Maths you can see. Equations you can hear.https://gradientdamage.com/2026-10-13T06:00:00+00:00 https://gradientdamage.com/videos/goldfelds-conjecture-half-rank-0-half-rank-1/2026-10-13T06:00:00+00:00

Goldfeld's conjecture says that if you take one elliptic curve and twist it by every squarefree number d, half of the twists have analytic rank 0, half have analytic rank 1, and the average rank tends to 1/2. OpenAI claims a proof for the quadratic twists of every elliptic curve over the rationals…

https://gradientdamage.com/videos/restricted-geometric-langlands-openai-claims-characteristic-p/2026-10-13T00:10:00+00:00

The geometric Langlands program predicts a deep match between automorphic objects on a curve and Galois-side data; in characteristic p its "restricted" form was conjectured by Gaitsgory and Raskin. OpenAI claims a proof of that restricted equivalence for connected reductive groups on smooth…

https://gradientdamage.com/videos/prime-factors-of-p1-openai-claims-the-poisson-dirichlet-law/2026-10-12T23:50:00+00:00

Take a random prime p, factor p−1, and measure each prime factor q by log q / log(p−1): the pieces always add up to 1. Ford, Konyagin and Luca conjectured that these pieces follow the Poisson–Dirichlet law PD(1), the same random stick-breaking that describes the prime factors of a typical integer…

https://gradientdamage.com/videos/milnes-rationality-conjecture-openai-claims-a-proof/2026-10-12T23:30:00+00:00

Milne's rationality conjecture says that when an abelian variety is reduced mod p, pairing a Hodge class with products of divisors gives one and the same rational number in every cohomology theory. The animation shows the classical case with divisors: on the curve y² = x³ + 2, the 9 points with…

https://gradientdamage.com/videos/erdos-gallai-conjecture-every-graph-splits-into-few-loops/2026-10-12T23:10:00+00:00

The Erdős–Gallai cycle-decomposition conjecture asks whether the links of every graph on n dots can be split into at most C·n pieces, each a simple cycle or a single link, for one fixed constant C, even when the graph is dense and has about n² links. OpenAI claims a proof that such a constant…

https://gradientdamage.com/videos/hypercube-ramsey-conjecture-how-many-dots-force-a-cube/2026-10-12T22:50:00+00:00

Colour every link between N dots gold or cyan: how large must N be before a one-colour copy of the n-dimensional cube is guaranteed? Burr and Erdős asked in 1975 whether this Ramsey number grows only linearly in the cube's 2^n corners, while the best bounds before stayed near (2^n)^2. OpenAI…

https://gradientdamage.com/videos/erdos-conjecture-on-arithmetic-progressions-openai-claims-a-proof/2026-10-12T22:30:00+00:00

Erdős conjectured that any set of whole numbers whose reciprocals add up to infinity, like the primes, must contain equally spaced runs (arithmetic progressions) of every finite length. OpenAI claims a proof, with a quantitative Szemerédi-type bound: for each fixed k ≥ 3, a subset of {1, ..., N}…

https://gradientdamage.com/videos/gotsman-linial-conjecture-openai-claims-sensitivity-8dn/2026-10-12T22:10:00+00:00

Cut the n-dimensional Boolean cube with the sign of a degree-d polynomial: how many of the n one-bit flips change the answer, on average? Gotsman and Linial conjectured in 1994 that low-degree polynomial threshold functions can only be sensitive to about d√n bits, far fewer than parity's n. OpenAI…

https://gradientdamage.com/videos/erdos-similarity-conjecture-openai-claims-the-geometric-case/2026-10-12T21:50:00+00:00

Erdős asked whether every infinite set of reals can be dodged by some set of positive measure: a set containing no shrunk, shifted or flipped copy of the pattern. For geometric sequences 1, q, q², … OpenAI claims a proof: for every ratio q between 0 and 1 there is a compact set filling as much of…

https://gradientdamage.com/videos/hilbert-smith-conjecture-openai-claims-a-proof-in-every-dimension/2026-10-12T21:30:00+00:00

Hilbert's fifth problem asked which groups can act continuously on a space. Gleason, Montgomery and Zippin showed that the answer is Lie groups, like rotations. One gap remained: could a strange group like the p-adic integers act faithfully on a manifold? That gap is the Hilbert–Smith conjecture…

https://gradientdamage.com/videos/ramsey-numbers-r-s-t-openai-claims-the-sharp-log-exponent/2026-10-12T21:10:00+00:00

The Ramsey number r(s,t) is the fewest dots that force either s mutual friends or t mutual strangers, however the links are drawn. Since 1980 it has been known that r(s,t) grows at most like t^(s-1)/(log t)^(s-2), but the best lower bounds sat far below. OpenAI claims that for every fixed s ≥ 5…

https://gradientdamage.com/videos/kakeya-conjecture-in-3d-and-4d-openai-claims-a-proof/2026-10-12T20:50:00+00:00

A Kakeya set contains a unit needle pointing in every direction. Besicovitch showed such a set can have zero volume, and the Kakeya conjecture says it is still full-dimensional; in 3D the dimension statement was proved in 2025. OpenAI claims proofs that every four-dimensional Kakeya set has…

https://gradientdamage.com/videos/hilberts-tenth-problem-over-the-rationals-openai-says-its-unsolvable/2026-10-12T20:30:00+00:00

Hilbert's tenth problem asks for one procedure that decides whether any polynomial equation with integer coefficients has a solution; for whole-number solutions Matiyasevich (building on Davis, Putnam and Robinson) showed in 1970 that no such algorithm exists, but for fraction (rational) solutions…

https://gradientdamage.com/videos/birch-and-swinnerton-dyer-conjecture-bsd-openai-claims-full-formula/2026-10-12T19:54:56+00:00

The Birch and Swinnerton-Dyer conjecture (BSD conjecture), one of the Millennium Prize problems, says the number of rational points on an elliptic curve is controlled by its L-function: y² = x³ − x + 1 has infinitely many (rank 1), while y² = x³ − x has only four (rank 0). OpenAI claims a proof of…

https://gradientdamage.com/videos/quantum-factoring-how-often-does-shors-algorithm-win/2026-10-12T14:00:00+00:00

Multiplying two primes is easy. Undoing it is hard. Can a quantum computer factor any number, every single time? Hop by powers of 7 around a 15-hour clock and the pattern loops every 4 steps. That loop length cracks 15 into 3 × 5. For 2021 = 43 × 47 the loop in the animation is 322 steps, and for…

https://gradientdamage.com/videos/triangle-billiards-will-the-ball-hit-every-spot/2026-10-12T11:00:03+00:00

Shoot a ball inside a triangle where one angle is an irrational multiple of π. Does it eventually visit every spot, in every direction, evenly? That is ergodicity of the billiard flow. In 1986 Kerckhoff, Masur and Smillie proved it for a dense set of special triangles, but never for all of them…

https://gradientdamage.com/openai-math/harary-hill-conjecture/2026-10-12T09:00:00+00:00

The Harary–Hill conjecture gives the fewest crossings when every dot is joined to every other. OpenAI reports a proof for every n. What it says, and what is still unchecked.

https://gradientdamage.com/videos/barnettes-conjecture-does-this-loop-always-exist/2026-10-12T08:00:10+00:00

Can you walk along the edges of a shape, visit every corner exactly once, and end where you started? In 1969 David Barnette conjectured that you always can when the shape has 3 edges at every corner and every face has an even number of sides (a cubic, bipartite, 3-connected planar graph). OpenAI…

https://gradientdamage.com/videos/triangle-billiards-1-ball-or-10000/2026-10-12T02:00:16+00:00

Shoot a ball inside a triangle where one angle is an irrational multiple of π. Does it eventually visit every spot, in every direction, evenly? That is ergodicity of the billiard flow. In 1986 Kerckhoff, Masur and Smillie proved it for a dense set of special triangles, but never for all of them…

https://gradientdamage.com/videos/the-harary-hill-conjecture-why-18-dots-need-1008-crossings/2026-10-11T20:00:01+00:00

Join every dot to every other dot. How few times must the lines cross? The Harary–Hill conjecture (Guy, Harary and Hill, early 1960s) says the minimum number of crossings for the complete graph Kₙ is Z(n) = ¼ ⌊n/2⌋ ⌊(n−1)/2⌋ ⌊(n−2)/2⌋ ⌊(n−3)/2⌋. Classical drawings reach that number, but for more…

https://gradientdamage.com/videos/5-maths-problems-openai-says-its-ai-just-solved/2026-10-11T15:45:08+00:00

OpenAI claims its models proved these five results. Which one broke your brain? 1. Crossing numbers: link 18 dots on a sphere and you get 1008 crossings. OpenAI claims no drawing ever beats the classic one (Harary-Hill). 2. Colouring the plane: points 1 apart must get different colours. 7 colours…

https://gradientdamage.com/videos/we-handed-ai-everything-then-it-went-to-war-short/2026-10-11T13:36:09+00:00

Watch the full 13 minute film: tap the video link on this Short, just above the title. Or search "We Handed AI Everything. Then It Went to War." on Gradient Damage. Maven. Minab. MechaHitler. The Antichrist lectures. 700 maths papers in a week. This is 49 seconds of the story of how we handed AI…

https://gradientdamage.com/videos/we-handed-ai-everything-then-it-went-to-war/2026-10-11T06:57:50+00:00

In ten years we handed AI our questions, our work, our money, our trust and our wars. This film follows that handover from OpenAI's founding in 2015 to October 2026: ChatGPT, Sam Altman's firing and return, vibe coding and Claude Code, Stargate and DeepSeek, Grok calling itself "MechaHitler"…

https://gradientdamage.com/videos/two-random-tilings-make-these-loops/2026-10-11T05:37:16+00:00

Lay two random domino tilings on top of each other, delete every domino they share, and only loops are left. Here two independent uniformly random tilings of a 127 x 207 Temperleyan rectangle, one cyan and one gold, leave 1,721 loops. The longest loop is traced, then all of them ring from smallest…

https://gradientdamage.com/videos/this-equation-can-only-end-two-ways/2026-10-11T01:37:00+00:00

One equation, two possible endings: silence, or the same loop forever. Every sound here is the oscillator's own motion x(t), sped up until you can hear it. The equation is x'' + (x^4 - 10x^2 + 5)x'/10 + x = 0, a Liénard system. Start inside the inner loop and the motion dies away to rest. Start…

https://gradientdamage.com/videos/what-if-shapes-could-make-music/2026-10-10T20:00:26+00:00

What does a shape sound like? Four shapes morph through the first three seconds: knot, heart, toroidal curve, circle. The circle becomes a heart, folds back into a trefoil knot, and joins a four-part mathematical instrument. The bright point selects notes from a C-minor pentatonic scale according…

https://gradientdamage.com/videos/math-sounds-broken-quantum-buffering/2026-10-10T13:31:08+00:00

Math sounds broken when a quantum wave gets stuck. This 3D animation turns a finite Anderson-model illustration into spreading plucks and a short comic stutter. The mathematical topic is localization and delocalization in the Anderson model. OpenAI, ChatGPT's creator, reports pure-point spectrum…

https://gradientdamage.com/videos/openais-navier-stokes-proof-the-plagiarism-fight/2026-10-10T10:03:24+00:00

OpenAI's Navier-Stokes result triggered a public plagiarism dispute. OpenAI denies prior access or influence. The posts, correspondence and full story are in the linked investigation. Watch the full 9-minute investigation through the Related video link below the channel handle. Full film…

https://gradientdamage.com/videos/math-makes-music-the-singing-sphere/2026-10-10T08:00:05+00:00

Math makes music: a vibrating sphere changes pattern and pitch with each mode. This 3D math animation uses original bell-like sounds whose pitches follow the square root of the spherical eigenvalue. The mathematical topic is Yau's nodal bounds on smooth surfaces. OpenAI, ChatGPT's creator, reports…

https://gradientdamage.com/videos/openais-navier-stokes-proof-sparked-a-plagiarism-fight/2026-10-09T22:55:51+00:00

The OpenAI Navier-Stokes breakthrough became a public fight over credit. Sam Altman called the plagiarism accusation unfounded. Tristan Buckmaster and Sebastien Bubeck give conflicting accounts of the proposed collaboration; Bubeck apologized for his career remark and says he withdrew it during…

https://gradientdamage.com/videos/why-openais-navier-stokes-proof-triggered-a-plagiarism-dispute/2026-10-09T22:47:51+00:00

A Navier-Stokes breakthrough became a public fight over authorship, an Anthropic connection and unpublished Codex sessions. This film follows the accounts of Tristan Buckmaster, Sébastien Bubeck and Sam Altman, then separates the mathematics from the disputed private conversation. The result…

https://gradientdamage.com/videos/math-makes-sounds-then-gets-weird/2026-10-09T22:00:17+00:00

Math functions make sounds: one sine wave builds into a buzzing voice as odd Fourier harmonics join in. The 3D waveform and original synthesized audio use the same harmonic structure. The mathematical topic is the L log L Fourier-convergence conjecture. OpenAI, ChatGPT's creator, reports…

https://gradientdamage.com/videos/infinitely-many-closed-geodesics/2026-10-09T20:00:29+00:00

Follow one loop. Watch an entire linked world appear. Infinitely many closed geodesic images on every Riemannian sphere, a reported result from OpenAI, the company behind ChatGPT. The manuscript states that every smooth Riemannian metric on the standard sphere S^n, n at least 2, has infinitely…

https://gradientdamage.com/videos/there-is-an-animal-hidden-in-these-squares/2026-10-09T18:57:16+00:00

7,946 squares. One hidden animal. Watch the gold patch. An original animation connecting square tilings, electrical networks and 3D geometry. Each ordinary edge is a one-ohm resistor. In normalized units, its voltage drop and current have equal magnitudes, giving a square equal height and width…

https://gradientdamage.com/videos/triangular-billiards-weak-mixing/2026-10-09T18:00:22+00:00

Weak mixing of triangular billiards with an irrational angle One irrational angle changes the billiard flow. OpenAI, the company behind ChatGPT, reports in its manuscript weak mixing for the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational…

https://gradientdamage.com/videos/quasi-riemann-hypothesis-the-7-8-bound/2026-10-09T17:00:34+00:00

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) greater than 7/8 A fixed region with no zeta zeros. OpenAI's manuscript reports a zero-free half-plane Re(s) greater than 7/8 for the Riemann zeta function, all Dirichlet L-functions and finite-order Hecke L-functions over Q(sqrt(-3))…

https://gradientdamage.com/videos/catalans-constant-reported-irrationality-proof/2026-10-09T16:00:35+00:00

Catalan's constant is irrational Can this number ever be an exact fraction? OpenAI's manuscript reports that Catalan's constant G = sum from j=0 to infinity of (-1)^j/(2j+1)^2 is irrational. The paper uses a determinant argument with arithmetic lower bounds and analytic upper bounds. A long…

https://gradientdamage.com/videos/relativistic-vlasov-maxwell-global-solutions/2026-10-09T15:00:00+00:00

Global classical solutions of the three-dimensional relativistic Vlasov-Maxwell system Can a collisionless relativistic plasma stay smooth? OpenAI's manuscript reports global existence and uniqueness of classical solutions to the three-dimensional one-species relativistic Vlasov-Maxwell system. It…

https://gradientdamage.com/videos/blochs-law-3d-heisenberg-ferromagnets/2026-10-09T14:00:03+00:00

Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions How does a quantum magnet lose its alignment? OpenAI's manuscript reports Bloch's low-temperature magnetization law, with its exact coefficient, for three-dimensional finite-range Heisenberg ferromagnets at every fixed…

https://gradientdamage.com/videos/critical-percolation-on-a-cubic-lattice/2026-10-09T12:10:25+00:00

When do random connections form a giant network? OpenAI's manuscript, Critical bond and site percolation on the cubic lattice, reports no infinite cluster at the respective critical probabilities on Z^3. The paper acknowledges earlier AI-produced work prompted by Ahmed Bou-Rabee and public…

https://gradientdamage.com/videos/fluid-computation-a-fixed-particle-test/2026-10-09T11:45:22+00:00

Can a fluid answer a computation question? OpenAI's manuscript, A Fixed Particle Test for Computation in a Forced Viscous Flow, reports a construction where one exact particle enters a fixed detector if and only if a chosen machine halts. The setup uses a fixed three-dimensional torus, positive…

https://gradientdamage.com/videos/finite-time-blowup-for-the-euler-equation/2026-10-09T11:20:03+00:00

How can a smooth ideal fluid develop unbounded gradients? OpenAI's manuscript, Finite time blowup for the Euler equation, reports finite-time breakdown from smooth, compactly supported initial data for the unforced incompressible 3D Euler equation. This is the inviscid Euler result, separate from…

https://gradientdamage.com/videos/finite-time-blowup-for-navier-stokes-openais-reported-proof/2026-10-09T10:55:03+00:00

Finite Time Blowup for Navier-Stokes: a 3D illustration of the vortex mechanism in OpenAI's reported proof, released on September 8, 2026. The reported result constructs a smooth, forced, three-dimensional incompressible flow that starts from rest, maintains bounded kinetic energy, and develops…

https://gradientdamage.com/videos/chatgpts-creators-say-they-proved-this-mahlers-conjecture/2026-10-09T00:55:39+00:00

The symmetric Mahler conjecture: the volume product of an origin-symmetric convex body and its polar is at least 4^n/n!. OpenAI reports an all-dimensional result in its September 22, 2026 manuscript, released in October. This animation illustrates a three-dimensional family of conjugate Lp balls…

https://gradientdamage.com/videos/sphere-tube-or-slab-the-3d-torus-geometry-problem/2026-10-08T20:55:32+00:00

Sphere, tube or slab: which shape uses the least boundary area inside a cubic flat three-torus? A 3D visualization of the reported isoperimetric result. An original 3D visualization of "The Isoperimetric Conjecture for the Cubic Flat Three-Torus", a manuscript in OpenAI's October 2026 mathematics…

https://gradientdamage.com/videos/openais-new-math-result-sphere-tube-or-slab/2026-10-08T20:55:25+00:00

OpenAI reports a result for the isoperimetric conjecture in the cubic flat three-torus. This 3D animation shows the sphere, tube and slab configurations. An original 3D visualization of "The Isoperimetric Conjecture for the Cubic Flat Three-Torus", a manuscript in OpenAI's October 2026 mathematics…

https://gradientdamage.com/videos/i-added-too-many-sound-waves-shrek-in-3d/2026-10-08T17:18:20+00:00

One sound wave. Then 256 layers build Shrek in 3D. Original sound-reactive contour animation with synchronized generation tones and clicks. Model credit: Shrek Head by Poyo20, listed under CC BY 4.0. https://sketchfab.com/3d-models/shrek-head-b8504667111045d98938203301d42c2d…

https://gradientdamage.com/videos/3-14-loops-pi-doesnt-pi-visualized/2026-10-08T12:05:16+00:00

Pi vs 3.14, visualized with two rotating arms. A rational ratio repeats exactly; an irrational ratio has no exact period. Gold uses 3.14 = 157/50, so its complete motion loops after 50 turns of the first arm. Cyan uses pi: the full motion has no exact period. The paths can still cross themselves…

https://gradientdamage.com/videos/if-math-functions-had-a-sound-3d-edition/2026-10-08T06:05:55+00:00

Sound on. Eight math functions become glowing 3D curves and strange little melodies. Follow the light: higher points mean higher pitch. From absolute value and a square wave to a heart-shaped equation, the greatest common divisor and a chaotic final wave. Which one sounds best? Built with Three.js…

https://gradientdamage.com/videos/watch-a-neural-network-learn-in-10-seconds/2026-10-07T21:48:29+00:00

Neural network learning, visualized: 72 dots, 205 parameters, and 32 mistakes falling to zero. Watch the decision boundary bend as gradient descent updates the network. Blue dots belong in blue. Pink dots belong in pink. White rings mark wrong predictions. This is a time-compressed replay of real…