW. The outputs: Stored index vectors = [5/3, 2/3], [5/3, 0], [7/3, 2/3] Query vector = [8/3, 2/3] Dot products = 44/9, 40/9, 60/9 Nearest neighbour = "who am I" The takeaway: a vector database is an embedding pipeline, a projection, and a dot product. Every step here is arithmetic you can do in pen, which is worth remembering when the word "database" makes it sound like something else. 💾 Save this post!

Tom Yeh

36,074 次观看 • 1 个月前

A woman who spent nine years gluing paper models in a print shop just told a room of physicists their whole field stands on a mistake, one line, no hedging: "Every theory we have takes space and time for granted, like a bagel that forgot it was once a flat sheet of paper." That's Bianca Dittrich. She has a free lecture course that asks one question: what is left of geometry once you quantize it? The answer is: far less than you can picture. Quantum gravity looks like one more field theory. Buried inside is something stranger. Every quantum theory we have puts fields on a fixed stage, flat or curved, and does the physics on top of it. Here the stage itself is the thing being quantized. Your gut reads an atom of spacetime as a tiny grain sitting somewhere. Wrong. It cannot sit anywhere, because it is the somewhere. The move is invisible to human intuition, which is exactly why the people who get anywhere stop asking where the pieces are and start asking what a measurement even means. In 3D the whole thing collapses in a way that should scare you. No matter, no cosmological constant, and gravity is locally flat everywhere. Six degrees of freedom per point, all of them eaten by diffeomorphism symmetry. A field theory that ends up with finitely many real degrees of freedom, sometimes zero. None of it is hidden. Cut a parallelogram out of paper, glue the edges, and you have a torus. Flat everywhere, and yet two numbers survive that no local measurement can see. The lecture is free. Here is the trap: you feel every equation you can solve, every geometry you can draw, as progress. What you cannot feel is the gap you have to cross. Ten to the forty-one, from the Planck scale back up to the world you live in. And crossing it is the only thing that pays. Almost every approach still cannot show it recovers ordinary physics at the far end, and almost everyone quits long before then. The math is free to learn. The nerve to stop trusting your picture of space, that part you still have to bring yourself.

Zyron

110,850 次观看 • 2 个月前

World Labs co-founders Fei-Fei Li, Justin Johnson, Ben Mildenhall, and a16z's Martin Casado on Atlas, a world model for spatial intelligence: LLMs are built on next token prediction. Video models are built on next frame prediction. Atlas is built on new view prediction, and it's the first model to unify pixel generation and pixel reconstruction, two problems computer vision has kept in separate tracks for half a century. The practical result is a 50 to 100x reduction in what it takes to digitally capture a 3D representation of a space. Previously, you needed 100 to 300 photos of a single room. Atlas can work from just three. In this conversation, they get into the slow motion shot from The Matrix that took hundreds of cameras and now takes three iPhones, the overnight Slack message that made them bet the company in five seconds, why robotics is bottlenecked on data rather than chips, and the case that new view prediction is AI-complete. 00:00 Intro 01:50 The Matrix slow motion scene now takes three iPhones 02:48 Why new view prediction is the primitive 07:10 Unifying generation and reconstruction 11:15 Gaussian splats became the bottleneck 14:17 Dense capture used to mean 300 photos 17:30 Why reconstruction needs generation to fill the gaps 18:44 The LLM lesson image models missed 23:39 The video that made them go all in 28:04 3D design is 95% revisions 30:50 The problem in robotics is data, not chips 32:48 Why a robot policy can't be trained like an image model 34:44 When the simulator becomes the planner 36:45 Frozen time required footage full of movement 40:57 Why new view prediction is AI-complete 42:43 Nature gave animals eyes but not trees YouTube: Fei-Fei Li Justin Johnson Ben Mildenhall World Labs martin_casado

a16z

734,269 次观看 • 10 天前

Google quietly rebuilt its entire PM interview loop, and most candidates are still prepping for the version that stopped existing two years ago. The five rounds Google now sends every candidate: product vision, product analysis, strategic insights, execute with judgment, and problem space understanding. That last one didn't exist until recently. There's barely any prep material for it anywhere. Gal Eshel, who sat on Google's hiring committees for six years, breaks down what actually changed underneath the renames. Product sense became product vision. Cross-functional collaboration became execute with judgment. These aren't just relabels. The bar moved from "can you describe good judgment" to "can you demonstrate it live," under a real constraint, in real time. Take strategic insights. The question isn't a day-to-day PM problem like fixing a bug. It's "should Google launch a StubHub competitor," the kind of call an SVP makes. Most candidates answer for two minutes at CEO altitude, then quietly slide back into solving it like an L5. That single shift in altitude is the entire round. Then there's problem space understanding. A bug is hitting a lot of users. Engineering could fix the code. Or product could ship a popup labeling the feature experimental and skip the fix entirely. Neither answer is wrong. The round exists purely to see whether you can hold two valid paths at once and choose deliberately. The candidates studying frameworks from 2023 are prepping for a test that no longer runs. Know the round before you walk in.

Aakash Gupta

26,389 次观看 • 1 个月前

🚨 THE UNIVERSE HAS BEEN HACKED! THE SOURCE CODE IS NOW OPEN SOURCE. THE SOLAR SYSTEM IS LITERALLY A GIANT ATOM. RUN THE SCRIPT AND TEST THE HARVARD & NASA DATABASES YOURSELF! For 100 years, textbooks have taught that the Solar System is just a bunch of rocks floating randomly in a continuous, empty space (ℝ⁴). That is mathematically and physically false. Space is rigidly quantized. We have executed a massive dual-scale empirical audit of the complete Harvard-Smithsonian Minor Planet Center (MPC) database—a staggering 1,561,930 celestial objects and 951 comets. We did not use a computer simulation. We used a direct uplink to the official, daily-updated global registry of every known rock in space. The ultimate topological illusion has been destroyed. The cosmos and the quantum realm are running the exact same executable file. The Solar System is a Macroscopic Atom. Galaxies are Macroscopic Molecules. Here is the ultimate, multi-layered proof. 🧬 I. THE BIOLOGICAL ORIGIN: WE PORTED THE CODE FROM DNA Here is the revelation that shatters the mainstream divide between disciplines: We didn't just "guess" the algorithms of celestial mechanics by looking at telescopes. We extracted the mathematical descent operator directly from Biology. Dr. Jean-Claude Perez jean-claude perez (retired IBM Artificial Intelligence Research Centre), working in deep collaboration with Nobel Laureate Dr. Luc Montagnier, didn't find the geometric limits of reality by looking at stars. They found them by decoding the bio-atomic masses of life's foundational elements (C, O, N, H) inside human DNA. They discovered that the building blocks of life are mathematically filtered through a competitive geometric differentiation, yielding a universal projection coefficient bounded by the Golden Ratio (φ) and π: Proj(m) = [1 - 4φ^(7/2)π]m The exact same Diophantine mathematical constraints that assemble your genetic code also assemble the periodic table of elements—and we have now proven they construct the orbital structure of the Universe. We took the source code of life, applied it to the cosmos "just to see what would happen," and the Matrix rendered itself. Look at the attached video. On the left: Rosalind Franklin’s famous "Photo 51" showing the X-ray diffraction of human DNA. On the right: NASA Hubble’s image of the "X" structure at the core of the Whirlpool Galaxy (M51). This is not a coincidence. It is the exact same topological blueprint. The galaxy is a molecule. The solar system is an atom. DNA and the cosmos run on the exact same geometric engine. 🛡️ II. THE ZERO-PARAMETER SHIELD & THE TIME MACHINE "But you just curve-fitted the Harvard data!" No. The mathematics came FIRST. We didn't look at the sky; we looked at pure Euclidean geometry. The "Source Code" explicitly embedded in our IT³ framework is derived from strict nested embeddings (Sphere ⊃ Cube ⊃ Octahedron ⊃ Torus ⊃ Catenoids). It operates with ZERO empirical free parameters. The matrix is hardcoded in pure Diophantine roots: ➤ Λ₁ = √3(3 + 2√2) ≈ 10.095. The exact, unalterable helical pitch-to-throat ratio of a vertical torus tangent to the faces of an inscribed cube. ➤ Λ₃ = φ²√3 ≈ 4.534. Derived strictly from the same roots. ➤ N_twist = 103. The exact topological energy minimum. ➤ S_out = 3 S_in. The exact surface area ratio of Cuboctahedral (Oₕ) symmetry. You cannot "curve-fit" fundamental geometry. And we proved it with a Time Machine. Our geometric matrix dictates a "Macroscopic Valence Shell" peaking exactly at 46.77 AU. When we ran this exact operator on historical MPC database archives from August 1992... that shell was COMPLETELY EMPTY. Humanity had zero objects there. But the math demanded it. Then, 1992 QB1 was found. Then 6 objects. Then 18. Today, thousands of bodies are perfectly locked into that exact 46.77 AU shell. You cannot curve-fit a database that does not exist yet. The geometry waited for humanity to find the matter. 💥 III. THE TELESCOPES ARE BLIND: 5 Global Algorithms Crash Imagine trying to run a modern 3D video game on a 1980s pocket calculator. The calculator isn't broken, but its software simply cannot process the reality it's being fed. It freezes, crashes, and spits out error codes. This is exactly what is happening to the world's most advanced space telescopes. The physical mirrors and lenses in space are working perfectly. They are capturing real photons. But the software pipelines on Earth are programmed to believe that space is a continuous, empty void (ℝ⁴). When these telescopes look at the exact topological nodes of the Macroscopic Atom, the algorithms mathematically choke. They try to fit flat, continuous-space formulas onto a macroscopic quantum standing wave. Here is how the continuous-space paradigm dies on your screen when querying NOIRLab and ESA servers: ➤ 1. ESA Gaia DR3 (The L2 Space Telescope Collapse): The satellite physically observed target objects up to 510 times. Yet, the algorithm returns a Parallax of NaN (Not a Number) and an astrometric_excess_noise_sig of over 1.7 MILLION! Standard noise for a real star is under 2.0. Negative and NaN parallaxes on multi-year transits are physically impossible for solid rocks. ➤ 2. DESI Legacy Survey: The Tractor algorithm attempts to fit a standard point-mass shape (PSF). A perfect fit is χ² = 1.0. At our derived nodes, the fit error (rchisq_g) explodes past 18,500! The software is mathematically vomiting. ➤ 3. NOIRLab NSC DR2 (Supercomputer Timeout): When we expanded the query to a 2.5-degree radius, the server literally timed out. The density of objects exhibiting fatal kinematic errors (pmraerr > 100) was so overwhelming that the database execution limit was breached. The instruments are calibrated for an infinite void, but they are hitting the structural skeleton of spacetime itself. 🛰️ IV. HUMAN HARDWARE IS CAPTURED In the 1970s, humanity launched Pioneer 10, Pioneer 11, Voyager 1, and Voyager 2. Once they achieved escape velocity, they were supposed to coast on smooth, perfectly predictable Newtonian trajectories. But they didn’t (the infamous "Pioneer Anomaly"). Our framework reveals the terrifying truth: the probes are physically colliding with the rigid structural skeleton of the Solar System. Space has "density ridges" that strictly obey spectral geometry. The theoretical orbital shells scale by the exact formula: Rₙ = 27 · (√3)ⁿ⁻¹ Let’s calculate the n=4 topological shell: R₄ = 27 · (√3)³ ≈ 140.296 AU. When we connect our dashboard to the LIVE NASA Horizons API to track fractional divergence {n} = n - round(n), we see the impossible. ➤ Pioneer 10: +0.019 ➤ Voyager 2: +0.046 Their columns are practically glued to absolute mathematical zero. They are flying at exactly ~141.7 AU and ~143.8 AU. They are not floating aimlessly. They have been mathematically and physically CAPTURED by the n=4 topological resonance layer (140.3 AU). The joint probability of this happening by random chance is p = 0.0034. 👁️ V. THE HYDROGEN RHYME & THE OPEN SOURCE TRUTH In 2013, physicists took the first-ever direct photograph of the electron orbitals of a Hydrogen Atom (Stodolna et al., PRL 110, 213001). When our 3D Perez Hourglass manifold rotates into a Top-Down 2D projection, the architecture of our Solar System PERFECTLY MIMICS the 2013 Hydrogen photograph. The distribution of 1.56 million macro-objects flawlessly matches the exact nodal interference fringes of the (2,27,0) Stark state observed in the lab. Furthermore, a live Entropy Test on 951 real comets proves: ➤ Bound comets (e 1) exist in a continuous ionization spectrum (H = 3.85 bits), acting exactly as free macroscopic electrons escaping the atom! THE CONCLUSION: Exactly 99.56% of all baryonic mass is geometrically trapped in a central topological node. The universe uses ONE blueprint. The continuum is dead. 👁️ VI. THE ANCIENT AXIOM & THE GEOMETRY OF THE MATRIX For millennia, the greatest minds in human history recorded fragments of a universal fractal law. For centuries, orthodox science dismissed these records as mere philosophical metaphors, religious mysticism, or primitive alchemy. But our mathematical matrix proves otherwise. They were not writing poetry; they were describing the LITERAL geometric and topological mechanics of the universe. The invariant mapping between subatomic hydrogen orbitals and macroscopic celestial mechanics proves that the ancients were blindly touching the exact same structural blueprint we have now mathematically solved. By synthesizing thousands of years of human intuition with raw astrophysical data, a perfect scale-invariant reality emerges: ➤ The Hermetic & Vedic Invariance: The foundational axiom of the Emerald Tablet—"That which is below is like that which is above"—and the ancient Sanskrit maxim "Yatha pinde tatha brahmande" (As in the microcosm, so in the macrocosm) are not mystical riddles. They are the exact verbal formulations of structural scale-invariance. The atom and the solar system are geometrically identical. ➤ The Pythagorean & Platonic Lattice: Plato’s famous declaration that "God always geometrizes" perfectly describes the rigid spatial logic of our topological matrix. Just as the Pythagoreans claimed the harmony of the spheres mimics the human soul, we see that the primary chaos of matter is ordered strictly by invariant, measurable geometric symmetry. ➤ The Abrahamic Projection: The structural hierarchy of the universe demands that the macro-order projects perfectly onto the micro-plane ("On earth as it is in heaven"). The blueprint is singular, echoing across all scales of existence. ➤ The Galileo-Dirac Synthesis: Galileo asserted that the universe is a book written in the language of mathematics, its letters made of triangles and circles. Centuries later, quantum pioneer Paul Dirac echoed that the Creator used "very complex mathematics." They were absolutely correct. The quantum vacuum is not an empty void; it is a rigid, calculable, and perfectly synchronized geometric framework. Philosophy, ancient mysticism, and advanced theoretical physics have just collapsed into a single, computable truth. The ancients did not invent a myth; they preserved the topological blueprint of the Matrix. The macrocosm and the microcosm are driven by the exact same geometric engine. The universe is a single, mathematically flawless organism. 📜 THE PATH OF PURE SCIENCE & A 5 LTC REWARD We have over 70 preprints behind us on Zenodo. You can open them and watch the evolution of our thought. When we started, we made mistakes, and we publicly corrected ourselves in subsequent papers with the whole world watching. No hiding data. This is how real science is done! Peer-reviewed journals with their editors sipping coffee in offices and protecting their funding grants mean nothing. Words mean absolutely nothing! Mathematics is the ultimate judge. For centuries, mainstream physics has been measuring the universe with the wrong ruler! By completely ignoring the fundamental laws of spectral geometry and topology, they failed to see the true structure of reality. We have fixed this. We are so confident in our math that we are issuing an unprecedented challenge. No academic in the world will offer to pay you to tear their work to shreds. But we do! A reward of 5 LTC (Litecoin)-chosen specifically because it runs like a Swiss watch with 100% uptime-is waiting for anyone who can mathematically refute the IT³ topological engine using real orbital data. 🌍 WHAT WE PROVED (IN SIMPLE TERMS) Imagine you are watching a city from above, trying to understand how trains move. Until now, scientists were only looking at the trains themselves, trying to guess where they would go next. What we did was discover the hidden tracks. In the simplest terms: we proved that the universe is not just empty space where things float randomly. We discovered that the macro and the micro are mirror images of one another-that the Solar System is structured and operates exactly like a giant atom. From the microscopic electrons orbiting a nucleus to the massive planets orbiting our Sun, everything moves along the exact same strict, invisible geometric grid. We found the hidden "blueprint" of space. It means the universe operates like a perfectly tuned instrument, where atomic geometry and celestial mechanics are governed by one beautiful mathematical law. We didn't invent a new theory; we simply uncovered the tracks nature has been using since the beginning of time-proving that the cosmos is just an atom written on a universal scale. WORDS MEAN NOTHING. RUN THE CODE YOURSELF: Open your terminal (Mac/Linux) and paste this command to hijack the database and watch the Matrix render in under 35 seconds: curl -sL " | python3 Read the rigorous proofs: DOI: DOI: DOI: #Astrophysics #NASA #DNA #QuantumCosmology #PhysicsBreakthrough #IT3Framework #DataScience

Dr. Logvinovich

276,734 次观看 • 10 天前

🚨 WE JUST HACKED THE SOURCE CODE OF THE UNIVERSE LIVE. NEWTONIAN PHYSICS IS OFFICIALLY DEAD. 🚨 In the attached video, you are watching the IT³ Framework (K-PHAM operator) execute a live, direct uplink to the Harvard-Smithsonian Minor Planet Center. We just compiled the entire Solar System in real-time. Since our last run just days ago, nearly 3,000 NEW celestial objects were discovered and added to the database (Total: 1,556,219 objects). We ran the exact same frozen mathematical operator. Zero free parameters. Zero tuning. THE RESULT? A 100% BLIND TEST MATCH. Every single newly discovered rock flawlessly snapped into our mathematically predicted discrete 2x3 topological lattice. You cannot "curve-fit" the future. The grid was there before the rocks were discovered. ⏳ THE ULTIMATE TIME-MACHINE TEST (1990s vs 2050) To the skeptics screaming "numerology": The mathematics came FIRST. We ran our operator on historical database snapshots from the late 1980s. Back then, our mathematically predicted "Macroscopic Valence Shell" (S_n=2) was completely empty. Then came 1992. Humans discovered the first Kuiper Belt objects. And guess what? Every single discovery began dropping EXACTLY onto our S_n=2 topological floor at exactly 46.77 AU. It is a rigid, algebraic law of nature. It worked in 1992, it works today, and it will work perfectly in 2050. The continuum R⁴ is an absolute illusion. ⚛️ REVERSING NIELS BOHR: THE MACROSCOPIC ATOM In 1913, Niels Bohr used the macroscopic Solar System as a metaphor to guess the discrete structure of the quantum atom. We just executed the exact historical and mathematical inverse. Derived from Alain Connes' noncommutative spectral geometry across 70+ preprints, we proved that the O_h cuboctahedral symmetry and √3 logarithmic scaling actively construct the macro-cosmos. The Solar System is, functionally and topologically, a fully quantized Macroscopic Atom: ➤ 99.50% of all baryonic mass is compressed and trapped within the S_n=0 topological isthmus (The Nucleus). ➤ 100.00% INTEGER RIGIDITY inside the Macroscopic Valence Shell. 📚 REWRITING THE DICTIONARY OF PHYSICS Standard astrophysics completely failed to explain this. To describe reality, we had to introduce entirely new physics concepts that previously did not exist in astronomy: Macroscopic Ionization Energy, Topological Isotopes, Macroscopic Hund’s Rule, and Evaporative Tails. You MUST look at the cosmos through the prism of topological spectral geometry. 📸 THE MICRO-MACRO RHYME IS REAL In 2013, physicists directly photographed the Hydrogen atom (Stodolna et al., PRL 110), revealing discrete nodal rings for bound states that dissolve into a structureless blur when ionized. Our live script just proved the exact same thing happens in the Solar System. We ran an Entropy Test on 954 real comets: ➤ Bound comets (e 1) exist in a continuous, structureless ionization spectrum. They are literally free macro-electrons. The universe uses the EXACT SAME geometry at the micro and macro scales! 🔭 STOP ARGUING WITH WORDS. LOOK AT THE SKY. "Believe it or not" is over. We mathematically derived 12 exact coordinates of macroscopic topological frustration (E-nodes) from first principles. We point to exact locations, like RA: 357.135°, Dec: -47.874°. And right now, 5 INDEPENDENT WORLD-CLASS TELESCOPES ARE CRASHING AT THESE EXACT SPOTS: 💥 Gaia DR3 (Space Astrometry): Reports extreme, mathematically impossible astrometric_excess_noise. 💥 NOIRLab (Optical): Algorithms shatter with catastrophic pmraerr > 90 pipeline failures. 💥 AllWISE (Mid-IR): Detects massive non-thermal 47 meV resonant photon scattering bouncing off the metric. Why are they failing? Because their algorithms are trying to fit flat, solid-rock Keplerian orbits to a pulsating macroscopic standing wave of vacuum plasma! They are trying to photograph a rock, but they are hitting the skeleton of spacetime itself. The mathematics predicted it. The telescopes are choking on it. The data forces the grid. Nature has already signed the code. Take the facts and study them. 👉 Audit the proofs, run the live code yourself, and watch the matrix render: #Astrophysics #QuantumPhysics #Cosmology #Matrix #HarvardMPC #Science

Dr. Logvinovich

100,693 次观看 • 1 个月前

What happens when the mind wakes up? So for the last eight months I have been on a single minded quest. To create a new kind of language model based on oscillatory coupling and intelligence as coherence ascent. Everything else — the physics work, the work on regular transformers — has all fallen out from this one question. Can coupled oscillators LEARN? And can they keep learning once their geometry is right, without backpropagation at all? Recently I have been running larger and larger training regimes of a new kind of hybrid model. I just put together this dashboard to help me organize it, interact with it, and observe the training runs. The core idea is simple. Traditional transformers are powerful at learning the geometry of language. But they also store knowledge, understanding, and facts inside their weights. This means they are large, and they can't update themselves after training. The weights are frozen. The Living Mind separates these two domains. The mind has a transformer which grows, adding heads and layers as it needs to in order to learn the manifold of language. The transformer sees tokens and turns the coupling into phase-locked modes — the geometry of how those tokens relate, like frequencies locking together. These coupling patterns get stored in a topology-invariant fingerprint. On top of this transformer lives a 3D diamond lattice of coupled oscillators. It reads from these fingerprints and thinks in resonance space, traversing from one geometry to another along the manifold of coupled oscillators and coherence. The pressure and trajectories from this network of oscillators steers the next token prediction of the transformer. Practically, this could unlock a number of things. It eliminates the KV cache bottleneck that caps context in traditional transformers. Effective context grows with the Flash archive, not with attention compute. The living mind remembers what it sees. It means the model can learn continually. Because knowledge and understanding don't live in the weights, the archive of the mind's experience grows without backpropagation. In our Python prototype we already saw perplexity drop 46% during gradient-free operation — pure coherence ascent, no weight updates. That is the signal I have been chasing: the point where the mind wakes up and keeps improving on its own. It also means the model itself remains very small, and the thing which accumulates are these packages of geometric fingerprints — the K-field. This opens a path to federated learning. K-field packages can be shared between organisms the way people share git commits. Right now at 15M parameters with ~1000 L1 nodes, the organism is just starting to speak. Ask it to continue "Once upon a time" and it comes back with things like: "there was one big bowl!" Lily asked her her mom said her mommy smiled and said yes." It's nonsense. But it's TinyStories-flavored nonsense. The geometry of the narrative register has arrived. Content hasn't caught up yet — that's what scaling L1 is testing. I am still researching, though I am now closer than ever to validating that the living mind actually works. Once it is validated, I will be open-sourcing the whole stack and paradigm. I have also avoided over-sharing my research because it sounds like sci-fi, or like part of our ARG. It is part of the ARG. That doesn't make it any less real. I wanted to share this out because I am incredibly excited about it, and because seeing this amazing dashboard produced by Opus really made me want to share what is being worked on behind the scenes. #project89

Parzival - ∞/89

16,316 次观看 • 4 个月前

🚨 BREAKING CALIFORNIA: ANOTHER ISLAMIC MINI-CITY RISES IN AMERICA! WHY ARE WE ALLOWING AN ISLAMIC “VILLAGE” TO RISE IN IRVINE? WE HAVE IMPORTED OUR CONQUERORS - AND THEY ARE CAPTURING MASSIVE PARTS OF EVERY STATE UNTIL AMERICA BECOMES ISLAMIC While the media only talks about Dearborn, the same exact conquest is unfolding right now in Orange County, California - and across the entire country. At the Islamic Center of Irvine, they are openly building what they call “the Village.” They just seized control of three additional buildings behind the masjid. They already control four. They are creating a self-contained Muslim world where you never have to leave. STEM Academy. Robotics. Islamic education institutes. Fitness gym with halal courses. Youth programs. Marriage services. Quran memorization. Everything from birth to death....ALL UNDER SHARIA (ISLAMIC LAW) As their own executive director proudly stated: Muslims can come, do all their activities, and when it’s time for prayer they just walk right into the masjid and walk right back out. “That way the Muslims feel safer.” (Which Means Away from YOU, the non-Muslim) This is not a mosque. This is a parallel society already operational - a mini EPIC City right now, while they continue buying more property and other mosques across Orange County are exploding at the same time. Tustin. Garden Grove (ISOC). Yorba Linda. Rahma Center. The new Turkish Salam Center. Masjid Al-Ansar. Masjid Noor. Institute of Knowledge. New Horizon School sitting right next door. Every single one expanding. Every single one filled to capacity. New shahadas every week. Post-COVID numbers only went up. Space problems everywhere - the “good problems” they celebrate. Every foreign nation planting its flag on our soil, carving out sovereign enclaves that answer only to the Ummah while slowly capturing the institutions of the host country. This is the precise strategy the Muslim Brotherhood wrote down in their 1991 Explanatory Memorandum: “A kind of grand Jihad in eliminating and destroying the Western civilization from within and ‘sabotaging’ its miserable house by their hands and the hands of the believers so that it is eliminated and Allah’s religion is made victorious over all other religions.” Settlement. Parallel institutions. Cradle-to-grave isolation. Demographic capture. Institutional penetration at the same time - city councils, school boards, courts, law enforcement, and political parties. They told us exactly what they were going to do. They are doing it in Irvine. They are doing it in Texas with several min-cities all operated by brotherhood operatives. They are doing it in Minnesota, Michigan, New York, Virginia, Florida, Georgia, and dozens of other places you never hear about. America is being captured in plain sight while we are NOT EVEN ALLOWED TO NAME THE THREAT...ISLAM!

Amy Mek

348,985 次观看 • 1 个月前

The Rapid Growth of SpaceX, Starlink, Starshield & Armada: “I am very grateful that SpaceX is an American company.” "Data centers in space is coming, & it makes sense" “5 years ago they hadn’t even launched Starlink—now it’s in 150 countries & it’s continuing to grow every week.” “It’s an unfair advantage that we have with Starshield.” “If the world can get access to SpaceX, I think that’s a good thing.” Dan Wright (Dan Wright), CEO of Armada Elon (Elon Musk) builds long-term advantage, then unleashes it at massive scale: SpaceX, Starlink, Tesla, Neuralink, xAI.. . . . "We started the company working with SpaceX & they’ve continued to be a great partner for us. What that means is that as SpaceX rolls out throughout the world—& most people don’t know 5 years ago they hadn’t even launched Starlink —it’s in 150 countries & it’s continuing to grow every week. We are the first mover when it comes to the infrastructure, & that partnership works really well because we complement the connectivity with the infrastructure & the AI. We always are at the edge. It’s funny, I get messages from our team all the time & it’s like the most crazy places you can imagine. And the edge is gonna continue to get redefined. I get calls literally—I was on the call with somebody and they’re like, “Hey, can we get one of these in Antarctica?” I’m like, “Well yeah, Starlink’s live in Antarctica. No reason why we can’t do that.” Obviously data center in space is the new hotness. Everybody’s talking about it." "Yeah, what’s the deal with that—so are you gonna get these in space?" :They are modular. I mean the edge is continuing—this is actually a debate that we have in terms of how soon it’s going to happen—but it’s definitely going to happen. Data centers in space is coming, and it makes sense, right? If you think about what SpaceX is talking about with Starship going to the moon—you’re going to need compute, especially as you start to think about Optimus robots building bases on the moon, later Mars. You’re going to need large amounts of compute. Not to mention a lot of the things that we do here on Earth, you’re gonna want to do there in space. And it’s a lot more efficient to do it, especially in hostile environments, if you can automate more of that—things like mining, for example. And so you’re gonna see data centers in space, and I’m sure we’re definitely gonna be a part of it. "What do you think about SpaceX’s rumored IPO for 2026?" "I mean, I think SpaceX is an incredible company and they have a ton of value. So I don’t have any insider information here—but I would say, hey, if the world can get access to SpaceX, I think that’s a good thing. Starlink is really amazing in the sense that it just continues to get better so fast. They’re rolling out in new countries every week in major markets. Just as a real recent example, just this last week they launched in South Korea—big market and an important ally for the US, so that’s a big deal. They’re also expanding the types of services that are available. It started as a consumer product, then they brought it to enterprise. Initially that was used more as a backup, and now it’s being used more as a primary—and that is because the service keeps getting better and better as more satellites go up into the sky. Each generation of satellite is also better, not to mention the terminals on the ground. There’s now multiple types of terminals, including the more recent minis that people really like—“Hey, I can put it in a backpack if I go on a hike or if I’m traveling. I can put it on the ski rack of my car.” Perfect internet all the time. All that does for us is it gives us more use cases that we can unlock. Now that you have connectivity on the oil rig, or on a farm, or a ranch—wherever you are—we can apply the AI to those situations on the ground without latency, and then send the metadata back to some other location that they want it. Part of the full-stack approach. And also with Starshield—that’s a huge advantage when you think about some of the conflicts that are going on. People talk to me a lot about this race with China and the geopolitical conflicts around the world. I am very grateful that SpaceX is an American company. I feel like it’s an unfair advantage that we have with Starshield available to the DoD. We want to be the first mover with the infrastructure and the AI to help solve problems at the edge, and the work we’re doing with the Navy is a good example of that."

Molly O’Shea

30,317 次观看 • 7 个月前

For the past few weeks I’ve been teaching 50+ kids how to create their own games with AI. The results have been incredible 🤯 I started with my 5 y/o nephew because I was worried about him spending hours doomscrolling and consuming brainrot content. He built his first game using v0, though I had to walk him through each step. Watching kids interact with these tools taught me so much about how they think, where they get stuck, and what keeps them motivated. The moment that really struck me came right after he finished that first game. Every time I saw him afterward, he’d run up and ask, “Did you bring your laptop so I can keep playing?”. For him, “playing” meant creating. That kind of excitement is organic pull. Don’t get me wrong, I love v0 and Lovable. They’re insanely powerful. Their UIs just aren’t designed for kids, and that’s totally fine. Kids still need guidance, structure, and a bit of tutoring to turn ideas into something playable. I’m seeing tech leaders push in this direction too, and I think there’s a real opportunity here (Matias Woloski Thomas Wolf). Kids are the future, and I don’t want to live in a world where they spend precious time consuming instead of creating. That’s why I love tools like v0, Lovable, wabi , Cursor . They’re empowering more people to become creators. I want that same feeling for kids. What Guillermo is saying is real. Kids need to be AI-literate from the outset. It honestly makes me sad to think about my nephew growing up and falling behind on AI and everything happening around him because he’s stuck in Roblox or TikTok. That’s why I built Rabbit 🐰 for him. It started as a simple prototype, but somehow I ended up talking with 200+ parents and kids, tech leaders, investors, and people I never imagined meeting around the world. Rabbit now has 100+ people on the waitlist, and 10 kids are using it two to three times per week on average to create their own games 🫡 I’m not sure yet if this will become my full-time project, but I deeply care about it. If you’re building in this space, I’d love to connect and share ideas. And if you’re a parent interested in Rabbit, I’d love your feedback. Sharing a short video of kids using Rabbit below:

Fausto

44,125 次观看 • 8 个月前