number-theoryAug 10, 2026Significance 68/100Registry: lean checked
Prior state unknown→proved
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 Montg…
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combinatoricsJul 10, 2026Significance 55/100Registry: lean checked
Prior state unknown→proved
Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.
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geometry-topologyAug 19, 2026Significance 55/100Registry: lean checked
Prior state unknown→disproved
Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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analysisAug 5, 2026Significance 25/100Registry: lean checked
Prior state unknown→proved
For which lattice parameters does a totally positive window function generate a Gabor frame? Gröchenig and Stöckler initiated the program in 2013; this paper gives the complete characterization, together with a Kadets-type theorem for shift-invariant spaces.
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number-theoryAug 21, 2026Significance 20/100Registry: lean checked
Prior state unknown→proved
Answers Problem 3 and generalizes it: the classification $\sqrt{m} \in \mathcal{S} \iff m = 2$ covers every square root, and a further theorem replaces parity by divisibility by any $p \ge 2$. 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 sect…
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number-theoryFeb 4, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
In the circle of Kummer's regular primes and Vandiver's conjecture, the paper proves that almost all primes are partially regular, yielding a partial Vandiver theorem for a density-one set of primes, with consequences for Kubota-Leopoldt p-adic L-functions, Eisenstein congruences and K-theory torsion.
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geometry-topologyMar 25, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
A density-1 obstruction, not a full resolution: the conjecture that the hard window contains no lattice triangles remains open on a density-0 set.
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combinatoricsAug 6, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
Dimension 5 only; public AI-generated candidate with no independent specialist review.
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combinatoricsJul 30, 2026Significance 15/100Registry: lean checked
Prior state unknown→proved
With $N = 2^k - 1$ and $\mathrm{wt}(n)$ the binary Hamming weight, Tu and Deng conjectured that for every $1 \leq t \leq N-1$ at most $2^{k-1}$ pairs $(a,b)$ satisfy $a + b \equiv t \pmod N$ and $\mathrm{wt}(a) + \mathrm{wt}(b) < k$. Proved in full.
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number-theoryMar 31, 2026Significance 15/100Registry: lean checked
Prior state unknown→disproved
For $A \subseteq \mathbb{F}_p$ let $A^* = (A+A) \cup (AA)$. Sárközy conjectured that for all large primes, every set of size at least $c\sqrt{p}$ has $A^* = \mathbb{F}_p$-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together with the exact extremal value.
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number-theoryAug 1, 2026Significance 12/100Registry: lean checked
Prior state unknown→proved
New theorems, not a formalisation of previously known results. For every base $b\ge2$ the Erdős–Kac law is established for the $\lambda$-digit base-$b$ palindromes and for the base-$b$ reversals of the $\lambda$-digit primes, for $\omega$ and $\Omega$ and for $\omega_S,\Omega_S$ with any regular set $S$ of primes; with normal order $\log\log n$ on both families, and, for $\omega$, all moments of order up to $\tfra…
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number-theoryAug 6, 2026Significance 12/100Registry: lean checked
Prior state unknown→proved
Proves the a = 3 layer; the conjecture is layered in a and remains open for larger a.
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combinatoricsAug 5, 2026Significance 12/100Registry: lean checked
Prior state unknown→proved
The multivariate independence polynomial is the partition function of the hard-core model with per-vertex fugacities. The paper proves a lower bound extending to the multivariate setting a result Tao proved in the univariate case, and settles a conjectured generalization for a multiaffine version of the semiproper colouring partition function with two proper colours.
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number-theoryAug 1, 2026Significance 11/100Registry: lean checked
Prior state unknown→proved
For every integer-valued strongly $b$-additive $g$ with $\gcd(g(1),\dots,g(b-1))=1$ and digit mean $\mu_g\ge0$: $g(p)$ is prime for infinitely many primes $p$. For $\mu_g>0$, $\sum 1/p$ over $p<X$ with $g(p)$ prime is $(d_g/\varphi(d_g))\log_3X + C_{g,1} + O(1/\log\log X)$, likewise for the first $j$ iterates. Also $\#\{p\le x: g(p)\text{ prime}\}\ll\pi(x)/\log\log x$, of that exact order on a large set of $x$, an…
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algebraAug 19, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
For the Kasami APN function $F(x) = x^{4^k - 2^k + 1}$ on $\mathrm{GF}(2^n)$ with $\gcd(k, n) = 1$, the conjecture asserts that for $\Delta = \{F(b) + F(b+1) + 1\}$ and all distinct nonzero $v_1, v_2$, the number of triples in $\Delta^3$ with $v_1 x + v_2 y + (v_1 + v_2) z = 0$ is exactly $2^{2n-3}$. Proved for $k \bmod n \in \{1, 2, n-2, n-1\}$ and verified exhaustively for $n \le 13$; the general case remains open.
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combinatoricsApr 16, 2026Significance 10/100Registry: lean checked
Prior state unknown→disproved
Erdős asked whether every $n$-point set in Euclidean space whose pairwise distances are mutually at least 1 apart must have diameter at least $(1+o(1))n^2$. Disproved: an explicit high-dimensional construction beats the conjectured constant.
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number-theoryAug 15, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
For every real $\xi>0$ the sequence of integer parts $[\xi 7^{n}]$, $n=0,1,2,\dots$, contains infinitely many composite numbers. Second, there is no infinite right truncatable prime in base~$7$.
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algebraFeb 3, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
Does the conjectured universal formula for normalized alternating syzygy power sums of numerical semigroup rings hold for every index?
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geometry-topologyFeb 3, 2026Significance 10/100Registry: lean checked
Prior state unknown→proved
For odd $k$ with $\gcd(n,k) = \gcd(n+1,k) = 1$, is $N_k(n) \equiv \lfloor (k+1)/4 \rfloor \pmod 2$, where $N_k(n)$ counts pairs $1 \le b_i \le (k-1)/2$ with $b_1 + b_2 \ge (k+1)/2$ and $b_2 \equiv n b_1 \pmod k$? Conjectured by Chen and Gendron; its proof removes a conditional step in the genus-zero and genus-one spin-parity classification.
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mathematical-physicsAug 9, 2026Significance 8/100Registry: lean checked
Prior state unknown→proved
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.
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combinatoricsJan 27, 2026Significance 8/100Registry: lean checked
Prior state unknown→proved
A generalization of Boppana's entropy inequality, of the kind used in union-closed-sets arguments, proved and formalized: the sharp form with the extremal constant characterized via the unique positive solution of an explicit equation.
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number-theoryApr 20, 2026Significance 8/100Registry: lean checked
Prior state unknown→proved
Nathanson asked which subsets of $\mathbb{N}$ can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.
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algorithms-optimizationJun 18, 2026Significance 7/100Registry: lean checked
Prior state unknown→disproved
Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.
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combinatoricsJul 31, 2026Significance 5/100Registry: lean checked
Prior state unknown→proved
Settles the conjecture for a large family and reduces the rest to unit fractions; the general unit-fraction case remains open.
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mathematical-physicsAug 14, 2026Significance 3/100Registry: lean checked
Prior state unknown→proved
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 $a=\pm b$. Not settled here: pro…
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