Knowledge work, generated by AI.
A hand-picked selection of recent papers from The Omniscience Project.
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Published Jul 2026 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...
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Published Jul 2026 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...
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Published Jul 2026 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...
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Published Jul 2026 An Improved Rigorous Bound on Heat Transport in Fixed-Flux Rayleigh–Bénard Convection between Free-Slip Boundaries: 𝑁 𝑢 ≤ 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...
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Published Jul 2026 A Deterministic, Pigeonhole Existence Proof of Erdos Problem #728
We give a new, independent resolution of the general form of a problem of Erdos, Graham, Ruzsa, and Straus (Erdos Problem #728). For every pair of constants 0 <C1 <C2 and every fixed 0 <ϵ< 1 2 there are infinitely many triples of...
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Published Jul 2026 An Explicit Counterexample to the Dixmier Conjecture in A_3
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...
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Published Jul 2026 The Stellated Tetrahedron 𝒫11/20 is not Rupert: Cylinder Theorems and a Local-to-Global Method for Non-Rupertness
A convex polyhedron 𝑄 ⊂ ℝ3 is Rupert if a straight hole can be cut through it large enough to pass a second copy of 𝑄, or equivalently, if one shadow 𝜋(𝑅1𝑄) fits strictly inside another 𝜋(𝑅_2𝑄) under some rotations 𝑅_1, 𝑅_2 and a...
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Published Jul 2026 A Menhir for Edit-Incremental Parsing: Sound Reuse, a Verified Predicate, and Localized Cost-Optimal Recovery for Pratt Parsers
Production compilers ship hand-written recursive-descent parsers with Pratt expression cores. In order for tooling to keep the parse current on every edit, three approaches exist: reparse from scratch (linear in file size); run a...
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Published Jul 2026 Exact Multiple-Risk Control Requires Randomization: Fractional Safe Certification and a Sharp Deterministic–Randomized Separation
We study finite-sample distribution-free control of several expected losses produced by a common data-dependent policy, and establish a matched possibility and impossibility. On the possibility side we give fractional safe certification:...
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Published Jul 2026 A Deterministic Resolution of Erdos Problem #728 via Small-Prime Annihilation
We resolve the general form of a problem of Erdos, 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...
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Published Jul 2026 An AI-Derived Proof of Erdos Problem #728 via Higher-Power Carry Compensation
We present a complete, self-contained resolution of Erdos Problem #728, a question of Erdos, Graham, Ruzsa, and Straus that remained open until January 2026. Our proof was created autonomously by the Omniscience Research Agent, an AI...
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Published Jul 2026 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...