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Claims an explicit compact complex threefold X, fibred over P1 by complex 2-tori via period functions on the (3,4,∞) orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues X is simply connected with H∗(X;Z)=H∗(S6;Z), hence diffeo…
The paper's appendix draws a distinction worth keeping: a counterexample may reduce to a finite certificate, checkable once the object is written down, or it may itself be a theorem quantified over all degenerations. This is the second kind. The method field records construction, because the resolution exhibits an explicit fivefold, but the difficulty lay elsewhere - candidate manifolds of this shape have been avail…
A record, not a resolution, and a small one by design. The bound moves from 2.371339 to 2.371177, about 1.6×10−4, and the authors describe it as a small step. Whether ω=2 is untouched, and nothing here suggests the laser method can reach it.
The interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper report…
Two tiers, and only the first is the record. Rank ≥31 is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been published…
VibeMathed recorded: Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region. Consult the linked registry entry for the original report and scope.
Two tiers, and only the first is the record. Rank ≥30 is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with Δ=4.25 gives an analytic rank of at most 31, and the root number is +1 so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial resu…
The paper constructs a real meromorphic function F on C satisfying F−1({0,1,∞})⊂R, with each of the three fibers F−1(0), F−1(1), and F−1(∞) infinite, such that for every a∈C∖{0,1,∞}, the a-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, F is not of bounded type in…
The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.
The theorem is the vector-valued endpoint bound ∥Rf∥L1,∞≤2∥f∥L1 for R=(R1,…,Rn), so the constant 2 also serves each component Rj uniformly in the dimension; the best previously known component bound grew like clogn.
The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative f∈L1∩L2 and every λ>0, write…
Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every 0≤m≤n−2, nonnegative Ricci plus positive (m+1)-inte…
The claim is the exact conjectured decay: ψμa(u)≤Ca/logu for every u>1 and every n, with Ca dimension-free - concretely ≲κa2(logκa−1κa)1/2 where κa=(1+a)/(1−a).
What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and X…
The authors construct a unitary oracle U such that
QMAU=QMA(2)U.
Their black-box problem is solvable by a QMA(2) verifier with one oracle query and linear-size unentangled proofs, whereas any QMA verifier must use either exponentially many queries or an exponentially large witness.
As a non-oracle consequence, they prove that for every fixed…
The paper constructs explicit R,T,D,S∈Q[x,q,p,z] defining a Poisson endomorphism of the canonical rank-two Poisson algebra P2 that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant 1, and has an explicit fiber of exactly three points. Thus PC(2) is false, and stabilization gives failure of PC(n) for…
VibeMathed recorded: Optimality of Greedy for Single-Pass Semi-Streaming Matching. Consult the linked registry entry for the original report and scope.
The new content is SOP2⇒SOP3; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in SOP3⇒SOP2⇒SOP1 reverses: Mutchnik answered the second (SOP1=SOP2), and this answers the first, collapsing the bottom of the hierarchy to SOP1=SOP2=SOP3. The SOPn hierarchy for n≥3 remains, as does everything above…
The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam h…
Two-copy distillable if and only if already one-copy distillable.
Four independent papers settled this within five days of each other, and no single one of them is the account of record. Fu, Gao and Park posted first on 23 July (arXiv:2607.21367), followed by Song and Chen on 26 July (arXiv:2607.23416), then on 27 July both Fraser, Huber, Pozsgay and Vona (arXiv:2607.24309) and Bharti, Gajjala and Haug (arXiv:2607.…
For the Sherrington-Kirkpatrick model with no external field, at zero temperature β=∞, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support [0,1) - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows qβ→1 as β→∞.
The positive-temperature input is Lopatto's: for every…
The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.
Claims the central limit theorem for the bipartite random assignment problem with bounded uniform costs: n(Cn−ζ(2))⇒N(0,4ζ(2)−4ζ(3)). The limiting constant is not itself new - Wästlund had computed exactly 4ζ(2)−4ζ(3) for the mean-one exponential model, and Malatesta, Parisi and Sicuro derived the non-bipartite analogue by replicas - but neither is a proof fo…
For every n≥3, let u∈C∞(B2) be an admissible solution of
κ[u]∈Γ2,σ2(κ[u])=1,
with
∥u∥L∞(B2)+∥Du∥L∞(B2)≤K.
Then
B1/2sup∣κ[u]∣≤C(n,K).
Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension n≥3. Prior unrestricted qua…