How Invisible Geometry Gives Online Choices a Fair Chance
A new online-selection theorem reveals how tracking every hidden subspace can protect the odds of choosing valuable, independent items.
Math
How a Hidden Averaging Trick Tames a Thorny Rectangle Problem
A new averaging trick preserves key rectangle averages while controlling overlap, helping prove a sharp bound for a broad family of geometric searches.
Why One More Step Can Shrink a Network’s Required Seed Set
A directed-graph puzzle reveals why one more step can make a tiny set reach every node—and why the two-step version remains open.
A Fixed-Point-Free Map Could Expose a Group’s Hidden Resistance
A map that never settles down may expose a group’s resistance to averaging—and illustrates the promise and cost of mining reusable proof tools.
Why Climate Predictions Need Memory, Not Just a Snapshot
A mathematical bridge from statistical physics shows when climate variables can predict one another—and when a snapshot hides the answer.
A Geometry Trick Lets Protein Models See the Hidden Interior
A tetrahedral geometry method maps buried protein volumes, improving computer benchmarks for binding pockets, interfaces, and binder design.
How a Random Ordering Turns Average Degree Into a Guaranteed Tree
A six-decade graph theory puzzle gets an elegant answer: random vertex orderings show when average density must contain every tree.
The Secret to Interesting Math May Be a Short Statement and a Long Proof
A formal-math system found that short statements with hard proofs can guide theorem discovery—and beat proof length alone.
A Volume-Based “Distance” Failed Its Simplest Three-Point Test
A proposed distance built from convex-body volumes passes a convexity test—but fails the triangle inequality in a certified 3D counterexample.
A Prime-2 Gap Challenges a Proof About Catalan’s Constant
Exact arithmetic exposes a prime-number accounting gap in a claimed proof about Catalan’s constant, while leaving its irrationality open.
How Pruning a Quantum Graph Can Make Its Dynamics Clearer
A graph-theory framework maps quantum dissipation, revealing which network links matter and helping predict quantum decay spectra.
A Seven-Manifold Shows Positive Curvature Needn’t Bring Symmetry
A seven-dimensional counterexample shows positive curvature need not bring circle symmetry, but the simply connected case remains open.
When Two Wave Fields Team Up, Their Rogue Peaks Arrive Early
A fluid-dynamics idea predicts when coupled optical waves may form rogue peaks—and why one wave can trigger another.
A Shape Between a Cube and an Associahedron Hides New Numbers
A new family of polytopes links cube vertices to polygon triangulations, revealing intermediate shapes counted by Pell–Catalan sequences.
A Random Geometry Game Breaks a Carefully Simplified Set Conjecture
A random point-marking game reveals why one tidy strategy for building set unions without cancellation must fail.
A Map With No Local Folds Still Sends 3 Points to the Same Place
A counterexample to the 1939 Jacobian conjecture shows how a map with no local folds can still send several points to one place.
Erase the Drawing, Keep the Order—and Count Every Crossing
A combinatorial method turns local edge orders into a sharp lower bound on crossings, settling a long-standing complete-graph conjecture.
The Hidden Interface That Forces a 3D Equation to Flatten
A proof turns the Allen–Cahn equation’s hidden interface into a geometric trap, ruling out curved stable fronts in three dimensions.
The Gaussian Shadow That Reveals Why Points Resist Balance
A Gaussian-based proof closes a major gap in point balancing and reveals why online sample thinning is harder before the final count.
A Hidden Tree Builds a Near-Full Set That Evades Infinite Patterns
A probabilistic tree routes points through nested grids to build a tiny barrier against every copy of an infinite geometric pattern.