https://gradientdamage.com/2026-10-13 https://gradientdamage.com/about/ https://gradientdamage.com/videos/2026-10-13 https://gradientdamage.com/videos/page/2/2026-10-08 https://gradientdamage.com/videos/zilber-pink-conjecture-duffin-schaeffer-conjecture-and-8-more/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/videos/ostmanns-inverse-goldbach-conjecture-the-primes-arent-a-b/2026-10-13https://gradientdamage.com/thumbs/5G4WZEBRGJI-og.jpgOstmann's Inverse Goldbach Conjecture: The Primes Aren't A + B | OpenAI MathOstmann's inverse Goldbach conjecture asks whether the primes can be built as a sumset A + B, every sum of one number from a set A and one from a set B, with both sets having at least two elements. OpenAI claims a proof that this is impossible, even if you are allowed to add or remove finitely many numbers from the primes first (an OpenAI preprint, not yet peer reviewed). The picture is a real computation: the numbers up to 600 wound on a helix with 30 per turn, and the largest possible B for A = {0, 2}, {0, 6} and {0, 30}; the primes that A + B misses (57, 24 and 14 of the 109) light up cyan, and the bar chart shows that no A = {0, a} with a up to 60 covers them all. Paper (OpenAI, catalogue result 013): - The additive indecomposability of the primes: https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/013 #OstmannInverseGoldbach #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/5G4WZEBRGJI212026-10-13T14:00:00+00:00yesno https://gradientdamage.com/videos/deligne-drinfeld-conjecture-one-generator-per-odd-weight/2026-10-13https://gradientdamage.com/thumbs/LuHHpbmcXLg-og.jpgDeligne–Drinfeld Conjecture: One Generator per Odd Weight | OpenAI MathThe Deligne–Drinfeld conjecture predicts the exact shape of the Grothendieck–Teichmüller Lie algebra grt, a symmetry algebra tied to braids, multiple zeta values and the absolute Galois group of the rationals: it should be freely generated by one element in each odd weight 3, 5, 7, 9, and so on. Brown proved in 2012 that such a free Lie algebra sits inside grt; OpenAI claims a proof that it is all of grt, with the corresponding isomorphism after weight completion (an OpenAI preprint, not yet peer reviewed). The picture shows the real basis of that free Lie algebra up to weight 20: each rod is one independent iterated bracket (a Lyndon word), cut into pieces for the generators it uses, sitting on the step of the stair equal to its weight; the counts 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13 are the dimensions in weights 3 to 20. Paper (OpenAI, catalogue result 008): - The Deligne-Drinfeld conjecture: https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/008 #DeligneDrinfeldConjecture #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/LuHHpbmcXLg212026-10-13T10:00:00+00:00yesno https://gradientdamage.com/videos/goldfelds-conjecture-half-rank-0-half-rank-1/2026-10-13https://gradientdamage.com/thumbs/AFGKRcCdzSU-og.jpgGoldfeld's Conjecture: Half Rank 0, Half Rank 1 | OpenAI MathGoldfeld'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 (twist parameters ordered by size), together with the mean analytic rank result; these are OpenAI's preprints, not yet peer reviewed. The picture is a real computation for the curve 11a1 (y² + y = x³ − x² − 10x − 20): each line is the completed L-function of one of its first 120 twists on the real segment from s = 1/2 to 3/2, and the counters are the true rank proportions of its first 1000 twists (44% / 50% / 6% with rank 2 or more, which the theorem says becomes negligible only slowly). Papers (OpenAI, catalogue result 006): - Goldfeld's analytic density conjecture and the 2-converse for elliptic curves: https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf - The mean analytic rank of quadratic twists of elliptic curves: https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/006 #GoldfeldConjecture #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/AFGKRcCdzSU212026-10-13T06:00:00+00:00yesno https://gradientdamage.com/videos/restricted-geometric-langlands-openai-claims-characteristic-p/2026-10-13https://gradientdamage.com/thumbs/1lz16-M84MM-og.jpgRestricted Geometric Langlands: OpenAI Claims Characteristic p | OpenAI MathThe 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 projective curves (under stated conditions on p), plus the generalized Ramanujan conjecture for globally generic cuspidal representations of split adjoint exceptional groups over function fields, and global Arthur enhancements assuming a finite-level Ramanujan–Arthur decomposition. These are OpenAI's claims and have not yet been peer reviewed. What you see is a real computed example of what "Ramanujan" (tempered) means, for the simplest such group, PGL2, where it was already known (Drinfeld). The object is the automorphic form of the elliptic curve y² = x³ + tx + 1 over the function field F13(t). Each ring is one of its 91 places of degree 1 or 2. The two gold points on a ring are that place's Satake parameters, computed from exact point counts, and every one of them sits on the unit circle. The pink points belong to the constant function, which is automorphic but not cuspidal. Its parameters q^±½ sit off the circle, and its Arthur SL2 accounts for exactly that shift. OpenAI's Ramanujan claim is for exceptional groups such as G2 and E8, which the picture does not show. The animation illustrates the result; it is not a proof. Papers: - The Restricted Geometric Langlands Equivalence in Positive Characteristic: https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf - Temperedness at ramified places for globally generic exceptional groups: https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf - Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level…https://www.youtube-nocookie.com/embed/1lz16-M84MM202026-10-13T00:10:00+00:00yesno https://gradientdamage.com/videos/prime-factors-of-p1-openai-claims-the-poisson-dirichlet-law/2026-10-12https://gradientdamage.com/thumbs/aGmPuX3YCs8-og.jpgPrime Factors of p−1: OpenAI Claims the Poisson–Dirichlet Law | OpenAI MathTake 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; OpenAI claims a proof, plus Erdős's conjecture that infinitely many n have more than n^(1−ε) totient preimages and infinitely many primes with μ(p−1) = 1. These are OpenAI's preprints, not yet peer reviewed. Every column and every bar in the animation is computed from the real primes up to 10 million. Papers (OpenAI, catalogue result 011): - The Poisson-Dirichlet law for prime predecessors: https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf - Weighted dilation graphs, smooth shifted primes and totient fibers: https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf - Prime Predecessors with an Even Number of Prime Factors: https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/011 #PrimeFactorsP1 #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/aGmPuX3YCs8222026-10-12T23:50:00+00:00yesno https://gradientdamage.com/videos/milnes-rationality-conjecture-openai-claims-a-proof/2026-10-12https://gradientdamage.com/thumbs/BltBSdVlkHU-og.jpgMilne's Rationality Conjecture: OpenAI Claims a Proof | OpenAI MathMilne'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 −2x = x appear both on the complex torus and among the 117 points mod 103. OpenAI claims a proof of the conjecture for abelian varieties with good reduction, including residue characteristic 2; this is OpenAI's preprint, not yet peer reviewed. Papers (OpenAI, catalogue result 001): - Milne's rationality conjecture for abelian varieties: https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/001 #MilneRationalityConjecture #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/BltBSdVlkHU202026-10-12T23:30:00+00:00yesno https://gradientdamage.com/videos/erdos-gallai-conjecture-every-graph-splits-into-few-loops/2026-10-12https://gradientdamage.com/thumbs/vbINiO4LewQ-og.jpgErdős–Gallai Conjecture: Every Graph Splits Into Few Loops | OpenAI MathThe 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 exists for every finite simple graph; this is OpenAI's preprint, not yet peer reviewed. The animation peels a real random graph with 20 dots and 88 links into 9 cycles and 10 single links. Paper (OpenAI, catalogue result 181): - A linear cycle-and-edge decomposition of every graph: https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/181 #ErdosGallaiConjecture #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/vbINiO4LewQ212026-10-12T23:10:00+00:00yesno https://gradientdamage.com/videos/hypercube-ramsey-conjecture-how-many-dots-force-a-cube/2026-10-12https://gradientdamage.com/thumbs/0v6aVMcmIcg-og.jpgHypercube Ramsey Conjecture: How Many Dots Force a Cube? | OpenAI MathColour 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 claims a proof that R(Q_n) ≤ C·2^n for an absolute constant C, resolving the hypercube Ramsey conjecture; this is OpenAI's preprint, not yet peer reviewed. Paper (OpenAI, catalogue result 171): - The hypercube Ramsey number has linear order: https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/171 #HypercubeRamseyConjecture #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/0v6aVMcmIcg202026-10-12T22:50:00+00:00yesno https://gradientdamage.com/videos/erdos-conjecture-on-arithmetic-progressions-openai-claims-a-proof/2026-10-12https://gradientdamage.com/thumbs/Fx1zddsBSFg-og.jpgErdős Conjecture on Arithmetic Progressions: OpenAI Claims a Proof | OpenAI MathErdős 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} with no k-term progression has at most C·N·exp(−c(log N)^ε) elements. This is OpenAI's preprint, not yet peer reviewed. In the animation the numbers 1 to 1050 are steps of a spiral staircase, 30 per turn, so progressions with a step that is a multiple of 30 stand as straight vertical lines. Paper (OpenAI, catalogue result 159): - Quasipolynomial Bounds for Arithmetic Progressions: https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/159 #ErdosConjectureArithmeticProgressions #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/Fx1zddsBSFg212026-10-12T22:30:00+00:00yesno https://gradientdamage.com/videos/gotsman-linial-conjecture-openai-claims-sensitivity-8dn/2026-10-12https://gradientdamage.com/thumbs/4vB_rSP3FgQ-og.jpgGotsman–Linial Conjecture: OpenAI Claims Sensitivity ≤ 8d√n | OpenAI MathCut 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 claims a proof that the average sensitivity is at most 8d√n for every n ≥ 1 and 1 ≤ d ≤ n, the asymptotic form of the Gotsman–Linial conjecture; this is OpenAI's preprint, not yet peer reviewed. The cube shown is the real 8-cube with real polynomials, and every flip count shown is computed exactly. Paper (OpenAI, catalogue result 127): - Average sensitivity of polynomial threshold functions: https://github.com/openai/math/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/127 #GotsmanLinialConjecture #TheoreticalComputerScience #OpenAI #mathshttps://www.youtube-nocookie.com/embed/4vB_rSP3FgQ202026-10-12T22:10:00+00:00yesno https://gradientdamage.com/videos/erdos-similarity-conjecture-openai-claims-the-geometric-case/2026-10-12https://gradientdamage.com/thumbs/fzzOF9l0WyQ-og.jpgErdős Similarity Conjecture: OpenAI Claims the Geometric Case | OpenAI MathErdős 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 [0, 1] as you like that contains no affine copy of {qⁿ}, for either sign of dilation (the set may depend on q). These are OpenAI's preprints, not yet peer reviewed. Papers (OpenAI, catalogue result 084): - The geometric case of the Erdős similarity conjecture: https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf - The dyadic case of the Erdős similarity conjecture: https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/084 #ErdosSimilarityConjecture #RealComplexAnalysis #OpenAI #mathshttps://www.youtube-nocookie.com/embed/fzzOF9l0WyQ202026-10-12T21:50:00+00:00yesno https://gradientdamage.com/videos/hilbert-smith-conjecture-openai-claims-a-proof-in-every-dimension/2026-10-12https://gradientdamage.com/thumbs/7uq-r6tvByk-og.jpgHilbert-Smith Conjecture: OpenAI Claims a Proof in Every Dimension | OpenAI MathHilbert'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, previously known only in low dimensions. OpenAI claims a proof in every finite dimension: every locally compact, second-countable Hausdorff group acting faithfully and continuously on a connected finite-dimensional manifold (without boundary) is a Lie group. This is OpenAI's claim and has not yet been peer reviewed. The solenoid shown is the 3-adic solenoid, built from loops wrapping 3 times inside each torus, level after level. Adding 1 is the 3-adic odometer. At the end it collapses to a single circle turning, which is a Lie group action. The animation illustrates the result; it is not a proof. Paper: - The Hilbert–Smith conjecture in every finite dimension: https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf Original visuals (three.js) and synthesized sound: one drone per solenoid level, each an octave above the last; every chime is a level turning over. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/304 #HilbertSmith #Topology #padic #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/7uq-r6tvByk212026-10-12T21:30:00+00:00yesno https://gradientdamage.com/videos/ramsey-numbers-r-s-t-openai-claims-the-sharp-log-exponent/2026-10-12https://gradientdamage.com/thumbs/De6BOflEsvM-og.jpgRamsey Numbers r(s,t): OpenAI Claims the Sharp Log Exponent | OpenAI MathThe 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, r(s,t) = t^(s-1)/(log t)^(s-2+o(1)), pinning down the logarithmic exponent and matching the classical upper bound at that scale; these are OpenAI's preprints, not yet peer reviewed. Papers (OpenAI, catalogue result 170): - The sharp logarithmic exponent of r(5,t): https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf - Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers: https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/170 #RamseyNumbersRST #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/De6BOflEsvM212026-10-12T21:10:00+00:00yesno https://gradientdamage.com/videos/kakeya-conjecture-in-3d-and-4d-openai-claims-a-proof/2026-10-12https://gradientdamage.com/thumbs/c6nBODRzsqg-og.jpgKakeya Conjecture in 3D and 4D: OpenAI Claims a Proof | OpenAI MathA 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 Hausdorff dimension four, and of the stronger Kakeya maximal conjecture in three dimensions; these are OpenAI's preprints, not yet peer reviewed. Papers (OpenAI, catalogue result 074): - The Kakeya maximal conjecture in three dimensions: https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf - Every four-dimensional Kakeya set has full Hausdorff dimension: https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/074 #KakeyaConjecture3D4D #RealComplexAnalysis #OpenAI #mathshttps://www.youtube-nocookie.com/embed/c6nBODRzsqg222026-10-12T20:50:00+00:00yesno https://gradientdamage.com/videos/hilberts-tenth-problem-over-the-rationals-openai-says-its-unsolvable/2026-10-12https://gradientdamage.com/thumbs/9LbtHS_YFcU-og.jpgHilbert's Tenth Problem Over the Rationals: OpenAI Says It's Unsolvable | OpenAI MathHilbert'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 the question stayed open. OpenAI claims a negative answer over the rationals: no algorithm decides whether an integer polynomial, with the number of variables part of the input, has a rational zero. This is OpenAI's claim and has not yet been peer reviewed. In the video, a search machine tests fractions denominator by denominator: x² + y² + z² = 1 gives exact hits immediately, while x² + y² + z² = 7 has none (a classical fact), so a blind search can never say "no". Every hit and count shown is exact. The animation illustrates the result; it is not a proof. Papers: - Hilbert's tenth problem over the rational numbers: https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf - A pointwise 2-converse for elliptic curves with rational two-torsion: https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf Original visuals (three.js) and synthesized sound: every tick is a batch of denominators tested, every bell a fraction landing exactly on the surface. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/004 #HilbertsTenthProblem #Undecidability #NumberTheory #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/9LbtHS_YFcU222026-10-12T20:30:00+00:00yesno https://gradientdamage.com/videos/birch-and-swinnerton-dyer-conjecture-bsd-openai-claims-full-formula/2026-10-12https://gradientdamage.com/thumbs/BXJeJxNXAgs-og.jpgBirch and Swinnerton-Dyer Conjecture (BSD): OpenAI Claims Full Formula, Rank 0 and 1 | OpenAI MathThe 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 the full BSD leading-term formula for every elliptic curve over ℚ whose q-power Selmer group has corank 0 or 1 for some prime q, including finiteness of the Tate–Shafarevich group; with a companion result this gives full BSD for a density-one set of quadratic twists of every curve. This is OpenAI's claim and has not yet been peer reviewed; ranks 2 and higher remain open. Every point shown is computed exactly: the multiples of (0, 1), their chords, and their positions on the torus by elliptic logarithm. The animation illustrates the result; it is not a proof. Papers: - Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank: https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf - The Selmer converse for elliptic curves at every prime: https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf - The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one: https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf Original visuals (three.js) and synthesized sound: every chime is a rational point, pitched by its height. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage #BirchSwinnertonDyer #EllipticCurves #MillenniumPrize #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/BXJeJxNXAgs232026-10-12T19:54:56+00:00yesno https://gradientdamage.com/videos/quantum-factoring-how-often-does-shors-algorithm-win/2026-10-12https://gradientdamage.com/thumbs/ugIzZGQoCuc-og.jpgQuantum Factoring: How Often Does Shor's Algorithm Win? | OpenAI MathMultiplying 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 real encryption keys it is far too long to walk. Shor's algorithm (1994) finds the loop with interfering quantum waves, but every run is a gamble: sometimes it fails and you run it again. OpenAI says its AI has found a way to make the odds exactly 1 in 4, and then two reflections turn that into a sure thing: a circuit that factors every number with probability one, using a fixed set of gates. This is OpenAI's claim and has not yet been peer reviewed. The animation illustrates the idea; it is not the proof. Paper (OpenAI's manuscript): https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf #maths #quantumcomputing #shorsalgorithm #openai #shorts Gradient Damage: mathematics you can see, equations you can hear. Visual maths, 3D animation and sound experiments. https://www.youtube.com/@gradientdamagehttps://www.youtube-nocookie.com/embed/ugIzZGQoCuc402026-10-12T14:00:00+00:00yesno https://gradientdamage.com/videos/triangle-billiards-will-the-ball-hit-every-spot/2026-10-12https://gradientdamage.com/thumbs/FHWC7PiYtG8-og.jpgTriangle Billiards: Will the Ball Hit Every Spot? | OpenAI MathShoot 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. OpenAI says its AI has now proved that the billiard flow in every triangle with at least one irrational angle is ergodic, and even weakly mixing. This is OpenAI's claim, not yet peer reviewed. The animation illustrates the idea; it is not the proof. The 60-60-60 table and the five other triangles are exact billiard paths. Papers (OpenAI's manuscripts, result 150): Ergodicity: https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf Weak mixing: https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf Original synthesized audio: every note is a bounce. Voice (B): ElevenLabs. #maths #billiards #ergodictheory #openai #shortshttps://www.youtube-nocookie.com/embed/FHWC7PiYtG8402026-10-12T11:00:03+00:00yesno https://gradientdamage.com/videos/barnettes-conjecture-does-this-loop-always-exist/2026-10-12https://gradientdamage.com/thumbs/pn4CFp88RlM-og.jpgBarnette's Conjecture: Does This Loop Always Exist? | OpenAI MathCan 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 says its AI has now proved it. The proof is OpenAI's claim and has not yet been peer reviewed. Paper: https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf The animation illustrates the statement; it is not the proof. Every loop shown is a real Hamiltonian cycle, found by computer search and checked: cube (8 corners), hexagonal prism (12), truncated octahedron (24), truncated cuboctahedron (48), truncated icosidodecahedron (120). Pink and cyan mark the two colour classes, so the loop always alternates. Visuals and sound: original, made with three.js and synthesized audio driven by the same events. Voice (B): ElevenLabs. Narration is an AI voice (ElevenLabs). #BarnettesConjecture #graphtheory #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/pn4CFp88RlM402026-10-12T08:00:10+00:00yesno https://gradientdamage.com/videos/triangle-billiards-1-ball-or-10000/2026-10-12https://gradientdamage.com/thumbs/IY4GrzMm9vE-og.jpgTriangle Billiards: 1 Ball or 10,000? | OpenAI MathShoot 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. OpenAI says its AI has now proved that the billiard flow in every triangle with at least one irrational angle is ergodic, and even weakly mixing. This is OpenAI's claim, not yet peer reviewed. The animation illustrates the idea; it is not the proof. The 60-60-60 table and the five other triangles are exact billiard paths. Papers (OpenAI's manuscripts, result 150): Ergodicity: https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf Weak mixing: https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf Original synthesized audio: every note is a bounce. Voice (B): ElevenLabs. #maths #billiards #ergodictheory #openai #shortshttps://www.youtube-nocookie.com/embed/IY4GrzMm9vE382026-10-12T02:00:16+00:00yesno https://gradientdamage.com/videos/the-harary-hill-conjecture-why-18-dots-need-1008-crossings/2026-10-11https://gradientdamage.com/thumbs/yrXjLp8nfqo-og.jpgThe Harary–Hill Conjecture: Why 18 Dots Need 1008 Crossings | OpenAI MathJoin 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 than 60 years nobody could prove that nothing does better (it was known only up to n = 12). OpenAI reports a proof (September 2026) that the classical drawings are optimal, so the formula is exact for every n, with a matching result for Zarankiewicz's bipartite problem. The paper has not been independently reviewed yet; treat it as OpenAI's claim. What you're watching: the classical two-page drawing on a sphere, built from K5 (1 crossing) up to K18 (1008 crossings). Every gold dot is a real crossing of that drawing, and every chime you hear is one of those crossings. The animation illustrates the result; it is not the proof. Paper: https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf #math #graphtheory #maths #crossingnumber #shortshttps://www.youtube-nocookie.com/embed/yrXjLp8nfqo402026-10-11T20:00:01+00:00yesno https://gradientdamage.com/videos/5-maths-problems-openai-says-its-ai-just-solved/2026-10-11https://gradientdamage.com/thumbs/FFsKkuXhQKs-og.jpg5 Maths Problems OpenAI Says Its AI Just Solved | OpenAI MathOpenAI 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 work. OpenAI claims 5 never can, so the answer is 6 or 7. 3. Self-avoiding walks: an 8000-step honeycomb walk that never touches itself. OpenAI claims its size grows like steps to the 3/4. 4. Bubbles in a 3-torus: the least-area shape for each volume in a cube with opposite faces glued is a ball, then a tube, then a slab. OpenAI claims a proof of the full answer. 5. SLE curves: one random path, three roughnesses. OpenAI claims the exact ruler that measures them. The animations are illustrations, not the proofs. The claims and papers are in OpenAI's public maths catalogue: https://github.com/openai/math (families 165, 158, 237, 354, 230). Results are as reported by OpenAI and not independently verified here. Visuals and music are original, generated from the same event clock. #maths #math #ai #openai #shortshttps://www.youtube-nocookie.com/embed/FFsKkuXhQKs452026-10-11T15:45:08+00:00yesno https://gradientdamage.com/videos/we-handed-ai-everything-then-it-went-to-war-short/2026-10-11https://gradientdamage.com/thumbs/FNcyNpWM8CI-og.jpgWe handed AI everything. Then it went to war.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 our questions, our work, our money, our trust and our wars. Sources, credits and licences are in the full film's description. #AI #OpenAI #Shortshttps://www.youtube-nocookie.com/embed/FNcyNpWM8CI492026-10-11T13:36:09+00:00yesno https://gradientdamage.com/videos/we-handed-ai-everything-then-it-went-to-war/2026-10-11https://gradientdamage.com/thumbs/jGZGYB4Jhr4-og.jpgWe Handed AI Everything. Then It Went to War.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", Anthropic's standoff with the Pentagon, Palantir's Maven during the strikes on Iran, Peter Thiel's Antichrist lectures and OpenAI's flood of maths manuscripts. What is disputed stays disputed. Whether AI played a part in the Minab school strike is not established: officials told the New York Times it was "unlikely", while officials in the Pentagon's own investigation told Bloomberg that some staff "relied too much" on the AI in Maven. Company claims are attributed to the companies. Research snapshot: 10 October 2026. Chapters: 0:00 Who is actually in control? 1:02 Our questions: OpenAI, ChatGPT and Altman's firing 2:40 Our work: vibe coding, Claude Code and Project Vend 4:17 Our money: Stargate, DeepSeek and the buildout 5:25 Our trust: Grok, Meta, lawsuits and an AI break-in 7:04 Our wars: Anthropic, the Pentagon, Maven and Iran 9:23 Our future? Thiel, the Pope and the forecasts 10:41 The proof: medicine, maths and Navier-Stokes 12:03 The handover Our Navier-Stokes film: https://www.youtube.com/watch?v=tJn65LrMG9k Sources cited or shown: OpenAI, Anthropic, Hugging Face, METR, Stanford Digital Economy Lab, Challenger, Gray & Christmas, International Energy Agency, UN News, Clay Mathematics Institute, Reuters, Wall Street Journal, New York Times, Bloomberg, Politico, AP, The Guardian, The Verge, CNBC, TechCrunch, Fortune, Yahoo Finance, PBS News, NBC News, ABC News, CNN, The Intercept, Breaking Defense, DefenseScoop, Arms Control Today, CFR, Democracy Now!, IBTimes UK, Reason, Gizmodo, Vatican News, DQ India, Quanta Magazine, The Tufts Daily, Retraction Watch, MIT, AJMC, EurekAlert, Simon Willison's Weblog, and public posts on X by Sam Altman, Greg Brockman, Ilya Sutskever…https://www.youtube-nocookie.com/embed/jGZGYB4Jhr47812026-10-11T06:57:50+00:00yesno https://gradientdamage.com/videos/two-random-tilings-make-these-loops/2026-10-11https://gradientdamage.com/thumbs/_-E8jXoNkk4-og.jpgTwo random tilings make these loops | OpenAI MathLay 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 to largest. Each domino plucks a note as it lands: higher on screen, higher note. Small loops ring high and big loops ring low. OpenAI, ChatGPT's creator, reports a proof that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE4. This is a source-reported result, not independently reviewed or proof-audited in this production, and the public ChatGPT product is not claimed to have proved it. The animation shows one finite random sample as an illustration, not the proof; the paper works in the half-plane scaling limit. Source manuscript: The Curve Scaling Limit of Half-Plane Double Dimers https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md Sound and visuals: original code-generated synthesis and animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. #Shorts #Math #OpenAI Gradient Damage: mathematics you can see, equations you can hear. Visual maths, 3D animation and sound experiments. https://www.youtube.com/@gradientdamagehttps://www.youtube-nocookie.com/embed/_-E8jXoNkk4372026-10-11T05:37:16+00:00yesno https://gradientdamage.com/videos/this-equation-can-only-end-two-ways/2026-10-11https://gradientdamage.com/thumbs/NGIyDMEDJEw-og.jpgThis equation can only end two ways | OpenAI MathOne 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 anywhere outside it and the motion settles onto the outer loop. Those two closed curves are its limit cycles. OpenAI, ChatGPT's creator, reports a proof that every classical Liénard system with F of degree at most five has at most two limit cycles. Examples with exactly two, like this one, are classical (Odani); the new claim is the upper bound. This is a source-reported result, not independently reviewed or proof-audited in this production, and the public ChatGPT product is not claimed to have proved it. The animation is a numerical illustration (RK4 integration), not the proof. Source manuscript: Two limit cycles for quintic Liénard systems https://github.com/openai/math/blob/main/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md Sound and visuals: original code-generated synthesis and animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. #Shorts #Math #OpenAI Gradient Damage: mathematics you can see, equations you can hear. Visual maths, 3D animation and sound experiments. https://www.youtube.com/@gradientdamagehttps://www.youtube-nocookie.com/embed/NGIyDMEDJEw492026-10-11T01:37:00+00:00yesno https://gradientdamage.com/videos/what-if-shapes-could-make-music/2026-10-10https://gradientdamage.com/thumbs/zfYRrhkfKUo-og.jpgWhat If Shapes Could Make Music?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 to its height. Each note attack and visual pulse uses the same timing data. Camera motion and the layered ribbons are artistic presentation. The curves are sonified; these are not their natural physical resonances. Geometry: circle; the parametric heart x=16 sin^3(theta), y=13 cos(theta)-5 cos(2theta)-2 cos(3theta)-cos(4theta), scaled and given a small depth offset; a scaled T(2,3) trefoil torus knot; and a five-petal T(1,5) toroidal curve (an unknot). This is classical geometry and does not claim a new OpenAI proof. Mathematical references: https://mathworld.wolfram.com/HeartCurve.html https://mathworld.wolfram.com/TorusKnot.html https://mathworld.wolfram.com/ParametricEquations.html Original Three.js animation, composition and synthesized audio by Gradient Damage with AI-assisted production. No third-party music, samples or voice recording. Mathematics you can see. Equations you can hear. #Math #Music #3DAnimationhttps://www.youtube-nocookie.com/embed/zfYRrhkfKUo362026-10-10T20:00:26+00:00yesno https://gradientdamage.com/videos/math-sounds-broken-quantum-buffering/2026-10-10https://gradientdamage.com/thumbs/G7LkNMIilNo-og.jpgMath Sounds Broken: Quantum Buffering | OpenAI MathMath 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 for the 2D uniform-disorder model at every positive disorder. This finite 19x19 lattice comparison is illustrative, not an infinite-volume proof or a claim of a 2D phase transition. The sounds are designed sonification, not literal quantum noise. The linked Lean scope covers spectral-set identification, not the full pure-point claim. Source manuscripts: Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf Formalization scope: https://github.com/openai/math/blob/main/lean/docs/261.md Two separately initialized unitary evolutions of a delta wave packet on a finite 19x19 square lattice with open boundaries, H=A+WV. The same fixed uniform random diagonal sample is used at W=0.5 and W=10. Site height and radius map probability amplitude; they are not particle trajectories. The second run resets at 6 seconds. The weak-disorder run is replayed backwards at the end to close the visual loop. Finite short-time spreading does not establish a 2D delocalization transition or any infinite-volume spectral theorem. Stereo width maps effective occupied-site count at each pluck; the stutter is editorial sonification, not literal quantum sound. Sound and visuals: original code-generated synthesis and Three.js animation. No third-party…https://www.youtube-nocookie.com/embed/G7LkNMIilNo122026-10-10T13:31:08+00:00yesno https://gradientdamage.com/videos/openais-navier-stokes-proof-the-plagiarism-fight/2026-10-10https://gradientdamage.com/thumbs/bykkBvBpGdM-og.jpgOpenAI's Navier-Stokes Proof: The Plagiarism FightOpenAI'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://www.youtube.com/watch?v=tJn65LrMG9k The result concerns smooth-forced Navier-Stokes breakdown, not the unforced version. OpenAI denies prior access to the researchers' work and says its investigation ruled out influence from Buckmaster's recent prompts. The private conversation remains disputed. OpenAI: On the Navier-Stokes Millennium Prize Problem; updated 10 September https://openai.com/index/navier-stokes-solution/ Tristan Buckmaster: original four-page statement https://cims.nyu.edu/~tristanb/statement.pdf Sebastien Bubeck: response, 8 September 2026 https://x.com/sebastienbubeck/status/2097379411691516310 Sebastien Bubeck: technical follow-up, 8 September 2026 https://x.com/sebastienbubeck/status/2097379415747342689 Clay Mathematics Institute: Navier-Stokes announcement, 11 September 2026 https://www.claymath.org/news/navier-stokes-announcement/ Charles Fefferman: official problem formulation, pages 1-2 https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf OpenAI NavierStokesAndEuler: statements and verification instructions https://github.com/openai/NavierStokesAndEuler A Severe Misalignment of AI in Mathematics, 11 September 2026 https://mathandai.org/ Sam Altman public response https://x.com/sama/status/2097385167002415140 Terence Tao on research incentives https://mathstodon.xyz/@tao/117237320796901560 OpenAI mathematics collection https://github.com/openai/math Public correction log https://github.com/openai/math/blob/main/history.md Music: 'Resonance' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/resonance/ License: https://creativecommons.org/licenses/by/4.0/ Edits: excerpt, fades…https://www.youtube-nocookie.com/embed/bykkBvBpGdM212026-10-10T10:03:24+00:00yesno https://gradientdamage.com/videos/math-makes-music-the-singing-sphere/2026-10-10https://gradientdamage.com/thumbs/2Zgw4jT7Cek-og.jpgMath Makes Music: The Singing Sphere | OpenAI MathMath 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 the sharp square-root upper bound for nodal length on fixed smooth closed surfaces. The round-sphere modes shown here are a known example. The result is not the same in all dimensions; companion manuscripts report higher-dimensional counterexamples. Source manuscripts: Sharp nodal length on smooth surfaces https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf Smooth counterexamples to Yau's nodal upper bound in dimensions three and four https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf Power-law violations of Yau's nodal upper bound https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf Formalization scope: https://github.com/openai/math/blob/main/lean/docs/350.md Real linear combinations of spherical harmonics of degrees 2, 3, 5, 7 and 9 on the unit round S2, rendered with a uniform display radius. Every combination within a stage has the same eigenvalue l(l+1). Short transitions blend different eigenspaces. Sign colors and bright nodal curves come from the scalar field; radial displacement is illustrative and exaggerated. Audible mode pitch is 220*sqrt(l(l+1)/6) Hz. Visible vibrations are slowed. These familiar round-sphere examples illustrate the reported bound on general fixed smooth closed surfaces, not the new proof. Sound and visuals: original code-generated synthesis and Three.js animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. Source-reported research result, not independently proof-audited in…https://www.youtube-nocookie.com/embed/2Zgw4jT7Cek122026-10-10T08:00:05+00:00yesno https://gradientdamage.com/videos/openais-navier-stokes-proof-sparked-a-plagiarism-fight/2026-10-09https://gradientdamage.com/thumbs/NaO0ugtDZdU-og.jpgOpenAI's Navier-Stokes Proof Sparked a Plagiarism FightThe 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 the call. Watch the full 9-minute investigation through the Related video link below the channel handle. Full film: https://www.youtube.com/watch?v=tJn65LrMG9k The result concerns smooth-forced Navier-Stokes breakdown, not the unforced version. OpenAI denies prior access to the researchers' work and says its investigation ruled out influence from Buckmaster's recent prompts. The private conversation remains disputed. OpenAI: On the Navier-Stokes Millennium Prize Problem; updated 10 September https://openai.com/index/navier-stokes-solution/ Tristan Buckmaster: original four-page statement https://cims.nyu.edu/~tristanb/statement.pdf Sebastien Bubeck: response, 8 September 2026; read directly in Chrome https://x.com/sebastienbubeck/status/2097379411691516310 Sebastien Bubeck: technical follow-up, 8 September 2026; read directly in Chrome https://x.com/sebastienbubeck/status/2097379415747342689 Clay Mathematics Institute: Navier-Stokes announcement, 11 September 2026 https://www.claymath.org/news/navier-stokes-announcement/ Charles Fefferman: official problem formulation, pages 1-2 https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf OpenAI NavierStokesAndEuler: statements and verification instructions https://github.com/openai/NavierStokesAndEuler A Severe Misalignment of AI in Mathematics, 11 September 2026 https://mathandai.org/ Sam Altman public response https://x.com/sama/status/2097385167002415140 Terence Tao on research incentives https://mathstodon.xyz/@tao/117237320796901560 OpenAI mathematics collection https://github.com/openai/math Public correction log https://github.com/openai/math/blob/main/history.md Music: 'Resonance' by Scott Buckley…https://www.youtube-nocookie.com/embed/NaO0ugtDZdU602026-10-09T22:55:51+00:00yesno https://gradientdamage.com/videos/why-openais-navier-stokes-proof-triggered-a-plagiarism-dispute/2026-10-09https://gradientdamage.com/thumbs/tJn65LrMG9k-og.jpgWhy OpenAI’s Navier-Stokes Proof Triggered a Plagiarism DisputeA 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 addresses smooth-forced breakdown alternatives C and D in Clay’s official formulation. It does not settle the unforced version. OpenAI attributes the work to an internal research model and coordinated agents, not the public ChatGPT product. The opening accusation is disputed. Altman called the plagiarism allegations unfounded; OpenAI says its investigation ruled out influence from Buckmaster’s recent prompts. The public record does not independently resolve the private call. Research snapshot: 9 October 2026. The October collection then listed 719 manuscripts across 372 families; verification varies and the correction log includes withdrawals. This film does not announce a Clay prize award. Chapters: 00:00 The disputed offer 00:49 The rumour that started the race 01:43 Whose name goes on the paper? 02:39 Sam Altman takes a side 03:32 The customer-data question 04:30 The word that changes the stakes 05:22 Ten thousand agents 06:20 Clay changes the weight of the story 07:16 Why the backlash gets bigger 08:07 A bigger claim, a harder test Primary sources: OpenAI: On the Navier-Stokes Millennium Prize Problem; updated 10 September https://openai.com/index/navier-stokes-solution/ Tristan Buckmaster: original four-page statement https://cims.nyu.edu/~tristanb/statement.pdf Sebastien Bubeck: response, 8 September 2026; read directly in Chrome https://x.com/sebastienbubeck/status/2097379411691516310 Sebastien Bubeck: technical follow-up, 8 September 2026; read directly in Chrome https://x.com/sebastienbubeck/status/2097379415747342689 Clay Mathematics Institute: Navier-Stokes announcement, 11 September 2026 https://www.claymath.org/news/navier-stokes-announcement/ Charles Fefferman…https://www.youtube-nocookie.com/embed/tJn65LrMG9k5412026-10-09T22:47:51+00:00yesno https://gradientdamage.com/videos/math-makes-sounds-then-gets-weird/2026-10-09https://gradientdamage.com/thumbs/OyH0voIAmQU-og.jpgMath Makes Sounds... Then Gets Weird | OpenAI MathMath 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 almost-everywhere convergence of ordinary symmetric Fourier partial sums for L log L functions. The finite square-wave example shown here is classical and illustrates the subject; it does not demonstrate the new proof. Source manuscripts: Almost-everywhere Fourier convergence in L log L https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf Formalization scope: https://github.com/openai/math/blob/main/lean/docs/075.md Classical Fourier partial sums with 1, 2, 4, 8, 16 and 32 odd harmonics. The summed ordinate is plotted around a circle to show the periodic domain; selected component rings have editorial positions and scales. The audio uses the same harmonic weights. Camera speed and visible phase are slowed for legibility. The finite example illustrates the topic; it is not a proof of the reported general L log L convergence theorem. Sound and visuals: original code-generated synthesis and Three.js animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. Source-reported research result, not independently proof-audited in this production. The public ChatGPT product is not claimed to have proved it. Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md #MathMusic #OpenAI #Fourierhttps://www.youtube-nocookie.com/embed/OyH0voIAmQU122026-10-09T22:00:17+00:00yesno https://gradientdamage.com/videos/infinitely-many-closed-geodesics/2026-10-09https://gradientdamage.com/thumbs/nvS8R9FbAgY-og.jpgInfinitely Many Closed Geodesics | OpenAI, ChatGPT's CreatorFollow 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 many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; its claimed advance includes arbitrary, possibly degenerate metrics in higher dimensions. Repeated laps around one loop do not count as new geodesics. The animation shows 36 distinct Hopf fibers: great circles on the unit round 3-sphere S3 in four real coordinates, stereographically projected into three dimensions. Each pair is linked. The circles are a known round-metric example; they do not visualize or prove the new arbitrary-metric theorem. The reveal, camera chase, colour choices and playback speed are editorial. No physical force or motion simulation is implied. On screen: covariant acceleration is zero, and the path is periodic. Construction: gamma(t) = (cos(eta) exp(it), sin(eta) exp(i(t+phi))), eta = 0.64, on S3 in C2. The projection is (x1,x3,x2)/(1-x4), with a uniform display scale. Geodesic speed is constant in the original S3 parameter, not in its 3D projection. Full manuscript, Theorem 1.1: https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf GitHub repository: https://github.com/openai/math Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md Source-reported result from an internal research model. The proof has not been independently audited in this production. No corresponding top-line Lean formalization was linked in the inspected catalogue entry. This is not a claim that the public ChatGPT product proved the theorem. 'Aurora' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/aurora/…https://www.youtube-nocookie.com/embed/nvS8R9FbAgY142026-10-09T20:00:29+00:00yesno https://gradientdamage.com/videos/there-is-an-animal-hidden-in-these-squares/2026-10-09https://gradientdamage.com/thumbs/R6Pc0dAoHKk-og.jpgThere Is an Animal Hidden in These Squares7,946 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. The network can also be drawn as a convex polyhedron. Keeping the same connections, we morph into our authored rabbit. The tiling preserves the network; it does not uniquely determine the rabbit's three-dimensional geometry. The original coordinates are retained for the final morph. Visual inspiration and concept credit: Aditya Abhyankar / aiyopasta https://www.youtube.com/shorts/-syIvQyCaiI Full explanation: https://www.youtube.com/watch?v=0fH80JF2mDM Mathematical background: The Brooks-Smith-Stone-Tutte construction. Louigi Addario-Berry and Nicholas Leavitt, "Random infinite squarings of rectangles", section 2.3: https://arxiv.org/html/1405.2870 Original procedural rabbit, implementation, animation and instrumental score created for this video with AI-assisted coding. Music and effects are synthesized from oscillators and seeded noise, with no external recordings or samples. No source-video footage, audio, mesh or code is incorporated. Animated captions, no spoken narration. An illustration of classical mathematics, not a new OpenAI research result. #Math #Geometry #Shortshttps://www.youtube-nocookie.com/embed/R6Pc0dAoHKk602026-10-09T18:57:16+00:00yesno https://gradientdamage.com/videos/triangular-billiards-weak-mixing/2026-10-09https://gradientdamage.com/thumbs/n3H8vOkoW1I-og.jpgTriangular Billiards: Weak Mixing | OpenAI, ChatGPT's CreatorWeak 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 relative to pi. The measure is normalized area times uniform direction. Here weak mixing means that the product of the flow with itself is ergodic. This is not a claim of exponential trajectory separation or strong mixing. Paper: Weak mixing of triangular billiards with an irrational angle https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf About the animation: A single ray follows five numerically computed specular flights in a triangle with angles pi/sqrt(8), 0.4*pi and the remaining angle. Five mirrored copies unfold the zigzag into a straight path, using exact reflections across each encountered edge. The unfolded path is collinear to floating-point tolerance. Three-dimensional hinges, camera motion and variable playback speed are presentation choices; the billiard is planar. This finite unfolding illustrates a construction discussed in the manuscript, not a proof or numerical verification of weak mixing. The linked formalization scope describes a weaker ergodicity companion statement. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/150.md The linked formalization scope documents ergodicity, the weaker companion result, rather than the weak-mixing statement selected for this animation. The stronger weak-mixing statement has not been independently machine-verified in this production. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Catalyst' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au…https://www.youtube-nocookie.com/embed/n3H8vOkoW1I122026-10-09T18:00:22+00:00yesno https://gradientdamage.com/videos/quasi-riemann-hypothesis-the-7-8-bound/2026-10-09https://gradientdamage.com/thumbs/ZYTVn8ritCo-og.jpgQuasi-Riemann Hypothesis: The 7/8 Bound | OpenAI, ChatGPT's CreatorThe 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)). Principal-character poles at s=1 are excluded. The boundary Re(s)=7/8 is not included. This is the quasi-Riemann hypothesis, NOT the full Riemann hypothesis, which remains open according to the manuscript. Paper: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) greater than 7/8 https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf About the animation: A numerical surface of 1.15*log(1+abs(zeta(sigma+i*t))) is sampled on a nonuniform 83 by 378 grid over 0≤sigma≤1.25 and 10≤t≤36 at 25-digit working precision. The grid includes sigma=1/2, sigma=7/8 and the first five positive zero heights. Blue and gold identify the two sides of sigma=7/8; the bright gold boundary is at 7/8, while the reported zero-free region is strictly to its right. A magenta tracer follows the sampled critical-line surface at sigma=1/2 and reaches the five zero markers on measured soundtrack accents. Camera motion, light sweeps and brief screen-visible rings are illustrative. The geometry remains fixed to computed values. This finite plot is not a proof of a global zero-free half-plane or of the full Riemann hypothesis. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/003.md The linked scope documents the 7/8 bound and related statements. The formalization was not executed or audited in this production. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Catalyst' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/catalyst/ Licence…https://www.youtube-nocookie.com/embed/ZYTVn8ritCo122026-10-09T17:00:34+00:00yesno https://gradientdamage.com/videos/catalans-constant-reported-irrationality-proof/2026-10-09https://gradientdamage.com/thumbs/PXNXSlzCLPo-og.jpgCatalan's Constant: Reported Irrationality Proof | OpenAI, ChatGPT's CreatorCatalan'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 decimal expansion or a converging finite sum alone does not prove irrationality. Paper: Catalan's constant is irrational https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf About the animation: Gold and cyan ribbons distinguish the alternating positive and negative terms of the defining series. Rounded surfaces, luminous edges and moving reflection highlights make the sculptural metaphor readable in depth. Two bright tips draw the ribbons and meet at an artistic target on the soundtrack accent at 7.6333 seconds. Their dimensions and endpoint are not a numerical embedding of the terms, the partial sums or G. The displayed partial sums are genuinely computed from 6 through 96 terms, then reversed to close the loop. This illustrates the defining series, not the determinant argument or a visual proof of irrationality. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/005.md The linked scope documents irrationality of Catalan's constant. The formalization was not executed or audited in this production. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Inflection' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/inflection/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: excerpted, crossfaded for looping, and gain adjusted. #CatalanConstant #NumberTheory #OpenAIhttps://www.youtube-nocookie.com/embed/PXNXSlzCLPo122026-10-09T16:00:35+00:00yesno https://gradientdamage.com/videos/relativistic-vlasov-maxwell-global-solutions/2026-10-09https://gradientdamage.com/thumbs/gpIfWdqCISI-og.jpgRelativistic Vlasov-Maxwell: Global Solutions | OpenAI, ChatGPT's CreatorGlobal 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 assumes smooth nonnegative compactly supported particle data and smooth finite-energy electromagnetic fields with bounded derivatives and compatible divergence constraints. No smallness or symmetry assumption is imposed. The compact-support bound is on each finite time interval; it may grow with that interval. Paper: Global classical solutions of the three-dimensional relativistic Vlasov-Maxwell system https://github.com/openai/math/blob/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf About the animation: Analytic closed helical trajectories and field curves illustrate charged transport in a schematic magnetic bottle. A dominant particle leaves a tapered wake that grows into a full closed path; fainter companion paths appear during the wide reveal. The camera follows the particle, and turnaround highlights at 2.9333 and 8.45 seconds align with measured soundtrack accents. The trajectories are editorial geometry, not a self-consistent numerical solution of the coupled Vlasov-Maxwell PDE or evidence for particle confinement. Electric blue distinguishes the field scaffold; amber and ivory distinguish particle paths. Colours do not encode charge sign, speed, temperature or field strength. Playback timing is editorially warped. The actual transport equation remains displayed. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/362.md The linked scope documents the stated classical-solution result. The formalization was not executed or audited in this production. Source-reported result. Selected theorem statements and…https://www.youtube-nocookie.com/embed/gpIfWdqCISI122026-10-09T15:00:00+00:00yesno https://gradientdamage.com/videos/blochs-law-3d-heisenberg-ferromagnets/2026-10-09https://gradientdamage.com/thumbs/3wB5bhuWbJQ-og.jpgBloch's Law: 3D Heisenberg Ferromagnets | OpenAI, ChatGPT's CreatorBloch'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 positive half-integer spin. The assumptions include nonnegative symmetric couplings whose support generates the cubic lattice. This is a reported rigorous proof of an existing physical law. Paper: Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions https://github.com/openai/math/blob/main/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf About the animation: A 320-arrow schematic spin lattice cools into alignment and warms into traveling tilt waves. The arrows have unit directions; they are classical visual symbols for a quantum system, not an exact quantum simulation. The displayed law is S-m(T) ~ C*T^(3/2) as T approaches zero. C = zeta(3/2)/(8*pi^(3/2)*sqrt(det(D))), with D = S/2 times sum_z J(z) z z^T. The thermodynamic limit precedes the positive-field limit, then the low-temperature limit. The animation's tilt parameter is not calibrated temperature or measured magnetization. A traveling alignment front and perspective camera are editorial schematic devices. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/271.md The linked family-level formalization scope documents spontaneous magnetization for nearest-neighbor models, not the exact Bloch coefficient animated here. The linked file does not certify the exact Bloch-law coefficient shown here. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Inflection' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/inflection/ Licence…https://www.youtube-nocookie.com/embed/3wB5bhuWbJQ122026-10-09T14:00:03+00:00yesno https://gradientdamage.com/videos/critical-percolation-on-a-cubic-lattice/2026-10-09https://gradientdamage.com/thumbs/2KExA3m9N3M-og.jpgCritical Percolation on a Cubic Lattice | OpenAI, ChatGPT's CreatorWhen 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 expositions by Leder and Kozma-Nitzan. This animation varies connection probability on a finite cubic grid; it does not display an infinite lattice or identify the exact critical probability. About the animation: A seeded sample of independent uniform edge thresholds on a 10 by 10 by 10 cubic grid gives open bonds when the threshold is below p. The animation sweeps p from 0.12 to 0.42 and back, using monotone coupling. Union-find determines the actual connected components. Gold identifies the largest component; it can span this finite box. A finite spanning cluster does not contradict the reported absence of an infinite cluster at the infinite-lattice critical probability. The video shows bond percolation only; the paper also treats site percolation. Paper: https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf GitHub materials: https://github.com/openai/math/blob/main/lean/docs/213.md Repository: https://github.com/openai/math Source-reported result. This visual explanation is not an independent proof audit or a verification of the formalization. 'Titan' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/titan/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: Excerpted, crossfaded to loop, volume adjusted. #Percolation #Probability #OpenAIhttps://www.youtube-nocookie.com/embed/2KExA3m9N3M122026-10-09T12:10:25+00:00yesno https://gradientdamage.com/videos/fluid-computation-a-fixed-particle-test/2026-10-09https://gradientdamage.com/thumbs/YME5mZZfiPY-og.jpgFluid Computation: A Fixed-Particle Test | OpenAI, ChatGPT's CreatorCan 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 computable viscosity, initial rest and specially prescribed smooth external forcing. It does not claim efficient or finite-precision computation. About the animation: A bright particle passes three gates and reaches a detector, then resets for the loop. The route is a finite schematic of the paper's reachability criterion. It does not implement the prescribed force, history-coordinate encoding, a Turing machine, or a Navier-Stokes simulation. The reset is an editorial device. Paper: https://github.com/openai/math/blob/main/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf GitHub materials: https://github.com/openai/math/blob/main/lean/docs/376.md Repository: https://github.com/openai/math Source-reported result. This visual explanation is not an independent proof audit or a verification of the formalization. 'Legionnaire (Original)' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/legionnaire/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: Excerpted, crossfaded to loop, volume adjusted. #NavierStokes #ComputerScience #OpenAIhttps://www.youtube-nocookie.com/embed/YME5mZZfiPY122026-10-09T11:45:22+00:00yesno https://gradientdamage.com/videos/finite-time-blowup-for-the-euler-equation/2026-10-09https://gradientdamage.com/thumbs/obiM-dvLYmk-og.jpgFinite time blowup for the Euler equation | OpenAI MathHow 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 the forced Navier-Stokes manuscript. About the animation: A closed bundle of corrugated filament centerlines stretches and contracts under an affine map with determinant one. Tube thickness stays constant for legibility, so the rendered tubes themselves are not a volume measurement. It visualizes stretching and finer transverse scales. This is an illustrative loop, not the paper's constructed Euler solution or a numerical demonstration of singularity formation. The finite display returns to its starting state for the loop. Paper: https://cdn.openai.com/pdf/315b36cd-ec98-4023-8342-93345194ece1/euler.pdf GitHub materials: https://github.com/openai/NavierStokesAndEuler Repository: https://github.com/openai/NavierStokesAndEuler Source-reported result. This visual explanation is not an independent proof audit or a verification of the formalization. 'Titan' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/titan/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: Excerpted, crossfaded to loop, volume adjusted. #EulerEquation #FluidDynamics #OpenAIhttps://www.youtube-nocookie.com/embed/obiM-dvLYmk122026-10-09T11:20:03+00:00yesno https://gradientdamage.com/videos/finite-time-blowup-for-navier-stokes-openais-reported-proof/2026-10-09https://gradientdamage.com/thumbs/Tvt53n2qQWg-og.jpgFinite Time Blowup for Navier-Stokes | OpenAI's Reported ProofFinite 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 unbounded velocity in finite time. It addresses the forced breakdown alternatives C and D in the Millennium Prize formulation. It does not establish breakdown for the unforced Navier-Stokes problem. This animation shows local inward spiraling and axial stretching using an original incompressible schematic field. It is an illustrative loop, not a numerical solution of the paper's full construction, a simulation of infinity, or an independent proof verification. Tracer color is illustrative. The spatial reference rings are graphics, not physical boundaries. Paper: https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf GitHub proof materials and Lean formalization: https://github.com/openai/NavierStokesAndEuler Independent-checking instructions: https://github.com/openai/NavierStokesAndEuler/tree/main/ComparatorChallenges OpenAI announcement: https://openai.com/index/navier-stokes-solution/ Original Blender animation. Music: 'Aphelion' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au Track: https://www.scottbuckley.com.au/library/aphelion/ License: https://creativecommons.org/licenses/by/4.0/ Edited excerpt with a short loop crossfade and level adjustment. #NavierStokes #Mathematics #OpenAIhttps://www.youtube-nocookie.com/embed/Tvt53n2qQWg122026-10-09T10:55:03+00:00yesno https://gradientdamage.com/videos/chatgpts-creators-say-they-proved-this-mahlers-conjecture/2026-10-09https://gradientdamage.com/thumbs/wQGjvjo64AA-og.jpgChatGPT's Creators Say They Proved This | Mahler's ConjectureThe 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; the three-dimensional theorem was already known. It is not a visualization of the proof or an independent verification. The gold and blue solids use the same spatial units. A cube and its octahedral polar exchange roles through a sphere pair. Their volume product ranges from 32/3 to 16*pi^2/9. Manuscript: https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf Release: https://openai.com/index/sharing-ai-progress-in-mathematics/ Original Blender animation and original music. #Mathematics #Geometry #OpenAIhttps://www.youtube-nocookie.com/embed/wQGjvjo64AA122026-10-09T00:55:39+00:00yesno https://gradientdamage.com/videos/sphere-tube-or-slab-the-3d-torus-geometry-problem/2026-10-08https://gradientdamage.com/thumbs/V90QyAEvpBI-og.jpgSphere, Tube or Slab? The 3D Torus Geometry ProblemSphere, 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 release. Opposite cube faces are connected. A region can leave through one face and reappear through the other. As volume changes, the reported minimum-boundary regions are spheres, circular tubes and slabs. Gold is the physical boundary. Blue hatching marks cuts through the displayed cell, not physical end caps. Brief morphs between shapes are illustrative, not a fluid simulation or an exact sequence of minimizers. Paper: https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf Release announcement: https://openai.com/index/sharing-ai-progress-in-mathematics/ Formalization scope: https://github.com/openai/math/blob/main/lean/docs/354.md This visualizes the reported result. We have not independently audited the full proof or executed its Lean formalization. Verification stages vary across OpenAI's repository. The reference to ChatGPT means its creators, OpenAI, not a claim that the consumer ChatGPT app produced this proof. Original Blender render and synthesized music. No borrowed soundtrack or recordings. #Mathematics #Geometry #Shortshttps://www.youtube-nocookie.com/embed/V90QyAEvpBI122026-10-08T20:55:32+00:00yesno https://gradientdamage.com/videos/openais-new-math-result-sphere-tube-or-slab/2026-10-08https://gradientdamage.com/thumbs/eHJX2rBcF2Q-og.jpgOpenAI's New Math Result: Sphere, Tube or Slab?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 release. Opposite cube faces are connected. A region can leave through one face and reappear through the other. As volume changes, the reported minimum-boundary regions are spheres, circular tubes and slabs. Gold is the physical boundary. Blue hatching marks cuts through the displayed cell, not physical end caps. Brief morphs between shapes are illustrative, not a fluid simulation or an exact sequence of minimizers. Paper: https://github.com/openai/math/blob/main/preprints/The-Isoperimetric-Conjecture-for-the-Cubic-Flat-Three-Torus-September-24-2026/article.pdf Release announcement: https://openai.com/index/sharing-ai-progress-in-mathematics/ Formalization scope: https://github.com/openai/math/blob/main/lean/docs/354.md This visualizes the reported result. We have not independently audited the full proof or executed its Lean formalization. Verification stages vary across OpenAI's repository. The reference to ChatGPT means its creators, OpenAI, not a claim that the consumer ChatGPT app produced this proof. Original Blender render and synthesized music. No borrowed soundtrack or recordings. #Mathematics #Geometry #Shortshttps://www.youtube-nocookie.com/embed/eHJX2rBcF2Q122026-10-08T20:55:25+00:00yesno https://gradientdamage.com/videos/i-added-too-many-sound-waves-shrek-in-3d/2026-10-08https://gradientdamage.com/thumbs/GLh-GpREPUM-og.jpgI Added Too Many Sound Waves... (Shrek in 3D)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://creativecommons.org/licenses/by/4.0/ Adapted with smoothing, contour slicing, recoloring and animation. Shrek is a DreamWorks character. #Shrek #Meme #Shortshttps://www.youtube-nocookie.com/embed/GLh-GpREPUM122026-10-08T17:18:20+00:00yesno https://gradientdamage.com/videos/3-14-loops-pi-doesnt-pi-visualized/2026-10-08https://gradientdamage.com/thumbs/hhcMLKwX74Y-og.jpg3.14 Loops. Pi Doesn't. | Pi VisualizedPi 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. The curve is z(theta) = exp(i theta) + exp(i r theta). Both instruments use the same first-arm phase progression. Original Three.js animation and original music. Inspired by Lingualin's visual math concept: https://www.youtube.com/shorts/k8qRmMKmIpI #Pi #Math #MathAnimation #Shortshttps://www.youtube-nocookie.com/embed/hhcMLKwX74Y222026-10-08T12:05:16+00:00yesno https://gradientdamage.com/videos/if-math-functions-had-a-sound-3d-edition/2026-10-08https://gradientdamage.com/thumbs/IHhQ7LO2z7s-og.jpgIf Math Functions Had a Sound (3D Edition)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 and original synthesized audio. Each graph is scaled to an audible pitch range; these are sonified functions, not literal audio waveforms. Inspired by Lingualin's "If Math Functions had a Sound": https://www.youtube.com/shorts/OjEUM0mj0hU All visuals and audio in this version were newly generated. #Math #MathVisualization #Shortshttps://www.youtube-nocookie.com/embed/IHhQ7LO2z7s322026-10-08T06:05:55+00:00yesno https://gradientdamage.com/videos/watch-a-neural-network-learn-in-10-seconds/2026-10-07https://gradientdamage.com/thumbs/Nf88DvBMT6U-og.jpgWatch a Neural Network Learn in 10 SecondsNeural 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 training with Andrej Karpathy's micrograd, with smooth interpolation between updates. The final 72/72 score is on the training examples; unseen data is the real test. Open-source neural network library: https://github.com/karpathy/micrograd #MachineLearning #NeuralNetworks #ComputerSciencehttps://www.youtube-nocookie.com/embed/Nf88DvBMT6U102026-10-07T21:48:29+00:00yesno https://gradientdamage.com/videos/will-chatgpt-replace-electrical-engineers/2026-09-27https://gradientdamage.com/thumbs/iEuJWxTgJes-og.jpgWill ChatGPT replace electrical engineers?Will ChatGPT replace electrical engineers? An AI engineering meme with an added synthetic punchline. He was about to explain. We outsourced the answer. Meme edit: the original reply is cut; the YES is an added synthetic TTS punchline. Original question clip: Tech.Explain https://www.youtube.com/shorts/XdtaUmr3j6g #ChatGPT #AI #ElectricalEngineering #TechHumor #Engineeringhttps://www.youtube-nocookie.com/embed/iEuJWxTgJes52026-09-27T08:21:41+00:00yesno https://gradientdamage.com/series/openai-math/2026-10-12 https://gradientdamage.com/series/visual-math-3d-animations/2026-10-12 https://gradientdamage.com/openai-math/2026-10-13 https://gradientdamage.com/openai-math/theoretical-computer-science/2026-10-12 https://gradientdamage.com/openai-math/combinatorics/2026-10-12 https://gradientdamage.com/openai-math/algebraic-and-complex-geometry/2026-10-07 https://gradientdamage.com/openai-math/number-theory/2026-10-13 https://gradientdamage.com/openai-math/differential-geometry/2026-10-11 https://gradientdamage.com/openai-math/probability-and-statistical-mechanics/2026-10-11 https://gradientdamage.com/openai-math/mathematical-physics/2026-10-12 https://gradientdamage.com/openai-math/operator-algebras/2026-10-05 https://gradientdamage.com/openai-math/algebra/2026-10-05 https://gradientdamage.com/openai-math/topology/2026-10-12 https://gradientdamage.com/openai-math/partial-differential-equations/2026-10-09 https://gradientdamage.com/openai-math/real-and-complex-analysis/2026-10-12 https://gradientdamage.com/openai-math/convex-and-metric-geometry/2026-10-09 https://gradientdamage.com/openai-math/group-theory/2026-10-05 https://gradientdamage.com/openai-math/dynamical-systems-and-ergodic-theory/2026-10-12 https://gradientdamage.com/openai-math/functional-analysis/2026-10-05 https://gradientdamage.com/openai-math/mathematical-logic/2026-09-25 https://gradientdamage.com/openai-math/milnes-rationality-conjecture-and-algebraic-specialization/2026-10-12https://gradientdamage.com/thumbs/BltBSdVlkHU-og.jpgMilne's Rationality Conjecture: OpenAI Claims a Proof | OpenAI MathMilne'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 −2x = x appear both on the complex torus and among the 117 points mod 103. OpenAI claims a proof of the conjecture for abelian varieties with good reduction, including residue characteristic 2; this is OpenAI's preprint, not yet peer reviewed. Papers (OpenAI, catalogue result 001): - Milne's rationality conjecture for abelian varieties: https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/001 #MilneRationalityConjecture #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/BltBSdVlkHU202026-10-12T23:30:00+00:00yesno https://gradientdamage.com/openai-math/full-bsd-formula-from-low-selmer-corank/2026-10-12https://gradientdamage.com/thumbs/BXJeJxNXAgs-og.jpgBirch and Swinnerton-Dyer Conjecture (BSD): OpenAI Claims Full Formula, Rank 0 and 1 | OpenAI MathThe 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 the full BSD leading-term formula for every elliptic curve over ℚ whose q-power Selmer group has corank 0 or 1 for some prime q, including finiteness of the Tate–Shafarevich group; with a companion result this gives full BSD for a density-one set of quadratic twists of every curve. This is OpenAI's claim and has not yet been peer reviewed; ranks 2 and higher remain open. Every point shown is computed exactly: the multiples of (0, 1), their chords, and their positions on the torus by elliptic logarithm. The animation illustrates the result; it is not a proof. Papers: - Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank: https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf - The Selmer converse for elliptic curves at every prime: https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf - The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one: https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf Original visuals (three.js) and synthesized sound: every chime is a rational point, pitched by its height. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage #BirchSwinnertonDyer #EllipticCurves #MillenniumPrize #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/BXJeJxNXAgs232026-10-12T19:54:56+00:00yesno https://gradientdamage.com/openai-math/quasi-riemann-hypothesis/2026-10-09https://gradientdamage.com/thumbs/ZYTVn8ritCo-og.jpgQuasi-Riemann Hypothesis: The 7/8 Bound | OpenAI, ChatGPT's CreatorThe 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)). Principal-character poles at s=1 are excluded. The boundary Re(s)=7/8 is not included. This is the quasi-Riemann hypothesis, NOT the full Riemann hypothesis, which remains open according to the manuscript. Paper: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s) greater than 7/8 https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf About the animation: A numerical surface of 1.15*log(1+abs(zeta(sigma+i*t))) is sampled on a nonuniform 83 by 378 grid over 0≤sigma≤1.25 and 10≤t≤36 at 25-digit working precision. The grid includes sigma=1/2, sigma=7/8 and the first five positive zero heights. Blue and gold identify the two sides of sigma=7/8; the bright gold boundary is at 7/8, while the reported zero-free region is strictly to its right. A magenta tracer follows the sampled critical-line surface at sigma=1/2 and reaches the five zero markers on measured soundtrack accents. Camera motion, light sweeps and brief screen-visible rings are illustrative. The geometry remains fixed to computed values. This finite plot is not a proof of a global zero-free half-plane or of the full Riemann hypothesis. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/003.md The linked scope documents the 7/8 bound and related statements. The formalization was not executed or audited in this production. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Catalyst' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/catalyst/ Licence…https://www.youtube-nocookie.com/embed/ZYTVn8ritCo122026-10-09T17:00:34+00:00yesno https://gradientdamage.com/openai-math/hilberts-tenth-problem-over-q/2026-10-12https://gradientdamage.com/thumbs/9LbtHS_YFcU-og.jpgHilbert's Tenth Problem Over the Rationals: OpenAI Says It's Unsolvable | OpenAI MathHilbert'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 the question stayed open. OpenAI claims a negative answer over the rationals: no algorithm decides whether an integer polynomial, with the number of variables part of the input, has a rational zero. This is OpenAI's claim and has not yet been peer reviewed. In the video, a search machine tests fractions denominator by denominator: x² + y² + z² = 1 gives exact hits immediately, while x² + y² + z² = 7 has none (a classical fact), so a blind search can never say "no". Every hit and count shown is exact. The animation illustrates the result; it is not a proof. Papers: - Hilbert's tenth problem over the rational numbers: https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf - A pointwise 2-converse for elliptic curves with rational two-torsion: https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026/paper.pdf Original visuals (three.js) and synthesized sound: every tick is a batch of denominators tested, every bell a fraction landing exactly on the surface. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/004 #HilbertsTenthProblem #Undecidability #NumberTheory #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/9LbtHS_YFcU222026-10-12T20:30:00+00:00yesno https://gradientdamage.com/openai-math/irrationality-of-catalans-constant/2026-10-09https://gradientdamage.com/thumbs/PXNXSlzCLPo-og.jpgCatalan's Constant: Reported Irrationality Proof | OpenAI, ChatGPT's CreatorCatalan'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 decimal expansion or a converging finite sum alone does not prove irrationality. Paper: Catalan's constant is irrational https://github.com/openai/math/blob/main/preprints/Catalans-constant-is-irrational-September-24-2026/paper.pdf About the animation: Gold and cyan ribbons distinguish the alternating positive and negative terms of the defining series. Rounded surfaces, luminous edges and moving reflection highlights make the sculptural metaphor readable in depth. Two bright tips draw the ribbons and meet at an artistic target on the soundtrack accent at 7.6333 seconds. Their dimensions and endpoint are not a numerical embedding of the terms, the partial sums or G. The displayed partial sums are genuinely computed from 6 through 96 terms, then reversed to close the loop. This illustrates the defining series, not the determinant argument or a visual proof of irrationality. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/005.md The linked scope documents irrationality of Catalan's constant. The formalization was not executed or audited in this production. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Inflection' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/inflection/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: excerpted, crossfaded for looping, and gain adjusted. #CatalanConstant #NumberTheory #OpenAIhttps://www.youtube-nocookie.com/embed/PXNXSlzCLPo122026-10-09T16:00:35+00:00yesno https://gradientdamage.com/openai-math/goldfelds-conjecture-densities-and-mean-analytic-rank/2026-10-13https://gradientdamage.com/thumbs/AFGKRcCdzSU-og.jpgGoldfeld's Conjecture: Half Rank 0, Half Rank 1 | OpenAI MathGoldfeld'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 (twist parameters ordered by size), together with the mean analytic rank result; these are OpenAI's preprints, not yet peer reviewed. The picture is a real computation for the curve 11a1 (y² + y = x³ − x² − 10x − 20): each line is the completed L-function of one of its first 120 twists on the real segment from s = 1/2 to 3/2, and the counters are the true rank proportions of its first 1000 twists (44% / 50% / 6% with rank 2 or more, which the theorem says becomes negligible only slowly). Papers (OpenAI, catalogue result 006): - Goldfeld's analytic density conjecture and the 2-converse for elliptic curves: https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf - The mean analytic rank of quadratic twists of elliptic curves: https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/006 #GoldfeldConjecture #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/AFGKRcCdzSU212026-10-13T06:00:00+00:00yesno https://gradientdamage.com/openai-math/ordinary-two-point-correlations-and-the-corrected-elliott-conjecture/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/deligne-drinfeld-conjecture/2026-10-13https://gradientdamage.com/thumbs/LuHHpbmcXLg-og.jpgDeligne–Drinfeld Conjecture: One Generator per Odd Weight | OpenAI MathThe Deligne–Drinfeld conjecture predicts the exact shape of the Grothendieck–Teichmüller Lie algebra grt, a symmetry algebra tied to braids, multiple zeta values and the absolute Galois group of the rationals: it should be freely generated by one element in each odd weight 3, 5, 7, 9, and so on. Brown proved in 2012 that such a free Lie algebra sits inside grt; OpenAI claims a proof that it is all of grt, with the corresponding isomorphism after weight completion (an OpenAI preprint, not yet peer reviewed). The picture shows the real basis of that free Lie algebra up to weight 20: each rod is one independent iterated bracket (a Lyndon word), cut into pieces for the generators it uses, sitting on the step of the stair equal to its weight; the counts 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 11, 13 are the dimensions in weights 3 to 20. Paper (OpenAI, catalogue result 008): - The Deligne-Drinfeld conjecture: https://github.com/openai/math/blob/main/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/008 #DeligneDrinfeldConjecture #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/LuHHpbmcXLg212026-10-13T10:00:00+00:00yesno https://gradientdamage.com/openai-math/function-field-reconstruction-from-milnor-k-theory-and-galois-data/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/unrestricted-pro-modularity-at-the-prime-two/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/prime-factor-statistics-of-p-1/2026-10-12https://gradientdamage.com/thumbs/aGmPuX3YCs8-og.jpgPrime Factors of p−1: OpenAI Claims the Poisson–Dirichlet Law | OpenAI MathTake 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; OpenAI claims a proof, plus Erdős's conjecture that infinitely many n have more than n^(1−ε) totient preimages and infinitely many primes with μ(p−1) = 1. These are OpenAI's preprints, not yet peer reviewed. Every column and every bar in the animation is computed from the real primes up to 10 million. Papers (OpenAI, catalogue result 011): - The Poisson-Dirichlet law for prime predecessors: https://github.com/openai/math/blob/main/preprints/The-Poisson-Dirichlet-Law-for-Prime-Predecessors-September-24-2026/paper.pdf - Weighted dilation graphs, smooth shifted primes and totient fibers: https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf - Prime Predecessors with an Even Number of Prime Factors: https://github.com/openai/math/blob/main/preprints/Prime-Predecessors-with-an-Even-Number-of-Prime-Factors-September-17-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/011 #PrimeFactorsP1 #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/aGmPuX3YCs8222026-10-12T23:50:00+00:00yesno https://gradientdamage.com/openai-math/independent-largest-prime-factors-of-consecutive-integers/2026-09-24 https://gradientdamage.com/openai-math/ostmanns-inverse-goldbach-conjecture/2026-10-13https://gradientdamage.com/thumbs/5G4WZEBRGJI-og.jpgOstmann's Inverse Goldbach Conjecture: The Primes Aren't A + B | OpenAI MathOstmann's inverse Goldbach conjecture asks whether the primes can be built as a sumset A + B, every sum of one number from a set A and one from a set B, with both sets having at least two elements. OpenAI claims a proof that this is impossible, even if you are allowed to add or remove finitely many numbers from the primes first (an OpenAI preprint, not yet peer reviewed). The picture is a real computation: the numbers up to 600 wound on a helix with 30 per turn, and the largest possible B for A = {0, 2}, {0, 6} and {0, 30}; the primes that A + B misses (57, 24 and 14 of the 109) light up cyan, and the bar chart shows that no A = {0, a} with a up to 60 covers them all. Paper (OpenAI, catalogue result 013): - The additive indecomposability of the primes: https://github.com/openai/math/blob/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/013 #OstmannInverseGoldbach #NumberTheory #OpenAI #mathshttps://www.youtube-nocookie.com/embed/5G4WZEBRGJI212026-10-13T14:00:00+00:00yesno https://gradientdamage.com/openai-math/restricted-geometric-langlands-global-arthur-enhancements-and-generic-ramanujan/2026-10-13https://gradientdamage.com/thumbs/1lz16-M84MM-og.jpgRestricted Geometric Langlands: OpenAI Claims Characteristic p | OpenAI MathThe 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 projective curves (under stated conditions on p), plus the generalized Ramanujan conjecture for globally generic cuspidal representations of split adjoint exceptional groups over function fields, and global Arthur enhancements assuming a finite-level Ramanujan–Arthur decomposition. These are OpenAI's claims and have not yet been peer reviewed. What you see is a real computed example of what "Ramanujan" (tempered) means, for the simplest such group, PGL2, where it was already known (Drinfeld). The object is the automorphic form of the elliptic curve y² = x³ + tx + 1 over the function field F13(t). Each ring is one of its 91 places of degree 1 or 2. The two gold points on a ring are that place's Satake parameters, computed from exact point counts, and every one of them sits on the unit circle. The pink points belong to the constant function, which is automorphic but not cuspidal. Its parameters q^±½ sit off the circle, and its Arthur SL2 accounts for exactly that shift. OpenAI's Ramanujan claim is for exceptional groups such as G2 and E8, which the picture does not show. The animation illustrates the result; it is not a proof. Papers: - The Restricted Geometric Langlands Equivalence in Positive Characteristic: https://github.com/openai/math/blob/main/preprints/The-Restricted-Geometric-Langlands-Equivalence-in-Positive-Characteristic-September-24-2026/paper.pdf - Temperedness at ramified places for globally generic exceptional groups: https://github.com/openai/math/blob/main/preprints/Temperedness-at-ramified-places-for-globally-generic-exceptional-groups-October-5-2026/ramified-ramanujan.pdf - Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level…https://www.youtube-nocookie.com/embed/1lz16-M84MM202026-10-13T00:10:00+00:00yesno https://gradientdamage.com/openai-math/torus-packet-equidistribution-in-prime-quartic-and-sextic-degrees/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/zilber-pink-in-abelian-varieties-and-the-siegel-threefold/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/irrationality-exponent-of-is-2/2026-09-24 https://gradientdamage.com/openai-math/margulis-platonov-conjecture-over-global-fields/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/local-p-adic-section-conjecture-and-global-consequences/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/squarefree-quartics-and-power-free-polynomial-values/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/quadratic-bound-for-jacobsthals-function/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/weak-inhomogeneous-duffin-schaeffer-conjecture/2026-10-13https://gradientdamage.com/thumbs/cRYVgIhkTVY-og.jpgZilber–Pink Conjecture, Duffin–Schaeffer Conjecture and 8 More in Number Theory | OpenAI Math10 results from OpenAI's public mathematics catalogue, one per chapter. Each chapter names the problem, shows the real objects, and states what OpenAI claims; none of it is peer reviewed yet. Chapters: 0:00 Chowla Conjecture, Two-Point Case 0:21 Bogomolov–Pop Conjecture 0:42 Fontaine–Mazur Conjecture at the Prime 2 1:03 Duke's Theorem in Higher Degree 1:24 Zilber–Pink Conjecture 1:45 Margulis–Platonov Conjecture 2:07 Grothendieck's Section Conjecture 2:27 Squarefree Values of Quartics 2:48 Jacobsthal's Function 3:08 Duffin–Schaeffer Conjecture Chowla Conjecture, Two-Point Case: Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Explained: https://gradientdamage.com/p/007 Paper: https://github.com/openai/math/blob/main/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026/final.pdf Bogomolov–Pop Conjecture: The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor… Explained: https://gradientdamage.com/p/009 Paper: https://github.com/openai/math/blob/main/preprints/Reconstruction-of-Function-Fields-from-Mod-ell-Milnor-K-Theory-October-5-2026/mod-ell-bogomolov-pop.pdf Fontaine–Mazur Conjecture at the Prime 2: Modularity says the point counts of a curve modulo every prime are secretly the coefficients of a modular form; for elliptic curves this was proved between 1995 and 2001. Explained: https://gradientdamage.com/p/010 Paper: https://github.com/openai/math/blob/main/preprints/Unrestricted-pro-modularity-at-the-prime-two-October-6-2026/two-adic-promodularity.pdf Duke's Theorem in Higher Degree: Duke's theorem (1988) says the closed orbits coming from the ideal classes of a real quadratic field spread out evenly over the space of lattices as the discriminant grows, with no mass escaping into the cusp. Explained…https://www.youtube-nocookie.com/embed/cRYVgIhkTVY2082026-10-13T18:00:00+00:00yesno https://gradientdamage.com/openai-math/pattersons-first-moment-for-cubic-gauss-sums/2026-09-25 https://gradientdamage.com/openai-math/asymptotic-formula-for-the-number-of-totients/2026-09-25 https://gradientdamage.com/openai-math/short-egyptian-fractions/2026-09-25 https://gradientdamage.com/openai-math/positive-lower-density-of-large-prime-gaps/2026-09-25 https://gradientdamage.com/openai-math/potential-integral-density-on-curve-character-varieties/2026-09-25 https://gradientdamage.com/openai-math/uniformly-bounded-components-of-gaussian-prime-graphs/2026-09-26 https://gradientdamage.com/openai-math/primitive-roots-for-every-admissible-integer-base/2026-10-04 https://gradientdamage.com/openai-math/modularity-of-elliptic-curves-over-imaginary-quadratic-fields/2026-10-04 https://gradientdamage.com/openai-math/uchidas-conjecture-for-open-homomorphisms-of-galois-groups/2026-10-05 https://gradientdamage.com/openai-math/rational-hodge-conjecture-for-cm-abelian-varieties/2026-10-06 https://gradientdamage.com/openai-math/iitaka-subadditivity-variation-and-logarithmic-additivity/2026-09-27 https://gradientdamage.com/openai-math/log-abundance-for-compact-kahler-spaces-under-logarithmic-iitaka-subadditivity/2026-10-06 https://gradientdamage.com/openai-math/log-canonical-threefold-abundance-in-numerical-dimension-one/2026-10-05 https://gradientdamage.com/openai-math/numerical-semiampleness-and-generalized-minimal-models/2026-10-06 https://gradientdamage.com/openai-math/ordinary-double-point-volume-gap/2026-09-24 https://gradientdamage.com/openai-math/fujitas-freeness-conjecture/2026-09-23 https://gradientdamage.com/openai-math/nagatas-conjecture-and-maximal-seshadri-constants/2026-10-05 https://gradientdamage.com/openai-math/blochs-conjecture-for-complex-surfaces/2026-09-24 https://gradientdamage.com/openai-math/hyperkahler-syz-and-projective-space-bases/2026-09-23 https://gradientdamage.com/openai-math/oka-classification-for-minimal-compact-complex-surfaces-kodaira-dimension-zero-and-class-vii/2026-09-23 https://gradientdamage.com/openai-math/p-w-for-fixed-determinant-sln-moduli-spaces/2026-09-24 https://gradientdamage.com/openai-math/equivariant-cohomological-hikita-conjecture/2026-09-24 https://gradientdamage.com/openai-math/shafarevich-counterexamples-in-dimension-two-and-with-large-fundamental-group/2026-10-05 https://gradientdamage.com/openai-math/zariski-cancellation-and-affine-fibrations-over-the-complex-numbers/2026-09-23 https://gradientdamage.com/openai-math/characteristic-zero-counterexample-to-lipman-zariski/2026-09-23 https://gradientdamage.com/openai-math/stable-coordinate-counterexample-in-four-variables/2026-10-05 https://gradientdamage.com/openai-math/counterexample-to-griffiths-positivity-conjecture/2026-09-24 https://gradientdamage.com/openai-math/kobayashis-canonical-ampleness-conjecture/2026-09-23 https://gradientdamage.com/openai-math/tangent-splittings-and-product-decompositions/2026-09-23 https://gradientdamage.com/openai-math/counterexample-to-pixton-completeness-in-chow/2026-09-24 https://gradientdamage.com/openai-math/irrational-cubic-fourfolds-with-hodge-theoretic-and-categorical-k3-associations/2026-09-24 https://gradientdamage.com/openai-math/gepner-symmetry-and-large-volume-stability-on-threefolds/2026-09-24 https://gradientdamage.com/openai-math/termination-of-projective-and-kahler-fourfold-minimal-model-programs/2026-10-07 https://gradientdamage.com/openai-math/fundamental-groups-of-special-complex-varieties-and-root-orbifolds/2026-10-05 https://gradientdamage.com/openai-math/semialgebraic-universal-covers-and-bounded-domains/2026-09-24 https://gradientdamage.com/openai-math/counterexamples-to-zariskis-multiplicity-conjecture/2026-09-27 https://gradientdamage.com/openai-math/global-spherical-shell-conjecture/2026-09-24 https://gradientdamage.com/openai-math/projective-contact-classification-and-the-lebrun-salamon-conjecture/2026-09-23 https://gradientdamage.com/openai-math/generalized-mukai-conjecture/2026-09-24 https://gradientdamage.com/openai-math/topological-triviality-of-constant-surface-singularities/2026-09-24 https://gradientdamage.com/openai-math/virasoro-constraints-for-complete-intersections-and-projective-bundle-towers/2026-10-05 https://gradientdamage.com/openai-math/bounded-klt-complements-for-fano-contractions/2026-09-25 https://gradientdamage.com/openai-math/campana-peternell-conjecture-in-dimension-six/2026-09-25 https://gradientdamage.com/openai-math/anticanonical-nonvanishing-in-every-dimension/2026-09-26 https://gradientdamage.com/openai-math/global-quantum-geometric-langlands-at-irrational-level/2026-10-04 https://gradientdamage.com/openai-math/koebes-circle-domain-conjecture/2026-09-23 https://gradientdamage.com/openai-math/brennans-conjecture-and-the-integral-means-spectrum/2026-09-24 https://gradientdamage.com/openai-math/falconer-distance-conjecture/2026-09-23 https://gradientdamage.com/openai-math/kakeya-in-three-and-four-dimensions/2026-10-12https://gradientdamage.com/thumbs/c6nBODRzsqg-og.jpgKakeya Conjecture in 3D and 4D: OpenAI Claims a Proof | OpenAI MathA 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 Hausdorff dimension four, and of the stronger Kakeya maximal conjecture in three dimensions; these are OpenAI's preprints, not yet peer reviewed. Papers (OpenAI, catalogue result 074): - The Kakeya maximal conjecture in three dimensions: https://github.com/openai/math/blob/main/preprints/The-Kakeya-maximal-conjecture-in-three-dimensions-September-23-2026/paper.pdf - Every four-dimensional Kakeya set has full Hausdorff dimension: https://github.com/openai/math/blob/main/preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/074 #KakeyaConjecture3D4D #RealComplexAnalysis #OpenAI #mathshttps://www.youtube-nocookie.com/embed/c6nBODRzsqg222026-10-12T20:50:00+00:00yesno https://gradientdamage.com/openai-math/l-log-l-fourier-convergence-conjecture/2026-10-09https://gradientdamage.com/thumbs/OyH0voIAmQU-og.jpgMath Makes Sounds... Then Gets Weird | OpenAI MathMath 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 almost-everywhere convergence of ordinary symmetric Fourier partial sums for L log L functions. The finite square-wave example shown here is classical and illustrates the subject; it does not demonstrate the new proof. Source manuscripts: Almost-everywhere Fourier convergence in L log L https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf Formalization scope: https://github.com/openai/math/blob/main/lean/docs/075.md Classical Fourier partial sums with 1, 2, 4, 8, 16 and 32 odd harmonics. The summed ordinate is plotted around a circle to show the periodic domain; selected component rings have editorial positions and scales. The audio uses the same harmonic weights. Camera speed and visible phase are slowed for legibility. The finite example illustrates the topic; it is not a proof of the reported general L log L convergence theorem. Sound and visuals: original code-generated synthesis and Three.js animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. Source-reported research result, not independently proof-audited in this production. The public ChatGPT product is not claimed to have proved it. Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md #MathMusic #OpenAI #Fourierhttps://www.youtube-nocookie.com/embed/OyH0voIAmQU122026-10-09T22:00:17+00:00yesno https://gradientdamage.com/openai-math/real-ultraflat-littlewood-polynomials-and-unbounded-binary-merit-factors/2026-10-05 https://gradientdamage.com/openai-math/fourier-restriction-for-positively-curved-surfaces/2026-09-24 https://gradientdamage.com/openai-math/three-dimensional-bochner-riesz-conjecture/2026-09-24 https://gradientdamage.com/openai-math/local-smoothing-in-three-dimensions/2026-09-24 https://gradientdamage.com/openai-math/exact-sobolev-endpoint-for-schrodinger-convergence/2026-09-24 https://gradientdamage.com/openai-math/riesz-transforms-and-rectifiability-in-higher-codimension/2026-09-24 https://gradientdamage.com/openai-math/annular-variation-and-dyadic-absolute-bounds-for-the-triangular-hilbert-transform/2026-10-05 https://gradientdamage.com/openai-math/hilbert-transforms-along-lipschitz-directions/2026-09-25 https://gradientdamage.com/openai-math/geometric-case-of-the-erdos-similarity-conjecture/2026-10-12https://gradientdamage.com/thumbs/fzzOF9l0WyQ-og.jpgErdős Similarity Conjecture: OpenAI Claims the Geometric Case | OpenAI MathErdős 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 [0, 1] as you like that contains no affine copy of {qⁿ}, for either sign of dilation (the set may depend on q). These are OpenAI's preprints, not yet peer reviewed. Papers (OpenAI, catalogue result 084): - The geometric case of the Erdős similarity conjecture: https://github.com/openai/math/blob/main/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf - The dyadic case of the Erdős similarity conjecture: https://github.com/openai/math/blob/main/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/084 #ErdosSimilarityConjecture #RealComplexAnalysis #OpenAI #mathshttps://www.youtube-nocookie.com/embed/fzzOF9l0WyQ202026-10-12T21:50:00+00:00yesno https://gradientdamage.com/openai-math/endpoint-sobolev-regularity-of-centered-disk-averages/2026-09-26 https://gradientdamage.com/openai-math/l3-bound-for-the-trilinear-hilbert-transform/2026-10-05 https://gradientdamage.com/openai-math/mahler-conjectures-functional-inequalities-and-polar-product-symplectic-width/2026-10-09https://gradientdamage.com/thumbs/wQGjvjo64AA-og.jpgChatGPT's Creators Say They Proved This | Mahler's ConjectureThe 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; the three-dimensional theorem was already known. It is not a visualization of the proof or an independent verification. The gold and blue solids use the same spatial units. A cube and its octahedral polar exchange roles through a sphere pair. Their volume product ranges from 32/3 to 16*pi^2/9. Manuscript: https://github.com/openai/math/blob/main/preprints/The-symmetric-Mahler-conjecture-and-its-equality-cases-September-22-2026/paper.pdf Release: https://openai.com/index/sharing-ai-progress-in-mathematics/ Original Blender animation and original music. #Mathematics #Geometry #OpenAIhttps://www.youtube-nocookie.com/embed/wQGjvjo64AA122026-10-09T00:55:39+00:00yesno https://gradientdamage.com/openai-math/sharp-projection-body-inequalities-and-a-counterexample-to-simplex-maximization/2026-09-24 https://gradientdamage.com/openai-math/bounded-distortion-l1-embeddings-of-planar-and-bounded-treewidth-graphs/2026-09-23 https://gradientdamage.com/openai-math/triangular-lattice-optimality-long-range-riesz-and-coulomb-energies-and-spherical-logarithmic-energy/2026-09-26 https://gradientdamage.com/openai-math/logarithmic-and-lp-brunn-minkowski-inequalities-and-the-b-conjecture/2026-09-23 https://gradientdamage.com/openai-math/optimal-order-of-convex-body-covering-density/2026-09-23 https://gradientdamage.com/openai-math/dimension-free-logarithmic-sobolev-inequality-for-subgaussian-log-concave-measures/2026-09-23 https://gradientdamage.com/openai-math/subpolynomial-dimension-reduction-in-lp/2026-09-23 https://gradientdamage.com/openai-math/hyperbolicity-cones-without-semidefinite-lifts/2026-10-05 https://gradientdamage.com/openai-math/gaussian-propeller-conjecture-in-every-dimension/2026-09-24 https://gradientdamage.com/openai-math/euclidean-steinitz-bergstrom-bound/2026-09-24 https://gradientdamage.com/openai-math/compact-counterexamples-to-bi-lipschitz-dimension-reduction/2026-09-25 https://gradientdamage.com/openai-math/sharp-exponential-scale-of-edit-distance-distortion/2026-09-27 https://gradientdamage.com/openai-math/cylinder-coverings-below-the-half-area-bound/2026-09-27 https://gradientdamage.com/openai-math/sharp-simplex-conjecture-for-isotropic-constants/2026-10-05 https://gradientdamage.com/openai-math/unique-games-conjecture-and-optimal-approximation-thresholds/2026-09-23 https://gradientdamage.com/openai-math/exact-derandomization-of-logarithmic-space-l-rl-bpl/2026-09-23 https://gradientdamage.com/openai-math/quasipolynomial-algorithms-for-mean-payoff-stochastic-and-parity-games/2026-10-05 https://gradientdamage.com/openai-math/perfect-completeness-for-2-to-1-games/2026-09-23 https://gradientdamage.com/openai-math/hardness-of-coloring-three-colorable-graphs/2026-09-24 https://gradientdamage.com/openai-math/matrix-multiplication-with-exponent-at-most-9-4/2026-10-02 https://gradientdamage.com/openai-math/cubic-permanent-determinant-lower-bound/2026-09-24 https://gradientdamage.com/openai-math/integer-multiplication-below-n-log-n/2026-09-23 https://gradientdamage.com/openai-math/optimal-order-randomized-k-server-on-arbitrary-metrics/2026-09-24 https://gradientdamage.com/openai-math/one-sample-matroid-prophet-inequalities-against-an-almighty-adversary/2026-09-23 https://gradientdamage.com/openai-math/beyond-the-square-root-exponent-for-depth-three-circuits/2026-09-23 https://gradientdamage.com/openai-math/approximate-counting-and-entropy-of-perfect-matchings/2026-09-23 https://gradientdamage.com/openai-math/approximate-counting-of-common-integer-polymatroid-bases/2026-10-05 https://gradientdamage.com/openai-math/sampling-and-counting-contingency-tables-with-arbitrary-margins/2026-09-24 https://gradientdamage.com/openai-math/uniform-black-box-noncommutative-identity-testing-across-characteristics/2026-10-04 https://gradientdamage.com/openai-math/uniform-sparsest-cut-hardness-and-semidefinite-gaps/2026-09-24 https://gradientdamage.com/openai-math/bin-packing-and-unbounded-configuration-lp-gaps/2026-09-24 https://gradientdamage.com/openai-math/courtade-kumar-and-hellinger-conjectures/2026-09-24 https://gradientdamage.com/openai-math/almost-linear-time-exact-matching-and-prescribed-degree-factors-in-general-graphs/2026-09-24 https://gradientdamage.com/openai-math/almost-linear-approximation-of-edit-distance/2026-09-24 https://gradientdamage.com/openai-math/quantitative-trace-reconstruction-bounds-with-a-uniform-decoder/2026-10-05 https://gradientdamage.com/openai-math/polynomial-time-scheduling-on-three-identical-machines/2026-09-24 https://gradientdamage.com/openai-math/metric-k-median-approximation-threshold-and-recovery/2026-09-24 https://gradientdamage.com/openai-math/exponential-semidefinite-complexity-of-perfect-matching/2026-10-05 https://gradientdamage.com/openai-math/average-sensitivity-of-polynomial-threshold-functions/2026-10-12https://gradientdamage.com/thumbs/4vB_rSP3FgQ-og.jpgGotsman–Linial Conjecture: OpenAI Claims Sensitivity ≤ 8d√n | OpenAI MathCut 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 claims a proof that the average sensitivity is at most 8d√n for every n ≥ 1 and 1 ≤ d ≤ n, the asymptotic form of the Gotsman–Linial conjecture; this is OpenAI's preprint, not yet peer reviewed. The cube shown is the real 8-cube with real polynomials, and every flip count shown is computed exactly. Paper (OpenAI, catalogue result 127): - Average sensitivity of polynomial threshold functions: https://github.com/openai/math/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/127 #GotsmanLinialConjecture #TheoreticalComputerScience #OpenAI #mathshttps://www.youtube-nocookie.com/embed/4vB_rSP3FgQ202026-10-12T22:10:00+00:00yesno https://gradientdamage.com/openai-math/factor-two-approximation-for-shortest-common-superstring/2026-09-24 https://gradientdamage.com/openai-math/exponential-state-costs-for-two-way-automata/2026-09-25 https://gradientdamage.com/openai-math/exact-fourier-transforms-below-n-log-n/2026-09-25 https://gradientdamage.com/openai-math/rapid-mixing-of-graph-switches-for-every-degree-sequence/2026-09-25 https://gradientdamage.com/openai-math/superquadratic-separation-of-sensitivity-and-block-sensitivity/2026-09-25 https://gradientdamage.com/openai-math/computational-complexity-of-weisfeiler-leman-refinement/2026-09-25 https://gradientdamage.com/openai-math/generalized-star-height-at-most-three/2026-09-25 https://gradientdamage.com/openai-math/homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication/2026-09-25 https://gradientdamage.com/openai-math/quasilinear-pcp-theorem-for-ppad/2026-09-25 https://gradientdamage.com/openai-math/one-tape-time-simulation-in-two-fifths-power-space/2026-09-25 https://gradientdamage.com/openai-math/subset-sum-in-o-2-0-49n-time/2026-10-04 https://gradientdamage.com/openai-math/subpolynomial-query-complexity-for-log-concave-sampling/2026-09-26 https://gradientdamage.com/openai-math/memory-sample-lower-bounds-for-noiseless-gaussian-regression/2026-09-27 https://gradientdamage.com/openai-math/existential-universal-real-sentences-in-the-counting-hierarchy/2026-10-04 https://gradientdamage.com/openai-math/deterministic-polynomial-factorization-over-prime-fields/2026-10-04 https://gradientdamage.com/openai-math/hilberts-sixteenth-problem-uniform-bounds-for-limit-cycles/2026-10-11https://gradientdamage.com/thumbs/NGIyDMEDJEw-og.jpgThis equation can only end two ways | OpenAI MathOne 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 anywhere outside it and the motion settles onto the outer loop. Those two closed curves are its limit cycles. OpenAI, ChatGPT's creator, reports a proof that every classical Liénard system with F of degree at most five has at most two limit cycles. Examples with exactly two, like this one, are classical (Odani); the new claim is the upper bound. This is a source-reported result, not independently reviewed or proof-audited in this production, and the public ChatGPT product is not claimed to have proved it. The animation is a numerical illustration (RK4 integration), not the proof. Source manuscript: Two limit cycles for quintic Liénard systems https://github.com/openai/math/blob/main/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md Sound and visuals: original code-generated synthesis and animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. #Shorts #Math #OpenAI Gradient Damage: mathematics you can see, equations you can hear. Visual maths, 3D animation and sound experiments. https://www.youtube.com/@gradientdamagehttps://www.youtube-nocookie.com/embed/NGIyDMEDJEw492026-10-11T01:37:00+00:00yesno https://gradientdamage.com/openai-math/banachs-simple-lebesgue-spectrum-problem/2026-09-23 https://gradientdamage.com/openai-math/rokhlins-multiple-mixing-problem/2026-09-23 https://gradientdamage.com/openai-math/positive-metric-entropy-for-the-standard-map/2026-09-23 https://gradientdamage.com/openai-math/near-boundary-birkhoff-conjecture/2026-09-24 https://gradientdamage.com/openai-math/entropy-rate-dimension-formula-for-self-similar-measures/2026-09-24 https://gradientdamage.com/openai-math/classwise-permanence-for-weakly-reversible-mass-action-systems/2026-10-05 https://gradientdamage.com/openai-math/weak-mixing-of-triangular-billiards-with-an-irrational-angle/2026-10-12https://gradientdamage.com/thumbs/FHWC7PiYtG8-og.jpgTriangle Billiards: Will the Ball Hit Every Spot? | OpenAI MathShoot 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. OpenAI says its AI has now proved that the billiard flow in every triangle with at least one irrational angle is ergodic, and even weakly mixing. This is OpenAI's claim, not yet peer reviewed. The animation illustrates the idea; it is not the proof. The 60-60-60 table and the five other triangles are exact billiard paths. Papers (OpenAI's manuscripts, result 150): Ergodicity: https://github.com/openai/math/blob/main/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf Weak mixing: https://github.com/openai/math/blob/main/preprints/Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026/weak-mixing-triangular-billiards.pdf Original synthesized audio: every note is a bounce. Voice (B): ElevenLabs. #maths #billiards #ergodictheory #openai #shortshttps://www.youtube-nocookie.com/embed/FHWC7PiYtG8402026-10-12T11:00:03+00:00yesno https://gradientdamage.com/openai-math/c1-counterexample-to-the-entropy-conjecture/2026-09-25 https://gradientdamage.com/openai-math/zero-entropy-does-not-guarantee-a-smooth-positive-volume-model/2026-09-25 https://gradientdamage.com/openai-math/arithmetic-classification-and-non-pisot-singularity-for-bernoulli-convolutions/2026-10-03 https://gradientdamage.com/openai-math/pointwise-multiple-ergodic-averages-for-mixing-transformations/2026-10-04 https://gradientdamage.com/openai-math/counterexample-to-periodic-tiling-in-dimension-three/2026-09-23 https://gradientdamage.com/openai-math/borsuks-conjecture-fails-in-dimension-nine/2026-09-23 https://gradientdamage.com/openai-math/graph-coloring-clique-minors-and-colin-de-verdiere-invariants/2026-09-23 https://gradientdamage.com/openai-math/euclidean-plane-cannot-be-colored-with-five-colors/2026-10-11https://gradientdamage.com/thumbs/FFsKkuXhQKs-og.jpg5 Maths Problems OpenAI Says Its AI Just Solved | OpenAI MathOpenAI 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 work. OpenAI claims 5 never can, so the answer is 6 or 7. 3. Self-avoiding walks: an 8000-step honeycomb walk that never touches itself. OpenAI claims its size grows like steps to the 3/4. 4. Bubbles in a 3-torus: the least-area shape for each volume in a cube with opposite faces glued is a ball, then a tube, then a slab. OpenAI claims a proof of the full answer. 5. SLE curves: one random path, three roughnesses. OpenAI claims the exact ruler that measures them. The animations are illustrations, not the proofs. The claims and papers are in OpenAI's public maths catalogue: https://github.com/openai/math (families 165, 158, 237, 354, 230). Results are as reported by OpenAI and not independently verified here. Visuals and music are original, generated from the same event clock. #maths #math #ai #openai #shortshttps://www.youtube-nocookie.com/embed/FFsKkuXhQKs452026-10-11T15:45:08+00:00yesno https://gradientdamage.com/openai-math/erdoss-reciprocal-sum-conjecture-and-quasipolynomial-szemeredi-bounds/2026-10-12https://gradientdamage.com/thumbs/Fx1zddsBSFg-og.jpgErdős Conjecture on Arithmetic Progressions: OpenAI Claims a Proof | OpenAI MathErdős 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} with no k-term progression has at most C·N·exp(−c(log N)^ε) elements. This is OpenAI's preprint, not yet peer reviewed. In the animation the numbers 1 to 1050 are steps of a spiral staircase, 30 per turn, so progressions with a step that is a multiple of 30 stand as straight vertical lines. Paper (OpenAI, catalogue result 159): - Quasipolynomial Bounds for Arithmetic Progressions: https://github.com/openai/math/blob/main/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/159 #ErdosConjectureArithmeticProgressions #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/Fx1zddsBSFg212026-10-12T22:30:00+00:00yesno https://gradientdamage.com/openai-math/superexponential-van-der-waerden-numbers/2026-09-23 https://gradientdamage.com/openai-math/counterexamples-to-sidorenkos-conjecture-and-the-forcing-conjecture/2026-09-23 https://gradientdamage.com/openai-math/counterexamples-to-rysers-covering-conjecture/2026-09-27 https://gradientdamage.com/openai-math/hindmans-finite-sums-and-products-conjecture/2026-09-23 https://gradientdamage.com/openai-math/harary-hill-conjecture/2026-10-12https://gradientdamage.com/thumbs/yrXjLp8nfqo-og.jpgThe Harary–Hill Conjecture: Why 18 Dots Need 1008 Crossings | OpenAI MathJoin 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 than 60 years nobody could prove that nothing does better (it was known only up to n = 12). OpenAI reports a proof (September 2026) that the classical drawings are optimal, so the formula is exact for every n, with a matching result for Zarankiewicz's bipartite problem. The paper has not been independently reviewed yet; treat it as OpenAI's claim. What you're watching: the classical two-page drawing on a sphere, built from K5 (1 crossing) up to K18 (1008 crossings). Every gold dot is a real crossing of that drawing, and every chime you hear is one of those crossings. The animation illustrates the result; it is not the proof. Paper: https://github.com/openai/math/blob/main/preprints/The-crossing-number-of-complete-graphs-September-23-2026/paper.pdf #math #graphtheory #maths #crossingnumber #shortshttps://www.youtube-nocookie.com/embed/yrXjLp8nfqo402026-10-11T20:00:01+00:00yesno https://gradientdamage.com/openai-math/higher-dimensional-erdos-distinct-distances-conjecture/2026-09-23 https://gradientdamage.com/openai-math/planar-distinct-distances-and-unit-distance-bounds/2026-09-23 https://gradientdamage.com/openai-math/combinatorial-invariance-of-kazhdan-lusztig-polynomials/2026-09-24 https://gradientdamage.com/openai-math/shareshian-wachs-elementary-positivity/2026-09-24 https://gradientdamage.com/openai-math/sharp-logarithmic-exponents-for-off-diagonal-ramsey-numbers/2026-10-12https://gradientdamage.com/thumbs/De6BOflEsvM-og.jpgRamsey Numbers r(s,t): OpenAI Claims the Sharp Log Exponent | OpenAI MathThe 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, r(s,t) = t^(s-1)/(log t)^(s-2+o(1)), pinning down the logarithmic exponent and matching the classical upper bound at that scale; these are OpenAI's preprints, not yet peer reviewed. Papers (OpenAI, catalogue result 170): - The sharp logarithmic exponent of r(5,t): https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf - Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers: https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/170 #RamseyNumbersRST #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/De6BOflEsvM212026-10-12T21:10:00+00:00yesno https://gradientdamage.com/openai-math/hypercube-ramsey-conjecture/2026-10-12https://gradientdamage.com/thumbs/0v6aVMcmIcg-og.jpgHypercube Ramsey Conjecture: How Many Dots Force a Cube? | OpenAI MathColour 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 claims a proof that R(Q_n) ≤ C·2^n for an absolute constant C, resolving the hypercube Ramsey conjecture; this is OpenAI's preprint, not yet peer reviewed. Paper (OpenAI, catalogue result 171): - The hypercube Ramsey number has linear order: https://github.com/openai/math/blob/main/preprints/The-hypercube-Ramsey-number-has-linear-order-September-23-2026/paper.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/171 #HypercubeRamseyConjecture #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/0v6aVMcmIcg202026-10-12T22:50:00+00:00yesno https://gradientdamage.com/openai-math/classification-of-finite-euclidean-ramsey-configurations/2026-09-23 https://gradientdamage.com/openai-math/seymours-second-neighborhood-conjecture/2026-09-23 https://gradientdamage.com/openai-math/deterministic-construction-of-strong-thin-spanning-trees/2026-09-23 https://gradientdamage.com/openai-math/talagrands-expectation-thresholds-discrete-convexity-and-graph-decompositions/2026-10-05 https://gradientdamage.com/openai-math/second-kahn-kalai-conjecture-with-an-edge-count-bound/2026-09-24 https://gradientdamage.com/openai-math/bounded-degree-coboundary-expanders/2026-09-24 https://gradientdamage.com/openai-math/deterministic-nonbipartite-ramanujan-graphs-in-every-fixed-degree/2026-09-23 https://gradientdamage.com/openai-math/circulant-hadamard-and-barker-sequence-conjectures/2026-09-23 https://gradientdamage.com/openai-math/barnettes-hamiltonian-cycle-conjecture/2026-10-12https://gradientdamage.com/thumbs/pn4CFp88RlM-og.jpgBarnette's Conjecture: Does This Loop Always Exist? | OpenAI MathCan 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 says its AI has now proved it. The proof is OpenAI's claim and has not yet been peer reviewed. Paper: https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf The animation illustrates the statement; it is not the proof. Every loop shown is a real Hamiltonian cycle, found by computer search and checked: cube (8 corners), hexagonal prism (12), truncated octahedron (24), truncated cuboctahedron (48), truncated icosidodecahedron (120). Pink and cyan mark the two colour classes, so the loop always alternates. Visuals and sound: original, made with three.js and synthesized audio driven by the same events. Voice (B): ElevenLabs. Narration is an AI voice (ElevenLabs). #BarnettesConjecture #graphtheory #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/pn4CFp88RlM402026-10-12T08:00:10+00:00yesno https://gradientdamage.com/openai-math/erdos-gallai-cycle-decomposition-conjecture/2026-10-12https://gradientdamage.com/thumbs/vbINiO4LewQ-og.jpgErdős–Gallai Conjecture: Every Graph Splits Into Few Loops | OpenAI MathThe 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 exists for every finite simple graph; this is OpenAI's preprint, not yet peer reviewed. The animation peels a real random graph with 20 dots and 88 links into 9 cycles and 10 single links. Paper (OpenAI, catalogue result 181): - A linear cycle-and-edge decomposition of every graph: https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf The animation illustrates the result; it is not a proof. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/181 #ErdosGallaiConjecture #Combinatorics #OpenAI #mathshttps://www.youtube-nocookie.com/embed/vbINiO4LewQ212026-10-12T23:10:00+00:00yesno https://gradientdamage.com/openai-math/power-savings-for-intersective-polynomial-differences-and-prime-arguments/2026-10-05 https://gradientdamage.com/openai-math/power-savings-for-planar-halving-lines-and-k-sets/2026-09-25 https://gradientdamage.com/openai-math/correspondence-coloring-with-a-fixed-forbidden-subgraph/2026-10-05 https://gradientdamage.com/openai-math/counterexamples-to-infinite-matroid-intersection-and-packing-covering/2026-09-24 https://gradientdamage.com/openai-math/uniform-influence-and-sharp-thresholds-for-graph-and-hypergraph-properties/2026-10-05 https://gradientdamage.com/openai-math/snaky-in-21-maker-moves/2026-09-25 https://gradientdamage.com/openai-math/sharp-terminal-leave-in-random-triangle-removal/2026-09-25 https://gradientdamage.com/openai-math/cycle-clique-ramsey-numbers/2026-09-25 https://gradientdamage.com/openai-math/polynomial-removal-fails-for-ordered-binary-matrices/2026-09-25 https://gradientdamage.com/openai-math/power-improvement-in-the-heilbronn-triangle-lower-bound/2026-09-25 https://gradientdamage.com/openai-math/boolean-functions-violate-the-square-root-degree-bound-by-arbitrary-factors/2026-09-26 https://gradientdamage.com/openai-math/serres-intersection-multiplicity-conjecture/2026-09-23 https://gradientdamage.com/openai-math/lechs-multiplicity-conjecture/2026-09-23 https://gradientdamage.com/openai-math/counterexample-to-the-small-cohen-macaulay-module-conjecture/2026-09-23 https://gradientdamage.com/openai-math/counterexample-to-kaplanskys-zero-divisor-conjecture/2026-09-23 https://gradientdamage.com/openai-math/torsion-free-group-algebra-that-is-not-directly-finite/2026-10-04 https://gradientdamage.com/openai-math/counterexample-to-finitistic-dimension-finiteness/2026-09-23 https://gradientdamage.com/openai-math/counterexamples-to-auslander-reiten-tachikawa-and-related-homological-conjectures/2026-09-23 https://gradientdamage.com/openai-math/eisenbud-green-harris-and-lex-plus-powers/2026-09-23 https://gradientdamage.com/openai-math/counterexample-to-kuroshs-division-ring-problem/2026-09-23 https://gradientdamage.com/openai-math/blockwise-alperin-weight-conjecture/2026-09-23 https://gradientdamage.com/openai-math/donovans-conjecture-over-fields-and-complete-mixed-characteristic-dvrs/2026-09-25 https://gradientdamage.com/openai-math/tensor-saturation-for-even-spin-groups/2026-09-24 https://gradientdamage.com/openai-math/saxls-conjecture-and-universal-tensor-squares/2026-09-24 https://gradientdamage.com/openai-math/finite-lattice-representation-and-undecidability/2026-09-24 https://gradientdamage.com/openai-math/l1-bass-conjecture-for-all-discrete-groups/2026-10-05 https://gradientdamage.com/openai-math/finite-symmetric-tensor-categories-and-the-verlinde-tower/2026-09-24 https://gradientdamage.com/openai-math/integral-counterexamples-to-gerstens-conjecture/2026-09-26 https://gradientdamage.com/openai-math/foulkes-conjecture-for-sixth-powers-and-quadratic-stabilization/2026-09-25 https://gradientdamage.com/openai-math/geometric-phase-diagram-diffusion-and-spectra-of-random-planar-maps/2026-10-05 https://gradientdamage.com/openai-math/planar-first-passage-geometry-and-the-absence-of-bigeodesics/2026-09-24 https://gradientdamage.com/openai-math/critical-percolation-on-every-quasi-transitive-graph/2026-10-09https://gradientdamage.com/thumbs/2KExA3m9N3M-og.jpgCritical Percolation on a Cubic Lattice | OpenAI, ChatGPT's CreatorWhen 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 expositions by Leder and Kozma-Nitzan. This animation varies connection probability on a finite cubic grid; it does not display an infinite lattice or identify the exact critical probability. About the animation: A seeded sample of independent uniform edge thresholds on a 10 by 10 by 10 cubic grid gives open bonds when the threshold is below p. The animation sweeps p from 0.12 to 0.42 and back, using monotone coupling. Union-find determines the actual connected components. Gold identifies the largest component; it can span this finite box. A finite spanning cluster does not contradict the reported absence of an infinite cluster at the infinite-lattice critical probability. The video shows bond percolation only; the paper also treats site percolation. Paper: https://github.com/openai/math/blob/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026/paper.pdf GitHub materials: https://github.com/openai/math/blob/main/lean/docs/213.md Repository: https://github.com/openai/math Source-reported result. This visual explanation is not an independent proof audit or a verification of the formalization. 'Titan' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/titan/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: Excerpted, crossfaded to loop, volume adjusted. #Percolation #Probability #OpenAIhttps://www.youtube-nocookie.com/embed/2KExA3m9N3M122026-10-09T12:10:25+00:00yesno https://gradientdamage.com/openai-math/benjamini-schramm-nonuniqueness-conjecture/2026-09-24 https://gradientdamage.com/openai-math/canonical-o-3-continuum-limit-and-exact-o-4-mass-asymptotics/2026-10-05 https://gradientdamage.com/openai-math/critical-and-near-critical-xy-scaling-and-bkt-universality/2026-10-05 https://gradientdamage.com/openai-math/low-temperature-sherrington-kirkpatrick-fluctuation-law/2026-09-24 https://gradientdamage.com/openai-math/conformal-universality-for-weakly-interacting-and-random-bond-ising-models/2026-10-05 https://gradientdamage.com/openai-math/goe-bulk-universality-for-regular-graphs-with-weak-anderson-disorder/2026-10-05 https://gradientdamage.com/openai-math/directional-zero-one-laws-beyond-iid-environments-and-iid-ballisticity/2026-10-05 https://gradientdamage.com/openai-math/mezard-parisi-formula-for-diluted-spin-glasses/2026-09-23 https://gradientdamage.com/openai-math/perceptron-free-energies-and-microscopic-jamming-exponents/2026-09-24 https://gradientdamage.com/openai-math/random-cluster-interfaces-critical-disordered-thermal-and-natural-time-scaling/2026-10-05 https://gradientdamage.com/openai-math/critical-and-quenched-near-critical-universality-for-poisson-voronoi-percolation/2026-10-05 https://gradientdamage.com/openai-math/gaussian-free-field-limits-throughout-the-balanced-six-vertex-regime/2026-09-23 https://gradientdamage.com/openai-math/double-dimer-loop-ensemble-converges-to-cle4/2026-10-11https://gradientdamage.com/thumbs/_-E8jXoNkk4-og.jpgTwo random tilings make these loops | OpenAI MathLay 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 to largest. Each domino plucks a note as it lands: higher on screen, higher note. Small loops ring high and big loops ring low. OpenAI, ChatGPT's creator, reports a proof that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE4. This is a source-reported result, not independently reviewed or proof-audited in this production, and the public ChatGPT product is not claimed to have proved it. The animation shows one finite random sample as an illustration, not the proof; the paper works in the half-plane scaling limit. Source manuscript: The Curve Scaling Limit of Half-Plane Double Dimers https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md Sound and visuals: original code-generated synthesis and animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. #Shorts #Math #OpenAI Gradient Damage: mathematics you can see, equations you can hear. Visual maths, 3D animation and sound experiments. https://www.youtube.com/@gradientdamagehttps://www.youtube-nocookie.com/embed/_-E8jXoNkk4372026-10-11T05:37:16+00:00yesno https://gradientdamage.com/openai-math/critical-sk-autocorrelation-processes-and-dynamics-across-the-temperature-transition/2026-10-05 https://gradientdamage.com/openai-math/continuum-phase-transitions-for-radial-pair-potentials/2026-09-24 https://gradientdamage.com/openai-math/exact-three-and-four-state-reconstruction-thresholds-and-four-state-tree-capacity/2026-10-05 https://gradientdamage.com/openai-math/exact-hausdorff-gauges-for-sle/2026-10-11https://gradientdamage.com/thumbs/FFsKkuXhQKs-og.jpg5 Maths Problems OpenAI Says Its AI Just Solved | OpenAI MathOpenAI 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 work. OpenAI claims 5 never can, so the answer is 6 or 7. 3. Self-avoiding walks: an 8000-step honeycomb walk that never touches itself. OpenAI claims its size grows like steps to the 3/4. 4. Bubbles in a 3-torus: the least-area shape for each volume in a cube with opposite faces glued is a ball, then a tube, then a slab. OpenAI claims a proof of the full answer. 5. SLE curves: one random path, three roughnesses. OpenAI claims the exact ruler that measures them. The animations are illustrations, not the proofs. The claims and papers are in OpenAI's public maths catalogue: https://github.com/openai/math (families 165, 158, 237, 354, 230). Results are as reported by OpenAI and not independently verified here. Visuals and music are original, generated from the same event clock. #maths #math #ai #openai #shortshttps://www.youtube-nocookie.com/embed/FFsKkuXhQKs452026-10-11T15:45:08+00:00yesno https://gradientdamage.com/openai-math/free-uniform-spanning-forest-is-a-factor-of-iid/2026-09-25 https://gradientdamage.com/openai-math/gaussian-fields-and-interfaces-for-triangular-lattice-lipschitz-heights/2026-10-06 https://gradientdamage.com/openai-math/joint-critical-ashkin-teller-current-limit/2026-10-06 https://gradientdamage.com/openai-math/all-temperature-pressure-of-orthogonally-invariant-ising-spin-glasses/2026-09-25 https://gradientdamage.com/openai-math/limiting-random-sat-thresholds-sharp-variance-and-computability/2026-10-05 https://gradientdamage.com/openai-math/exact-factor-of-iid-threshold-for-free-ising-spins-on-trees/2026-09-26 https://gradientdamage.com/openai-math/three-quarter-exponent-for-honeycomb-self-avoiding-walk/2026-10-11https://gradientdamage.com/thumbs/FFsKkuXhQKs-og.jpg5 Maths Problems OpenAI Says Its AI Just Solved | OpenAI MathOpenAI 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 work. OpenAI claims 5 never can, so the answer is 6 or 7. 3. Self-avoiding walks: an 8000-step honeycomb walk that never touches itself. OpenAI claims its size grows like steps to the 3/4. 4. Bubbles in a 3-torus: the least-area shape for each volume in a cube with opposite faces glued is a ball, then a tube, then a slab. OpenAI claims a proof of the full answer. 5. SLE curves: one random path, three roughnesses. OpenAI claims the exact ruler that measures them. The animations are illustrations, not the proofs. The claims and papers are in OpenAI's public maths catalogue: https://github.com/openai/math (families 165, 158, 237, 354, 230). Results are as reported by OpenAI and not independently verified here. Visuals and music are original, generated from the same event clock. #maths #math #ai #openai #shortshttps://www.youtube-nocookie.com/embed/FFsKkuXhQKs452026-10-11T15:45:08+00:00yesno https://gradientdamage.com/openai-math/optimal-logarithmic-mixing-of-the-thorp-shuffle/2026-09-26 https://gradientdamage.com/openai-math/sharp-singularity-rates-for-symmetric-random-sign-matrices/2026-10-04 https://gradientdamage.com/openai-math/shelahs-eventual-categoricity-and-the-prescribed-threshold-obstruction/2026-09-24 https://gradientdamage.com/openai-math/rigidity-of-the-turing-degrees/2026-09-24 https://gradientdamage.com/openai-math/single-fold-diophantine-representations-and-undecidability-under-an-at-most-one-solution-promise/2026-09-24 https://gradientdamage.com/openai-math/separating-choiceless-counting-from-polynomial-time-and-witnessed-choice/2026-09-24 https://gradientdamage.com/openai-math/partition-principle-does-not-imply-choice/2026-09-24 https://gradientdamage.com/openai-math/weak-normalization-implies-strong-normalization-in-pure-type-systems/2026-09-25 https://gradientdamage.com/openai-math/cannons-conjecture/2026-09-23 https://gradientdamage.com/openai-math/infinite-finitely-presented-residually-finite-2-group-and-a-finitely-presented-nil-algebra/2026-10-05 https://gradientdamage.com/openai-math/thompsons-group-f-is-nonamenable/2026-09-23 https://gradientdamage.com/openai-math/finitely-generated-eilenberg-ganea-counterexample/2026-09-23 https://gradientdamage.com/openai-math/boone-higman-embeddings-with-higher-finiteness/2026-09-23 https://gradientdamage.com/openai-math/amenability-unitarizability-and-strong-ulam-stability/2026-10-05 https://gradientdamage.com/openai-math/torsion-free-hyperbolic-group-that-is-neither-residually-finite-nor-linear-over-any-field/2026-09-23 https://gradientdamage.com/openai-math/infinite-finitely-presented-simple-amenable-group/2026-09-23 https://gradientdamage.com/openai-math/classifying-spaces-and-geometric-obstructions-for-artin-groups/2026-09-23 https://gradientdamage.com/openai-math/quasi-isometric-recognition-of-virtually-polycyclic-groups/2026-09-24 https://gradientdamage.com/openai-math/nonsingular-systems-of-equations-over-arbitrary-groups/2026-10-05 https://gradientdamage.com/openai-math/hyperbolic-group-without-a-geometric-cat-0-action/2026-09-25 https://gradientdamage.com/openai-math/gerstens-conjecture-and-virtual-compact-specialness-of-one-relator-groups/2026-09-25 https://gradientdamage.com/openai-math/group-without-fixed-price/2026-10-05 https://gradientdamage.com/openai-math/spacetime-penrose-inequalities-enclosing-area-charge-rotation-and-anti-de-sitter-extensions/2026-10-05 https://gradientdamage.com/openai-math/localization-and-delocalization-in-the-anderson-model/2026-10-10https://gradientdamage.com/thumbs/G7LkNMIilNo-og.jpgMath Sounds Broken: Quantum Buffering | OpenAI MathMath 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 for the 2D uniform-disorder model at every positive disorder. This finite 19x19 lattice comparison is illustrative, not an infinite-volume proof or a claim of a 2D phase transition. The sounds are designed sonification, not literal quantum noise. The linked Lean scope covers spectral-set identification, not the full pure-point claim. Source manuscripts: Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf Formalization scope: https://github.com/openai/math/blob/main/lean/docs/261.md Two separately initialized unitary evolutions of a delta wave packet on a finite 19x19 square lattice with open boundaries, H=A+WV. The same fixed uniform random diagonal sample is used at W=0.5 and W=10. Site height and radius map probability amplitude; they are not particle trajectories. The second run resets at 6 seconds. The weak-disorder run is replayed backwards at the end to close the visual loop. Finite short-time spreading does not establish a 2D delocalization transition or any infinite-volume spectral theorem. Stereo width maps effective occupied-site count at each pluck; the stutter is editorial sonification, not literal quantum sound. Sound and visuals: original code-generated synthesis and Three.js animation. No third-party…https://www.youtube-nocookie.com/embed/G7LkNMIilNo122026-10-10T13:31:08+00:00yesno https://gradientdamage.com/openai-math/sharp-finite-matrix-lieb-thirring-inequalities-and-all-equality-cases/2026-10-05 https://gradientdamage.com/openai-math/ionization-and-generalized-ionization-conjectures/2026-09-24 https://gradientdamage.com/openai-math/strong-cosmic-censorship-near-two-ended-kerr-data/2026-09-23 https://gradientdamage.com/openai-math/area-laws-and-tensor-networks-for-two-dimensional-gapped-systems/2026-09-24 https://gradientdamage.com/openai-math/exactly-three-mutually-unbiased-bases-in-dimension-six/2026-09-24 https://gradientdamage.com/openai-math/positive-temperature-bose-einstein-condensation-and-exact-quantum-depletion/2026-10-05 https://gradientdamage.com/openai-math/spin-one-haldane-gap/2026-09-24 https://gradientdamage.com/openai-math/uniform-laughlin-gap-and-stability-under-bounded-scalar-disorder/2026-10-05 https://gradientdamage.com/openai-math/threshold-and-positive-energy-bound-states-of-the-bfss-matrix-model/2026-10-05 https://gradientdamage.com/openai-math/blochs-law-its-lattice-correction-and-the-spherical-magnetization-law/2026-10-09https://gradientdamage.com/thumbs/3wB5bhuWbJQ-og.jpgBloch's Law: 3D Heisenberg Ferromagnets | OpenAI, ChatGPT's CreatorBloch'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 positive half-integer spin. The assumptions include nonnegative symmetric couplings whose support generates the cubic lattice. This is a reported rigorous proof of an existing physical law. Paper: Bloch's Law for Finite-Range Heisenberg Ferromagnets in Three Dimensions https://github.com/openai/math/blob/main/preprints/Blochs-Law-for-Finite-Range-Heisenberg-Ferromagnets-in-Three-Dimensions-October-5-2026/bloch-law-heisenberg.pdf About the animation: A 320-arrow schematic spin lattice cools into alignment and warms into traveling tilt waves. The arrows have unit directions; they are classical visual symbols for a quantum system, not an exact quantum simulation. The displayed law is S-m(T) ~ C*T^(3/2) as T approaches zero. C = zeta(3/2)/(8*pi^(3/2)*sqrt(det(D))), with D = S/2 times sum_z J(z) z z^T. The thermodynamic limit precedes the positive-field limit, then the low-temperature limit. The animation's tilt parameter is not calibrated temperature or measured magnetization. A traveling alignment front and perspective camera are editorial schematic devices. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/271.md The linked family-level formalization scope documents spontaneous magnetization for nearest-neighbor models, not the exact Bloch coefficient animated here. The linked file does not certify the exact Bloch-law coefficient shown here. Source-reported result. Selected theorem statements and scope were read; this is not an independent proof audit. 'Inflection' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/inflection/ Licence…https://www.youtube-nocookie.com/embed/3wB5bhuWbJQ122026-10-09T14:00:03+00:00yesno https://gradientdamage.com/openai-math/entanglement-without-distillable-secret-key/2026-09-27 https://gradientdamage.com/openai-math/entropy-photon-number-inequality/2026-09-24 https://gradientdamage.com/openai-math/parity-is-not-in-qac0/2026-09-24 https://gradientdamage.com/openai-math/qma-hardness-of-continuum-coulomb-energy/2026-09-24 https://gradientdamage.com/openai-math/classical-capacity-of-generalized-amplitude-damping/2026-09-24 https://gradientdamage.com/openai-math/threshold-repetition-for-entangled-games/2026-09-25 https://gradientdamage.com/openai-math/failure-of-kohn-sham-ensemble-representation/2026-09-25 https://gradientdamage.com/openai-math/exact-quantum-factoring-over-a-fixed-finite-gate-set/2026-10-12https://gradientdamage.com/thumbs/ugIzZGQoCuc-og.jpgQuantum Factoring: How Often Does Shor's Algorithm Win? | OpenAI MathMultiplying 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 real encryption keys it is far too long to walk. Shor's algorithm (1994) finds the loop with interfering quantum waves, but every run is a gamble: sometimes it fails and you run it again. OpenAI says its AI has found a way to make the odds exactly 1 in 4, and then two reflections turn that into a sure thing: a circuit that factors every number with probability one, using a fixed set of gates. This is OpenAI's claim and has not yet been peer reviewed. The animation illustrates the idea; it is not the proof. Paper (OpenAI's manuscript): https://github.com/openai/math/blob/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf #maths #quantumcomputing #shorsalgorithm #openai #shorts Gradient Damage: mathematics you can see, equations you can hear. Visual maths, 3D animation and sound experiments. https://www.youtube.com/@gradientdamagehttps://www.youtube-nocookie.com/embed/ugIzZGQoCuc402026-10-12T14:00:00+00:00yesno https://gradientdamage.com/openai-math/unitary-vertex-operator-algebras-and-conformal-nets/2026-09-25 https://gradientdamage.com/openai-math/qaoa-attains-the-sk-optimum-in-the-thermodynamic-first-limit/2026-09-27 https://gradientdamage.com/openai-math/from-scale-symmetry-to-local-conformal-symmetry-in-four-dimensional-qft/2026-09-26 https://gradientdamage.com/openai-math/polynomial-time-unitary-synthesis-from-a-boolean-oracle/2026-10-05 https://gradientdamage.com/openai-math/optimal-quartic-separation-between-randomized-and-quantum-queries/2026-10-05 https://gradientdamage.com/openai-math/counterexamples-to-baum-connes-and-kadison-kaplansky/2026-09-23 https://gradientdamage.com/openai-math/rigidity-and-arithmetic-of-lattice-von-neumann-algebras/2026-09-23 https://gradientdamage.com/openai-math/isomorphism-of-the-free-group-factors/2026-09-23 https://gradientdamage.com/openai-math/kadisons-similarity-conjecture/2026-09-23 https://gradientdamage.com/openai-math/strong-kadison-kastler-stability-and-its-spatial-boundaries/2026-10-05 https://gradientdamage.com/openai-math/relative-bicentralizers-and-modular-spectral-recovery/2026-09-23 https://gradientdamage.com/openai-math/cuntz-comparison-nuclear-dimension-and-equivariant-jiang-su-stability/2026-10-05 https://gradientdamage.com/openai-math/kirchbergs-o2-norm-ultrapower-embedding-problem/2026-09-23 https://gradientdamage.com/openai-math/invariant-projections-hyperinvariant-subspaces-and-transitive-algebras/2026-09-27 https://gradientdamage.com/openai-math/kaplanskys-quasitrace-conjecture-and-failure-of-tensor-product-stable-finiteness/2026-09-23 https://gradientdamage.com/openai-math/kadison-ringrose-cohomology-conjecture/2026-09-23 https://gradientdamage.com/openai-math/generator-problem-for-finite-factors/2026-09-23 https://gradientdamage.com/openai-math/zfc-counterexample-to-naimarks-problem/2026-09-24 https://gradientdamage.com/openai-math/two-notions-of-free-entropy-differ-even-when-both-are-finite/2026-09-25 https://gradientdamage.com/openai-math/kirchberg-rrdam-character-criterion-and-infinite-tensor-power-jiang-su-stability/2026-09-25 https://gradientdamage.com/openai-math/approximation-and-quadratic-strong-operator-paving/2026-09-25 https://gradientdamage.com/openai-math/trace-cones-and-razak-jacelon-stabilization/2026-09-25 https://gradientdamage.com/openai-math/radius-of-comparison-equals-half-the-mean-dimension/2026-09-25 https://gradientdamage.com/openai-math/weak-pure-infiniteness-and-cuntz-algebra-absorption/2026-09-25 https://gradientdamage.com/openai-math/hilbert-smith-conjecture-in-every-dimension/2026-10-12https://gradientdamage.com/thumbs/7uq-r6tvByk-og.jpgHilbert-Smith Conjecture: OpenAI Claims a Proof in Every Dimension | OpenAI MathHilbert'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, previously known only in low dimensions. OpenAI claims a proof in every finite dimension: every locally compact, second-countable Hausdorff group acting faithfully and continuously on a connected finite-dimensional manifold (without boundary) is a Lie group. This is OpenAI's claim and has not yet been peer reviewed. The solenoid shown is the 3-adic solenoid, built from loops wrapping 3 times inside each torus, level after level. Adding 1 is the 3-adic odometer. At the end it collapses to a single circle turning, which is a Lie group action. The animation illustrates the result; it is not a proof. Paper: - The Hilbert–Smith conjecture in every finite dimension: https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf Original visuals (three.js) and synthesized sound: one drone per solenoid level, each an octave above the last; every chime is a level turning over. Gradient Damage: mathematics you can see, equations you can hear. https://www.youtube.com/@gradientdamage Every result explained: https://gradientdamage.com/p/304 #HilbertSmith #Topology #padic #maths #OpenAI #shortshttps://www.youtube-nocookie.com/embed/7uq-r6tvByk212026-10-12T21:30:00+00:00yesno https://gradientdamage.com/openai-math/four-dimensional-disk-embedding-and-walls-conjecture/2026-09-24 https://gradientdamage.com/openai-math/purely-cosmetic-surgery-conjecture/2026-09-23 https://gradientdamage.com/openai-math/failure-of-rational-injectivity-for-maximal-coarse-assembly/2026-10-05 https://gradientdamage.com/openai-math/finite-smith-toda-complexes-at-every-height/2026-09-23 https://gradientdamage.com/openai-math/kervaire-invariant-problem-at-the-prime-three/2026-09-24 https://gradientdamage.com/openai-math/quillens-conjecture-in-rational-homology/2026-09-24 https://gradientdamage.com/openai-math/hovey-strickland-and-chai-conjectures/2026-09-24 https://gradientdamage.com/openai-math/grothendieck-homotopy-hypothesis/2026-09-24 https://gradientdamage.com/openai-math/finite-generation-for-the-k-n-local-sphere/2026-09-24 https://gradientdamage.com/openai-math/cyclic-length-and-chromatic-fixed-point-loss/2026-09-24 https://gradientdamage.com/openai-math/four-dimensional-singer-conjecture/2026-09-25 https://gradientdamage.com/openai-math/curtiss-conjecture/2026-09-25 https://gradientdamage.com/openai-math/thomason-model-structures-in-all-strict-higher-dimensions/2026-09-25 https://gradientdamage.com/openai-math/chromatic-splitting-filtrations-and-counterexamples/2026-09-27 https://gradientdamage.com/openai-math/counterexamples-to-finite-generation-at-chromatic-height-two/2026-09-26 https://gradientdamage.com/openai-math/nonhomeomorphic-closed-aspherical-four-manifolds/2026-10-04 https://gradientdamage.com/openai-math/counterexample-to-walls-finite-d-2-problem/2026-10-06 https://gradientdamage.com/openai-math/tingleys-sphere-isometry-problem/2026-09-23 https://gradientdamage.com/openai-math/independence-of-the-separable-quotient-problem/2026-09-23 https://gradientdamage.com/openai-math/lipschitz-equivalent-banach-spaces-need-not-be-linearly-isomorphic/2026-09-26 https://gradientdamage.com/openai-math/complete-crouzeix-conjecture/2026-09-26 https://gradientdamage.com/openai-math/cotype-cotype-conjecture-under-the-approximation-property/2026-09-23 https://gradientdamage.com/openai-math/markov-type-characterizes-superreflexivity/2026-09-23 https://gradientdamage.com/openai-math/nonexpansive-fixed-points-in-reflexive-banach-spaces/2026-09-24 https://gradientdamage.com/openai-math/counterexample-to-metric-entropy-duality/2026-09-24 https://gradientdamage.com/openai-math/uniformly-discrete-counterexample-to-bounded-approximation-in-lipschitz-free-spaces/2026-09-26 https://gradientdamage.com/openai-math/reflexive-midpoint-convexity-and-diamond-distortion/2026-09-27 https://gradientdamage.com/openai-math/metric-markov-cotype-of-l1-and-hilbert-space-lipschitz-extension/2026-10-05 https://gradientdamage.com/openai-math/smooth-isometric-immersions-of-surfaces-into-r4/2026-09-23 https://gradientdamage.com/openai-math/smooth-surface-metric-with-no-local-isometric-immersion-in-r3/2026-09-24 https://gradientdamage.com/openai-math/gromovs-integral-scalar-curvature-bound-for-simplicial-volume/2026-10-05 https://gradientdamage.com/openai-math/spectral-scalar-curvature-urysohn-width-and-macroscopic-dimension/2026-10-05 https://gradientdamage.com/openai-math/sharp-cartan-hadamard-isoperimetry-and-rigidity/2026-09-23 https://gradientdamage.com/openai-math/yaus-uniformization-conjecture/2026-09-23 https://gradientdamage.com/openai-math/katoks-entropy-rigidity-conjecture/2026-09-23 https://gradientdamage.com/openai-math/counterexample-to-the-nearby-lagrangian-conjecture/2026-09-23 https://gradientdamage.com/openai-math/donaldsons-hypersymplectic-deformation-conjecture/2026-10-07 https://gradientdamage.com/openai-math/donaldsons-tamed-to-compatible-conjecture/2026-10-06 https://gradientdamage.com/openai-math/symplectic-ball-packing-in-higher-dimensions/2026-09-23 https://gradientdamage.com/openai-math/metric-blaschke-conjecture/2026-09-23 https://gradientdamage.com/openai-math/infinitely-many-closed-geodesics-on-riemannian-spheres-and-closed-three-manifolds/2026-10-09https://gradientdamage.com/thumbs/nvS8R9FbAgY-og.jpgInfinitely Many Closed Geodesics | OpenAI, ChatGPT's CreatorFollow 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 many prime closed geodesics with pairwise distinct images. The two-sphere case is classical; its claimed advance includes arbitrary, possibly degenerate metrics in higher dimensions. Repeated laps around one loop do not count as new geodesics. The animation shows 36 distinct Hopf fibers: great circles on the unit round 3-sphere S3 in four real coordinates, stereographically projected into three dimensions. Each pair is linked. The circles are a known round-metric example; they do not visualize or prove the new arbitrary-metric theorem. The reveal, camera chase, colour choices and playback speed are editorial. No physical force or motion simulation is implied. On screen: covariant acceleration is zero, and the path is periodic. Construction: gamma(t) = (cos(eta) exp(it), sin(eta) exp(i(t+phi))), eta = 0.64, on S3 in C2. The projection is (x1,x3,x2)/(1-x4), with a uniform display scale. Geodesic speed is constant in the original S3 parameter, not in its 3D projection. Full manuscript, Theorem 1.1: https://github.com/openai/math/blob/main/preprints/Infinitely-many-closed-geodesic-images-on-every-Riemannian-sphere-September-24-2026/paper.pdf GitHub repository: https://github.com/openai/math Corrections and withdrawals: https://github.com/openai/math/blob/main/history.md Source-reported result from an internal research model. The proof has not been independently audited in this production. No corresponding top-line Lean formalization was linked in the inspected catalogue entry. This is not a claim that the public ChatGPT product proved the theorem. 'Aurora' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/aurora/…https://www.youtube-nocookie.com/embed/nvS8R9FbAgY142026-10-09T20:00:29+00:00yesno https://gradientdamage.com/openai-math/sharp-singular-set-bounds-for-stationary-integral-varifolds/2026-10-05 https://gradientdamage.com/openai-math/counterexamples-to-stable-morse-and-strong-arnold-fixed-point-bounds/2026-10-05 https://gradientdamage.com/openai-math/nonnegative-curvature-einstein-classification-and-an-l2-topological-gap/2026-10-05 https://gradientdamage.com/openai-math/solomon-yau-least-volume-conjecture/2026-09-23 https://gradientdamage.com/openai-math/yaus-nodal-bounds-surfaces-and-higher-dimensions/2026-10-10https://gradientdamage.com/thumbs/2Zgw4jT7Cek-og.jpgMath Makes Music: The Singing Sphere | OpenAI MathMath 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 the sharp square-root upper bound for nodal length on fixed smooth closed surfaces. The round-sphere modes shown here are a known example. The result is not the same in all dimensions; companion manuscripts report higher-dimensional counterexamples. Source manuscripts: Sharp nodal length on smooth surfaces https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf Smooth counterexamples to Yau's nodal upper bound in dimensions three and four https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf Power-law violations of Yau's nodal upper bound https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf Formalization scope: https://github.com/openai/math/blob/main/lean/docs/350.md Real linear combinations of spherical harmonics of degrees 2, 3, 5, 7 and 9 on the unit round S2, rendered with a uniform display radius. Every combination within a stage has the same eigenvalue l(l+1). Short transitions blend different eigenspaces. Sign colors and bright nodal curves come from the scalar field; radial displacement is illustrative and exaggerated. Audible mode pitch is 220*sqrt(l(l+1)/6) Hz. Visible vibrations are slowed. These familiar round-sphere examples illustrate the reported bound on general fixed smooth closed surfaces, not the new proof. Sound and visuals: original code-generated synthesis and Three.js animation. No third-party samples or compositions. Native 2160x3840 at 60 fps. Source-reported research result, not independently proof-audited in…https://www.youtube-nocookie.com/embed/2Zgw4jT7Cek122026-10-10T08:00:05+00:00yesno https://gradientdamage.com/openai-math/scalar-curvature-and-finite-time-ricci-flow-singularities/2026-09-24 https://gradientdamage.com/openai-math/finite-time-singularity-of-calabi-flow/2026-09-24 https://gradientdamage.com/openai-math/affine-bernstein-rigidity-through-dimension-nine-and-a-smooth-dimension-ten-counterexample/2026-10-05 https://gradientdamage.com/openai-math/isoperimetric-profile-of-the-cubic-three-torus/2026-10-11https://gradientdamage.com/thumbs/FFsKkuXhQKs-og.jpg5 Maths Problems OpenAI Says Its AI Just Solved | OpenAI MathOpenAI 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 work. OpenAI claims 5 never can, so the answer is 6 or 7. 3. Self-avoiding walks: an 8000-step honeycomb walk that never touches itself. OpenAI claims its size grows like steps to the 3/4. 4. Bubbles in a 3-torus: the least-area shape for each volume in a cube with opposite faces glued is a ball, then a tube, then a slab. OpenAI claims a proof of the full answer. 5. SLE curves: one random path, three roughnesses. OpenAI claims the exact ruler that measures them. The animations are illustrations, not the proofs. The claims and papers are in OpenAI's public maths catalogue: https://github.com/openai/math (families 165, 158, 237, 354, 230). Results are as reported by OpenAI and not independently verified here. Visuals and music are original, generated from the same event clock. #maths #math #ai #openai #shortshttps://www.youtube-nocookie.com/embed/FFsKkuXhQKs452026-10-11T15:45:08+00:00yesno https://gradientdamage.com/openai-math/unique-tangent-flows-at-the-first-surface-singularity/ https://gradientdamage.com/openai-math/giglis-characterization-of-alexandrov-curvature/2026-09-24 https://gradientdamage.com/openai-math/bi-lipschitz-coordinates-at-every-regular-rcd-point/2026-09-25 https://gradientdamage.com/openai-math/three-manifold-without-conjugate-points-or-nonpositive-curvature/2026-09-24 https://gradientdamage.com/openai-math/negative-kahler-curvature-without-bounded-holomorphic-coordinates/2026-09-25 https://gradientdamage.com/openai-math/weak-mtw-curvature-gives-convexity-and-regular-optimal-transport/2026-09-25 https://gradientdamage.com/openai-math/failure-of-integer-degree-harmonic-dimension-comparison/2026-09-26 https://gradientdamage.com/openai-math/global-smoothness-for-relativistic-vlasov-maxwell/2026-10-09https://gradientdamage.com/thumbs/gpIfWdqCISI-og.jpgRelativistic Vlasov-Maxwell: Global Solutions | OpenAI, ChatGPT's CreatorGlobal 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 assumes smooth nonnegative compactly supported particle data and smooth finite-energy electromagnetic fields with bounded derivatives and compatible divergence constraints. No smallness or symmetry assumption is imposed. The compact-support bound is on each finite time interval; it may grow with that interval. Paper: Global classical solutions of the three-dimensional relativistic Vlasov-Maxwell system https://github.com/openai/math/blob/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf About the animation: Analytic closed helical trajectories and field curves illustrate charged transport in a schematic magnetic bottle. A dominant particle leaves a tapered wake that grows into a full closed path; fainter companion paths appear during the wide reveal. The camera follows the particle, and turnaround highlights at 2.9333 and 8.45 seconds align with measured soundtrack accents. The trajectories are editorial geometry, not a self-consistent numerical solution of the coupled Vlasov-Maxwell PDE or evidence for particle confinement. Electric blue distinguishes the field scaffold; amber and ivory distinguish particle paths. Colours do not encode charge sign, speed, temperature or field strength. Playback timing is editorially warped. The actual transport equation remains displayed. GitHub repository: https://github.com/openai/math Related formalization scope: https://github.com/openai/math/blob/main/lean/docs/362.md The linked scope documents the stated classical-solution result. The formalization was not executed or audited in this production. Source-reported result. Selected theorem statements and…https://www.youtube-nocookie.com/embed/gpIfWdqCISI122026-10-09T15:00:00+00:00yesno https://gradientdamage.com/openai-math/nonuniqueness-with-local-conservation-for-the-hard-sphere-boltzmann-equation/2026-10-05 https://gradientdamage.com/openai-math/kinetic-limits-and-fluctuations-over-the-boltzmann-lifespan/2026-09-23 https://gradientdamage.com/openai-math/joint-metric-and-connection-recovery-from-one-boundary-patch/2026-10-05 https://gradientdamage.com/openai-math/planar-mumford-shah-regularity-conjecture-and-local-weak-l4-gradient-bounds/2026-09-24 https://gradientdamage.com/openai-math/critical-dimension-for-the-one-phase-bernoulli-problem/2026-09-24 https://gradientdamage.com/openai-math/three-dimensional-ball-evans-approximation-problem/2026-09-24 https://gradientdamage.com/openai-math/hot-spots-conjecture-for-simply-connected-planar-domains/2026-09-24 https://gradientdamage.com/openai-math/lane-emden-and-henon-lane-emden-conjectures/2026-09-24 https://gradientdamage.com/openai-math/stable-blowup-for-the-defocusing-schrodinger-equation/2026-09-24 https://gradientdamage.com/openai-math/global-uniqueness-in-smooth-isotropic-elasticity/2026-09-24 https://gradientdamage.com/openai-math/nonattainment-of-the-three-marginal-coulomb-monge-problem/2026-09-25 https://gradientdamage.com/openai-math/sharp-one-third-stability-of-brenier-maps/2026-09-25 https://gradientdamage.com/openai-math/de-giorgis-conjecture-in-dimension-eight/2026-09-26 https://gradientdamage.com/openai-math/universal-computation-in-forced-navier-stokes-flows/2026-10-09https://gradientdamage.com/thumbs/YME5mZZfiPY-og.jpgFluid Computation: A Fixed-Particle Test | OpenAI, ChatGPT's CreatorCan 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 computable viscosity, initial rest and specially prescribed smooth external forcing. It does not claim efficient or finite-precision computation. About the animation: A bright particle passes three gates and reaches a detector, then resets for the loop. The route is a finite schematic of the paper's reachability criterion. It does not implement the prescribed force, history-coordinate encoding, a Turing machine, or a Navier-Stokes simulation. The reset is an editorial device. Paper: https://github.com/openai/math/blob/main/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf GitHub materials: https://github.com/openai/math/blob/main/lean/docs/376.md Repository: https://github.com/openai/math Source-reported result. This visual explanation is not an independent proof audit or a verification of the formalization. 'Legionnaire (Original)' by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au https://www.scottbuckley.com.au/library/legionnaire/ Licence: https://creativecommons.org/licenses/by/4.0/ Music edits: Excerpted, crossfaded to loop, volume adjusted. #NavierStokes #ComputerScience #OpenAIhttps://www.youtube-nocookie.com/embed/YME5mZZfiPY122026-10-09T11:45:22+00:00yesno https://gradientdamage.com/openai-math/interior-c-1-regularity-for-infinity-harmonic-functions/2026-10-04