The problem is open, and the repository says so: what is proved are criteria for it and two instances, not the general case.
Every compared statement that concludes Problem 10.61 does so for a quadratic setup - a real root α>1 of X2−aX−b whose conjugate has modulus below one - and both instances are quadratic. The arbitrary-degree material is conditional ingredients: for the family…
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 manuscript proves the upper bound ∣A∣≤1280 for every admissible subset of the fixed ambient code D. Together with Ho’s existing construction, this determines the exact maximum. The proof partitions D into 256 cosets of a 16-word subspace whose induced forbidden-distance graph is the Clebsch graph; each coset contributes at most five words. This does not determine the unrestricted kissing number…
The manuscript gives an explicit centrally symmetric simplicial
7-polytope with 88 integer vertices. Its antipodal boundary quotient
is a 44-vertex triangulation of real projective 6-space, with f-vector
(44, 938, 7024, 22555, 34936, 25914, 7404).
This improves the 45-vertex construction of Guyer, Steinerberger and
Yang and gives an affirmative answer to their Question 3.2. Integer
coordinates, facet lists and exac…
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…
Answered in the negative for every n≥2. The preprint constructs a bijection F:Rn→Rn that maps every connected set to a connected set, is continuous exactly off the closed ray [0,∞)×{0}n−1, and pulls the straight segment {(1,0,…,0)}×[0,1] back to the middle-thirds Cantor set on that ray, so F−1 is not connectedness-preserving. The construction extends a th…
Let H be the Berlekamp–van Lint–Seidel graph on 243 vertices, the Cayley graph of Z35 with strongly regular parameters
(243,22,1,2),
and let G=H.
The proof establishes
γ(G)=3
because every pair has a common neighbor in H, while an H-triangle gives a dominating triple in G.
It then proves
γ∞(G)=3
by showing that the family of all dominating triples…
Let T be a complete first-order theory in discrete or continuous logic, let M≺U, and let μ∈Mx(U) be Borel-definable over M. The paper proves that the following three conditions are equivalent:
(i)μ is a frequency interpretation measure (fim) over M;
(ii)μ is definable over M and its canonical random extension rμ is generically stable over…
Fixed four-terminal planar DAG; positive route-cost differences for m≥2, with non-attained suprema; signed/zero only for m=1, whose attainment is unstated. No topology-wide/unrestricted-planar claim.
We prove HGP(K8−e)=14 using an explicit PGL(2,13)-equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the n=5,6,7 constructions…
The preprint claims an exact classification. Write C5(G) for the residues modulo five represented by cycle lengths in G. Let
E5=K6,K5,5∪H5,n;t:2≤t≤5<n,
where H5,n;t is obtained from K5,n by deleting 5−t edges incident with one vertex in the part of size n.
For every finite simple graph G with minimum degree at least five, exactly one alternative holds:…
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 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 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…
The repository gives a Lean proof of Conway's 1976 refinement conjecture for omnific integers: whenever
ab=cd
with a,b,c,d∈Oz, there exist e,f,g,h∈Oz satisfying
a=ef,b=gh,c=eg,d=fh.
The proof is formalized twice: once using CombinatorialGames' surreal-number implementation and once with the required surreal definitions inlined over Mathlib. The development also…
We prove HGP(K6−e)=10. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The r…
We prove HGP(K7−e)=12. One lower-bound proof uses two explicit block-disjoint S(5,6,12) Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also…
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…
The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 for…
VibeMathed recorded: Mauri and Moraga's Question on Log Calabi-Yau Pairs with Big Boundary. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Ciliberto et al. Question on Effective Divisors of Positive Self-Intersection. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication. Consult the linked registry entry for the original report and scope.
For n <= 2, we proof that the Generalized Vanishing Conjecture holds, and we have a failure at dimension 3 (including a counter-example).
This improves the original failure at dimension 5.
The exact value is HGP(K5−e)=8. The lower bound uses explicit legal twin-player rules over F23. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion…
The paper proves that a tensor-product ensemble of independently chosen single-qubit Clifford gates is negl(n)-distinct when the number of queries is polynomial in n. This disproves the authors' conjecture that negligibly distinct ensembles must necessarily be entangling. As an application, the depth-logn global Clifford/unitary-2-design layer used in the PFC pseudorandom-unitary construct…
Knauer proves that every finite n-vertex multigraph satisfies bn(G)=O(logn), improving the previous general O(log2n) bound and matching the known Ω(logn) order. He also proves the sharper cycle-rank bound bn(G)=O(logβ(G)). Combined with a reduction of Lehner and Miraftab, this yields bn(G)=O(logg) for graphs of Euler genus g, improving the previous…
For every nonnegative n×n matrix A whose bipartite support graph has girth at least an even integer g≥4, Dong and Jain prove the sharp inequality
Bethe(A)≤per(A)≤22n/gBethe(A).
The factor 22n/g is optimal whenever g∣2n, attained by matrices whose support graphs are disjoint unions of g-cycles. Thus the result confirms Anari's conjectur…
Let P⊆Qn be convex. Williams proves the stronger statement that for every k≥0,
w(P×Qk)≥w(Qn+k)∣P∣2−n,
where w denotes poset width.
Taking k=0 gives
w(P)≥∣P∣(⌊n/2⌋n)2−n,
which is exactly the Daykin-Frankl conjecture.
The proof proceeds by induction on n, reducing the step to a structural lemma for a convex subset of R×Q1 and…
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…
For every neighborhood U of the origin in C×R and every ε>0, Curry constructs a smooth nonnegative perturbation ϕ, supported in U with ∥ϕ∥C2<ε, such that
Tϕ0,1=spanC{Lϕ}
is strongly pseudoconvex, its canonical bundle admits a nowhere-zero closed section, yet every C1 CR function near the origin satisfie…
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…
For the space-homogeneous Boltzmann equation in dimension 3 with collision kernels
B=4π1∣v−v∗∣γ,γ∈[0,1],
the paper constructs smooth, positive, radial mixtures fR=(1−p)M1+pMR for which ∂tD(fR)>0 for sufficiently large R. Thus entropy production need not decrease even for Maxwell molecules (γ=0) or hard spheres (γ=1), negatively resolving McKean's…
The paper disproves both the Burness–Giudici common neighbour conjecture for primitive groups of base size 2 and its later generalisation to arbitrary base size. For every integer B≥2, it constructs infinitely many primitive permutation groups of base size B whose generalised Saxl graphs contain two nonadjacent vertices with no common neighbour. At base size 2, it gives infinite counterexample families of…
The paper constructs a simple theory in which forking cannot always be witnessed by a stable formula. Precisely, what fails is: in a simple theory, if a↓Cb then there is φ(x,bˉ)∈tp(a/Cb) forking over C whose parameter-free form φ(x,y) is stable.
The counterexample is an infinite-dimensional vector space over the division ring of fractions of the quantum graph alg…
VibeMathed recorded: (ξαn)n≥1 is not uniformly distributed modulo one for Pisot α and ξ in the Cantor set C(α). Consult the linked registry entry for the original report and scope.
Claims the last open case of Dean's conjecture: every finite simple graph with minimum degree at least five contains a cycle of length divisible by five. Verified independently against Luo, Ma and Zhao (arXiv:2601.13552), whose abstract confirms the landscape - the conjecture was known for k∈{3,4} and they proved every k≥6, leaving k=5 open. If this proof stands, Dean's conjecture holds for all k≥3.…
The paper proves that there exists a randomized voting rule using only ordinal rankings with metric distortion at most 11641/5000=2.3282. This improves the previous best upper bound of 2.5 and closes about 44% of the gap to the known asymptotic lower bound of approximately 2.1126. It does not determine the optimal randomized metric distortion: for m≥4, the exact optimum and its asymptotic limit remain o…
For
f(n)=∣A∣=nmin∣{p prime:p∣a+b for some distinct a,b∈A}∣,
Astra formally proves
lognf(n)→∞.
The repository contains substantially stronger proofs. In particular, one verified alternate resolution establishes
n≤3r2,
where r is the number of supporting primes, yielding
f(n)≫n1/2.
Other independent resolutions give…
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…
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…
If Γ is a non-coHopfian property-(T) group and G is its ascending HNN extension, then the full group algebra C∗(G) is not finite. Indeed, Kazhdan projections pH<pΓ become unitarily equivalent under the stable letter, which cannot occur inside a finite C∗-algebra.
For
W=G/Γ⨁Z/2Z⋊G,
the paper proves more strongly that every homomorphism…
No such pair exists: for every positive integer m and every n, (Dn,2m) is rate-monotonic (Theorem A), and more generally so is every inversion extension of a finite abelian group by an involution, a family containing the generalized dihedral, dicyclic and generalized quaternion groups (Theorem B). The picture is completed in the other direction: for every real p≥1 that is not an even integer there is an…
The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant 2/9 to 1. The every-genus ladder it climbs is Katz-Sabourau's 19/120 and then Liu-Petri's 2/9, the latter also by a random construction. Constant 1 was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather than…
For two qubits, the paper considers
ρ=21∣Φ+⟩⟨Φ+∣+21∣01⟩⟨01∣,
where ∣Φ+⟩=(∣00⟩+∣11⟩)/2. This rank-2 state has no global supporting affine functional for Entanglement of Formation.
Setting ρt=(1−t)ρ+t∣10⟩⟨10∣, Wootters’ formula gives
C(ρt)=21−2t+O(t).
Consequently,…
The paper proves that not every Heyting algebra can occur as the lattice of subterminal objects of an elementary topos. Specifically, the free Heyting algebra F2 on two generators cannot occur.
Using Bellissima’s representation F2↪O↑(K2), the authors construct an upward-closed subset A⊆K2 with A∈/F2. They show that if some elementary topos E satis…
Answered in full: for every k≥4, a target with exactly two nonadjacent zero digits has Fk(t)=(k+23)3k−4, independently of the distance between the zeros. Exact at every width, no error term, no hypothesis on k (Theorem 1.1). This is an evaluation, not an extremal result, and the paper is explicit about the difference: the plateau value is not maximal. At k=12 it reads 35⋅38=229,635 while…
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…
For fixed δ∈(0,1) and all sufficiently large Δ depending only on δ, Glauber dynamics for proper q-colorings mixes rapidly on every graph of girth at least 5 whenever q≥(1+δ)Δ: spectral gap Ωδ(1/n) and tmix(ε)=Oδ(n2logq+nlog(1/ε)). An analogous theorem holds for the anti-ferromagnetic Potts model at…
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 repository supplies an executable verifier in exact integer arithmetic; for n=51 an integer vector x satisfies xTAx>11xTx, so λ1>11=D. The deficit λ1(n)−D(n) grows through n=5000 in the repository's computations, but no asymptotic theorem proving divergence is claimed, so "false for every constant C" is a computed pattern, not a proved one.
One anomaly, which the…
VibeMathed recorded: Equivalence of generic stability notions for Keisler measures. Consult the linked registry entry for the original report and scope.
If a bilinear form admits an (r,s)-sparse bound, its coordinate-wise extension to Cn-valued functions admits an (r,s)-convex body sparse bound, for 1≤r,s<∞ with r1+s1>1. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form.
On scope, two thin…
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…
Let S=k[x1,x2,x3,x4] over any characteristic-zero field. For each d∈{5,7} and every r≥1, the paper proves that r general degree-d forms satisfy Fröberg’s predicted Hilbert series
HSS/(F1,…,Fr)(t)=[(1−t)4(1−td)r]+.
The genuinely new ranges are 6≤r≤11 for quintics and 6≤r≤21 for septics. These are reduced to finitely many exac…
For a Bernoulli(p) source with 0<p<21, the paper proves that the best lossy compression achievable by a linear encoder is exactly the simple time-sharing strategy that losslessly compresses a fraction of the bits and estimates the rest as zero. Equivalently, every full-row-rank H∈F2k×n satisfies
nk≥h(p)(1−pD(H)).
This resolves Massey's question affirma…
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…
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…
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…
The manuscript claims Ca,b is transcendental for every a≥1 and b≥1−a, settling the positive integer-valued affine subclass of Erdős Problem 270. Two pieces of context matter. Problem 270 as Erdős and Graham posed it, for every f(n)→∞, was already answered no by Crmarić and Kovač in 2025: for any α>0 some such f makes the series sum to α. What survives is the non-decreasing case,…
Proves Haglund's Conjecture 4 for k=1: every non-real first-quadrant zero of Φ1+tΦ2 is simple with strictly decreasing imaginary part, no branch escapes forward, and every finite-multiplicity real collision stays real afterwards. The cases k≥2 remain open. Two readings worth separating: Conjecture 4 asserts the monotone descent alone, so the no-escape and stays-real statements are this paper's own ad…
Answers Problem 3 and generalizes it: the classification m∈S⟺m=2 covers every square root, and a further theorem replaces parity by divisibility by any p≥2. Note the scope of the machine-checking, which is narrower than the paper: the author states that the case m=3 is what is verified in Lean, and the repository flags the thickness computation of section 4.1 and all of sectio…
Disproved on the Hopf threefold X=(C3∖{0})/⟨z↦e−1z⟩: Xia and Zhang construct a smooth Hermitian form ω and smooth functions φj with ω+ddcφj>0 whose Monge-Ampère masses tend to infinity, so the universal bounded mass property fails already in complex dimension three. The construction uses the Hopf threefold's elliptic fibration, an exact mass…
A negative answer to Heuer and Löh's question: the isomorphism type of a one-relator group ⟨S∣r⟩ does not determine sclS(r). The witnesses are r=aabABabABBAbaabABBAb and r′=aabABabABabABBAbaBAb, both length 20 in F2′, with ⟨a,b∣r⟩≅⟨a,b∣r′⟩ but scl(r)=1 against scl(r′)=1/2.
The mechanism is what ma…
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment…
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…
VibeMathed recorded: The Fractional Colouring Conjecture for Triply Efficient Pauli Shadow Tomography. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique. Consult the linked registry entry for the original report and scope.
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.
Independent of ZFC, which is why this entry is the first to carry that result rather than proved or disproved. Both directions are formalized: Hechler's 1972 construction gives a model where the answer is no, and adding c+ random reals over a model of CH gives one where it is yes.
The credit is shared and mostly human. Newelski, Pawlikowski and Seredynski settled the problem's second question in 1987,…
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…
The claim is R(c)=40c+41 for every c≥2, reduced to three finite facts: the base value R(2)=121, and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every D≥1; the two unsatisfiability results carry DRAT proofs.
This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a sc…
For pairwise distinct real numbers x1,…,xN and an integer D≥1, Ma proves the complete classification
det[(xj−xi)D]i,j=1N=0⟺D≥N−1and(N is even or D is even).
This fully resolves Colombo’s original conjecture for even N. The new part is the even-size, odd-exponent branch, strengthened to the strict Pfaffian…
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…
Four results, and the first is partly a refutation. Jakimiuk conjectured cp=μp−1 is optimal for every p≥3; the paper proves that for p≥4 and gives a counterexample for every 2<p<4, so the conjecture is false as posed and the corrected range is p≥4. The witness is the two-coordinate vector S2=(ε1+ε2)/2.
The Baranski-Murawski-Nayar-Oleszkiewicz flat-po…
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…
Disproved. An explicit 32-leaf tree has least isometric ℓ∞-dimension six, not the conjectured five, and keeps dimension six under every assignment of positive edge lengths, so the weighted sharp-threshold conjecture falls too. Every tree with at most 31 leaves does attain ⌈log2t⌉, so 32 is the first failure; Brigham et al. had checked through 21. The proof is finite and exact: six explici…
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…
VibeMathed recorded: Composites Among [ξ7n] and Right-Truncatable Primes in Base 7. Consult the linked registry entry for the original report and scope.
Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and ever…
Nothing in the paper's proof is affected. At the witness, inequality (4) holds with slack +27.0, inequality (5) holds with equality (f is inner), and the theorem itself holds with slack 2 - kappa = +0.80. Only the shortcut Remark 2 floats is refuted.
VibeMathed recorded: Tarizadeh's Conjecture on the Maximality of Purely-Prime Ideals. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Cone Theorem for Effective Fourfold Pairs in Characteristic p>5. Consult the linked registry entry for the original report and scope.
Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber a=±b. Not settled here: promo…
VibeMathed recorded: Score-Determined Induced Tournament Statistics: an All-Orders Classification. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Pavez-Signe's Length-Control Question for Spanning Subdivisions. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Neuen-Grohe Problem: Isomorphism of Tournaments with Bounded VC Dimension. Consult the linked registry entry for the original report and scope.
The dihedral case only, for every a≥4 and b≥1; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim 1+(a−1)(b−1) for all a≥3; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue Rcyc(Paalt,Kb) for a≥4 remains open. The engin…
Nineteen individual exact values, each decided by SAT certificate: unsatisfiable at the claimed n, witnessed satisfiable at n−1. They close cells in DD26's Tables 3-13 but settle no infinite family - the sibling entries do that for the K column. The three overlap cells are Rdih(P4alt,K6)=16, Rdih(P3alt,K9)=17 and Rdih(P9alt,K3)=17, each an instance of a sibling theorem; the remain…
Part III of a series, and the first unconditional positive results in it. Settled exactly: the local envelope ladder E(2)=1, E(3)=9/8 and E(4)=(299−4141)/32=1.13974707…, which recasts the earlier record constants as exact envelopes of the general theory rather than isolated instances, plus exact constants for four classes - out-trees 0, two-layer hubs 1, outerplanar two-exit interval spines 1 (shar…
The authors construct smooth Am and smooth solutions um in B2⊂R3 with one fixed ellipticity bound
I≤Am≤281I,
common boundary data and ∥um∥L∞(B2)≤1, but
∥Dum∥L1(B1)→∞.
Thus no interior W1,1 estimate can depend only on dimension and ellipticity. Consequently, no such W1,p estimate exists for any p≥1.
They further obtain a u…
For every 0≤β≤21, Bondarenko and Seip construct a nonzero real-valued continuous even function
fβ∈L1(R)∩L2(R)
such that
fβ=fβ
and
fβ([log(e+n)]βn)=0(n≥0).
They normalize the construction by requiring fβ(1/2)=1, so the function is genuinely nontrivial.
For β=0, this gives…
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.
Courtade conjectured that for convex bodies B,C⊂Rn,
(∣B∣∣C∣)1/n+(∣B2n∣∣B2n+B+C∣)1/n≤∣B2n+B∣1/n∣B2n+C∣1/n.
The paper proves that this inequality is false in every dimension n≥3. In dimension 3, it gives an explicit geometric counterexample using two orthogonal double bodies of revolution. It then strengthens this by constructing zonoid counterexamples in every d…
Establishes that every PPT channel is eventually entanglement-breaking (finite EB index), in full generality, and bounds the index by 3 uniformly in dimension for a family strictly containing the 2-superpositive maps. The PPT-squared conjecture itself - index at most 2 - remains open; the paper presents its results as strong evidence toward the cubed version.
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 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…
Disproves the Bowler-Brown-Fenner bound of 2*floor((n-1)/3) on common cards between nonisomorphic graphs: an explicit connected 78-vertex pair shares at least 51 cards against the predicted 50, and for every even r >= 4 there are families with overlap fraction asymptotically at least 1 - 1/r, so the attainable fraction approaches the full deck. The Kelly-Ulam reconstruction conjecture itself is untouched.
Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existenc…
Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both conj…
The first explicit counterexample rather than a first suspicion: the abstract is clear that the failure was widely expected and that what was missing was a witness. It gives an infinite family, one for each d >= 2, all Lamplighter groups over free groups.
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on th…
The honest reading, which the paper gives itself: the reduction is implicit in earlier work of Aboulker, Oijid, Petit, Rocton and Simon, and the model itself surfaced that reference when asked about originality. So this establishes the conjecture and supplies a polynomial-time algorithm, while the underlying idea is a rediscovery rather than a first. It is a striking record of a model producing an argument and then…
The manuscript claims a polynomial-time reduction from 3-SAT proving (2,1)-C1P NP-hard; together with membership in NP, this establishes NP-completeness and closes the sole unresolved (k,δ) case from the earlier classification. It also implies NP-completeness of the equivalent completion problem.
The proof package further shows that, within its specific nested-prefix/internal-local gadget architecture, no…
The a = 3 slice is settled outright. The parent conjecture's dihedral side has since been resolved for every a >= 4 as well (see the related entry), so Conjecture 4.9's claim 1 + (a-1)(b-1) now stands proved for all a >= 3; the trivial a = 1, 2 cases and the cyclic analogue for a >= 4 remain formally unaddressed.
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…
VibeMathed recorded: The Five-Dimensional Geode Challenge of Amdeberhan, Kauers and Zeilberger. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Second Computational Chomp Challenge of Ekhad and Zeilberger. Consult the linked registry entry for the original report and scope.
Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.
The general principle is the interesting part: every semialgebraic property of a bounded fixed-dimensional mean parameter is eventually almost surely predictable. Against merely integrable matrix laws it fails from dimension two.
Tensorisation and parabolic rescaling carry the one-dimensional example to real symmetric isotropic counterexamples on R^d and on every bounded domain, in every dimension.
VibeMathed recorded: Spahn and Zeilberger's Third Challenge: Holonomicity of the Restricted Permutation Counts. Consult the linked registry entry for the original report and scope.
Strict convexity and real-analytic boundaries are what make this sharp: the classical Gordon-Webb-Wolpert drums are non-convex polygons, so the obvious escape routes are closed off.
The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.
The key step is a lower bound on the truncated imbalance sum, which yields every Erdos-Gallai inequality for the sorted imbalance list; a parity computation finishes it.
Teschner's universal bound b(G) <= (3/2)Delta(G) is false: the 18-vertex cubic bipartite graph has b(G) = 5 against a bound of 4.5. What survives is the restricted statement Teschner actually proved, that the bound holds for graphs of domination number at most three, and Gagarin and Zverovich's 2013 result that it holds for almost all graphs. The counterexample does not suggest a replacement bound, and the correct g…
Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.
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…
Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: N2/3E⟨R1,22⟩ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of N1/3R1,2 converges to an explicit random probability measure defined from the reflected Airy1 point process. The same…
Answers the obstruction rather than the whole question: it says exactly when the Matrix-Tree Theorem can be applied and classifies the chambers, and gives a compact five-graviton formula outside the decay region. Simplifying the general solution, which is what the source paper left to future work, remains open.
The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.
Partial deliberately: the NPT bound entanglement problem itself is untouched. What falls is the conjecture about the canonical family, and the paper is explicit that a substantial neighbouring region remains unresolved while another is known two-copy undistillable.
An average-case claim about the Gaussian model, not a contradiction of the worst-case NP-hardness of integer least squares. If it holds, no computational-statistical gap separates polynomial-time detection from exhaustive maximum likelihood at first order in this model.
Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support U30 and conjectured that it was not a complete classification because it did not use the full field Q(ζ30). This work proves the complete classification. The larger field enlarges the common amplitude α, while every relative ambiguity remains a norm-one parameter in Q(ζ6). Th…
The paper also proves the conjecture in the unimodular case and characterizes equality there, so the boundary between true and false is drawn rather than just crossed.
Three immediate consequences follow for undirected unweighted planar graphs: better compression of the Okamura-Seymour metric, less space for constant-time exact distance oracles, and a faster distributed algorithm.
One conjecture each way: the second proved, the first disproved. Two further Chen-Lawrencenko conjectures remain open and are flagged as such in the paper.
VibeMathed recorded: A counterexample to the Howland-Kato problem for positive commutators. Consult the linked registry entry for the original report and scope.
Five existence statements, all by explicit construction: AME(12,5), AME(17,11), AME(18,11), AME(17,13) and AME(18,13). The [12,6,7]25 code came from a direct search with no symmetry imposed; its automorphism group turned out to have a regular Z32 coordinate orbit, and imposing that translation symmetry on two nine-coordinate orbits collapses an…
One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time. Ve…
For the Ising pure p-spin glass where p≥3, it was predicted by Gardner that there exists critical inverse temperatures 0<β1p<β2p<∞ such that: (1) When 0<β≤β1p, the Parisi measure is replica symmetric (RS); (2) When β1p<β≤β2p, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When β>βp2, the Parisi measure is full rep…
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…
The paper records how the collaboration actually went: the authors first aimed at a counterexample showing EF1 and PO incompatible under matroid constraints, and when the model surfaced fundamental difficulties with that plan they redirected toward proving the positive result instead. The paper also extends the technique to category constraints and leaves a pseudopolynomial-time algorithm open.
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.
Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den Heuv…
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…
For every n>2, the paper constructs a bounded map U∈W1,n(Bn,Rn+2), smooth on Bn∖{0} but discontinuous at the origin, together with an antisymmetric potential
Ω∈Ln(Bn,so(n+2)⊗Rn)
such that
−Div(∣∇U∣n−2∇U)=Ω⋅∣∇U∣n−2∇Uin D′(Bn).
Moreover, the potential satisfies the sh…
VibeMathed recorded: Completeness of Maximally Entangled States for Pseudo-Telepathy. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Nazarov's Conjecture on Truncations for Fractional Laplacians. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Perfectly Complete Quantum Key Agreement from One-Way Functions. Consult the linked registry entry for the original report and scope.
Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.
Chen, Nie and Xu prove a Nakai–Moishezon-type numerical criterion for a broad class of complex Hessian-type equations on compact projective manifolds. In particular, Corollary 1.3 gives a uniform version of Székelyhidi’s conjecture for complex Hessian quotient equations, while Corollary 1.5 proves the uniform version, formulated by Murakami, for complex k-Hessian equations. However, the results assume projectivity a…
subDL satisfies the signed depth relevance property, answering an open question posed by Øgaard (2026). More precisely, every valid inference in subDL contains a propositional variable that occurs in both the premises and conclusion with matching sign and at matching implicational depth.
VibeMathed recorded: Exact SSUF scenario-count ladder on the four-terminal planar gadget. Consult the linked registry entry for the original report and scope.
The paper records how stuck this was: the first author had discussed the question with Caucher Birkar, Osamu Fujino, Christopher D. Hacon, Junpeng Jiao, Vladimir Lazic and Lingyao Xie, and writes that despite a general feeling that a negative answer was likely, no precise counterexample could be found. The authors also note that, given the limitations of generative AI, they may have missed related literature and wel…
The Formal Conjectures pull request flipping this from open to solved is still open rather than merged, so the canonical repository has not yet accepted it.
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling ent…
Claimed in a self-published research draft; a standalone by-product is the transcendence of the integral of exp(q) between distinct algebraic endpoints for nonconstant algebraic q
For every integer-valued strongly b-additive g with gcd(g(1),…,g(b−1))=1 and digit mean μg≥0: g(p) is prime for infinitely many primes p. For μg>0, ∑1/p over p<X with g(p) prime is (dg/φ(dg))log3X+Cg,1+O(1/loglogX), likewise for the first j iterates. Also #{p≤x:g(p) prime}≪π(x)/loglogx, of that exact order on a large set of x, and…
VibeMathed recorded: Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication. Consult the linked registry entry for the original report and scope.
New theorems, not a formalisation of previously known results. For every base b≥2 the Erdős–Kac law is established for the λ-digit base-b palindromes and for the base-b reversals of the λ-digit primes, for ω and Ω and for ωS,ΩS with any regular set S of primes; with normal order loglogn on both families, and, for ω, all moments of order up to…
New theorems, not a formalisation of known results. In base ten the bare existence of a square with any admissible digit sum was recorded in the recreational literature (Murthy–Ashbacher 2005); the theorem here is the quantitative all-base version — roots coprime to b (excluding their trailing-zero trick), exp(c_b√q) representations of controlled size — and that statement is what is machine-checked, with effective b…
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
Record lower bound only. The sub-2 ceiling is the codimension-two case and does not bound the record ladder (k=17 has complement mass 11). 4/3 and 2 are conjectures; the proved gap is [1.28249, 2].
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.
The sharp valuation bounds and optimal γ(m) are proved for ALL loop quivers m≥2, hence for the extremal BPS invariants of all twist knots (both rows, matching every twist-knot entry of GKS Table 1). Scope limits: the m=3/figure-eight divisibility 2nr/r∈Z was previously proved by Basor–Conrey–Morrison (arXiv:1703.00990), whose per-r2-adic characterization for m=3 is finer than the uni…
This is the MODIFIED conjecture, not the original Lyons-Sidorova one, and it is proved for continuous bounded-variation paths. Prior work had a line-image result under the stronger assumption of infinite radius on every subinterval; this removes that assumption.
VibeMathed recorded: Mixed Partition Functions and Exponentially Bounded Edge-Connection Rank. Consult the linked registry entry for the original report and scope.
A partial result. Fuglede's conjecture remains open for finite cyclic groups generally; this settles an infinite family and, for the spectral-to-tiling direction, only under rapid growth of the prime factors.
VibeMathed recorded: Positivity on Deligne–Mumford Stacks Without the Torsion-Free Hypothesis. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Stanley's Rankwise Lower-Bound Conjecture for Differential Posets. Consult the linked registry entry for the original report and scope.
Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma f…
VibeMathed recorded: Infinitely Many Components in Auslander–Reiten Quivers over Perfect Fields. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Chafai-Dadoun-Youssef Questions on Logarithmic Energy Monotonicity. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Powers of the Vandermonde Determinant Are Eventually Non-SNP. Consult the linked registry entry for the original report and scope.
the PIR consequence is conditional on a number-theoretic conjecture implied by either the generalized repunit conjecture or Schinzel's hypothesis H, and is unconditional for s <= 15
Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: "The proof turns conjugation, near s, into base-p arithmetic." A cycle near the fixed point splits into a coarse base permutation and a vector of carries in Z/p,…
VibeMathed recorded: Four-Terminal Planar Case of the Dinitz-Garg-Goemans Cost Conjecture. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Commutator Relators Do Not Force Hopficity, Residual Finiteness or Automaticity. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Target-Free Clique Conjecture for Threshold-Linear Networks. Consult the linked registry entry for the original report and scope.
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…
VibeMathed recorded: Dual Sequential Fat-Shattering and Tight Threshold Extraction. Consult the linked registry entry for the original report and scope.
The evenly-divided reading of WOWII Conjecture 72 holds:
⌈(A+L)/3⌉≤t,
where t= tree(G) (order of a largest induced tree), A= average eccentricity and L= maximum neighbourhood independence number.
A stronger reading that divides only L by three is false. The argument rests on two elementary observations (a diametral path is chordless and therefore induces a tre…
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.…
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: Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2). Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Pak-Slonim Conjecture on Stretched Schubert Structure Constants. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Completeness of Canonical Closure Representations Is coNP-Complete. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Optimality of Greedy for Single-Pass Semi-Streaming Matching. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Kourovka Problem 3.46 - Maximal Locally Soluble Normal Subgroups. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Kourovka Problem 18.50 - Prescribed Permuted-Product Cardinality. Consult the linked registry entry for the original report and scope.
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…
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
Proves positive lower density. The Formal Conjectures encoding asks for Set.HasPosDensity, a density that exists and is positive; erdosproblems.com says Erdos most likely meant lower density.
VibeMathed recorded: Kourovka Problem 21.150 - Rank Inequality for p-Group Extensions. Consult the linked registry entry for the original report and scope.
The headline is the constant, and its two halves have different histories. The lower bound, liminf(f(n)−2n)/(n/logn)≥4029639598/25970038185, is not new here: it is Mausberg's thirteen-layer valuation cut, posted to the erdosproblems.com forum in May 2026 and credited as such in the paper. Its author wrote there that it "does not prove an upper bound, nor does it prove that an asymptotic constant exists."…
VibeMathed recorded: Kaul-Mudrock Conjecture on the Unlabeled List Color Function. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Ji-Zhang Question on the Power Set of a Quasinilpotent Operator. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Oracle-Complexity Gap in Derivative-Free Convex Optimization. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: A Universal Leading-Residue Formula for Witten Zeta Functions. Consult the linked registry entry for the original report and scope.
exact largest modulus 3·2^{k-3} for k ≥ 5, near-linear least maximum, reciprocal mass Θ(log k), and an infinite divisor family; the counting asymptotic rests on the cited BBMST theorem
VibeMathed recorded: Klopp-Zadik Question on Polynomial-Time Node-Private Recovery. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: A Counterexample to Nivat's Conjecture for a Non-Convex Window. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Strichartz's Question on Fourier Frames for the Cantor Measure. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Stable Phase Retrieval for Spans of Independent Random Variables. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Counterexample to the Odd-Dimensional Rank Bound for Abelian p-Group Actions. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: K-Polystable Toric Fano Varieties With Small Alpha Invariants. Consult the linked registry entry for the original report and scope.
uniqueness proved for stable Harrison-Reiman systems with a nonsingular M-matrix reflection; an infinite-dimensional obstruction is shown in the larger completely-S class
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
Both bounds move, and the gap stays enormous: the lower bound rises from exp(Ω(log2k)) to exp(Ω(k1/3)) and the upper falls from exp(O(k4)) to exp(O(k2)), so A(k) is still undetermined between an exponent of k1/3 and one of k2. The paper's own closing discussion argues its lower-bound construction is near the limit of the method and that beating it needs additional randomness,…
VibeMathed recorded: Ziegler's Cross-Polytope Conjecture (simplicial 0/1-polytopes). Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Erdos's Question on the Independence Ratio of Unit-Distance Graphs. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Quadratic-Time Hardness of Furthest Pair in Superconstant Dimension. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Kontsevich's Asphericity Conjecture for Strata of Differentials. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Baker's Question on Smooth Hyperplane Sections over Finite Fields. Consult the linked registry entry for the original report and scope.
Both parts claimed answered affirmatively, with a Lean formalization; two proof claims are filed on erdosproblems.com but the problem is still listed open
VibeMathed recorded: Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case). Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Zhi-Wei Sun's Conjecture 3.4 on a Truncated Legendre-Symbol Determinant. Consult the linked registry entry for the original report and scope.
The problem statement is ambiguous: the limit-exists reading is claimed proved (Lean), while the convergence-from-hypotheses reading was disproved by a Lean-checked construction of Price that the community classes as a variant
VibeMathed recorded: Erdos's Conjecture on Consecutive Integers Free of Certain Prime Factors. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Ramachandra-Natarajan Pairwise Independent Correlation Gap Conjecture. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Papamanthou-Tollis Conjecture on Parameterized st-Orientations. Consult the linked registry entry for the original report and scope.
An existence question settled by exhibiting an object, not a general theorem: one model in the family has no exponential degeneracies for generic couplings, and nothing here says which others do.
The route is worth recording because it is not the one anyone was looking down. The author had tried and failed to find such a model directly. It surfaced instead from an unrelated classification of medium-range spin chain…
Resolved (if correct) by an independence result rather than a proof or disproof in ZFC: consistency of the positive answer is equivalent to a measurable cardinal, of the negative to ZFC alone
Both problems are answered affirmatively, in every dimension at once: every connectedness-preserving injection from a connected closed n-manifold into an n-manifold is a homeomorphism onto a component, so in particular every Darboux self-bijection of Sn and of every closed manifold is a homeomorphism. This removes the finite-H1 hypothesis of the 2020 theorem for 3-manifolds and extends it above dim…
VibeMathed recorded: Steurer's Conjecture on Vectors with Small Average Correlation. Consult the linked registry entry for the original report and scope.
Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitter…
VibeMathed recorded: Strong Convergence of Cesaro Means of Firmly Nonexpansive Iterates. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Frankl-Peng-Rodl-Talbot Question on Turan Density Intervals. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Brezis's Problems on Degenerate Constants in Degree Inequalities. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Optimal Approximation Ratio for Permanents of PSD Matrices. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Mauri and Moraga's Question on Log Calabi-Yau Pairs with Big Boundary. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Ciliberto et al. Question on Effective Divisors of Positive Self-Intersection. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Conjectures of Hopkins, Sagan-Wilson and Defant et al. on Lattices, Parking Functions and the Plactic Monoid. Consult the linked registry entry for the original report and scope.
This settles Steinerberger's third open question and separates the two models: in the disk, full chords cost you a factor growing like logL over what is achievable without the restriction. It also improves the Steinhaus-type O(L1/3) to polylogarithmic within the full-chord class.
It does not settle the Buffon discrepancy problem itself. Steinerberger's first question - whether every convex body admits a se…
VibeMathed recorded: Maximum Entropy of Sums of Independent Ternary Random Variables. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Jauslin-Kreiss-Moser Vanishing-Viscosity Selection Problem. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Escobar-Klein-Weigandt Conjecture on Cohen-Macaulay ASM Varieties. Consult the linked registry entry for the original report and scope.
the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture
VibeMathed recorded: Garamvölgyi-Jackson-Jordán Conjecture on Cliques in Minimally Globally Rigid Graphs. Consult the linked registry entry for the original report and scope.
An elementary solution via a primitive-row decomposition of the Chebyshev-node measures; the main theorem is formalized in Lean, but erdosproblems.com still lists the problem open
As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open
VibeMathed recorded: Hamilton Decompositions of the Directed 5-Torus, Odd Modulus. Consult the linked registry entry for the original report and scope.
Two independent affirmative claims (Adriano's, posted first, and a GPT-5.5 Pro note); Erdős himself wrote in 1982 that the problem had been solved affirmatively long before, without a locatable reference
For even n, let q(L)=det(L)/bn. The work proves that q(L) is even exactly when the stronger centered divisibility n2∣det(Estd) holds. For n≡2(mod4), this is equivalent to rankF2(Amod2)<n−1; for n≡0(mod4), it is equivalent to adj(Amod2)1=0. An explicit family gives odd q(L) for every n≡2(mod4), n≥6. This rem…
VibeMathed recorded: Sharp Convergence Rates for Viscous Hamilton-Jacobi Homogenization. Consult the linked registry entry for the original report and scope.
Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance
an escape path dominating every power of |z| with bounded initial length is constructed, and universal positive-power lower bounds are ruled out; the broader variant remains open
VibeMathed recorded: Lower Bounds for Lebesgue Constants and an Erdős-Turán Interpolation Problem. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: The Leading Constant for Large-Order Davenport-Schinzel Sequences. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Minimax Rate for Density Estimation under Wasserstein Contamination. Consult the linked registry entry for the original report and scope.
VibeMathed recorded: Curto et al.'s Minimality Conjecture for Threshold-Linear Networks. Consult the linked registry entry for the original report and scope.