{"id":"c2c1518e-c8ed-4d6b-9dd9-c2097c59d040","arxiv_id":"2607.23980","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For independent nonnegative variables with mean ≤1, the sum stays below mean plus δ with probability at least (1-1/(n+δ))^n for δ≥1, proving Feige's conjecture there.","lead":"This paper proves a sharp lower bound on the probability that a sum of independent nonnegative random variables, each with mean at most 1, stays below its mean plus a positive slack δ, for δ at least 1. The result settles the long-open δ≥1 case of Feige's conjecture, a 20-year-old question in probability with links to combinatorics and algorithm analysis.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is a direct corollary of the unproved Vlassis–Thomas calibration theorem (Theorem 2.1); the geometric reduction is internally sound, but the central claim's validity is contingent on this external input.","rationale":"The reader identified the Vlassis–Thomas calibration theorem as the weakest assumption; my re-derivation of the geometric part supports that. The manuscript's internal argument is coherent: the sharpness example is correct, the shift argument is valid, and the application of Letwin–Yaskin is algebraically sound for all δ. The paper explicitly labels Theorem 2.1 as an external input, so the lack of proof is transparent but still creates a correctness risk. The claimed Lean formalization is a positive signal, but without a commit hash or artifact it is not auditable, and it only covers δ=1. Therefore the conditional verdict is appropriate, and no verdict change is needed. The concrete test to settle the concern is to verify Theorem 2.1 independently, either via the formalization or a targeted numerical search. This is a genuine, single, load-bearing point: if Theorem 2.1 fails, the entire theorem fails; if it holds, the proof is complete.","tokens_in":4868,"tokens_out":22768,"duration_ms":187984,"concrete_test":"Verify Theorem 2.1 by (a) extracting the formalized Vlassis–Thomas statement from the linked Lean repository (github.com/pengzhang91/Feige) and running the Lean compiler; if the formalization is absent or fails, (b) independently re-derive Theorem 2.1 from [VT26] and check the exact hypotheses (mean ≤1, nonnegative, finite support?), or (c) run a numerical search for counterexamples with n=2 over two-point distributions with EY_i≤1, testing whether P(K_n(Y)≤α)≤α for α∈{0.1,0.25,0.5,0.75,0.9}; any violation would refute Theorem 1.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 in Section 2.2 has two components: a deterministic geometric bound and a probabilistic calibration step. I checked the geometric bound in detail: the support-function computation h_K(-ξ)=(1/(n+1))∑y_i is correct; the projection onto 1⊥ does not alter the halfspace volume; the algebra for both branches (0<δ<1 and δ≥1) matches Theorem 2.2 exactly; and the inequality direction P(∑y_iD_i≥1) ≥ P(∑y_iD_i≥(∑y_i)/(n+δ)) is valid because (∑y_i)/(n+δ) ≥ 1 under ∑y_i≥n+δ. The shift Y_i=X_i+1-μ_i is nonnegative with mean 1, and the event inclusion is correct. Thus the only load-bearing assumption is Theorem 2.1, stated without proof and cited to [VT26], a preprint posted 18 days before this manuscript. If Theorem 2.1 is false, or has an unstated condition (e.g., a support or continuity restriction), the final inequality P(S≥ES+δ) ≤ P(K_n(Y)≤1-b_{n,δ}) ≤ 1-b_{n,δ} collapses. No internal inconsistency is apparent, but the central claim is exactly as strong as this external theorem, and the manuscript provides no proof or auditable formal certificate for it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a sharp small-deviation lower bound for sums of independent nonnegative random variables with mean at most one. For n≥1 and δ>0, it proves P(S<ES+δ) ≥ b_{n,δ}, with b_{n,δ}=δ(n/(n+δ))^n for 0<δ<1 and b_{n,δ}=(1-1/(n+δ))^n for δ≥1. Since b_{n,δ}≥e^{-1} for δ≥1, the paper claims Feige's conjecture is resolved in the affirmative for δ≥1. The proof shifts X_i to Y_i=X_i+1-μ_i, which are nonnegative with mean 1, then bounds a Dirichlet tail K_n(Y) geometrically via a generalized Grünbaum inequality, and finally applies an external calibration theorem of Vlassis and Thomas to convert the tail bound into a probability inequality. The sharpness example for δ≥1 is explicit and gives equality. The geometric part is internally sound, but the central probability inequality depends entirely on Theorem 2.1, which is stated without proof and cited to a preprint posted only 18 days before the manuscript.","tokens_in":5190,"tokens_out":9747,"duration_ms":97176,"significance":"If the external Theorem 2.1 is valid, this is an elegant and significant reduction: it settles Feige's conjecture for the entire range δ≥1, a long-standing open problem, and it also gives a new bound for 0<δ<1. The proof has several strengths: the centering shift Y_i=X_i+1-μ_i is natural and preserves the hypotheses; the support-function computation h_K(-ξ)=(1/(n+1))∑ y_i is correct; the projection of the shifted simplex to R^n is handled correctly; the algebra matching the two branches of Theorem 2.2 to b_{n,δ} checks out; and the sharpness example attains equality for all n and δ≥1. There is no evident circularity: the derivation does not use Feige's conjecture and no parameters are fitted to force the result. The paper does not itself contain a machine-checked proof; it references an external Lean formalization, but the artifact is not included in the submission. The decisive caveat is that Theorem 2.1 — the calibration theorem of Vlassis and Thomas — is used as a black box. The presented proof is therefore conditional on an unpublished, unpeer-reviewed external theorem.","major_comments":[{"comment":"Theorem 2.1 is the load-bearing ingredient of the paper. In Section 2.2, the proof applies it with α=1-b_{n,δ} to conclude P(S≥ES+δ)≤1-b_{n,δ}. If Theorem 2.1 is false, or if it has an unstated support/continuity restriction, the final inequality (1.2) collapses, regardless of the correctness of the geometric estimate. The manuscript provides no proof of Theorem 2.1 and cites only [VT26], a preprint posted 18 days before this paper. For a serious journal, this is not sufficient for a central theorem whose statement is at least as strong as the paper's main result. The authors must include a self-contained proof of Theorem 2.1 in an appendix, or give a peer-reviewed reference, or provide a machine-checked certificate as part of the submission. Without this, the paper cannot independently claim to prove Feige's conjecture for δ≥1.","section":"Section 2.1 (Theorem 2.1) and Section 2.2"},{"comment":"The manuscript states that an accompanying Lean formalization 'provides an end-to-end formal proof of Feige's e^{-1} conjecture' and that the initial proof was found by ChatGPT 5.6 Pro. The Lean artifact is not included in the submission; only a URL is mentioned. If the formalization is intended to address the unproved status of Theorem 2.1, it must be supplied and its scope stated precisely, including which parts of the proof are formalized and which external theorems are assumed. As written, the claim about a machine-checked proof is not verifiable from the manuscript and should be either substantiated or removed.","section":"Statement on AI use and abstract"}],"minor_comments":[{"comment":"The sentence 'The proof is found by ChatGPT 5.6 Pro' is unusual and, if kept, should be moved to the AI-use statement rather than appearing in the abstract. The mathematical content is unaffected, but the wording is inappropriate for a formal research paper.","section":"Abstract and Section 1"},{"comment":"The reference [VT26] is a preprint posted 9 July 2026, and [MRS+26] is posted 21 July 2026. These are very recent and not yet peer-reviewed. Please provide stable publication data when available, and for [LY24] cite the final journal version if it has appeared.","section":"References"},{"comment":"The geometric step is correct but quite compressed. The transition from the event {∑ y_iD_i ≥ (1/(n+δ))∑ y_i} to the halfspace intersection H_{(1-δ)/(n+δ),ξ} is not shown in detail. Adding two lines of algebra, and explicitly checking that α=(1-δ)/(n+δ) lies in the required range (-1,1/n) for all δ>0, would improve readability and make the proof easier to audit.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a reduction of Feige's conjecture to the Vlassis–Thomas calibration theorem, plus a sharp geometric estimate. The geometric reduction is correct and the sharpness example is a nice asset. However, the central claim is conditional on an unproved, unpublished external theorem. I would not reject outright because the dependency is explicit and fixable: a proof of Theorem 2.1, or a formalized certificate, would make the paper self-contained. The novelty, once the external theorem is supplied, may be moderate but the result would be significant. I am not concerned about circularity, and I found no internal inconsistency."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper gives a sharp small-deviation bound for sums of independent nonnegative variables and claims to prove Feige's conjecture for δ≥1. The internal math is clean and the reduction is elegant; the whole result, however, rests on a single external theorem stated without proof. Right now the paper is a conditional corollary of that theorem.\n\nWhat is actually new: Theorem 1.1 states a sharp two-branch bound b_{n,δ}. The δ≥1 branch resolves Feige's conjecture with the optimal constant e^{-1}. Previous work only handled i.i.d. or log-concave cases or weaker constants. The sharpness construction is correct: a two-point distribution achieves the bound. The proof itself is a neat shift: Y_i = X_i + 1 - μ_i turns the event S ≥ E S + δ into ΣY_i ≥ n + δ, and then the Dirichlet calibration theorem plus the Letwin–Yaskin generalization of Grünbaum's inequality gives the bound. I checked the geometry: the support function computation h_K(-ξ) = (1/(n+1))Σy_i is right, the projection onto 1^⊥ is harmless, and the algebra matches Theorem 2.2 both for 0<δ<1 and δ≥1. No circularity or fitting anywhere; the citation pattern looks honest.\n\nThe soft spot, and it's a real one: Theorem 2.1 (Vlassis–Thomas) is a preprint posted 18 days before this one, and it is used as a black box with no proof. The final inequality is exactly P(K_n(Y) ≤ 1 - b_{n,δ}) ≤ 1 - b_{n,δ}; if that theorem has a hidden condition or is false, the whole result collapses. The Lean formalization is mentioned but no commit hash or artifact is provided, and it covers only δ=1. The authors need to supply a proof or a precise formal certificate for Theorem 2.1 before the result is usable. The Letwin–Yaskin theorem is less concerning—it's to appear in Israel J. Math.\n\nThe paper is a genuine contribution in outline, and deserves a serious referee. Send it out, but the referees should be asked to verify the Vlassis–Thomas theorem independently, and the authors should provide the missing proof or a machine-checkable version of it. Then, if it holds up, this is a major within-field result.\n\nReading group: maybe—worth discussing if someone wants to see the reduction. I wouldn't cite it in my own work until the external theorem is checked.","headline":"A clean reduction that would settle Feige's conjecture for δ≥1, if the unproved Vlassis–Thomas calibration theorem holds—currently a conditional result.","tokens_in":5715,"tokens_out":3552,"would_cite":false,"duration_ms":31321,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60E15","52A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a sharp lower bound for the probability that a sum of independent nonnegative random variables falls below its expected value plus a slack δ, confirming the e^{-1} conjecture for all δ≥1.","keywords":["small deviations","sums of nonnegative random variables","e^{-1} conjecture","sharp inequality","first moment bound","simplex calibration","halfspace volume","centroid inequality"],"falsifier":"Find independent nonnegative variables Y_i with E Y_i ≤ 1 and a number α∈(0,1) for which P(K_n(Y) ≤ α) exceeds α, where K_n uses the uniform simplex distribution. Such a counterexample would refute the calibration theorem and hence the paper's main inequality. The paper does not supply its own proof of the calibration theorem, so this is the most direct test.","tokens_in":4745,"feed_emoji":"🎲","tokens_out":11209,"duration_ms":100233,"temperature":0.7,"pith_summary":"The paper establishes a sharp small-deviation inequality for sums of independent nonnegative random variables subject only to first-moment constraints. Specifically, if E X_i ≤ 1 and S = ∑ X_i, then the chance that S < E S + δ is at least b_{n,δ}, a closed-form expression that for δ≥1 never falls below e^{-1} and is exactly attained by a two-point distribution for every n. This settles a long-standing open conjecture in the range δ≥1 and provides the best possible universal constant. For the remaining range 0<δ<1, the paper obtains the explicit but weaker bound δ e^{-δ}. The proof reduces the probabilistic question to a volume estimate on the simplex via a calibration theorem for simplex-weighted sums.","feed_headline":"e^{-1} bound proven sharp for δ≥1","feed_subtitle":"For δ≥1, any such sum stays below mean+δ with probability at least e^{-1}, and the bound is achieved.","key_machinery":"The key object is the function K_n(y) = P_D(∑ y_i D_i ≤ 1), with D uniform on the standard n-simplex. It serves as both a probability and a geometry: it is the normalized volume of a halfspace slice of the simplex, and it is a calibrated statistic for independent nonnegative random variables with unit means. The proof's machinery is the interplay between these two roles — the event {S ≥ E S + δ} is mapped to {K_n(Y) ≤ 1 - b_{n,δ}}, and the generalized centroid inequality bounds the volume of the corresponding slice from below. In effect, the problem becomes a convex-geometric one, with the calibration theorem supplying the final probabilistic step.","core_discovery":"The central claim is Theorem 1.1: for independent nonnegative X_i with E X_i ≤ 1, the small-deviation probability P(S < E S + δ) is at least b_{n,δ} = δ(n/(n+δ))^n for 0<δ<1 and b_{n,δ} = (1 - 1/(n+δ))^n for δ≥1, with equality attained for every n and δ≥1. The proof has two independent ingredients. A calibration theorem, taken as an external input, states that for independent nonnegative variables with unit means, the function K_n(Y) = P_D(∑ Y_i D_i ≤ 1), where D is uniform on the standard simplex, is stochastically no smaller than uniform. The geometric ingredient bounds K_n(y) by 1 - b_{n,δ} whenever ∑ y_i ≥ n+δ, using the classical centroid inequality for convex bodies and its recent exte","pith_inferences":["The gap between the new bound and the conjectured sharp bound in the range 0<δ<1 stems entirely from the geometric estimate; a tighter centroid-type inequality for halfspaces that do not contain the centroid would likely close the gap.","The two-point sharpness example for δ≥1 suggests that for δ<1 the extremal distribution may require more than two atoms; the paper's method gives no characterisation of the extremal law in that regime.","The calibration theorem at the core of the proof originates in distribution-free testing; read in that direction, the e^{-1} bound may translate into a finite-sample coverage guarantee for a nonparametric confidence interval, a consequence the paper leaves implicit.","Because the paper's main theorem inherits the calibration theorem as an unproved input, the result's full strength depends on that theorem's validity; any weakening of its hypotheses would extend the inequality to broader classes of random variables."],"forward_implications":["For every n and δ≥1, the inequality P(S < E S + δ) ≥ e^{-1} holds, matching the sharp constant; this confirms the e^{-1} conjecture in that range.","For δ≥1, the bound b_{n,δ} is optimal for each fixed n, as demonstrated by the two-point example that attains it.","For 0<δ<1, the paper establishes the explicit bound P(S < E S + δ) ≥ δ e^{-δ}, weaker than the conjectured min{δ/(1+δ), e^{-1}}.","The proof shows that the probabilistic inequality follows from a halfspace volume estimate for the simplex, so the result holds for any first-moment-constrained independent nonnegative variables.","The paper reports a fully formalized proof of the e^{-1} case, so the main argument can be checked mechanically."],"fun_headline_variants":["Feige's conjecture true for δ≥1","Sharp e^{-1} bound for sum tails","Probability ≥ e^{-1} is optimal for δ≥1","Small-deviation inequality proven sharp","δ≥1: Feige's conjecture resolved"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is the calibration theorem taken as external input: for independent nonnegative variables with unit means, the simplex-weighted probability K_n(Y) satisfies P(K_n(Y) ≤ α) ≤ α for all α. If this fails, the final step of the proof collapses even though the geometric halfspace bound is correct.","fun_headline_variants_meta":{"raw":{"variants":["Feige's conjecture true for δ≥1","Sharp e^{-1} bound for sum tails","Probability ≥ e^{-1} is optimal for δ≥1","Small-deviation inequality proven sharp","δ≥1: Feige's conjecture resolved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000337,"raw_usage":{"total_tokens":1755,"prompt_tokens":852,"completion_tokens":903,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":832}},"tokens_in":596,"tokens_out":903,"duration_ms":8794,"temperature":1.0,"reasoning_tokens":832,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T23:22:33.431768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find independent nonnegative variables Y_i with E Y_i ≤ 1 and a number α∈(0,1) for which P(K_n(Y) ≤ α) exceeds α, where K_n uses the uniform simplex distribution. Such a counterexample would refute the calibration theorem and hence the paper's main inequality. The paper does not supply its own proof of the calibration theorem, so this is the most direct test.","supporting_citations":[],"review_version":1}