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Fractional Safe Certification: Exact Multiple-Risk Control and a Sharp Boundary Separation

Can a calibration rule satisfy exact expected-risk constraints while retaining the utility of a feasible policy at an atomic risk boundary? We study this question for bounded policy losses under exchangeability. Given a candidate...

A certified lower bound μ ≥ 0.380020 for Erdos’ minimum overlap problem

Let μ denote the limiting constant in Erdos’ minimum overlap problem. We prove μ ≥ 0.380020, improving the previous certified lower bound 0.37912; the best known upper bound is 0.380868, so this closes about 51% of the remaining gap. The...

The closed curve of length 4π closest to the sphere

Kimberling’s tenth unsolved problem asks for optimal parametric equations for a simple closed curve of arclength 4πon the unit sphere minimizing the mean spherical distance from the sphere to the curve. Precisely, for a simple closed...

Ten new cases of a conjecture of Chen and Xu on primes plus sparse power sums

Let (F_n) be the Fibonacci sequence. Chen and Xu conjectured that if r_1,...,r_t are positive integers with r−1 1 +···+ r−1 t = 1, then there is an infinite arithmetic progression of positive integers none of which is representable as p+...

Integer Tiles with Unsupported Cyclotomic Divisors at Three Primes

Let A(X) = a∈A Xa denote the mask polynomial of a finite subset of a cyclic group. Kiss, Łaba, Marshall, and Somlai asked whether a set satisfying the Coven–Meyerowitz conditions (T1) and (T2) can have an unsupported cyclotomic divisor....

Quadratic Generation for the Canonical-Form Embeddings of X_cf(3, 6) and Y(3,6), and Kozulness for del Pezzo Complements

We resolve Conjecture 5.5 of Hollering–Pavlov–Pratt (Log Canonical Models and Positive Geometries, arXiv:2607.28368) in the affirmative, for both of its assertions. First, the homogeneous vanishing ideal of the canonical-form embedding...

Linear independence of additive convolutions of multiplicative coset indicators: an existence theorem for index-p cyclotomy

Let q = fp + 1 with p, q odd primes, let H ≤ F × q be the subgroup of order f (index p), and let Ai = gki H (i = 1, 2, 3) be three of its cosets. We study the 23 = 8 subset convolutions μS = ∗i∈S 1Ai (S ⊆ {1, 2, 3}) as elements of the...

The Turán density of H_r_t is Θ_t(1/r)

Let H_r_t be the r-uniform hypergraph with r + 1 vertices and t edges, also known as the (t, t−1)-daisy. We determine its Turán density up to a constant depending only on t: π(Hr t ) = Θt(1/r) for every fixed t ≥ 3, with an explicit...

New Bounds for Sequences with Distinct Consecutive Sums in Erdős Problem 357

In Erdős Problem 357, let 𝑓(𝑛) be the largest integer 𝑘 for which there are integers 1 ≤ 𝑎_1 < ⋯ < 𝑎_𝑘 ≤ 𝑛 such that all sums of consecutive terms 𝑎_𝑢 + ⋯ + 𝑎_𝑣 are distinct. Erdős and Harzheim asked for the order of 𝑓(𝑛) and, in...

The Non‑Commutative Rung of Correlated‑Secret LWE: A Division-Algebra Classifier and a Cyclic-Algebra-LWE Reduction, with a Conditional Worst-Case Foundation

A decade of lattice constructions — GGH15 encodings, matrix PRFs, and most recently the Diamond-iO all-product assumption — rest on a pattern nobody has managed to reduce to standard LWE: many samples 𝑐𝑤 = 𝑠𝑃𝑤𝐴 + 𝑒𝑤 from one shared...

An Explicit High-Degree Threshold for Sendov’s Conjecture

Let p be a polynomial of degree n ≥2 whose zeros lie in the closed unit disk. Sendov’s conjecture asserts that every zero aof plies within distance 1 of a zero of p′. Tao proved that the conjecture holds for all sufficiently large...

An Improved Rigorous Bound on Heat Transport in Fixed-Flux Rayleigh–Bénard Convection between Free-Slip Boundaries: 𝑁𝑢 ≤ 0.5500 𝑅^1/3

The background-field method of Doering and Constantin yields rigorous upper bounds on turbulent heat transport, and improving the explicit constants in these bounds for canonical flows is a long-standing open problem. For...

Irreducible Covering Sets A Solution of Erdős Problem 1189

A finite set of distinct integers 1 <n1 <···<nk is called a covering set if one can choose one residue class modulo each ni so that the resulting classes cover Z. It is irreducible if no proper subset is a covering set. We determine the...

An AI-Derived Proof of Erdos Problem #728 via Higher-Power Carry Compensation

We present a complete, self-contained resolution of Erd˝os Problem #728, a question of Erdos, Graham, Ruzsa, and Straus that remained open until January 2026. Our proof was created autonomously by the Paratelligent Research Agent, an AI...

A Deterministic, Pigeonhole Existence Proof of Erdos Problem #728

We give a new, independent resolution of the general form of a problem of Erd˝os, Graham, Ruzsa, and Straus (Erd˝os Problem #728). For every pair of constants 0 <C1 <C2 and every fixed 0 <ϵ< 1 2 there are infinitely many triples of...

A Deterministic Resolution of Erdos Problem #728 via Small-Prime Annihilation

We resolve the general form of a problem of Erd˝os, Graham, Ruzsa, and Straus (Erdos Problem #728) in the affirmative. For every pair of constants 0 < C1 < C2 and every fixed 0 <ϵ< 1 2 , there are infinitely many triples of positive...

Erdos Problem 394 and the average order of t_2: a counterexample to the strong form

For n≥1 let t2(n) be the least m≥1 with n|m(m+ 1). Erd˝os Problem 394 [3] asks, in its weak form, whether n≤x t2(n) ≪x2/(log x)c for some c > 0; the remarks to the problem record a strong form due to Erd˝os and Hall, that the sum is o...

An Explicit Counterexample to the Dixmier Conjecture in A3

We exhibit an explicit injective, non-surjective endomorphism of the third Weyl algebra A3(k) over an arbitrary field k of characteristic zero, disproving the Dixmier conjecture in di- mension three. The input is a polynomial map F: A3 k...

Certified Finite-Key Minimum-Distance Bounds for Sparse QC-MDPC Ensembles: The BIKE Case and a Cyclotomic-Kernel Extension

We first prove an exact, non-asymptotic upper bound on the probability that a BIKE-type quasi-cyclic moderate-density parity-check code contains a low-weight codeword. For the BIKE parameter condition that the block length r is prime and...

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