Closed Form — AI mathematics, explained https://nishachay.github.io/closed-form/ Newly explained entries from Closed Form, newest first. 369 · On smooth flat shapes without holes, the lasting hot and cold spots lie on the edge. https://nishachay.github.io/closed-form/e/openai-math-2026-369/ https://nishachay.github.io/closed-form/e/openai-math-2026-369/ Thu, 08 Oct 2026 17:12:00 GMT 334 · Some smooth rules for measuring length cannot be built as a surface in space, even near one point. https://nishachay.github.io/closed-form/e/openai-math-2026-334/ https://nishachay.github.io/closed-form/e/openai-math-2026-334/ Thu, 08 Oct 2026 17:12:00 GMT 322 · Every distance-keeping map between unit spheres of two real Banach spaces extends to a linear map of the spaces. https://nishachay.github.io/closed-form/e/openai-math-2026-322/ https://nishachay.github.io/closed-form/e/openai-math-2026-322/ Thu, 08 Oct 2026 17:12:00 GMT 312 · Grothendieck's algebraic ∞-groupoids capture all shapes of spaces, a paper claims, with its key step checked in Lean. https://nishachay.github.io/closed-form/e/openai-math-2026-312/ https://nishachay.github.io/closed-form/e/openai-math-2026-312/ Thu, 08 Oct 2026 17:12:00 GMT 297 · An algebra with one irreducible representation that is not the compact operators, built without extra axioms. https://nishachay.github.io/closed-form/e/openai-math-2026-297/ https://nishachay.github.io/closed-form/e/openai-math-2026-297/ Thu, 08 Oct 2026 17:12:00 GMT 296 · Every type II₁ factor with separable predual is generated by a single operator. https://nishachay.github.io/closed-form/e/openai-math-2026-296/ https://nishachay.github.io/closed-form/e/openai-math-2026-296/ Thu, 08 Oct 2026 17:12:00 GMT 288 · Every bounded map from a C*-algebra to operators can be straightened to respect the mirror operation. https://nishachay.github.io/closed-form/e/openai-math-2026-288/ https://nishachay.github.io/closed-form/e/openai-math-2026-288/ Thu, 08 Oct 2026 17:12:00 GMT 266 · Dimension six fits only three mutually unbiased bases, by a computer-assisted search. https://nishachay.github.io/closed-form/e/openai-math-2026-266/ https://nishachay.github.io/closed-form/e/openai-math-2026-266/ Thu, 08 Oct 2026 17:12:00 GMT 263 · A nucleus can hold only a fixed number of extra electrons, however large its charge. https://nishachay.github.io/closed-form/e/openai-math-2026-263/ https://nishachay.github.io/closed-form/e/openai-math-2026-263/ Thu, 08 Oct 2026 17:12:00 GMT 248 · Thompson's group F, a famous group of interval stretchings, is not amenable. https://nishachay.github.io/closed-form/e/openai-math-2026-248/ https://nishachay.github.io/closed-form/e/openai-math-2026-248/ Thu, 08 Oct 2026 17:12:00 GMT 246 · A hyperbolic group whose boundary is a sphere acts geometrically on hyperbolic three-space. https://nishachay.github.io/closed-form/e/openai-math-2026-246/ https://nishachay.github.io/closed-form/e/openai-math-2026-246/ Thu, 08 Oct 2026 17:12:00 GMT 244 · Fitting sets back inside their images does not give you the Axiom of Choice. https://nishachay.github.io/closed-form/e/openai-math-2026-244/ https://nishachay.github.io/closed-form/e/openai-math-2026-244/ Thu, 08 Oct 2026 17:12:00 GMT 213 · At the exact tipping point, random networks of a broad, symmetric kind form no endless cluster. https://nishachay.github.io/closed-form/e/openai-math-2026-213/ https://nishachay.github.io/closed-form/e/openai-math-2026-213/ Thu, 08 Oct 2026 17:12:00 GMT 196 · A group with no looping moves still lets two nonzero sums multiply to zero. https://nishachay.github.io/closed-form/e/openai-math-2026-196/ https://nishachay.github.io/closed-form/e/openai-math-2026-196/ Thu, 08 Oct 2026 17:12:00 GMT 180 · Every Barnette graph has a round trip through all its points. https://nishachay.github.io/closed-form/e/openai-math-2026-180/ https://nishachay.github.io/closed-form/e/openai-math-2026-180/ Thu, 08 Oct 2026 17:12:00 GMT 179 · Circulant Hadamard matrices exist only in sizes 1 and 4. https://nishachay.github.io/closed-form/e/openai-math-2026-179/ https://nishachay.github.io/closed-form/e/openai-math-2026-179/ Thu, 08 Oct 2026 17:12:00 GMT 173 · In every network of one-way arrows, some point reaches at least as many points in two steps as in one. https://nishachay.github.io/closed-form/e/openai-math-2026-173/ https://nishachay.github.io/closed-form/e/openai-math-2026-173/ Thu, 08 Oct 2026 17:12:00 GMT 165 · The classic crossing formulas for drawing complete networks are exact. https://nishachay.github.io/closed-form/e/openai-math-2026-165/ https://nishachay.github.io/closed-form/e/openai-math-2026-165/ Thu, 08 Oct 2026 17:12:00 GMT 162 · Ryser's covering conjecture is false: some intersecting hypergraphs with r parts need r vertices to cover. https://nishachay.github.io/closed-form/e/openai-math-2026-162/ https://nishachay.github.io/closed-form/e/openai-math-2026-162/ Thu, 08 Oct 2026 17:12:00 GMT 161 · A 35-vertex bipartite pattern appears less often in some graph than random would predict. https://nishachay.github.io/closed-form/e/openai-math-2026-161/ https://nishachay.github.io/closed-form/e/openai-math-2026-161/ Thu, 08 Oct 2026 17:12:00 GMT 159 · Any set of whole numbers whose reciprocals add to infinity holds evenly spaced runs of every length. https://nishachay.github.io/closed-form/e/openai-math-2026-159/ https://nishachay.github.io/closed-form/e/openai-math-2026-159/ Thu, 08 Oct 2026 17:12:00 GMT 158 · No way of coloring the plane with five colors keeps every unit-distance pair apart. https://nishachay.github.io/closed-form/e/openai-math-2026-158/ https://nishachay.github.io/closed-form/e/openai-math-2026-158/ Thu, 08 Oct 2026 17:12:00 GMT 156 · A set in nine-dimensional space cannot be cut into ten smaller pieces, so Borsuk's conjecture fails there. https://nishachay.github.io/closed-form/e/openai-math-2026-156/ https://nishachay.github.io/closed-form/e/openai-math-2026-156/ Thu, 08 Oct 2026 17:12:00 GMT 154 · For mixing systems, averages along k, 2k, …, nk are claimed to settle at almost every starting point, for every n. https://nishachay.github.io/closed-form/e/openai-math-2026-154/ https://nishachay.github.io/closed-form/e/openai-math-2026-154/ Thu, 08 Oct 2026 17:12:00 GMT 153 · Claimed singular Bernoulli convolutions beyond Pisot reciprocals, plus an arithmetic test for every parameter. https://nishachay.github.io/closed-form/e/openai-math-2026-153/ https://nishachay.github.io/closed-form/e/openai-math-2026-153/ Thu, 08 Oct 2026 17:12:00 GMT 152 · An abstract zero-entropy system with no smooth, volume-keeping copy in any dimension. https://nishachay.github.io/closed-form/e/openai-math-2026-152/ https://nishachay.github.io/closed-form/e/openai-math-2026-152/ Thu, 08 Oct 2026 17:12:00 GMT 151 · A smooth-enough map can double a shape's homology while its orbits stay orderly. https://nishachay.github.io/closed-form/e/openai-math-2026-151/ https://nishachay.github.io/closed-form/e/openai-math-2026-151/ Thu, 08 Oct 2026 17:12:00 GMT 150 · A paper claims billiards in every triangle with an irrational angle forget their start: they are weakly mixing. https://nishachay.github.io/closed-form/e/openai-math-2026-150/ https://nishachay.github.io/closed-form/e/openai-math-2026-150/ Thu, 08 Oct 2026 17:12:00 GMT 149 · With fixed rates, every weakly reversible chemical network herds each class of states into one common window. https://nishachay.github.io/closed-form/e/openai-math-2026-149/ https://nishachay.github.io/closed-form/e/openai-math-2026-149/ Thu, 08 Oct 2026 17:12:00 GMT 148 · A single formula gives the dimension of every self-similar measure on the line, even when its pieces collide. https://nishachay.github.io/closed-form/e/openai-math-2026-148/ https://nishachay.github.io/closed-form/e/openai-math-2026-148/ Thu, 08 Oct 2026 17:12:00 GMT 147 · Two papers claim that a smooth billiard table with an unbroken band of caustics along its edge must be an ellipse. https://nishachay.github.io/closed-form/e/openai-math-2026-147/ https://nishachay.github.io/closed-form/e/openai-math-2026-147/ Thu, 08 Oct 2026 17:12:00 GMT 146 · The standard map is chaotic on a region of positive area for every large enough parameter. https://nishachay.github.io/closed-form/e/openai-math-2026-146/ https://nishachay.github.io/closed-form/e/openai-math-2026-146/ Thu, 08 Oct 2026 17:12:00 GMT 145 · If long waits make any two moments independent, they make every finite group of moments independent. https://nishachay.github.io/closed-form/e/openai-math-2026-145/ https://nishachay.github.io/closed-form/e/openai-math-2026-145/ Thu, 08 Oct 2026 17:12:00 GMT 144 · A smooth map of a 3D torus where one measurement’s past and future build every average-zero measurement. https://nishachay.github.io/closed-form/e/openai-math-2026-144/ https://nishachay.github.io/closed-form/e/openai-math-2026-144/ Thu, 08 Oct 2026 17:12:00 GMT 143 · Claims a finite cap, set by degree alone, on limit cycles of planar polynomial systems. https://nishachay.github.io/closed-form/e/openai-math-2026-143/ https://nishachay.github.io/closed-form/e/openai-math-2026-143/ Thu, 08 Oct 2026 17:12:00 GMT 142 · A claimed coin-free algorithm that factors any polynomial modulo a prime in time polynomial in the input size. https://nishachay.github.io/closed-form/e/openai-math-2026-142/ https://nishachay.github.io/closed-form/e/openai-math-2026-142/ Thu, 08 Oct 2026 17:12:00 GMT 141 · Deciding “there is x so that for every y” questions about real numbers claimed at one fixed counting level. https://nishachay.github.io/closed-form/e/openai-math-2026-141/ https://nishachay.github.io/closed-form/e/openai-math-2026-141/ Thu, 08 Oct 2026 17:12:00 GMT 140 · Exact clues still cost more reading for each extra digit of accuracy when memory is limited. https://nishachay.github.io/closed-form/e/openai-math-2026-140/ https://nishachay.github.io/closed-form/e/openai-math-2026-140/ Thu, 08 Oct 2026 17:12:00 GMT 139 · Sampling from a gently curved bowl needs no fixed power of the dimension in queries. https://nishachay.github.io/closed-form/e/openai-math-2026-139/ https://nishachay.github.io/closed-form/e/openai-math-2026-139/ Thu, 08 Oct 2026 17:12:00 GMT 138 · A randomized algorithm claims to decide every Subset Sum instance in O(2^(0.49n)) time, below the 2^(n/2) scale. https://nishachay.github.io/closed-form/e/openai-math-2026-138/ https://nishachay.github.io/closed-form/e/openai-math-2026-138/ Thu, 08 Oct 2026 17:12:00 GMT 137 · A one-tape machine's run can be replayed with far less memory than the old square-root rule allowed. https://nishachay.github.io/closed-form/e/openai-math-2026-137/ https://nishachay.github.io/closed-form/e/openai-math-2026-137/ Thu, 08 Oct 2026 17:12:00 GMT 135 · Five-layer recipes for multiplying n matrices need n to a power that grows like √n. https://nishachay.github.io/closed-form/e/openai-math-2026-135/ https://nishachay.github.io/closed-form/e/openai-math-2026-135/ Thu, 08 Oct 2026 17:12:00 GMT 134 · Three nested stars are enough to describe any regular language once complement is allowed. https://nishachay.github.io/closed-form/e/openai-math-2026-134/ https://nishachay.github.io/closed-form/e/openai-math-2026-134/ Thu, 08 Oct 2026 17:12:00 GMT 133 · Deciding Weisfeiler–Leman equivalence at dimension k needs time n to a power proportional to k. https://nishachay.github.io/closed-form/e/openai-math-2026-133/ https://nishachay.github.io/closed-form/e/openai-math-2026-133/ Thu, 08 Oct 2026 17:12:00 GMT 132 · Block sensitivity can outgrow any constant times sensitivity squared. https://nishachay.github.io/closed-form/e/openai-math-2026-132/ https://nishachay.github.io/closed-form/e/openai-math-2026-132/ Thu, 08 Oct 2026 17:12:00 GMT 131 · Random rewiring reaches a fair random network quickly, for every possible list of degrees. https://nishachay.github.io/closed-form/e/openai-math-2026-131/ https://nishachay.github.io/closed-form/e/openai-math-2026-131/ Thu, 08 Oct 2026 17:12:00 GMT 130 · In exact arithmetic, the discrete Fourier transform can be computed in slightly fewer than n log n steps. https://nishachay.github.io/closed-form/e/openai-math-2026-130/ https://nishachay.github.io/closed-form/e/openai-math-2026-130/ Thu, 08 Oct 2026 17:12:00 GMT 129 · Removing guesses from simple two-way reading machines can cost exponentially many states. https://nishachay.github.io/closed-form/e/openai-math-2026-129/ https://nishachay.github.io/closed-form/e/openai-math-2026-129/ Thu, 08 Oct 2026 17:12:00 GMT 128 · A fast method builds a string containing all given pieces, never more than twice the shortest possible. https://nishachay.github.io/closed-form/e/openai-math-2026-128/ https://nishachay.github.io/closed-form/e/openai-math-2026-128/ Thu, 08 Oct 2026 17:12:00 GMT 127 · Yes-or-no rules built from degree-d polynomials on n inputs have average sensitivity at most 8d√n. https://nishachay.github.io/closed-form/e/openai-math-2026-127/ https://nishachay.github.io/closed-form/e/openai-math-2026-127/ Thu, 08 Oct 2026 17:12:00 GMT 126 · Exact semidefinite descriptions of perfect matching need exponential size, a new paper claims. https://nishachay.github.io/closed-form/e/openai-math-2026-126/ https://nishachay.github.io/closed-form/e/openai-math-2026-126/ Thu, 08 Oct 2026 17:12:00 GMT 125 · The best possible guarantee for placing k facilities is exactly 1 + 2/e, assuming P ≠ NP. https://nishachay.github.io/closed-form/e/openai-math-2026-125/ https://nishachay.github.io/closed-form/e/openai-math-2026-125/ Thu, 08 Oct 2026 17:12:00 GMT 124 · The best schedule for one-hour jobs with order rules on three machines can be found in polynomial time. https://nishachay.github.io/closed-form/e/openai-math-2026-124/ https://nishachay.github.io/closed-form/e/openai-math-2026-124/ Thu, 08 Oct 2026 17:12:00 GMT 122 · Rebuilding a string from randomly thinned copies needs more than polynomially many copies. https://nishachay.github.io/closed-form/e/openai-math-2026-122/ https://nishachay.github.io/closed-form/e/openai-math-2026-122/ Thu, 08 Oct 2026 17:12:00 GMT 121 · Edit distance can be estimated to within any fixed accuracy in barely more than linear time. https://nishachay.github.io/closed-form/e/openai-math-2026-121/ https://nishachay.github.io/closed-form/e/openai-math-2026-121/ Thu, 08 Oct 2026 17:12:00 GMT 119 · No yes-or-no summary of random bits survives noise better than simply keeping one bit. https://nishachay.github.io/closed-form/e/openai-math-2026-119/ https://nishachay.github.io/closed-form/e/openai-math-2026-119/ Thu, 08 Oct 2026 17:12:00 GMT 118 · Rounding up the fractional bin count can be wrong by any fixed number of bins. https://nishachay.github.io/closed-form/e/openai-math-2026-118/ https://nishachay.github.io/closed-form/e/openai-math-2026-118/ Thu, 08 Oct 2026 17:12:00 GMT 117 · Finding a network's weakest split even roughly is claimed NP-hard, and a key relaxation can miss by almost √log n. https://nishachay.github.io/closed-form/e/openai-math-2026-117/ https://nishachay.github.io/closed-form/e/openai-math-2026-117/ Thu, 08 Oct 2026 17:12:00 GMT 116 · One fixed set of matrices exposes every nonzero small formula whose variables do not commute. https://nishachay.github.io/closed-form/e/openai-math-2026-116/ https://nishachay.github.io/closed-form/e/openai-math-2026-116/ Thu, 08 Oct 2026 17:12:00 GMT 115 · Number grids with fixed row and column sums can be sampled fairly and counted fast, whatever their size. https://nishachay.github.io/closed-form/e/openai-math-2026-115/ https://nishachay.github.io/closed-form/e/openai-math-2026-115/ Thu, 08 Oct 2026 17:12:00 GMT 114 · Shared solutions of two capacity rules can be counted approximately, at a cost set by digits, not size. https://nishachay.github.io/closed-form/e/openai-math-2026-114/ https://nishachay.github.io/closed-form/e/openai-math-2026-114/ Thu, 08 Oct 2026 17:12:00 GMT 113 · A fast coin-flipping method estimates the number of perfect matchings in any network, to any chosen accuracy. https://nishachay.github.io/closed-form/e/openai-math-2026-113/ https://nishachay.github.io/closed-form/e/openai-math-2026-113/ Thu, 08 Oct 2026 17:12:00 GMT 112 · One fast-to-decide problem needs three-layer circuits bigger than any 2^(A√n) for large inputs. https://nishachay.github.io/closed-form/e/openai-math-2026-112/ https://nishachay.github.io/closed-form/e/openai-math-2026-112/ Thu, 08 Oct 2026 17:12:00 GMT 111 · One sample per item lets an online rule keep a fixed share of the best, even against an all-seeing opponent. https://nishachay.github.io/closed-form/e/openai-math-2026-111/ https://nishachay.github.io/closed-form/e/openai-math-2026-111/ Thu, 08 Oct 2026 17:12:00 GMT 110 · Randomized k-server costs at most a squared-logarithm factor on every space, matching the known lower bound. https://nishachay.github.io/closed-form/e/openai-math-2026-110/ https://nishachay.github.io/closed-form/e/openai-math-2026-110/ Thu, 08 Oct 2026 17:12:00 GMT 109 · A tape-based computer can multiply huge whole numbers slightly faster than the long-conjectured n log n limit. https://nishachay.github.io/closed-form/e/openai-math-2026-109/ https://nishachay.github.io/closed-form/e/openai-math-2026-109/ Thu, 08 Oct 2026 17:12:00 GMT 108 · Disguising the permanent as a determinant needs a matrix that grows at least like the cube of its size. https://nishachay.github.io/closed-form/e/openai-math-2026-108/ https://nishachay.github.io/closed-form/e/openai-math-2026-108/ Thu, 08 Oct 2026 17:12:00 GMT 107 · Huge n-by-n complex matrices can be multiplied in about n^(9/4) arithmetic steps, up to any tiny slack. https://nishachay.github.io/closed-form/e/openai-math-2026-107/ https://nishachay.github.io/closed-form/e/openai-math-2026-107/ Thu, 08 Oct 2026 17:12:00 GMT 106 · Even a three-colorable graph can be NP-hard to tell from one with only tiny independent sets. https://nishachay.github.io/closed-form/e/openai-math-2026-106/ https://nishachay.github.io/closed-form/e/openai-math-2026-106/ Thu, 08 Oct 2026 17:12:00 GMT 105 · Telling fully solvable 2-to-1 games from nearly worthless ones is NP-hard. https://nishachay.github.io/closed-form/e/openai-math-2026-105/ https://nishachay.github.io/closed-form/e/openai-math-2026-105/ Thu, 08 Oct 2026 17:12:00 GMT 104 · Who wins a mean-payoff game can be found in quasipolynomial time, however large the scores. https://nishachay.github.io/closed-form/e/openai-math-2026-104/ https://nishachay.github.io/closed-form/e/openai-math-2026-104/ Thu, 08 Oct 2026 17:12:00 GMT 103 · Coin flips give no extra power to computers with tiny memory: L = RL = BPL. https://nishachay.github.io/closed-form/e/openai-math-2026-103/ https://nishachay.github.io/closed-form/e/openai-math-2026-103/ Thu, 08 Oct 2026 17:12:00 GMT 102 · Telling almost-solvable puzzles from hopeless ones can be impossibly hard. https://nishachay.github.io/closed-form/e/openai-math-2026-102/ https://nishachay.github.io/closed-form/e/openai-math-2026-102/ Thu, 08 Oct 2026 17:12:00 GMT 100 · Cylinders can cover a regular tetrahedron for less than half its smallest shadow. https://nishachay.github.io/closed-form/e/openai-math-2026-100/ https://nishachay.github.io/closed-form/e/openai-math-2026-100/ Thu, 08 Oct 2026 17:12:00 GMT 099 · Edit distance between words can be drawn as ℓ1 distance only with distortion of size exp(√(log d·log log d)). https://nishachay.github.io/closed-form/e/openai-math-2026-099/ https://nishachay.github.io/closed-form/e/openai-math-2026-099/ Thu, 08 Oct 2026 17:12:00 GMT 098 · Some uncrowded point sets in infinite dimensions cannot be copied faithfully into any finite dimension. https://nishachay.github.io/closed-form/e/openai-math-2026-098/ https://nishachay.github.io/closed-form/e/openai-math-2026-098/ Thu, 08 Oct 2026 17:12:00 GMT 097 · One choice of signs keeps every running total of unit vectors within a constant times √d, at any length. https://nishachay.github.io/closed-form/e/openai-math-2026-097/ https://nishachay.github.io/closed-form/e/openai-math-2026-097/ Thu, 08 Oct 2026 17:12:00 GMT 096 · Three 120-degree wedges are the best way to cut a bell-curve cloud, in every dimension. https://nishachay.github.io/closed-form/e/openai-math-2026-096/ https://nishachay.github.io/closed-form/e/openai-math-2026-096/ Thu, 08 Oct 2026 17:12:00 GMT 095 · A claimed proof that some hyperbolicity cones have no semidefinite description, even with hidden variables. https://nishachay.github.io/closed-form/e/openai-math-2026-095/ https://nishachay.github.io/closed-form/e/openai-math-2026-095/ Thu, 08 Oct 2026 17:12:00 GMT 094 · Points in Lp space can be squeezed into far fewer coordinates if distances may stretch slightly. https://nishachay.github.io/closed-form/e/openai-math-2026-094/ https://nishachay.github.io/closed-form/e/openai-math-2026-094/ Thu, 08 Oct 2026 17:12:00 GMT 092 · The worst convex shape needs covering density about n log n in dimension n, lattice or not. https://nishachay.github.io/closed-form/e/openai-math-2026-092/ https://nishachay.github.io/closed-form/e/openai-math-2026-092/ Thu, 08 Oct 2026 17:12:00 GMT 091 · Blending two symmetric shapes the thrifty way never loses volume below their geometric mean. https://nishachay.github.io/closed-form/e/openai-math-2026-091/ https://nishachay.github.io/closed-form/e/openai-math-2026-091/ Thu, 08 Oct 2026 17:12:00 GMT 090 · No arrangement of points in the plane beats the triangular lattice for any steadily fading repulsion. https://nishachay.github.io/closed-form/e/openai-math-2026-090/ https://nishachay.github.io/closed-form/e/openai-math-2026-090/ Thu, 08 Oct 2026 17:12:00 GMT 089 · Every planar road network can be redrawn in taxicab distance with stretch below one fixed number. https://nishachay.github.io/closed-form/e/openai-math-2026-089/ https://nishachay.github.io/closed-form/e/openai-math-2026-089/ Thu, 08 Oct 2026 17:12:00 GMT 088 · In four or more dimensions, ellipsoids have the smallest shadow body for their volume. https://nishachay.github.io/closed-form/e/openai-math-2026-088/ https://nishachay.github.io/closed-form/e/openai-math-2026-088/ Thu, 08 Oct 2026 17:12:00 GMT 087 · Symmetric convex bodies cannot beat Mahler's volume bound. https://nishachay.github.io/closed-form/e/openai-math-2026-087/ https://nishachay.github.io/closed-form/e/openai-math-2026-087/ Thu, 08 Oct 2026 17:12:00 GMT 085 · Taking the best disk average at each point never multiplies a surface's total slope by more than a fixed factor. https://nishachay.github.io/closed-form/e/openai-math-2026-085/ https://nishachay.github.io/closed-form/e/openai-math-2026-085/ Thu, 08 Oct 2026 17:12:00 GMT 084 · A set covering almost all of [0, 1] can dodge every copy of a geometric list like 1/2, 1/4, 1/8, … https://nishachay.github.io/closed-form/e/openai-math-2026-084/ https://nishachay.github.io/closed-form/e/openai-math-2026-084/ Thu, 08 Oct 2026 17:12:00 GMT 083 · Hilbert transforms along any Lipschitz direction field in the plane obey one energy bound at short lengths. https://nishachay.github.io/closed-form/e/openai-math-2026-083/ https://nishachay.github.io/closed-form/e/openai-math-2026-083/ Thu, 08 Oct 2026 17:12:00 GMT 082 · A triangle-shaped singular integral stays bounded at the symmetric L³ point, whatever cutoffs each point picks. https://nishachay.github.io/closed-form/e/openai-math-2026-082/ https://nishachay.github.io/closed-form/e/openai-math-2026-082/ Thu, 08 Oct 2026 17:12:00 GMT 081 · A bounded Riesz transform forces surfaces in higher codimension to contain big flat-like pieces. https://nishachay.github.io/closed-form/e/openai-math-2026-081/ https://nishachay.github.io/closed-form/e/openai-math-2026-081/ Thu, 08 Oct 2026 17:12:00 GMT 076 · Polynomials with only +1 and −1 coefficients can be almost perfectly flat on the unit circle. https://nishachay.github.io/closed-form/e/openai-math-2026-076/ https://nishachay.github.io/closed-form/e/openai-math-2026-076/ Thu, 08 Oct 2026 17:12:00 GMT 075 · Fourier sums of every L log L function converge to it at almost every point of the circle. https://nishachay.github.io/closed-form/e/openai-math-2026-075/ https://nishachay.github.io/closed-form/e/openai-math-2026-075/ Thu, 08 Oct 2026 17:12:00 GMT 074 · Claimed proofs: the Kakeya maximal conjecture in 3D, and full dimension for every Kakeya set in 4D. https://nishachay.github.io/closed-form/e/openai-math-2026-074/ https://nishachay.github.io/closed-form/e/openai-math-2026-074/ Thu, 08 Oct 2026 17:12:00 GMT 073 · Any closed, bounded set whose dimension is more than half that of its space has distances with positive total length. https://nishachay.github.io/closed-form/e/openai-math-2026-073/ https://nishachay.github.io/closed-form/e/openai-math-2026-073/ Thu, 08 Oct 2026 17:12:00 GMT 072 · Reshaping a region into a disk can never squeeze too hard: Brennan's 1978 conjecture is proved. https://nishachay.github.io/closed-form/e/openai-math-2026-072/ https://nishachay.github.io/closed-form/e/openai-math-2026-072/ Thu, 08 Oct 2026 17:12:00 GMT 071 · Every region of the sphere, however many holes it has, can be reshaped so that each hole is round. https://nishachay.github.io/closed-form/e/openai-math-2026-071/ https://nishachay.github.io/closed-form/e/openai-math-2026-071/ Thu, 08 Oct 2026 17:12:00 GMT 066 · One complement index, fixed by dimension and singularity bound, works for every mildly singular Fano contraction. https://nishachay.github.io/closed-form/e/openai-math-2026-066/ https://nishachay.github.io/closed-form/e/openai-math-2026-066/ Thu, 08 Oct 2026 17:12:00 GMT 058 · Unrolled shapes with a polynomial description split into just three kinds of pieces. https://nishachay.github.io/closed-form/e/openai-math-2026-058/ https://nishachay.github.io/closed-form/e/openai-math-2026-058/ Thu, 08 Oct 2026 17:12:00 GMT 052 · Two integrable families of directions on a compact Kähler manifold make its universal cover a product. https://nishachay.github.io/closed-form/e/openai-math-2026-052/ https://nishachay.github.io/closed-form/e/openai-math-2026-052/ Thu, 08 Oct 2026 17:12:00 GMT 050 · Algebraic positivity need not come with positive curvature, even on the simple surface built from two spheres. https://nishachay.github.io/closed-form/e/openai-math-2026-050/ https://nishachay.github.io/closed-form/e/openai-math-2026-050/ Thu, 08 Oct 2026 17:12:00 GMT 049 · A polynomial that becomes a coordinate with one extra variable, but not without it, disproves two conjectures. https://nishachay.github.io/closed-form/e/openai-math-2026-049/ https://nishachay.github.io/closed-form/e/openai-math-2026-049/ Thu, 08 Oct 2026 17:12:00 GMT 047 · A four-dimensional shape that is not ordinary space, until you add one line. https://nishachay.github.io/closed-form/e/openai-math-2026-047/ https://nishachay.github.io/closed-form/e/openai-math-2026-047/ Thu, 08 Oct 2026 17:12:00 GMT 039 · Nagata's 1959 rule for curves through random points holds for every count from 10 points up. https://nishachay.github.io/closed-form/e/openai-math-2026-039/ https://nishachay.github.io/closed-form/e/openai-math-2026-039/ Thu, 08 Oct 2026 17:12:00 GMT 038 · On every smooth shape, n + 1 copies of a positive bundle are always enough for Fujita's freeness. https://nishachay.github.io/closed-form/e/openai-math-2026-038/ https://nishachay.github.io/closed-form/e/openai-math-2026-038/ Thu, 08 Oct 2026 17:12:00 GMT 033 · Twisting a family of shapes never destroys the forms its fibers and base contribute. https://nishachay.github.io/closed-form/e/openai-math-2026-033/ https://nishachay.github.io/closed-form/e/openai-math-2026-033/ Thu, 08 Oct 2026 17:12:00 GMT 028 · No walk through Gaussian primes with steps of bounded length can go on forever. https://nishachay.github.io/closed-form/e/openai-math-2026-028/ https://nishachay.github.io/closed-form/e/openai-math-2026-028/ Thu, 08 Oct 2026 17:12:00 GMT 027 · On every curve, whole-number matrix rules spread across the character variety after one enlargement of the integers. https://nishachay.github.io/closed-form/e/openai-math-2026-027/ https://nishachay.github.io/closed-form/e/openai-math-2026-027/ Thu, 08 Oct 2026 17:12:00 GMT 026 · For any size bar, a fixed share of prime gaps clears it, all the way out. https://nishachay.github.io/closed-form/e/openai-math-2026-026/ https://nishachay.github.io/closed-form/e/openai-math-2026-026/ Thu, 08 Oct 2026 17:12:00 GMT 025 · Every fraction below 1 is a sum of about log log b distinct unit fractions, as Erdős conjectured. https://nishachay.github.io/closed-form/e/openai-math-2026-025/ https://nishachay.github.io/closed-form/e/openai-math-2026-025/ Thu, 08 Oct 2026 17:12:00 GMT 024 · Counting totient values up to x gets an exact leading formula, and doubling x doubles the count in the limit. https://nishachay.github.io/closed-form/e/openai-math-2026-024/ https://nishachay.github.io/closed-form/e/openai-math-2026-024/ Thu, 08 Oct 2026 17:12:00 GMT 023 · Cubic Gauss sums over primes lean positive at exactly the rate Patterson predicted in 1978. https://nishachay.github.io/closed-form/e/openai-math-2026-023/ https://nishachay.github.io/closed-form/e/openai-math-2026-023/ Thu, 08 Oct 2026 17:12:00 GMT 021 · Every run of about k² whole numbers holds one sharing no factor with a number that has k prime factors. https://nishachay.github.io/closed-form/e/openai-math-2026-021/ https://nishachay.github.io/closed-form/e/openai-math-2026-021/ Thu, 08 Oct 2026 17:12:00 GMT 020 · Whole-number quartics like n⁴ + 2 take squarefree values at exactly the share the local guess predicts. https://nishachay.github.io/closed-form/e/openai-math-2026-020/ https://nishachay.github.io/closed-form/e/openai-math-2026-020/ Thu, 08 Oct 2026 17:12:00 GMT 017 · Fractions can never get unusually close to π. https://nishachay.github.io/closed-form/e/openai-math-2026-017/ https://nishachay.github.io/closed-form/e/openai-math-2026-017/ Thu, 08 Oct 2026 17:12:00 GMT 015 · Number-field grids spread evenly over all grid shapes in prime degrees from five, and in degrees four and six. https://nishachay.github.io/closed-form/e/openai-math-2026-015/ https://nishachay.github.io/closed-form/e/openai-math-2026-015/ Thu, 08 Oct 2026 17:12:00 GMT 013 · The primes cannot be built by adding two sets of whole numbers. https://nishachay.github.io/closed-form/e/openai-math-2026-013/ https://nishachay.github.io/closed-form/e/openai-math-2026-013/ Thu, 08 Oct 2026 17:12:00 GMT 012 · The largest prime factors of n and n + 1 act like independent dice, so each wins half the time. https://nishachay.github.io/closed-form/e/openai-math-2026-012/ https://nishachay.github.io/closed-form/e/openai-math-2026-012/ Thu, 08 Oct 2026 17:12:00 GMT 010 · Odd two-dimensional 2-adic Galois representations are claimed to come from modular forms, with no condition mod 2. https://nishachay.github.io/closed-form/e/openai-math-2026-010/ https://nishachay.github.io/closed-form/e/openai-math-2026-010/ Thu, 08 Oct 2026 17:12:00 GMT 009 · A field of functions can be rebuilt from its multiplication, blurred by powers, plus one trace of addition. https://nishachay.github.io/closed-form/e/openai-math-2026-009/ https://nishachay.github.io/closed-form/e/openai-math-2026-009/ Thu, 08 Oct 2026 17:12:00 GMT 008 · A short list of braid equations has exactly the solutions Deligne and Drinfeld predicted, in every size. https://nishachay.github.io/closed-form/e/openai-math-2026-008/ https://nishachay.github.io/closed-form/e/openai-math-2026-008/ Thu, 08 Oct 2026 17:12:00 GMT 007 · Neighboring numbers stop agreeing on prime-factor parity. https://nishachay.github.io/closed-form/e/openai-math-2026-007/ https://nishachay.github.io/closed-form/e/openai-math-2026-007/ Thu, 08 Oct 2026 17:12:00 GMT 006 · Among the quadratic twists of any elliptic curve, half are claimed to have analytic rank 0 and half rank 1. https://nishachay.github.io/closed-form/e/openai-math-2026-006/ https://nishachay.github.io/closed-form/e/openai-math-2026-006/ Thu, 08 Oct 2026 17:12:00 GMT 005 · Catalan's constant, 1 − 1/9 + 1/25 − 1/49 + …, is not a fraction. https://nishachay.github.io/closed-form/e/openai-math-2026-005/ https://nishachay.github.io/closed-form/e/openai-math-2026-005/ Thu, 08 Oct 2026 17:12:00 GMT 004 · No algorithm can decide whether a whole-number polynomial equation has a solution in fractions, the paper claims. https://nishachay.github.io/closed-form/e/openai-math-2026-004/ https://nishachay.github.io/closed-form/e/openai-math-2026-004/ Thu, 08 Oct 2026 17:12:00 GMT 003 · No zero of Riemann's zeta function, or of any Dirichlet L-function, lies to the right of real part 7/8. https://nishachay.github.io/closed-form/e/openai-math-2026-003/ https://nishachay.github.io/closed-form/e/openai-math-2026-003/ Thu, 08 Oct 2026 17:12:00 GMT 002 · The full Birch–Swinnerton-Dyer formula is claimed for every elliptic curve with a small Selmer group at one prime. https://nishachay.github.io/closed-form/e/openai-math-2026-002/ https://nishachay.github.io/closed-form/e/openai-math-2026-002/ Thu, 08 Oct 2026 17:12:00 GMT 001 · After reduction mod p, Hodge classes on abelian varieties are claimed to give one rational number in every cohomology. https://nishachay.github.io/closed-form/e/openai-math-2026-001/ https://nishachay.github.io/closed-form/e/openai-math-2026-001/ Thu, 08 Oct 2026 17:12:00 GMT