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An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate.
What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montgom…
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…
Assuming three explicit input statements, the project proves DHL[40,2]: every admissible 40-tuple contains at least two primes infinitely often after translation. An explicit admissible 40-tuple of diameter 186 then gives
n→∞liminf(pn+1−pn)≤186.
The Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimat…
H1≤212, improving Stadlmann's 240 of three days earlier and the 246 of Polymath8b that had stood since 2014. The twin prime conjecture, H1=2, is untouched. Held the record for hours at most: OpenAI's paper claiming 186 is dated 30 August, four days before this one, though its Lean development appeared on 2 September.
GPT-6 Astra constructs a unital algebra R=k⊕A over the countable field
k=F2,
with I=A a nil two-sided ideal, together with a matrix
W∈M2(I)
that is not nilpotent.
The algebra A is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of A is nilpotent. At the same time, a suitable polynomial combination of the…
The paper proves
G(X)≫(log3X)2logX(log2X)2log4X
for all sufficiently large X. Its main new ingredient is a short-translates theorem: for any sufficiently small set S⊆[1,H] with ∣S∣≤δx, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the clai…
Claims G(T)≫logTlog2T/log4T against FGKMT's logTlog2Tlog4T/log3T, a gain of log3T/(log4T)2, together with Y(X)≫XlogX/log3X for the covering problem behind it.
Ben Green resists calling it incremental: "The sieving procedure is different to the Erdos-Rankin one which underpinned all bounds on the problem since 1938, and it wins log3N over that procedure. The [FGKMT] p…
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…
For every n,k with k+1≤n, every simple graph G on n vertices satisfying
∣E(G)∣≥2k−1n+1
contains every tree on k+1 vertices.
The proof counts pairs (π,j) where π=(v1,…,vn) is an ordering of the host vertices and v1vj is an edge. There are exactly
2∣E(G)∣(n−1)!
such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus…
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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…
Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.
For every rational α satisfying
1≤α<2,
the Lean theorem constructs some finite q and a bipartite graph
G:SimpleGraph(Finq)
such that
ex(n;G)=Θ(nα)
as n→∞.
This resolves the single-graph rational-exponents conjecture. Earlier work of Bukh and Conlon proved the corresponding statement only for a finite family of forbid…
The formal theorem proves that no universal constant C>0 can satisfy
N>C2∣A∣
for every nonempty interval bound N and every sum-distinct A⊆{1,…,N}.
Equivalently, for every ε>0 there are arbitrarily large n and sum-distinct n-element sets contained in {1,…,N} with
N≤ε2n.
The proof is ineffective: it establishes the existence of arbitrarily la…
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…
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.
The formal proof constructs a complex polynomial p such that
p(0)=0,p′(0)=1,
and for every critical point c of p,
cp(c)>1.
Thus at z=0 there is no critical point satisfying
∣c∣∣p(0)−p(c)∣≤∣p′(0)∣=1,
which disproves Smale's conjectured universal constant K=1.
The counterexample has very large unspecified degree and violates the bound only by a smal…
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…
Astra constructs a strictly stationary real process
Xt=ξt+K(ξt−1,ξt−2,…),
where the innovations ξt are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and K is bounded, continuous, and odd. The process is φ-mixing, centered, square-integrable, and satisfies
Var(Sn)→∞.
Nevertheless there are times nj→∞ such that…
This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are ind…
Sendov's conjecture is resolved for every degree n >= 2, closing a gap that had stood since 1959: degrees up to eight were settled piecemeal between 1969 and 1999, and Tao's 2020 result covered all sufficiently large degrees without ever specifying the threshold, leaving the middle range open. Tao's digestion establishes the stronger interior form of the statement, which resolves the Phelps-Rodriguez conjecture in f…
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.
Astra proves that there exists a divergent function
f:N→N,f(n)→∞,
such that no graph G of infinite chromatic number can satisfy
dbip(H)≤f(n)
for every finite n-vertex subgraph H⊆G, where dbip(H) is the minimum number of edges that must be deleted to make H bipartite.
In fact, every included resolution proves the stronger state…
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…