Found by the engine.
Bounds and exact values found by the Wirkung Engine, each with the value it replaces and the paper that contains it.
Tensor rank
Pratt’s balanced tripartition tensors Pn connect tensor rank to the Set Cover Conjecture. Previous values are the partial-Fourier bounds of Flavi, Jelisiejew and Michałek unless stated.
Pratt’s tensor P3, rank
Upper bound from one closed form for every tripartition tensor.
Pratt’s tensor P4
Rank from the closed form; border rank from depth-one decompositions and torus ladders.
Pratt’s tensor P5
Rank from the closed form; border rank from depth-one decompositions and torus ladders.
Pratt’s tensor P2
No abelian group of order below 32 realises P2, so Pratt’s group construction is optimal among abelian groups; this bound lies beyond that method.
The tensor T2,2,3
Upper bound from torus chambers, lower bound from flattenings.
Every tripartition tensor Ta,b,c
An explicit rank decomposition over any field of characteristic not 2: one term for every vertex of the Boolean cube outside three disjoint Hamming balls.
Exact rank of T1,1,c
The upper bound is sharp for every c.
Exact ranks of T1,2,2 and T1,2,3
With the optimal bilinear algorithms that follow from them, and border-rank lower bounds 40, 47 and 45 for T1,2,4, T1,3,3 and T2,2,3.
Random matrices
The Birkhoff polytope Bn is the set of n × n doubly stochastic matrices, the convex hull of the permutation matrices.
Real spectra in B3
The probability that a uniformly random 3 × 3 doubly stochastic matrix has only real eigenvalues. The spectrum is real exactly when a point lies outside a light cone, β² + γ² ≥ δ².
Real spectra in conv D4, conv D6, conv O(2)
The same method, exact for the convex hulls of dihedral groups and for the operator-norm unit ball.
Real spectra in B4
An exact reduction to an integral over the 7-sphere. The conjectured value one third does not hold.
Singularities in fluids
The Córdoba–Córdoba–Fontelos model, ωt − u ωx = ω ux with ux = Hω, is a one-dimensional model for singularity formation in fluids.
Second unstable blow-up rate λ2
Ten digits. The eight published digits of Wang, Léger, Lai and Buckmaster agree; their longer value is 6.4 × 10⁻⁷ too high.
Is there a fourth rate?
No. The three known rates are all the smoothness crossings of the kink branch; below λ2 the branch ends at a sonic singularity with exponent −1/2.
The engine
Every exact result goes back into the Wirkung Engine as a new instruction and a better search order.
Learning from its own solutions
Four instructions learned from the engine’s own solutions, tested on targets it had never seen. Deeper search without them reaches 26 at 7.7 times the effort.
