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In 1859, Riemann wrote one sentence about where a function equals zero. 10 trillion cases checked, not a single counterexample - and it's still unproven. Worth $1 million. The zeros of the zeta function encode the exact position of every prime number as a sum of waves - the...

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Eron

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27,642 görüntüleme • 2 gün önce •via X (Twitter)

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I never realized the connection between the Riemann Hypothesis and transformer positional encoding before, it's really interesting.

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All of them will inevitably crash before the launch.

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The first 10 trillion non-trivial zeros of the Riemann zeta function all sit exactly on the line where the real part equals 1/2. Every single one. And yet after 165 years nobody can prove it must be true - the Clay Institute offers $1 million for a proof. The prime numbers hidden inside these zeros are what make RSA encryption unbreakable - the same math that protects every LLM API key, bank transfer, and password on the internet. 38 minutes. Bookmark & watch today. The most beautiful unsolved problem in mathematics.
45:57

The first 10 trillion non-trivial zeros of the Riemann zeta function all sit exactly on the line where the real part equals 1/2. Every single one. And yet after 165 years nobody can prove it must be true - the Clay Institute offers $1 million for a proof. The prime numbers hidden inside these zeros are what make RSA encryption unbreakable - the same math that protects every LLM API key, bank transfer, and password on the internet. 38 minutes. Bookmark & watch today. The most beautiful unsolved problem in mathematics.

Eron

33,509 görüntüleme • 2 ay önce

Primes Numbers Are Not Random Noise Prime numbers look scattered when you meet them one by one, but their disorder has structure. In this animation, every integer is placed on a growing spiral, primes ignite as sharp golden events, and the hidden oscillations come from the first Riemann zeta zeros. The scene is based on the von Mangoldt prime signal Λ(n) and the explicit formula for ψ(x), where primes appear as impulses and the zeta zeros behave like frequencies correcting the smooth drift of x. #PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #MathAnimation
0:32

Primes Numbers Are Not Random Noise Prime numbers look scattered when you meet them one by one, but their disorder has structure. In this animation, every integer is placed on a growing spiral, primes ignite as sharp golden events, and the hidden oscillations come from the first Riemann zeta zeros. The scene is based on the von Mangoldt prime signal Λ(n) and the explicit formula for ψ(x), where primes appear as impulses and the zeta zeros behave like frequencies correcting the smooth drift of x. #PrimeNumbers #RiemannHypothesis #NumberTheory #Mathematics #MathAnimation

Mathelirium

16,018 görüntüleme • 4 ay önce

The Moore curve of order 4 is a loop of 256 unit steps through every cell of a 16 × 16 grid. Each coordinate has its own real Fourier series, and one chain of arms per axis draws it. The second half of the loop is the first half turned 180°, so every even harmonic is zero, and its mirror leaves x only sines and y only cosines. A curve that wiggles everywhere converges slowly: 32.3 % of the height at N = 1, still 0.47 % at N = 255.
0:32

The Moore curve of order 4 is a loop of 256 unit steps through every cell of a 16 × 16 grid. Each coordinate has its own real Fourier series, and one chain of arms per axis draws it. The second half of the loop is the first half turned 180°, so every even harmonic is zero, and its mirror leaves x only sines and y only cosines. A curve that wiggles everywhere converges slowly: 32.3 % of the height at N = 1, still 0.47 % at N = 255.

matharium

72,548 görüntüleme • 19 gün önce

In 1958, the US Navy stood Frank Rosenblatt in front of a $2 million IBM computer and told reporters it was the embryo of a machine that would someday be conscious of its own existence. His entire idea fit on one line of math. In 1969, two MIT researchers published a single puzzle that killed the funding for a decade. The line of math survived anyway - it runs inside every neuron in every LLM deployed today, copied billions of times and stacked in layers. The formula is a weighted sum followed by a nonlinearity: step of W dot X plus B. Draw it geometrically and it's a straight line dividing space into ones and zeros. The learning rule nudges the weights whenever it's wrong. Minsky and Papert proved it could never solve XOR - four points on a square where no straight line separates the classes. The fix was stacking layers and training them with backpropagation, published in Nature in 1986 by Rumelhart, Hinton, and Williams. Hinton shared the 2024 Nobel Prize in Physics for it. Rosenblatt drowned in a boating accident in 1971 on his 43rd birthday, 15 years before the solution arrived. 5 minutes. Bookmark & watch today. The one line of 1958 math that every transformer, every LLM, and every GPU on earth still runs on.
5:23

In 1958, the US Navy stood Frank Rosenblatt in front of a $2 million IBM computer and told reporters it was the embryo of a machine that would someday be conscious of its own existence. His entire idea fit on one line of math. In 1969, two MIT researchers published a single puzzle that killed the funding for a decade. The line of math survived anyway - it runs inside every neuron in every LLM deployed today, copied billions of times and stacked in layers. The formula is a weighted sum followed by a nonlinearity: step of W dot X plus B. Draw it geometrically and it's a straight line dividing space into ones and zeros. The learning rule nudges the weights whenever it's wrong. Minsky and Papert proved it could never solve XOR - four points on a square where no straight line separates the classes. The fix was stacking layers and training them with backpropagation, published in Nature in 1986 by Rumelhart, Hinton, and Williams. Hinton shared the 2024 Nobel Prize in Physics for it. Rosenblatt drowned in a boating accident in 1971 on his 43rd birthday, 15 years before the solution arrived. 5 minutes. Bookmark & watch today. The one line of 1958 math that every transformer, every LLM, and every GPU on earth still runs on.

Deliriumm

11,638 görüntüleme • 20 gün önce

Every prime number — 2, 3, 5, 7, 11, 13 — leaves a fingerprint in the gap between itself and the next prime. Those gaps aren't random. They follow a strict set of rules that can be mapped onto just six positions on a circle, forming a machine that governs where every prime can go next. This is the Prime Gap State Machine A 6 state automaton running on modular arithmetic. The six positions are the only residues modulo 9 that prime numbers can occupy {1, 2, 4, 5, 7, 8}. The Spine positions — 0, 3, and 6 — are permanently excluded. No prime above 3 ever lands there. What you're watching is every prime number moving through those six positions as it flows along the number line. The glowing yellow dots are primes in motion. Each dot travels from one node to the next according to the gap rule: add the gap, take the result mod 9, land on the next state. Zero violations across tens of millions of primes tested. Those six states don't arrange themselves randomly on the circle. They split into exactly two groups, the Hard Wall states {1, 4, 7} and the Temporal states {2, 5, 8}. Each group sits at perfectly equal 120° spacing around the circle. One forms a red triangle. The other forms a blue triangle. Two equilateral triangles, rotated relative to each other, superimposed on the same circle. That is a hexogram Emerging from pure prime number arithmetic. Not designed. Not chosen. Forced by the structure of mod-9 arithmetic and the constraints that govern where primes can live. This is from the Prime Lattice Coherence Framework original independent mathematical research showing that prime numbers organize themselves on a {2,3}-based lattice with a Hard Wall boundary at the prime 7. The hexagram you see is the geometric shadow of that boundary, cast onto a circle. The universe keeps writing the same shapes.
0:24

Every prime number — 2, 3, 5, 7, 11, 13 — leaves a fingerprint in the gap between itself and the next prime. Those gaps aren't random. They follow a strict set of rules that can be mapped onto just six positions on a circle, forming a machine that governs where every prime can go next. This is the Prime Gap State Machine A 6 state automaton running on modular arithmetic. The six positions are the only residues modulo 9 that prime numbers can occupy {1, 2, 4, 5, 7, 8}. The Spine positions — 0, 3, and 6 — are permanently excluded. No prime above 3 ever lands there. What you're watching is every prime number moving through those six positions as it flows along the number line. The glowing yellow dots are primes in motion. Each dot travels from one node to the next according to the gap rule: add the gap, take the result mod 9, land on the next state. Zero violations across tens of millions of primes tested. Those six states don't arrange themselves randomly on the circle. They split into exactly two groups, the Hard Wall states {1, 4, 7} and the Temporal states {2, 5, 8}. Each group sits at perfectly equal 120° spacing around the circle. One forms a red triangle. The other forms a blue triangle. Two equilateral triangles, rotated relative to each other, superimposed on the same circle. That is a hexogram Emerging from pure prime number arithmetic. Not designed. Not chosen. Forced by the structure of mod-9 arithmetic and the constraints that govern where primes can live. This is from the Prime Lattice Coherence Framework original independent mathematical research showing that prime numbers organize themselves on a {2,3}-based lattice with a Hard Wall boundary at the prime 7. The hexagram you see is the geometric shadow of that boundary, cast onto a circle. The universe keeps writing the same shapes.

CTFTHEORY

68,774 görüntüleme • 4 ay önce

Each coordinate of a closed outline has its own real Fourier series: x(t) = ∑ (aₖ cos kt + bₖ sin kt), and the same for y. A cosine and sine of frequency k make one arm of length √(aₖ² + bₖ²) turning k times per lap. One chain sets x, one sets y, and the pen sits where their guides cross. With N harmonics per axis the knight is off by 30.2 % of its height at N = 1, 0.70 % at N = 50. At N = 1 both coordinates are pure sinusoids: an ellipse.
0:26

Each coordinate of a closed outline has its own real Fourier series: x(t) = ∑ (aₖ cos kt + bₖ sin kt), and the same for y. A cosine and sine of frequency k make one arm of length √(aₖ² + bₖ²) turning k times per lap. One chain sets x, one sets y, and the pen sits where their guides cross. With N harmonics per axis the knight is off by 30.2 % of its height at N = 1, 0.70 % at N = 50. At N = 1 both coordinates are pure sinusoids: an ellipse.

matharium

1,374,703 görüntüleme • 23 gün önce

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!
0:32

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

13,602 görüntüleme • 1 ay önce

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!
0:32

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 görüntüleme • 3 ay önce

In 1830, Liouville proved the area under the most important curve in mathematics has no formula. Not because nobody was clever enough - because the formula was never there to be found. Then somebody squared the problem, lifted it into a second dimension, and a circle walked into the exponent entirely on its own. The area is the square root of π - the same √π that normalizes every probability distribution inside every LLM and every diffusion model on every GPU cluster today. The curve is e to the minus x squared - the bell curve De Moivre discovered in 1733 solving a coin-flip problem. One dimension and every integration method slides off it. Square the integral, rename one variable to y, and x² + y² appears in the exponent - Pythagoras hiding inside a function with no circles anywhere. Switch to polar coordinates and geometry hands you a free factor of r. The infinite hill collapses to π. That factor now sits inside every softmax normalization, every Gaussian noise layer in stable diffusion training, every loss function a transformer optimizes during backpropagation. 20 minutes. Bookmark & watch today. The proof that a circle was hiding inside a straight line the entire time.
19:47

In 1830, Liouville proved the area under the most important curve in mathematics has no formula. Not because nobody was clever enough - because the formula was never there to be found. Then somebody squared the problem, lifted it into a second dimension, and a circle walked into the exponent entirely on its own. The area is the square root of π - the same √π that normalizes every probability distribution inside every LLM and every diffusion model on every GPU cluster today. The curve is e to the minus x squared - the bell curve De Moivre discovered in 1733 solving a coin-flip problem. One dimension and every integration method slides off it. Square the integral, rename one variable to y, and x² + y² appears in the exponent - Pythagoras hiding inside a function with no circles anywhere. Switch to polar coordinates and geometry hands you a free factor of r. The infinite hill collapses to π. That factor now sits inside every softmax normalization, every Gaussian noise layer in stable diffusion training, every loss function a transformer optimizes during backpropagation. 20 minutes. Bookmark & watch today. The proof that a circle was hiding inside a straight line the entire time.

Eron

14,069 görüntüleme • 10 gün önce

The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated
47:28

The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated

Zyphor

123,759 görüntüleme • 2 ay önce

"Nearly 7 million people a year die from air pollution, that's about the same number of people that die every half an hour as died in Chernobyl in its entirety." - Tim Gregory (Nuclear Scientist) New podcast where we discuss the lunacy of Net Zero. Link to show 👇
0:50

"Nearly 7 million people a year die from air pollution, that's about the same number of people that die every half an hour as died in Chernobyl in its entirety." - Tim Gregory (Nuclear Scientist) New podcast where we discuss the lunacy of Net Zero. Link to show 👇

Peter McCormack 🏴‍☠️🇬🇧🇮🇪

36,646 görüntüleme • 7 ay önce

Fibonacci numbers are the sequence where each one is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… Every so often one of them is prime these are called Fibonacci primes, and only 21 are known. Here's the new discovery, take any Fibonacci prime (past the first few small ones) and reduce it to modulo 144. You'd expect the leftover remainder could be almost anything , 144 different possibilities. It never is. It only ever lands on one of just four numbers: 1, 5, 13, or 89. Mod 144... This just means finding the remainder after dividing by 144. Here's the three-step process, using the actual Fibonacci prime 1597 as an example. 1. Divide the number by 144. 1597÷144=11.09. 2. Keep only the whole number part, then multiply back by 144. 11×144=1584 3. Subtract to find what's left over. 1597 - 1584 = 13 So 1597 "reduced mod 144" is 13 one of exactly the four allowed numbers. Try it with any of the other 20 known Fibonacci primes and you'll always land on 1, 5, 13, or 89. Never anything else. That's a 97% reduction. Out of 144 possible remainders, 140 of them are simply forbidden to Fibonacci primes. We proved this happens every single time, for every Fibonacci prime known, with zero exceptions. It gets stranger. Those four allowed remainders 1, 5, 13, 89 are themselves smaller numbers from the same Fibonacci sequence. Every Fibonacci prime, when you shrink it down this way, lands back on another Fibonacci prime. The sequence points back at itself. And the positions that produce those four numbers the 1st, 5th, 7th, and 11th spots in the sequence turn out to be exactly the same four numbers that mark critical boundaries in a completely separate math system,(PLCT) one based on multiplying the numbers 2 and 3 instead of adding golden-ratio powers. Two totally different number systems, built from different operations, share the same four "checkpoint" numbers. The animation traces 233 different ways to build the number 144 purely out of powers of the golden ratio (phi ≈ 1.618, the number where a whole equals its bigger part divided by its smaller part). 233 is itself a Fibonacci number the sequence even shows up in how many ways you can build it. Every flash you see is one valid combination, spiraling and glowing as it cycles through all of them. The Golden Lattice ϕ-Power Representations, the General Count Law, and the Fibonacci Prime Mod-144 Signature Theorem
0:33

Fibonacci numbers are the sequence where each one is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233… Every so often one of them is prime these are called Fibonacci primes, and only 21 are known. Here's the new discovery, take any Fibonacci prime (past the first few small ones) and reduce it to modulo 144. You'd expect the leftover remainder could be almost anything , 144 different possibilities. It never is. It only ever lands on one of just four numbers: 1, 5, 13, or 89. Mod 144... This just means finding the remainder after dividing by 144. Here's the three-step process, using the actual Fibonacci prime 1597 as an example. 1. Divide the number by 144. 1597÷144=11.09. 2. Keep only the whole number part, then multiply back by 144. 11×144=1584 3. Subtract to find what's left over. 1597 - 1584 = 13 So 1597 "reduced mod 144" is 13 one of exactly the four allowed numbers. Try it with any of the other 20 known Fibonacci primes and you'll always land on 1, 5, 13, or 89. Never anything else. That's a 97% reduction. Out of 144 possible remainders, 140 of them are simply forbidden to Fibonacci primes. We proved this happens every single time, for every Fibonacci prime known, with zero exceptions. It gets stranger. Those four allowed remainders 1, 5, 13, 89 are themselves smaller numbers from the same Fibonacci sequence. Every Fibonacci prime, when you shrink it down this way, lands back on another Fibonacci prime. The sequence points back at itself. And the positions that produce those four numbers the 1st, 5th, 7th, and 11th spots in the sequence turn out to be exactly the same four numbers that mark critical boundaries in a completely separate math system,(PLCT) one based on multiplying the numbers 2 and 3 instead of adding golden-ratio powers. Two totally different number systems, built from different operations, share the same four "checkpoint" numbers. The animation traces 233 different ways to build the number 144 purely out of powers of the golden ratio (phi ≈ 1.618, the number where a whole equals its bigger part divided by its smaller part). 233 is itself a Fibonacci number the sequence even shows up in how many ways you can build it. Every flash you see is one valid combination, spiraling and glowing as it cycles through all of them. The Golden Lattice ϕ-Power Representations, the General Count Law, and the Fibonacci Prime Mod-144 Signature Theorem

CTFTHEORY

12,946 görüntüleme • 3 ay önce

The Golden Centrifuge Take the golden ratio 1.618…, the number behind spiral shells and sunflower seeds and raise it to every power from −12 to 12. How many combinations of those powers add up to exactly 144? The answer is 233. We found that a few weeks ago and treated it as a curiosity. We just proved it isn't one. THE LAW Run the same question at every size of window, not just 12, and a rule appears with zero exceptions, an odd numbered window always counts itself , its answer is just its own target number. An even numbered window always counts the nextnumber in the sequence instead. 144 is the twelfth number in that sequence. Twelve is even. That single fact is the entire reason the count comes out to 233 and not 144. Once you know a number's position in the sequence, you know its count instantly no searching required, for numbers of any size. What the spiral is. Every particle is placed using the same angle a sunflower uses to arrange its seeds —the golden angle, about 137.5° which is why distinct spiral arms appear rather than a random spray. As the field descends it narrows and spins faster, the way the mathematics itself narrows toward an answer with each step. The pulsing rings are the rhythm of the count, repeating every two steps, the way the law says it should. And the glowing throat is the moment 89 and 144 stop being separate and become one: 89 + 144 = 233, not asserted from outside but produced by what's already happening inside the funnel two branches of the same window turn out to be the two previous Fibonacci numbers, because they are smaller copies of the same problem. The golden ratio has relatives the silver and bronze ratios, numbers that build their own Fibonacci-like sequences the same way φ does. If our law were just "numbers like this behave this way," it should show up for them too. It doesn't. Not weaker absent. The target is unreachable in those windows, every time we tried. And everything those windows can reach, they reach exactly once, with zero redundancy. The golden window is the only member of its family capable of folding back on itself and we found the one line of algebra responsible: a relation between consecutive powers of φ that holds only when the governing equation is exactly φ² = φ + 1, and breaks for every relative it has. Why this matters going forward. A proof like this doesn't just explain what already happened it predicts what hasn't been checked yet. We can now state, in advance and without running a single computation, the exact count for any Fibonacci number no matter how large, just from whether its position in the sequence is odd or even. And we can predict, before testing it, that the identical experiment run on any other metallic ratio will fail completely because the failure isn't a pattern we noticed, it's a consequence of one piece of algebra that only balances for φ. That's the difference between a coincidence and a law: a coincidence describes the past. A law tells you what's true somewhere you haven't looked yet. The Parity Theorem for the Golden Window The Prime Lattice Coherence Framework: A Unified Master Document The Golden Lattice ϕ-Power Representations, the General Count Law, and the Fibonacci Prime Mod-144 Signature Theorem
0:30

The Golden Centrifuge Take the golden ratio 1.618…, the number behind spiral shells and sunflower seeds and raise it to every power from −12 to 12. How many combinations of those powers add up to exactly 144? The answer is 233. We found that a few weeks ago and treated it as a curiosity. We just proved it isn't one. THE LAW Run the same question at every size of window, not just 12, and a rule appears with zero exceptions, an odd numbered window always counts itself , its answer is just its own target number. An even numbered window always counts the nextnumber in the sequence instead. 144 is the twelfth number in that sequence. Twelve is even. That single fact is the entire reason the count comes out to 233 and not 144. Once you know a number's position in the sequence, you know its count instantly no searching required, for numbers of any size. What the spiral is. Every particle is placed using the same angle a sunflower uses to arrange its seeds —the golden angle, about 137.5° which is why distinct spiral arms appear rather than a random spray. As the field descends it narrows and spins faster, the way the mathematics itself narrows toward an answer with each step. The pulsing rings are the rhythm of the count, repeating every two steps, the way the law says it should. And the glowing throat is the moment 89 and 144 stop being separate and become one: 89 + 144 = 233, not asserted from outside but produced by what's already happening inside the funnel two branches of the same window turn out to be the two previous Fibonacci numbers, because they are smaller copies of the same problem. The golden ratio has relatives the silver and bronze ratios, numbers that build their own Fibonacci-like sequences the same way φ does. If our law were just "numbers like this behave this way," it should show up for them too. It doesn't. Not weaker absent. The target is unreachable in those windows, every time we tried. And everything those windows can reach, they reach exactly once, with zero redundancy. The golden window is the only member of its family capable of folding back on itself and we found the one line of algebra responsible: a relation between consecutive powers of φ that holds only when the governing equation is exactly φ² = φ + 1, and breaks for every relative it has. Why this matters going forward. A proof like this doesn't just explain what already happened it predicts what hasn't been checked yet. We can now state, in advance and without running a single computation, the exact count for any Fibonacci number no matter how large, just from whether its position in the sequence is odd or even. And we can predict, before testing it, that the identical experiment run on any other metallic ratio will fail completely because the failure isn't a pattern we noticed, it's a consequence of one piece of algebra that only balances for φ. That's the difference between a coincidence and a law: a coincidence describes the past. A law tells you what's true somewhere you haven't looked yet. The Parity Theorem for the Golden Window The Prime Lattice Coherence Framework: A Unified Master Document The Golden Lattice ϕ-Power Representations, the General Count Law, and the Fibonacci Prime Mod-144 Signature Theorem

CTFTHEORY

10,623 görüntüleme • 2 ay önce

This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.
43:46

This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.

Lupen

738,828 görüntüleme • 2 ay önce

Everyone memorized 21 million. The number I'd study is 4,000. That is roughly the most transactions that fit in one Bitcoin block, and a block comes about every 10 minutes. Give 8 billion people one on-chain transaction each and the line takes 38 years to clear. So the coin supply is capped, and the room to move those coins is capped too. Miners are the ones who sell that room, one block at a time. Today fees are about 2% of what miners earn and the block subsidy is the other 98%. The subsidy gets cut in half every four years until it reaches zero. As it shrinks, blockspace becomes the product a miner has left to sell, with 8 billion people sharing 4,000 slots every 10 minutes.
0:22

Everyone memorized 21 million. The number I'd study is 4,000. That is roughly the most transactions that fit in one Bitcoin block, and a block comes about every 10 minutes. Give 8 billion people one on-chain transaction each and the line takes 38 years to clear. So the coin supply is capped, and the room to move those coins is capped too. Miners are the ones who sell that room, one block at a time. Today fees are about 2% of what miners earn and the block subsidy is the other 98%. The subsidy gets cut in half every four years until it reaches zero. As it shrinks, blockspace becomes the product a miner has left to sell, with 8 billion people sharing 4,000 slots every 10 minutes.

Billy ₿oone

11,689 görüntüleme • 6 gün önce

Gems of URDUWOOD 🙄 An Indian RAW agent rescuing the Prime Minister of Pakistan 😭 while Pakistanis sing the Indian National Anthem. Their intentions are obvious ,portraying the Pakistani Prime Minister and its citizens in a positive light. In truth, every single one of them harbors animosity towards India and Hindus, even the children.
1:30

Gems of URDUWOOD 🙄 An Indian RAW agent rescuing the Prime Minister of Pakistan 😭 while Pakistanis sing the Indian National Anthem. Their intentions are obvious ,portraying the Pakistani Prime Minister and its citizens in a positive light. In truth, every single one of them harbors animosity towards India and Hindus, even the children.

Sheetal Chopra 🇮🇳

15,112 görüntüleme • 7 ay önce

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.
19:29

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 görüntüleme • 2 ay önce

For thousands of years, the number 144,000 has been a deep mystery, famously appearing in ancient texts like the Book of Revelation. what if it wasn’t just a spiritual metaphor? What if it was an advanced mathematical blueprint? To understand it, we look to Buckminster Fuller, the famous architect who popularized the geodesic dome (think of the giant sphere at EPCOT). Fuller found a simple formula to count the exact number of intersecting points (nodes) on a spherical grid , 10 times the "frequency" squared. Frequency is just how many times you slice up the triangles to make the sphere smoother. If we use a frequency of exactly 120, something cool happens. Why 120? In math, 120 is "5 factorial" (written as 5!, which just means 5 x 4 x 3 x 2 x 1). It is a highly foundational number that also represents the exact number of perfect symmetries in a 20-sided 3D shape. Plug 120 into Fuller's formula: 120 squared is 14,400. Multiply that by 10, and you get exactly 144,000 points. This specific 144,000-node sphere does the impossible. It unifies the two fundamental "languages" of the universe. Normally, the math of adding (like the Golden Ratio) and multiplying (like Prime numbers) do not align. They are two different grids. But at this exact 144,000-node geometry, they perfectly overlap. The physical coordinates of the sphere land flawlessly on a Golden Ratio grid, while the overall structure perfectly obeys the complex laws of prime numbers. They didn't just pick a big number to sound impressive. 144,000 is a mathematical singularity where geometry and number theory become one. The Golden Geodesic Fibonacci Structure in the 144,000-Node CTF Manifold The Prime Lattice Coherence Framework: A Unified Master Document
0:30

For thousands of years, the number 144,000 has been a deep mystery, famously appearing in ancient texts like the Book of Revelation. what if it wasn’t just a spiritual metaphor? What if it was an advanced mathematical blueprint? To understand it, we look to Buckminster Fuller, the famous architect who popularized the geodesic dome (think of the giant sphere at EPCOT). Fuller found a simple formula to count the exact number of intersecting points (nodes) on a spherical grid , 10 times the "frequency" squared. Frequency is just how many times you slice up the triangles to make the sphere smoother. If we use a frequency of exactly 120, something cool happens. Why 120? In math, 120 is "5 factorial" (written as 5!, which just means 5 x 4 x 3 x 2 x 1). It is a highly foundational number that also represents the exact number of perfect symmetries in a 20-sided 3D shape. Plug 120 into Fuller's formula: 120 squared is 14,400. Multiply that by 10, and you get exactly 144,000 points. This specific 144,000-node sphere does the impossible. It unifies the two fundamental "languages" of the universe. Normally, the math of adding (like the Golden Ratio) and multiplying (like Prime numbers) do not align. They are two different grids. But at this exact 144,000-node geometry, they perfectly overlap. The physical coordinates of the sphere land flawlessly on a Golden Ratio grid, while the overall structure perfectly obeys the complex laws of prime numbers. They didn't just pick a big number to sound impressive. 144,000 is a mathematical singularity where geometry and number theory become one. The Golden Geodesic Fibonacci Structure in the 144,000-Node CTF Manifold The Prime Lattice Coherence Framework: A Unified Master Document

CTFTHEORY

39,962 görüntüleme • 3 ay önce

Each coordinate of a closed butterfly outline has its own real Fourier series: x(t) = Σₖ (aₖ cos kt + bₖ sin kt), and similarly for y. A sine–cosine pair at frequency k corresponds to an arm of length √(aₖ² + bₖ²), turning k times per lap. One chain sets x, another sets y, and the pen sits where their guides cross. With N harmonics per axis, the maximum sampled reconstruction error is 30.45% of the butterfly’s height at N = 1, 1.15% at N = 50, and 0.52% at N = 100. At N = 1, both coordinates are pure sinusoids: an ellipse. The higher harmonics recover the wings and narrow body, turning that ellipse into a butterfly.
0:26

Each coordinate of a closed butterfly outline has its own real Fourier series: x(t) = Σₖ (aₖ cos kt + bₖ sin kt), and similarly for y. A sine–cosine pair at frequency k corresponds to an arm of length √(aₖ² + bₖ²), turning k times per lap. One chain sets x, another sets y, and the pen sits where their guides cross. With N harmonics per axis, the maximum sampled reconstruction error is 30.45% of the butterfly’s height at N = 1, 1.15% at N = 50, and 0.52% at N = 100. At N = 1, both coordinates are pure sinusoids: an ellipse. The higher harmonics recover the wings and narrow body, turning that ellipse into a butterfly.

Philosophy Of Physics

18,433 görüntüleme • 21 gün önce