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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…
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…
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…
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…
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…
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…
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.
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…
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…
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…
VibeMathed recorded: Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case). 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: Kourovka Problem 3.46 - Maximal Locally Soluble Normal Subgroups. 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.