Sharp resolution of the 20-year-old Feige conjecture via a novel bridge to distribution-free testing, with broad downstream applications, tempered by heavy reliance on recent external results and the still-open general case.
Abstract
We present a short proof of Feige's conjecture: for independent nonnegative random variables with expectation one, the probability that their sum is less than is at least . The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthrough of Vlassis and Thomas establishing Gaffke's conjecture on the finite-sample validity of a distribution-free -value. We also discuss the implications of the subsequent work of Ming, Ramdas, Shen, Wang, and Waudby-Smith.
AI Impact Assessments
(1 models)Scientific Impact Assessment
Core Contribution
This paper presents a short, self-contained proof of Feige's conjecture: for independent nonnegative random variables with expectation one, . This resolves a conjecture open since Feige's 2006 paper, where the original bound of was successively improved to , , and over fifteen years. The paper achieves the *sharp* constant , with matching extremal examples. More generally, Theorem 1.3 establishes for , and Section 3 recovers the sharp two-variable form for all via an admissible merger .
The central novelty is a conceptual bridge: the authors reformulate the small-deviation problem in terms of "mergers" — functions arising from recent work on distribution-free hypothesis testing (Vlassis–Thomas's resolution of Gaffke's conjecture) — and reduce Feige's conjecture to a pointwise bound on the Dirichlet-based merger . That bound in turn follows from a Grünbaum-type geometric inequality on half-space sections of a simplex (Letwin–Yaskin). This unexpected route connects previously separate literatures (finite-sample statistics, convex geometry, and probabilistic concentration).
Methodological Rigor
As a pure-theory paper the argument is clean and the reduction (Lemma 2.2) is elementary and correct. The main proof leans heavily on three external results: the merger property of (VT26), the geometric inequality (LY24), and the merger (MRS+26). Conditional on those, the derivations here are transparent and verifiable in a few pages. The sharpness claim is substantiated with explicit extremal distributions. The chief methodological caveat is dependency: the paper's correctness is contingent on very recent, still-arXiv results that have not fully settled into the literature.
Potential Impact
Feige's inequality is a workhorse dimension-free tail bound cited across randomized graph algorithms, extremal combinatorics (Erdős matching, nonnegative -sums), runtime analysis of evolutionary/estimation-of-distribution algorithms, and models of innovation diffusion in economics. Sharpening the constant to its optimal value tightens all downstream applications and closes the headline case of a well-known problem. Beyond the specific result, the merger-based methodology is likely to be reused: it offers a template for attacking the still-open arbitrary- conjecture (Conjecture 1.1) and possibly other small-deviation questions, and it demonstrates a productive transfer of tools from the anytime-valid inference / e-value community into classical probability.
Timeliness & Relevance
The paper is highly timely, exploiting a cluster of 2026 breakthroughs (Vlassis–Thomas, Ming et al.) almost immediately. It also documents AI-assisted mathematical discovery (GPT-assisted, human-verified), which is itself a topical development in how research mathematics is conducted — though the mathematical merit stands independently.
Strengths & Limitations
Strengths: (1) resolves a famous conjecture with the sharp constant; (2) the proof is remarkably short and elegant given the fifteen-year history of incremental improvements; (3) a genuinely novel and non-obvious cross-disciplinary connection; (4) explicit sharpness. Limitations: (1) the general version remains open, and the authors note that higher-dimensional analogues of are substantially harder; (2) the result is heavily scaffolded on other authors' recent breakthroughs — much of the "heavy lifting" (the merger property, the geometric inequality) is imported, so the marginal contribution is the clever assembly rather than a new deep tool; (3) Section 3 largely reproduces already-known two-variable results for expository comparison.
Additional Observations
The paper is a good example of a "keystone" contribution — modest in length but placing the final stone on a long-standing structure. Its foundationality is moderate-to-high: the merger reduction is a reusable primitive, but the specific theorem is a closed result. Resource intensity is minimal (pencil-and-paper plus specialist insight). The unusual future-dated references and AI-assistance framing are notable but do not affect the mathematical assessment.
Generated Jul 28, 2026
Comparison History (15)
Paper 2 likely has higher impact: resolving Feige’s conjecture is a crisp, long-standing result in probability/combinatorics with broad downstream implications (e.g., concentration/anti-concentration, randomized algorithms, statistical inference via p-values) and immediate citability. The connection to recent distribution-free inference work increases timeliness and cross-field relevance. Paper 1 is rigorous and valuable for McKean–Vlasov SDE theory, but appears more specialized; its advances (well-posedness, functional inequalities, convergence rates) are impactful mainly within stochastic analysis and mean-field dynamics.
Paper 1 offers a broader, more generalized mathematical contribution by establishing sharp small-deviation inequalities for any positive delta, rather than focusing solely on the specific threshold of Feige's conjecture (delta = 1) like Paper 2. By bridging statistical breakthroughs with convex geometry, Paper 1 introduces a richer methodological framework. Its generalized, sharp bounds will likely yield wider applications in probability theory, concentration of measure, and randomized algorithms compared to the shorter, narrower proof presented in Paper 2.
Paper 2 resolves Feige's conjecture, a long-standing open problem in probability theory with broad implications for concentration inequalities and statistical inference (connections to distribution-free p-values, Gaffke's conjecture). Resolving a named conjecture has clear, high-visibility impact across probability and statistics. Paper 1 is technically rigorous but incremental, extending known hydrodynamic fluctuation results to a specific two-species exclusion model with slow boundaries—valuable within its subfield but narrower in reach and timeliness compared to settling a famous conjecture with cross-disciplinary relevance.
Paper 2 introduces a genuinely novel mathematical object (Brownian loop-catchers) with deep connections to SLE, loop soups, and conformal field theory across central charges. It develops new tools (entangled multipath LERW, Green function test) with sharp results and even extends to 3D. This represents substantial, foundational contributions to probability and mathematical physics. Paper 1, while providing a clean proof of a known conjecture, is narrower in scope and builds heavily on recent existing breakthroughs. Paper 2's breadth, methodological innovation, and lasting theoretical significance give it higher potential impact.
Paper 1 resolves a fundamental problem (Feige's conjecture) in probability theory, offering broad theoretical implications for statistics, randomized algorithms, and distribution-free inference. Furthermore, its use of an advanced AI model represents a significant methodological milestone in automated theorem proving. In contrast, while Paper 2 provides rigorous technical advancements in stochastic analysis and mathematical finance, its reliance on a regularized square root makes it a specialized, incremental contribution compared to the foundational, interdisciplinary breakthrough presented in Paper 1.
Paper 1 delivers major, technically deep advances in high-dimensional percolation: it proves long-standing conjectures (Aizenman–Newman; Hara–Slade; van der Hofstad–Slade), establishes super-Brownian/CRT scaling limits in multiple intrinsic metrics, and derives sharp one-arm asymptotics. This is a substantial, broadly influential contribution to probability, statistical physics, and geometric scaling limits with high methodological rigor. Paper 2 resolves an important conjecture with a short proof and has relevance to probability/statistics, but its scope and downstream theoretical reach appear narrower than the comprehensive structural results in Paper 1.
Paper 2 likely has higher impact because it claims a proof of Feige’s conjecture, a well-known open problem in probabilistic combinatorics/inequalities; resolving such a conjecture is highly novel, timely, and broadly influential, with downstream implications in theoretical CS, probability, and statistics (via links to distribution-free inference). If correct and rigorously vetted, it would be a major result. Paper 1 is strong and applicable (dimension-free rates for Markov/SPDE empirical measures via sliced Wasserstein), but is more incremental within an active area and likely narrower in transformative impact.
Paper 1 resolves a named mathematical conjecture with broad implications for statistical hypothesis testing and distribution-free p-values. Furthermore, its integration of advanced AI (GPT-5.6 Sol) for theorem-proving represents a highly novel, timely, and disruptive methodological paradigm shift in mathematics, giving it significantly higher cross-disciplinary impact than Paper 2's purely theoretical reframing of a classical inequality.
Paper 1 resolves a major, widely recognized mathematical problem (Feige's conjecture). Furthermore, its pioneering use of advanced AI for theorem proving represents a paradigm shift in mathematical research, offering immense novelty, timeliness, and cross-disciplinary relevance. While Paper 2 presents a rigorous and highly specialized advancement in stochastic processes with solid financial or statistical applications, Paper 1's fundamental probabilistic breakthrough and innovative AI methodology guarantee a significantly broader and deeper scientific impact.
Paper 2 resolves Feige's conjecture, a well-known open problem in probability with connections to distribution-free inference and statistics, giving it broad appeal across probability, statistics, and combinatorics. Solving a named conjecture typically has higher visibility and impact than an incremental convergence-rate result. Paper 1 is a solid, technically rigorous contribution to numerical analysis of fractional SDEs, but its impact is narrower and more specialized. The resolution of a conjecture with implications for statistical methodology gives Paper 2 broader significance.
Sharp resolution of the 20-year-old Feige conjecture via a novel bridge to distribution-free testing, with broad downstream applications, tempered by heavy reliance on recent external results and the still-open general case.