1033 matching entries · Page 12 of 21
For a balanced probability law ν on the 24 permutations of four labels at total-variation distance at most r≤1/24 from uniform, and nonnegative rows fᵢ normalized by the uniform mean of fᵢ² equal to one, 1−Eν∏ᵢfᵢ(σ(i))≥(1−24r)/9 ΣᵢVar(fᵢ). The coefficient is sharp at each radius and 1/24 is the exact admissible radius.
On a connected open subset of the Euclidean plane, if the canonical local geodesic midpoint maps of the dual torsion-free connections D+KC and D−KC are complementary near every diagonal point, then their totally symmetric cubic tensor field C is constant. The formal proof constructs the short geodesic branches and also proves a differential obstruction criterion and a five-direction test.
For a finite open chain with positive strong-bond lower bounds βᵢ, boundary parameter s≥0 and weak-to-strong ratios qᵢ∈[ℓᵢ,ρᵢ], the minimum scalar squared gap minq λmin(DβQ(q)ᵀQ(q)Dβ) is positive and equals the minimum over the ordered one-switch endpoint vectors q⁽ᵏ⁾, with lower endpoints before k and upper endpoints after k. The same value sharply bounds the squared nonzero gap for arbitrary finite-dimensional complex matrix channels satisfying the stated singular-value constraints.
For every N≥0 and h≥0, constant-step gradient descent xᵢ₊₁=xᵢ−(h/L)∇f(xᵢ) on a differentiable convex function with L-Lipschitz gradient in a real Hilbert space satisfies ‖∇f(xN)‖≤L‖x0−x*‖max{1/(1+Nh),|1−h|^N}. One-dimensional Huber and quadratic examples attain the two branches. The compared Lean contract also classifies sampled equality trajectories and proves the same sharp factor for the terminal residual of relaxed proximal minimization with actual minimizers.
For any finite-Kraus 1→2 qubit channel whose mean fidelity over the six Pauli eigenstates is at least 5/6−ε, the Kraus operators have total squared Frobenius distance at most 6ε from the span of the two Bužek–Hillery Kraus operators. For every input matrix ρ, the channel output is within 200√ε‖ρ‖F of the Bužek–Hillery channel in Frobenius norm.
For every repetition threshold α with (8/3)^+≤α<3, the lengths of α-free uniform binary morphisms are exactly the positive integers other than 3 and 6; each admitted length has a complementary witness. At the strict threshold 8/3, length 5 is impossible.
For Thue–Morse digit-sum parity t, maximum monochromatic progression length A(d), and earliest maximizing start i(d): i(2^n+1)=3·2^(2n)−2^n−1 for n≥2; i(2^(2n)−1)=3·2^(4n)−2^(2n)+1 for n≥1; and i(2^(2n+1)−1)=2^(2n+1)−1 for n≥0.
The decimal Copeland–Erdős constant formed by concatenating the primes is not simply strongly normal in base 10, and therefore is not strongly normal in base 10.
Conditional on R₄(3)≤61, the sixth Schur number satisfies S(6)≤1801; the general Ramsey-to-Schur bound and the stated frontier restrictions are proved with the Ramsey assertion left as an explicit premise.
The specified seven-vertex induced Paley tournament has transitive number 4 and capacity between √18 and 5; every tournament on at most six vertices has capacity equal to its transitive number, so seven is minimal.
For every s≥3, allowed base q and k≥1, the explicit N=q^(sk) digital net has covering radius ≤√s N^(−1/s); when q≥5a*_s+4 its point separation is ≥(1−a*_s/q)N^(−1/s), with the stated bound on a*_s.
For every convex P in the Boolean lattice Q_n and all k≥0, w(P×Q_k)≥w(Q_{n+k})|P|/2^n. The k=0 case proves the Daykin–Frankl relative-width conjecture.
For positive-integer two-size Gasoline instances with deliveries in {1,K}, the position-first iterative-rounding algorithm under the doubly stochastic assignment-LP interpretation has capacity at most OPT+K−2. For K≥3, the original-index tie rule has matching instances with OPT=K and returned capacity 2K−2, giving worst-case ratio 2−2/K.
For every linear code of length n and encoded coordinate i, the expected uniform-with-replacement recovery reads satisfy a_i(C)+a_i(C⊥)=n under the time-zero convention; therefore C is recovery balanced exactly when its dual is.
In Arnosti’s random bipartite matching model with independent uniform job neighbourhoods of prescribed sizes and a common positive integer bin capacity, RANDOM-VERTEX stochastically dominates RANKING in the number of accepted jobs.
Every proper unit-distance colouring of the plane whose unit ball is the regular 14-gon uses at least five colours.
Project-defined Volkenborn-normalized zeta2(7) and zeta2(9) are not both rational in Q₂, and every defining J(s) limit exists.
At least one of project-defined Volkenborn-normalized zeta2(7), zeta2(9), zeta2(11) is not rational in Q₂, and every defining J(s) limit exists.
AlphaGeometry solved 25/30 specialized IMO-AG-30 classical Euclidean geometry translations using learned auxiliary-point constructions plus a symbolic deduction and algebra engine; the paper does not claim all natural-language IMO geometry conditions are represented.
Deterministic integer 3SUM in O(n^1.99896) and APSP in O(n^2.99791) in the pinned upstream word RAM model, for polynomially bounded input numbers.
At least one of the eight classical Riemann zeta values ζ(7),ζ(9),…,ζ(21) is irrational; the theorem names no particular value.
The project-defined Volkenborn-normalized 2-adic value zeta2(7)=J(6)/768 is outside the image of Q in Q₂, with convergence of the defining Riemann sums proved in Lean.
There are absolute constants A,C>0 such that, for every odd prime p, the least positive quadratic nonresidue n(p) is at most C(log p)^A; hence n(p)≪δ p^δ for each δ>0.
There is an explicit trace-preserving PPT channel Θ:M21(C)→M21(C) such that Θ∘Θ is not entanglement breaking.
There exists a countable domain R that is left PCI and left V but neither right Ore, right PCI nor right V. Thus right PCI does not imply left PCI, and left V-domain does not imply right V-domain.
L(4)=16 and L(5)≥49 for the Schur degree of block sums. The two results refute Eliahou–Revuelta Conjecture 5.6 at n=4 and n=5.
The selected contribution is the model’s empirical identification of cusp translations as salient for knot signature, not the human-defined natural slope or the later theorem.
The selected contribution is GPT-5-Thinking’s original rational-trace strategy for the completeness barrier; the current published proof uses Phillip Harris’s determinant instead.
The selected contribution is the 48-vertex tree of §2.4, not Wagner’s subsequent infinite-family theorem.
Every finite-alphabet configuration c:Z²→A with P_c(m,n)≤mn for some positive m,n has a nonzero global period.
Erdős problem #36 lower bound c_E > 0.38055470 for the asymptotic minimum-overlap constant.
GPT 6 Astra’s manuscript claims lim inf of consecutive prime gaps is at most 186, by proving DHL[40,2] and applying it to an explicit admissible 40-tuple of diameter 186.
For the switch–walk–switch lamplighter walk with Z₂ lamps over the infinite d-regular tree (d≥3), p₂ₙ(e,e)=ρ_d^{2n} exp[−(π²(log(d−1))²+o(1))n/(log n)²], where ρ_d=2√(d−1)/d.
There exists a weakly quasi-complete Noetherian local ring that is not quasi-complete.
For every depth p and even ring size N with 2p+2≤N, the optimal QAOA approximation ratio for MaxCut on the ring of disagrees is exactly (2p+1)/(2p+2).
Every finite loopless bridgeless undirected multigraph has a multiset of cycles covering each edge exactly twice.
Kimura reports exactly 16,453,493 imaginary quadratic fields with class number at most 2000; the largest absolute field discriminant on the list is 1,062,350,083.
For n ≥ 2 and 1 ≤ v ≤ π(n), the count C_{n,v} of minimal generators of the truncation ideal I_n with least support index v equals Legendre's sifting function Φ(n,p_v).
For each integer a ≥ 1, the affine Erdős Problem 270 constants C(a,0), …, C(a,a−1), C(a,a+1) are algebraically independent; every other C(a,b) with b ≥ 1−a is a nonconstant rational affine combination of them and is transcendental.
Harmonic reports that Aristotle generated machine-verified solutions for 2025 IMO Problems 1–5 from human-formalized problem statements, with the Lean proofs published.
An advanced Gemini Deep Think model produced natural-language solutions to five of the six 2025 International Mathematical Olympiad problems, earning 35 of 42 points under the official coordinators’ grading.
AlphaEvolve found a step-function construction giving the Erdős minimum-overlap constant C5 an upper bound of 0.380924.
The constant ξ=Σ_{a,c≥0}1/(2^a3^c·2^{2^a3^c}) is normal in base two: every finite binary word has its expected limiting frequency when overlapping occurrences are counted.
For integers b,c,d≥2 with gcd(b,c)=gcd(b,d)=1, the sum of the two Stoneham constants α(b,c)+α(b,d) is normal in base b, including when c=d or the two parameters share prime factors.
For integers b,c≥2, the Stoneham number α(b,c)=Σ_{k≥1}1/(c^k b^{c^k}) is normal in base b exactly when some prime divisor of c does not divide b. In particular α(6,4) is not 6-normal despite neither parameter dividing the other.
Every even integer N>2 is a sum of positive integers a and b for which the Liouville function satisfies λ(a)=λ(b)=−1.
For each integer d≥3 there are αd∈(0,1/3] and Cd<∞ such that every bounded infinity-harmonic function on the unit ball has gradient supremum and αd-Hölder seminorm on the half ball bounded by Cd times its oscillation on the unit ball.
For every deterministic one-tape machine and finite input, an effective smooth divergence-free mean-zero force on the flat three-torus gives a unique classical forced Navier–Stokes flow from rest, with zero pressure, whose fixed material particle enters one fixed open strip exactly when the machine halts. The force and velocity have bounded mixed derivatives and become one-periodic after initialization.
Every entire C² solution u:ℝ⁸→(−1,1) of Δu=u³−u with a strictly positive derivative in one fixed direction is a planar tanh heteroclinic, without an assumed limit at either end of that direction.
For the uniform measure on any compact convex body with nonempty interior in ℝᵈ, d≥2, and targets supported in one fixed compact set, the L² distance between their quadratic optimal maps is at most C times the one-third power of the targets’ 2-Wasserstein distance. No larger uniform exponent works in general, even for three-atom targets on a fixed cube.
These entries have recorded screening decisions and are not counted as curated contributions. Importance, evidence, interest, and date filters apply only to curated contributions.
View all matching screening decisionsThese dates show source checks, not the latest research. Failed checks can leave gaps.
arXiv · algebra-geometry — last successful metadata check: 21/10/2026, 07:20:46 UTC
arXiv · algorithms-formal — last successful metadata check: 21/10/2026, 07:19:48 UTC
arXiv · analysis — last successful metadata check: 21/10/2026, 07:21:51 UTC
arXiv · combinatorics — last successful metadata check: 21/10/2026, 07:18:49 UTC
arXiv · geometry-topology — last successful metadata check: 21/10/2026, 07:22:23 UTC
arXiv · logic-foundations — last successful metadata check: 21/10/2026, 07:23:04 UTC
arXiv · optimization-information — last successful metadata check: 21/10/2026, 07:23:10 UTC
arXiv · probability-dynamics-physics — last successful metadata check: 21/10/2026, 07:22:36 UTC
arXiv · theoretical-physics — last successful metadata check: 21/10/2026, 07:22:50 UTC
Allow optional Google Analytics to help improve AIProofWatch? Privacy details.
