{"schema_version":"claim-reader-v1","record":{"available_versions":[2,1],"current_version":2,"assessment_summary":"consistent","claim":{"statement":"For every n ≥ 1, let X₁,…,Xₙ be independent nonnegative integrable random variables with E Xᵢ ≤ 1, and put S = Σᵢ Xᵢ. For every δ > 0, P(S < E S + δ) ≥ δ(n/(n+δ))^n when 0 < δ < 1, and P(S < E S + δ) ≥ (1−1/(n+δ))^n when δ ≥ 1. The latter bound is sharp for each fixed n and δ ≥ 1.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every n ≥ 1, let X₁,…,Xₙ be independent nonnegative integrable random variables with E Xᵢ ≤ 1, and put S = Σᵢ Xᵢ. For every δ > 0, "},{"kind":"math","text":"P(S < E S + δ) ≥ δ(n/(n+δ))^n","latex":"\\mathbb P(S<\\mathbb ES+\\delta)\\ge\\delta\\left(\\frac n{n+\\delta}\\right)^n"},{"kind":"text","text":" when 0 < δ < 1, and "},{"kind":"math","text":"P(S < E S + δ) ≥ (1−1/(n+δ))^n","latex":"\\mathbb P(S<\\mathbb ES+\\delta)\\ge\\left(1-\\frac1{n+\\delta}\\right)^n"},{"kind":"text","text":" when δ ≥ 1. The latter bound is sharp for each fixed n and δ ≥ 1."}]},"id":"claim-feige-july-deviation-bounds","slug":"feige-july-deviation-bounds","version":2,"kind":"bound","title":"The July bounds in two deviation ranges","assumptions":["Independent nonnegative integrable variables, each mean at most one.","Positive n and δ; the event uses a strict inequality."],"scope":"The first branch is a lower bound, without a sharpness claim. Nie–Wei prove the first expression for unit means and 0 < δ ≤ 1; shifting each variable by 1 minus its mean extends that statement to means at most one.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"],"formal_declarations":[]},"sha256":"35a6058bf3bb1a185e5b03d392eef25d282113de31450e8879cbbb68e8ef6482","contexts":[{"assessment_history":[{"report_id":"feige-conjecture:report:ranges:v1","source_id":"feige-conjecture-fu","source_version_id":"feige-conjecture:source:fu:v1","source_title":"Sharp small-deviation inequalities for sums of independent nonnegative random variables","source_label":"Two parameter-range bounds proved in the cited paper","state":"resolved","scope":"Lower bound δ(n/(n+δ))^n for 0 < δ < 1, and (1−1/(n+δ))^n for δ ≥ 1. 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For every δ > 0, "},{"kind":"math","text":"P(S < E S + δ) ≥ δ(n/(n+δ))^n","latex":"\\mathbb P(S<\\mathbb ES+\\delta)\\ge\\delta\\left(\\frac n{n+\\delta}\\right)^n"},{"kind":"text","text":" when 0 < δ < 1, and "},{"kind":"math","text":"P(S < E S + δ) ≥ (1−1/(n+δ))^n","latex":"\\mathbb P(S<\\mathbb ES+\\delta)\\ge\\left(1-\\frac1{n+\\delta}\\right)^n"},{"kind":"text","text":" when δ ≥ 1. The latter bound is sharp for each fixed n and δ ≥ 1."}]},"id":"claim-feige-july-deviation-bounds","slug":"feige-july-deviation-bounds","version":2,"kind":"bound","title":"The July bounds in two deviation ranges","assumptions":["Independent nonnegative integrable variables, each mean at most one.","Positive n and δ; the event uses a strict inequality."],"scope":"The first branch is a lower bound, without a sharpness claim. 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An atom exactly at the threshold does not belong to this event.","source_entry_ids":["feige-conjecture:entry:fu:v1"]}],"proofs":[{"id":"feige-conjecture-account-1","claim":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"title":"Fu–Han–Wang–Yan–Zhang–Zhou account","method":"Sharp small-deviation inequalities","summary":"This account gives the sharp finite-dimensional unit-slack constant and a broader deviation inequality.","limitation":"Its general deviation bound is sharp on the δ ≥ 1 branch; the later all-δ proposal is assessed separately.","dependencies":[],"source_entry_ids":["feige-conjecture:entry:fu:v1"],"report_ids":["feige-conjecture:report:unit-fu:v1"]},{"id":"feige-conjecture-account-2","claim":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"title":"Nie–Wei account","method":"A separate sharp-bound argument","summary":"A second author team establishes the same sharp finite-dimensional unit-slack constant.","limitation":"The Wang–Zhang formal project has its own scope; credit for this mathematical argument remains separate.","dependencies":[],"source_entry_ids":["feige-conjecture:entry:nie:v1"],"report_ids":["feige-conjecture:report:unit-nie:v1"]}],"bounds":[{"id":"feige-unit-slack-bound-detail","claim":{"id":"claim-feige-unit-slack","version":1},"quantity":"Unit-slack probability","direction":"lower","value":"1/e","conditions":"All positive finite dimensions; independent nonnegative variables with means at most one, strict threshold.","comparison_group":"feige-dimension-free","date":{"value":"2026-07-27"},"source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"]},{"id":"feige-sharp-finite-unit-slack-bound-detail","claim":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"quantity":"Unit-slack probability","direction":"exact","value":"(n/(n+1))^n","conditions":"Optimal lower bound for fixed finite n; tends to 1/e without attaining it at finite n.","comparison_group":"feige-fixed-dimension","date":{"value":"2026-07-27"},"source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1","feige-conjecture:entry:lean:v1"]}],"witnesses":[{"id":"feige-sharp-finite-unit-slack-witness-detail","claim":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"kind":"distribution","title":"A finite-dimensional extremizer","fields":[{"label":"Distribution","value":"Independent Xᵢ: value 0 with probability n/(n+1), and value n+1 with probability 1/(n+1)."},{"label":"Mean","value":"Each mean is one; E S = n."},{"label":"Strict event","value":"S < n+1 occurs exactly when every variable is zero, with probability (n/(n+1))^n."}],"limitation":"This attains the fixed-n bound. The dimension-free constant 1/e is approached as n grows and is not attained by this family at finite n.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1","feige-conjecture:entry:lean:v1"]}],"contributors":{"summary":"Fu, Han, Wang, Yan, Zhang and Zhou and Nie–Wei receive separate proof credits. The papers attribute different roles to GPT-5.6 Pro and GPT-5.6 Sol. Wang–Zhang’s Codex formalization is separately credited. Ling owns the later Samuels proposal; its source does not attribute a specific AI role.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1","feige-conjecture:entry:ling:v1","feige-conjecture:entry:lean:v1"]},"scopes":[],"connections":{"statements":[{"claim":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"label":"Sharp finite-dimensional bound"},{"claim":{"id":"claim-feige-unit-slack","version":1},"label":"Dimension-free unit-slack bound"}],"relationships":[{"connection_id":"feige-conjecture-finite-to-limit","title":"Finite-dimensional and universal bounds","purpose":"comparison"}]}},"representations":[{"schema_version":"visual-representation-v1","id":"feige-conjecture-setup","version":4,"subject":{"kind":"problem","id":"problem-feige-conjecture","sha256":"79a84a063ffefd12f320922004842cdd83a09262e268c377fb1042e82b22a2ad"},"purpose":"setup","title":"Why the strict threshold matters","caption":"Three independent variables, each 0 or 4, attain the finite-dimensional unit-slack bound (3/4)³ = 27/64.","description":"The sum has values 0,4,8,12 with probabilities 27/64,27/64,9/64,1/64. 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Nie–Wei prove the first expression for unit means and 0 < δ ≤ 1; shifting each variable by 1 minus its mean extends that statement to means at most one.","result":"proved","report_ids":["feige-conjecture:report:ranges:v1"],"independent_review":"pending"},{"claim":{"id":"claim-feige-all-deviations-sharp","version":1},"relation":"related_variant","relationship_note":"Ling’s later Corollary 1.3 proposes the fully sharp formula; it is not part of the linked unit-slack Lean proof.","result":"proposed","report_ids":["feige-conjecture:report:all-delta:v1"],"independent_review":"pending"},{"claim":{"id":"claim-feige-conjecture-consequence","version":1},"relation":"conditional_implication","relationship_note":"Finite-dimensional equality examples attain their own larger constants. The limiting value 1/e is approached as n grows, rather than attained in one finite dimension.","result":"proved","report_ids":["feige-conjecture:report:consequence:v1"],"independent_review":"pending","reading":{"placement":"consequence","note":"Finite-dimensional equality examples attain their own larger constants. The limiting value 1/e is approached as n grows, rather than attained in one finite dimension."}},{"claim":{"id":"claim-feige-bound-one-thirteenth","version":1},"relation":"adjacent_result","relationship_note":"An earlier lower bound on the unit-slack probability, before the sharp 1/e.","result":"proved","report_ids":["feige-conjecture:report:one-thirteenth:v1"],"independent_review":"pending","credit":{"name":"Feige","year":"2004","entry_id":"feige-conjecture:entry:fu-history:v1"}},{"claim":{"id":"claim-feige-bound-one-eighth","version":1},"relation":"adjacent_result","relationship_note":"An earlier lower bound on the unit-slack probability, before the sharp 1/e.","result":"proved","report_ids":["feige-conjecture:report:one-eighth:v1"],"independent_review":"pending","credit":{"name":"He, Zhang & Zhang","year":"2010","entry_id":"feige-conjecture:entry:fu-history:v1"}},{"claim":{"id":"claim-feige-bound-0-1798","version":1},"relation":"adjacent_result","relationship_note":"An earlier lower bound on the unit-slack probability, before the sharp 1/e.","result":"proved","report_ids":["feige-conjecture:report:0-1798:v1"],"independent_review":"pending","credit":{"name":"Guo, He, Ling & Liu","year":"2020","entry_id":"feige-conjecture:entry:fu-history:v1"}}],"connections":[{"id":"feige-conjecture-finite-to-limit","source":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"target":{"id":"claim-feige-unit-slack","version":1},"relation":"related_to","assertion":null,"explanation":"The finite constants (n/(n+1))^n decrease to 1/e; each is strictly larger than 1/e.","limitation":"Finite-n sharpness and the limiting universal bound are not identical statements.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"],"independent_review":"pending"}],"milestones":[{"id":"feige-conjecture-question","date":{"value":"2006"},"date_basis":"publication","category":"foundation","title":"Feige’s probability question","summary":"The July papers cite Uriel Feige’s journal formulation of a lower-tail bound using independence and first moments.","limitation":"The conference version dates to 2004; this milestone uses the journal year.","claims":[{"id":"claim-feige-unit-slack","version":1}],"source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"],"independent_review":"pending"},{"id":"feige-conjecture-july","date":{"value":"2026-07-27"},"date_basis":"preprint","category":"bound","title":"Separately authored proofs of the unit-slack bound","summary":"Fu and coauthors and Nie–Wei post arguments reaching the sharp finite-n unit-slack constant.","limitation":"Authorship, AI contributions and mathematical input credits remain separate.","claims":[{"id":"claim-feige-unit-slack","version":1},{"id":"claim-feige-sharp-finite-unit-slack","version":1},{"id":"claim-feige-july-deviation-bounds","version":2}],"source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"],"independent_review":"pending"},{"id":"feige-conjecture-general","date":{"value":"2026-08-18"},"date_basis":"preprint","category":"proposal","title":"A broader Samuels proof is proposed","summary":"Ling’s later paper claims a formula for arbitrary mean vectors and all positive deviations.","limitation":"The general formula has its own assessment and is not covered by the unit-slack Lean theorem.","claims":[{"id":"claim-feige-all-deviations-sharp","version":1}],"source_entry_ids":["feige-conjecture:entry:ling:v1"],"independent_review":"pending"},{"id":"feige-conjecture-formal","date":null,"date_basis":"formalization","category":"formalization","title":"Unit-slack optimality has a public Lean proof","summary":"Wang and Zhang’s project publishes the exact fixed-dimensional optimality declaration.","limitation":"The pinned source is inspected, without an OpenMath proof execution or an inferred first-publication date.","claims":[{"id":"claim-feige-sharp-finite-unit-slack","version":1}],"source_entry_ids":["feige-conjecture:entry:lean:v1"],"independent_review":"pending"}],"activities":[{"id":"feige-conjecture-general-assessment","title":"Assess the later general formula","kind":"research_direction","source_entry_ids":["feige-conjecture:entry:ling:v1"],"summary":"Compare Ling’s reduction and weighted Bernoulli argument with the July bounds and the exact Samuels statement.","limitation":"No independent assessment or formal proof of this later proposal is recorded.","related_claim_ids":["claim-feige-all-deviations-sharp"]}],"curation":{"checked_on":"2026-10-13","selection":"Original question, distinct results, proof provenance, selected history and an explanatory schematic.","search_scope":"Selected primary papers, exact formal declarations, project provenance and official descriptions checked on 13 October 2026; Fu et al. §1.1 for the earlier lower bounds.","formal_status":"The pinned Lean source reports complete optimality for the unit-slack event. Deliberate challenge placeholders are separate from the solution imports. OpenMath has not built the project.","ai_status":"Fu, Han, Wang, Yan, Zhang and Zhou and Nie–Wei receive separate proof credits. The papers attribute different roles to GPT-5.6 Pro and GPT-5.6 Sol. Wang–Zhang’s Codex formalization is separately credited. Ling owns the later Samuels proposal; its source does not attribute a specific AI role.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, P(S < E S + 1) ≥ 1/e, where S = Σᵢ Xᵢ.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, "},{"kind":"math","text":"P(S < E S + 1) ≥ 1/e","latex":"\\mathbb P(S<\\mathbb ES+1)\\ge e^{-1}"},{"kind":"text","text":", where "},{"kind":"math","text":"S = Σᵢ Xᵢ","latex":"S=\\sum_i X_i"},{"kind":"text","text":"."}]},"id":"claim-feige-unit-slack","slug":"feige-unit-slack","version":1,"kind":"bound","title":"The dimension-free unit-slack inequality","assumptions":["Positive finite number of independent variables.","Nonnegative variables with each mean at most one; strict threshold."],"scope":"The conventional unit-slack conjecture. The stronger finite-n theorem yields this bound and approaches 1/e as n grows.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"],"formal_declarations":[]},{"statement":"For every n ≥ 1, the optimal universal lower bound for P(S < E S + 1) is (n/(n+1))^n, over independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every n ≥ 1, the optimal universal lower bound for "},{"kind":"math","text":"P(S < E S + 1)","latex":"\\mathbb P(S<\\mathbb ES+1)"},{"kind":"text","text":" is "},{"kind":"math","text":"(n/(n+1))^n","latex":"\\left(\\frac n{n+1}\\right)^n"},{"kind":"text","text":", over independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1."}]},"id":"claim-feige-sharp-finite-unit-slack","slug":"feige-sharp-finite-unit-slack","version":1,"kind":"theorem","title":"The optimal bound for a fixed number of variables","assumptions":["Fix positive finite n.","Strict unit-slack event and individual means at most one."],"scope":"Both July arguments support the same mathematical assertion. The pinned Lean declaration formalizes this fixed-dimensional optimality statement.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1","feige-conjecture:entry:lean:v1"],"formal_declarations":["PalomarFeige.sharp_unit_slack_feige_complete"]},{"statement":"For every n ≥ 1, let X₁,…,Xₙ be independent nonnegative integrable random variables with E Xᵢ ≤ 1, and put S = Σᵢ Xᵢ. For every δ > 0, P(S < E S + δ) ≥ δ(n/(n+δ))^n when 0 < δ < 1, and P(S < E S + δ) ≥ (1−1/(n+δ))^n when δ ≥ 1. The latter bound is sharp for each fixed n and δ ≥ 1.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every n ≥ 1, let X₁,…,Xₙ be independent nonnegative integrable random variables with E Xᵢ ≤ 1, and put S = Σᵢ Xᵢ. For every δ > 0, "},{"kind":"math","text":"P(S < E S + δ) ≥ δ(n/(n+δ))^n","latex":"\\mathbb P(S<\\mathbb ES+\\delta)\\ge\\delta\\left(\\frac n{n+\\delta}\\right)^n"},{"kind":"text","text":" when 0 < δ < 1, and "},{"kind":"math","text":"P(S < E S + δ) ≥ (1−1/(n+δ))^n","latex":"\\mathbb P(S<\\mathbb ES+\\delta)\\ge\\left(1-\\frac1{n+\\delta}\\right)^n"},{"kind":"text","text":" when δ ≥ 1. The latter bound is sharp for each fixed n and δ ≥ 1."}]},"id":"claim-feige-july-deviation-bounds","slug":"feige-july-deviation-bounds","version":2,"kind":"bound","title":"The July bounds in two deviation ranges","assumptions":["Independent nonnegative integrable variables, each mean at most one.","Positive n and δ; the event uses a strict inequality."],"scope":"The first branch is a lower bound, without a sharpness claim. Nie–Wei prove the first expression for unit means and 0 < δ ≤ 1; shifting each variable by 1 minus its mean extends that statement to means at most one.","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1"],"formal_declarations":[]},{"statement":"For every n ≥ 1 and δ > 0, the infimum of P(S < E S + δ) over independent nonnegative integrable variables with E Xᵢ ≤ 1 equals min{δ/(1+δ), (1−1/(n+δ))^n}. This infimum is attained. Taking the infimum over n gives min{δ/(1+δ), 1/e}.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For every n ≥ 1 and δ > 0, the infimum of P(S < E S + δ) over independent nonnegative integrable variables with E Xᵢ ≤ 1 equals "},{"kind":"math","text":"min{δ/(1+δ), (1−1/(n+δ))^n}","latex":"\\min\\left\\{\\frac\\delta{1+\\delta},\\left(1-\\frac1{n+\\delta}\\right)^n\\right\\}"},{"kind":"text","text":". This infimum is attained. Taking the infimum over n gives "},{"kind":"math","text":"min{δ/(1+δ), 1/e}","latex":"\\min\\left\\{\\frac\\delta{1+\\delta},e^{-1}\\right\\}"},{"kind":"text","text":"."}]},"id":"claim-feige-all-deviations-sharp","slug":"feige-all-deviations-sharp","version":1,"kind":"bound","title":"The later proposal for every positive deviation","assumptions":["Strict threshold, fixed positive finite n and positive δ.","The optimization ranges over all admissible distributions."],"scope":"Ling’s later Corollary 1.3 proposes the fully sharp formula; it is not part of the linked unit-slack Lean proof.","source_entry_ids":["feige-conjecture:entry:ling:v1"],"formal_declarations":[]},{"statement":"If the sharp fixed-dimensional unit-slack constant is (n/(n+1))^n for every n ≥ 1, then the infimum of P(S < E S + 1) over all positive finite n and admissible distributions is 1/e. No fixed finite n attains this dimension-free infimum.","id":"claim-feige-conjecture-consequence","slug":"feige-conjecture-consequence","version":1,"kind":"theorem","title":"The universal constant is a limit across dimensions","assumptions":["Independent nonnegative integrable variables, each mean at most one.","Allow the positive finite number of variables to vary."],"scope":"Finite-dimensional equality examples attain their own larger constants. The limiting value 1/e is approached as n grows, rather than attained in one finite dimension.","source_entry_ids":["feige-conjecture:entry:fu:v1"],"formal_declarations":[]},{"statement":"For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, P(S < E S + 1) ≥ 1/13, where S = Σᵢ Xᵢ.","id":"claim-feige-bound-one-thirteenth","slug":"feige-bound-one-thirteenth","version":1,"kind":"bound","title":"Feige’s first universal lower bound: 1/13","assumptions":["Independent nonnegative integrable variables with means at most one.","Reported in the 2026 paper’s account of earlier work."],"scope":"An earlier universal lower bound, superseded by the sharp constant 1/e.","source_entry_ids":["feige-conjecture:entry:fu-history:v1"],"formal_declarations":[]},{"statement":"For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, P(S < E S + 1) ≥ 1/8, where S = Σᵢ Xᵢ.","id":"claim-feige-bound-one-eighth","slug":"feige-bound-one-eighth","version":1,"kind":"bound","title":"A fourth-moment lower bound: 1/8","assumptions":["Independent nonnegative integrable variables with means at most one.","Reported in the 2026 paper’s account of earlier work."],"scope":"An earlier universal lower bound, superseded by the sharp constant 1/e.","source_entry_ids":["feige-conjecture:entry:fu-history:v1"],"formal_declarations":[]},{"statement":"For every n ≥ 1 and independent nonnegative integrable X₁,…,Xₙ with E Xᵢ ≤ 1, P(S < E S + 1) ≥ 0.1798, where S = Σᵢ Xᵢ.","id":"claim-feige-bound-0-1798","slug":"feige-bound-0-1798","version":1,"kind":"bound","title":"A moment and Berry–Esseen lower bound: 0.1798","assumptions":["Independent nonnegative integrable variables with means at most one.","Reported in the 2026 paper’s account of earlier work."],"scope":"An earlier universal lower bound, superseded by the sharp constant 1/e.","source_entry_ids":["feige-conjecture:entry:fu-history:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"proved","extent":"full_statement","summary":"The unit-slack question is proved in two separately credited July papers, with finite-dimensional optimality reported complete in Lean. A later Samuels paper proposes the full sharp formula for arbitrary positive deviations.","report_ids":["feige-conjecture:report:unit-fu:v1","feige-conjecture:report:unit-nie:v1","feige-conjecture:report:unit-lean:v1"],"progress_claim_ids":[],"remaining":null},"statement_kind":"conjecture","problem_id":"problem-feige-conjecture","slug":"feige-conjecture","title":"Feige’s probability conjecture","question":"How small can P(S < E S + 1) be for a sum S of independent nonnegative random variables whose individual means are at most one?","question_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"How small can "},{"kind":"math","text":"P(S < E S + 1)","latex":"\\mathbb P(S<\\mathbb ES+1)"},{"kind":"text","text":" be for a sum S of independent nonnegative random variables whose individual means are at most one?"}]},"scope":"Fix a positive finite integer n. The variables X₁,…,Xₙ are independent, nonnegative and integrable with E Xᵢ ≤ 1, and S = Σᵢ Xᵢ. The event is strictly below the mean plus one; a mass at the threshold is excluded. The finite-dimensional sharp constant and the limit over all n are different bounds. Other positive deviations δ have their own Claims.","aliases":["Feige conjecture","Samuels conjecture","Sharp unit-slack inequality"],"fields":["Probability"],"msc":["60E15","60G50"],"topics":[],"background":"undergraduate","scale":"major conjecture","depth":"maintained","source_entry_ids":["feige-conjecture:entry:fu:v1","feige-conjecture:entry:nie:v1","feige-conjecture:entry:ling:v1","feige-conjecture:entry:lean:v1","feige-conjecture:entry:fu-history:v1"],"status_report_ids":["feige-conjecture:report:unit-fu:v1","feige-conjecture:report:unit-nie:v1","feige-conjecture:report:unit-lean:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"known","scope":"Source comparison and reader curation; independent mathematical assessment remains pending.","rationale":"The unit-slack question is proved in two separately credited July papers, with finite-dimensional optimality reported complete in Lean. A later Samuels paper proposes the full sharp formula for arbitrary positive deviations.","decided_on":"2026-10-13","review_due_on":"2027-01-13","report_ids":["feige-conjecture:report:unit-fu:v1","feige-conjecture:report:unit-nie:v1","feige-conjecture:report:unit-lean:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"Sharp small-deviation inequalities for sums of independent nonnegative random variables","url":"https://arxiv.org/abs/2607.23980v1","note":"Theorem 1.1, Corollary 1.2 and AI disclosure"},{"title":"On Feige’s conjecture","url":"https://arxiv.org/abs/2607.24528v1","note":"Theorem 1.3 and AI acknowledgement; arXiv submitted 27 July, HTML manuscript dated 24 August"},{"title":"On Samuels’ Conjecture","url":"https://arxiv.org/abs/2608.18392v1","note":"Theorem 1.1 and Corollary 1.3; arXiv submitted 18 August, manuscript dated 19 August"},{"title":"Lean proof of the sharp unit-slack Feige bound","url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","note":"PalomarFeige.sharp_unit_slack_feige_complete; README and formalization metadata"}],"formalizations":[{"title":"Sharp finite-dimensional Feige inequality","url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","note":"Lean 4 proof reports complete fixed-n optimality for δ = 1. It does not certify the July δ-dependent theorem or Ling’s Samuels proposal. Source definitions quantify over measurable, integrable, pointwise nonnegative independent variables on probability spaces.","target":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"proof_status":"reported_complete","source_entry_ids":["feige-conjecture:entry:lean:v1"],"declaration":"PalomarFeige.sharp_unit_slack_feige_complete","source_commit":"19c61d6cb556d1e967dbb0ada9f293c290837867","alignment":"staff_linked","checked_on":"2026-10-13"}],"related":[],"tasks":[],"insights":[{"id":"feige-conjecture-extremal","title":"An exact two-point extremizer","summary":"For δ ≥ 1, let each variable be 0 with probability 1−1/(n+δ), and n+δ otherwise. Each mean is one; the strict event occurs exactly when all variables are zero.","review":"pending","kind":"approach","source_entry_ids":["feige-conjecture:entry:fu:v1"],"claims":[{"id":"claim-feige-sharp-finite-unit-slack","version":1},{"id":"claim-feige-july-deviation-bounds","version":2}],"limitation":"This construction shows sharpness in that parameter range. It does not establish optimality for 0 < δ < 1."},{"id":"feige-conjecture-impact","title":"The universal constant is a limit across dimensions","summary":"If the sharp fixed-dimensional unit-slack constant is (n/(n+1))^n for every n ≥ 1, then the infimum of P(S < E S + 1) over all positive finite n and admissible distributions is 1/e. No fixed finite n attains this dimension-free infimum.","review":"pending","kind":"impact","source_entry_ids":["feige-conjecture:entry:fu:v1"],"claims":[{"id":"claim-feige-conjecture-consequence","version":1}],"scenario":{"id":"claim-feige-sharp-finite-unit-slack","version":1},"limitation":"Finite-dimensional equality examples attain their own larger constants. The limiting value 1/e is approached as n grows, rather than attained in one finite dimension."}],"coverage_gaps":["Independent mathematical review of the cited arguments is pending.","Ling’s all-deviation proposal has no linked formal proof in this source selection."],"checked_on":"2026-10-21"},"sources":[{"work":{"authors":["Weibo Fu","Yanjun Han","Guanyang Wang","Jun Yan","Peng Zhang","Zhengqing Zhou"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"The authors report GPT-5.6 Pro assistance in proof discovery; the calibration theorem of Vlassis and Thomas and Grünbaum-type inequalities are mathematical inputs, not authors of this paper.","artifacts":[],"published":{"value":"2026-07-27"}},"id":"feige-conjecture:source:fu:v1","source_id":"feige-conjecture-fu","title":"Sharp small-deviation inequalities for sums of independent nonnegative random variables","owner":"Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou","url":"https://arxiv.org/abs/2607.23980v1","subject_scope":"Proves two parameter-range lower bounds, with the finite-n bound sharp when δ ≥ 1.","revision":"2026-07-27","content_sha256":"29b7a68622eee0c46a86dc6e712c47e66e7b1821fd23255af9fe878f6957c842","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/2607.23980v1","attribution":"Weibo Fu, Yanjun Han, Guanyang Wang, Jun Yan, Peng Zhang, Zhengqing Zhou","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Zipei Nie","Jiaye Wei"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"self_reported","ai_note":"Nie and Wei credit GPT-5.6 Sol for exploration and discovery, state that they checked the arguments and authored the text, and retain responsibility. Their argument uses the Vlassis–Thomas calibration theorem.","artifacts":[],"published":{"value":"2026-07-27"}},"id":"feige-conjecture:source:nie:v1","source_id":"feige-conjecture-nie","title":"On Feige’s conjecture","owner":"Zipei Nie, Jiaye Wei","url":"https://arxiv.org/abs/2607.24528v1","subject_scope":"Gives a separate argument for the unit-slack inequality and the bound δ(n/(n+δ))^n for 0 < δ ≤ 1 under unit means.","revision":"2026-07-27","content_sha256":"e579dbffa9befce4f0094a587e7037bfd1fd45bd89de69c1298b6d146135114f","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/2607.24528v1","attribution":"Zipei Nie, Jiaye Wei","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Zhi Ling"],"kind":"research_paper","publication":"preprint","publication_note":"Source publication and attributed assessments are distinct from independent OpenMath review.","ai_role":"not_reported","ai_note":"No specific AI contribution is attributed by this source record.","artifacts":[],"published":{"value":"2026-08-18"}},"id":"feige-conjecture:source:ling:v1","source_id":"feige-conjecture-ling","title":"On Samuels’ Conjecture","owner":"Zhi Ling","url":"https://arxiv.org/abs/2608.18392v1","subject_scope":"Claims Samuels’ sharp mean-vector formula and the full sharp Feige formula for every δ > 0.","revision":"2026-08-18","content_sha256":"bc0ac69664c21b2508d3bdcefc4391b37d5b2a949d5eddb8beab4e89e7daef99","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://arxiv.org/abs/2608.18392v1","attribution":"Zhi Ling","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90},{"work":{"authors":["Guanyang Wang","Peng Zhang"],"kind":"research_project","publication":"project","publication_note":"Public formal source; execution and statement correspondence are separate checks.","ai_role":"self_reported","ai_note":"The project credits GPT-5.6 Sol Codex for formalization. This is a different contribution from the July discovery model. Authors report verification; OpenMath has not executed it.","artifacts":[{"kind":"formal_code","title":"Lean proof of the sharp unit-slack Feige bound — pinned source","url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","version":"19c61d6cb556d1e967dbb0ada9f293c290837867","assessment":"not_executed","note":"Exact source inspected; OpenMath has not executed this formal proof project.","sha256":"61807ad8500eca72545c86ad34ae2e3569bff83f713d1251b97b19b59e425ec1"}]},"id":"feige-conjecture:source:lean:v1","source_id":"feige-conjecture-lean","title":"Lean proof of the sharp unit-slack Feige bound","owner":"Guanyang Wang, Peng Zhang","url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","subject_scope":"The headline theorem asserts optimality of (n/(n+1))^n for every positive finite n in the strict unit-slack event.","revision":"19c61d6cb556d1e967dbb0ada9f293c290837867","content_sha256":"61807ad8500eca72545c86ad34ae2e3569bff83f713d1251b97b19b59e425ec1","update_method":"Compare the exact source and publish a new immutable curation revision.","identifier_policy":"Stable source identity with exact versioned passages.","observed_on":"2026-10-13","source_updated_on":null,"rights":"metadata-link","license":"Original source license and document rights apply.","terms_url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","attribution":"Guanyang Wang, Peng Zhang","rights_note":"OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.","check_interval_days":90}],"entries":[{"id":"feige-conjecture:entry:fu:v1","source_version_id":"feige-conjecture:source:fu:v1","item_id":"fu","url":"https://arxiv.org/abs/2607.23980v1","wording":"Proves two parameter-range lower bounds, with the finite-n bound sharp when δ ≥ 1.","wording_kind":"paraphrase","locator":"Theorem 1.1, Corollary 1.2 and AI disclosure","supersedes":null},{"id":"feige-conjecture:entry:nie:v1","source_version_id":"feige-conjecture:source:nie:v1","item_id":"nie","url":"https://arxiv.org/abs/2607.24528v1","wording":"Gives a separate argument for the unit-slack inequality and the bound δ(n/(n+δ))^n for 0 < δ ≤ 1 under unit means.","wording_kind":"paraphrase","locator":"Theorem 1.3 and AI acknowledgement; arXiv submitted 27 July, HTML manuscript dated 24 August","supersedes":null},{"id":"feige-conjecture:entry:ling:v1","source_version_id":"feige-conjecture:source:ling:v1","item_id":"ling","url":"https://arxiv.org/abs/2608.18392v1","wording":"Claims Samuels’ sharp mean-vector formula and the full sharp Feige formula for every δ > 0.","wording_kind":"paraphrase","locator":"Theorem 1.1 and Corollary 1.3; arXiv submitted 18 August, manuscript dated 19 August","supersedes":null},{"id":"feige-conjecture:entry:lean:v1","source_version_id":"feige-conjecture:source:lean:v1","item_id":"lean","url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","wording":"The headline theorem asserts optimality of (n/(n+1))^n for every positive finite n in the strict unit-slack event.","wording_kind":"paraphrase","locator":"PalomarFeige.sharp_unit_slack_feige_complete; README and formalization metadata","supersedes":null},{"id":"feige-conjecture:entry:fu-history:v1","source_version_id":"feige-conjecture:source:fu:v1","item_id":"§1.1, Background and related work; references [Fei04], [HZZ10], [GHLL20]","url":"https://arxiv.org/abs/2607.23980v1","wording":"Feige proved the first universal lower bound, 1/13 (2004); it was later improved to 1/8 by He, Zhang and Zhang (2010), to approximately 0.14 by Garnett (2020), and to 0.1798 by Guo, He, Ling and Liu (2020).","wording_kind":"paraphrase","locator":"§1.1, Background and related work; references [Fei04], [HZZ10], [GHLL20]","supersedes":null}],"reports":[{"id":"feige-conjecture:report:unit-fu:v1","entry_id":"feige-conjecture:entry:fu:v1","state":"resolved","source_label":"Sharp unit-slack bound proved by Fu and coauthors","scope":"Theorem 1.1 at δ = 1, including finite-dimensional sharpness.","source_date":"2026-07-27","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.23980v1","supersedes":null},{"id":"feige-conjecture:report:unit-nie:v1","entry_id":"feige-conjecture:entry:nie:v1","state":"resolved","source_label":"Unit-slack bound proved by Nie and Wei","scope":"Their separately authored proof reaches the same finite-dimensional constant.","source_date":"2026-07-27","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.24528v1","supersedes":null},{"id":"feige-conjecture:report:unit-lean:v1","entry_id":"feige-conjecture:entry:lean:v1","state":"resolved","source_label":"Unit-slack optimality reported complete in Lean","scope":"The pinned source targets δ = 1 only; no general-δ formal proof is attributed to it.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://github.com/pengzhang91/Feige/blob/19c61d6cb556d1e967dbb0ada9f293c290837867/Solution.lean","supersedes":null},{"id":"feige-conjecture:report:ranges:v1","entry_id":"feige-conjecture:entry:fu:v1","state":"resolved","source_label":"Two parameter-range bounds proved in the cited paper","scope":"Lower bound δ(n/(n+δ))^n for 0 < δ < 1, and (1−1/(n+δ))^n for δ ≥ 1. Sharpness is asserted only in the second range.","source_date":"2026-07-27","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.23980v1","supersedes":null},{"id":"feige-conjecture:report:all-delta:v1","entry_id":"feige-conjecture:entry:ling:v1","state":"proposed_solution","source_label":"Full sharp deviation formula proposed by Ling","scope":"Corollary 1.3 covers every δ > 0 via the proposed proof of Samuels’ conjecture; independent assessment is pending.","source_date":"2026-08-18","observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2608.18392v1","supersedes":null},{"id":"feige-conjecture:report:consequence:v1","entry_id":"feige-conjecture:entry:fu:v1","state":"resolved","source_label":"Conditional mathematical implication in the cited source","scope":"Finite-dimensional equality examples attain their own larger constants. The limiting value 1/e is approached as n grows, rather than attained in one finite dimension.","source_date":null,"observed_on":"2026-10-13","evidence_url":"https://arxiv.org/abs/2607.23980v1","supersedes":null},{"id":"feige-conjecture:report:one-thirteenth:v1","entry_id":"feige-conjecture:entry:fu-history:v1","state":"resolved","source_label":"Earlier lower bound 1/13","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/2607.23980v1","supersedes":null},{"id":"feige-conjecture:report:one-eighth:v1","entry_id":"feige-conjecture:entry:fu-history:v1","state":"resolved","source_label":"Earlier lower bound 1/8","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/2607.23980v1","supersedes":null},{"id":"feige-conjecture:report:0-1798:v1","entry_id":"feige-conjecture:entry:fu-history:v1","state":"resolved","source_label":"Earlier lower bound 0.1798","scope":"Only the exact statement and passage cited here; not an independent OpenMath assessment.","source_date":null,"observed_on":"2026-10-21","evidence_url":"https://arxiv.org/abs/2607.23980v1","supersedes":null}]},"status":{"state":"proved","label":"Proved","extent":"full_statement","summary":"The unit-slack question is proved in two separately credited July papers, with finite-dimensional optimality reported complete in Lean. A later Samuels paper proposes the full sharp formula for arbitrary positive deviations.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-13","progress_label":null,"progress_claim_ids":[],"remaining":null,"views":["resolved"]}},"selected_context":null}}