{"schema_version":"claim-reader-v1","record":{"available_versions":[1],"current_version":1,"assessment_summary":"consistent","claim":{"statement":"For a positive normalized factor g with bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s), the standard Chernoff density has f(x) = C_f g(x)g(−x), where C_f = 2^(−1/3)/Ai(0)², and κ(x) = v(x)+v(−x) for v = −(log g)″.","id":"claim-chernoff-airy-factor","slug":"chernoff-airy-factor","version":1,"kind":"theorem","title":"Airy factorization of the density and curvature","assumptions":["The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives."],"scope":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.","source_entry_ids":["chernoff-density:entry:factor:v1"],"formal_declarations":[]},"sha256":"2bae29f81fe38915f4a3f071735e7baed518ac97c9cdcf124050a92ff4d9a9c8","contexts":[{"assessment_history":[{"report_id":"chernoff-density:report:factor:v1","source_id":"chernoff-density-paper","source_version_id":"chernoff-density:source:paper:v1","source_title":"Chernoff’s Density Is Strongly Log-Concave","source_label":"Airy factorization and curvature identity","state":"resolved","scope":"The normalized one-sided factor and fixed Brownian/drift convention.","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.18619v1","entry_id":"chernoff-density:entry:factor:v1","source_date":"2026-07-21"}],"problem_id":"problem-chernoff-density","slug":"chernoff-density","title":"Strong log-concavity of the Chernoff density","publication_version":8,"relationship":{"claim":{"id":"claim-chernoff-airy-factor","version":1},"relation":"supporting_result","relationship_note":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.","result":"proved","report_ids":["chernoff-density:report:factor:v1"],"independent_review":"pending","reading":{"placement":"background","note":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives."}},"assessments":[{"report_id":"chernoff-density:report:factor:v1","source_id":"chernoff-density-paper","source_version_id":"chernoff-density:source:paper:v1","source_title":"Chernoff’s Density Is Strongly Log-Concave","source_label":"Airy factorization and curvature identity","state":"resolved","scope":"The normalized one-sided factor and fixed Brownian/drift convention.","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.18619v1","entry_id":"chernoff-density:entry:factor:v1","source_date":"2026-07-21"}],"connections":[],"related_claims":[],"progress":[],"tasks":[],"statements":[{"statement":"For a positive normalized factor g with bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s), the standard Chernoff density has f(x) = C_f g(x)g(−x), where C_f = 2^(−1/3)/Ai(0)², and κ(x) = v(x)+v(−x) for v = −(log g)″.","id":"claim-chernoff-airy-factor","slug":"chernoff-airy-factor","version":1,"kind":"theorem","title":"Airy factorization of the density and curvature","assumptions":["The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives."],"scope":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.","source_entry_ids":["chernoff-density:entry:factor:v1"],"formal_declarations":[],"relation":"supporting_result","relationship_note":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.","result":"proved","report_ids":["chernoff-density:report:factor:v1"],"independent_review":"pending","reading":{"placement":"background","note":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives."}}],"entries":[{"id":"chernoff-density:entry:factor:v1","source_version_id":"chernoff-density:source:paper:v1","item_id":"factor","url":"https://arxiv.org/html/2607.18619v1","wording":"The Chernoff density factors as f(x) = C_f g(x)g(−x), where the normalized one-sided factor has reciprocal-Airy bilateral Laplace transform. The curvature is v(x)+v(−x).","wording_kind":"paraphrase","locator":"Equations (2)–(6); Sections 2–4","supersedes":null}],"sources":[{"work":{"authors":["Xianyang Zhang","Quan Zhou"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"The authors attribute the proof to GPT-5.6 Sol, describe an initial algebraic failure and later corrected approaches, and report carefully checking the argument and revising the exposition.","artifacts":[],"published":{"value":"2026-07-21"}},"id":"chernoff-density:source:paper:v1","source_id":"chernoff-density-paper","title":"Chernoff’s Density Is Strongly Log-Concave","owner":"Xianyang Zhang, Quan Zhou","url":"https://arxiv.org/html/2607.18619v1","subject_scope":"Theorem 1; reciprocal-Airy representation, exponential peeling, convolution identity, compactness and AI Usage.","revision":"v1","content_sha256":"0929362634e98d635b7e7df132bbfc8b741d55b625f47df0162a54f395830a0c","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2607.18619v1","attribution":"Xianyang Zhang, Quan Zhou","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}]}],"assessments":[{"report_id":"chernoff-density:report:factor:v1","source_id":"chernoff-density-paper","source_version_id":"chernoff-density:source:paper:v1","source_title":"Chernoff’s Density Is Strongly Log-Concave","source_label":"Airy factorization and curvature identity","state":"resolved","scope":"The normalized one-sided factor and fixed Brownian/drift convention.","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.18619v1","entry_id":"chernoff-density:entry:factor:v1","source_date":"2026-07-21"}],"sources":[{"work":{"authors":["Xianyang Zhang","Quan Zhou"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"The authors attribute the proof to GPT-5.6 Sol, describe an initial algebraic failure and later corrected approaches, and report carefully checking the argument and revising the exposition.","artifacts":[],"published":{"value":"2026-07-21"}},"id":"chernoff-density:source:paper:v1","source_id":"chernoff-density-paper","title":"Chernoff’s Density Is Strongly Log-Concave","owner":"Xianyang Zhang, Quan Zhou","url":"https://arxiv.org/html/2607.18619v1","subject_scope":"Theorem 1; reciprocal-Airy representation, exponential peeling, convolution identity, compactness and AI Usage.","revision":"v1","content_sha256":"0929362634e98d635b7e7df132bbfc8b741d55b625f47df0162a54f395830a0c","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2607.18619v1","attribution":"Xianyang Zhang, Quan Zhou","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}],"entries":[{"id":"chernoff-density:entry:factor:v1","source_version_id":"chernoff-density:source:paper:v1","item_id":"factor","url":"https://arxiv.org/html/2607.18619v1","wording":"The Chernoff density factors as f(x) = C_f g(x)g(−x), where the normalized one-sided factor has reciprocal-Airy bilateral Laplace transform. The curvature is v(x)+v(−x).","wording_kind":"paraphrase","locator":"Equations (2)–(6); Sections 2–4","supersedes":null}],"formalizations":[],"tasks":[],"assessment_state":"proved","publication":{"version":8,"sha256":"62ef078768365582eb41aceaf86e39d2ee1abda3b26f65550aaa8941b947dbb1","published_at":"2026-10-21T11:59:54.628059+00:00","reason":"Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.","withdrawn":false,"draft":{"reader_guide":{"reading":[{"entry_id":"chernoff-density:entry:question:v1","role":"original_question","note":"Read the published theorem and the stronger conjecture."},{"entry_id":"chernoff-density:entry:theorem:v1","role":"proof","note":"Read Theorem 1 and the compactness argument."},{"entry_id":"chernoff-density:entry:factor:v1","role":"proof","note":"Inspect the normalized factor, transform domain and prior Airy identity."}],"definitions":[{"id":"chernoff-density-brownian","term":"Two-sided standard Brownian motion","explanation":"A continuous random path on both sides of zero, with W(0) = 0 and independent standard Brownian motions on the positive and negative half-lines. Subtracting t² penalizes locations far from zero.","source_entry_ids":["chernoff-density:entry:theorem:v1"]},{"id":"chernoff-density-curvature","term":"Strong log-concavity","explanation":"The potential −log f has second derivative at least a single positive constant everywhere. This is stronger than convexity, and stronger than merely positive curvature at every point.","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1"]}],"proofs":[{"id":"chernoff-density-analytic","claim":{"id":"claim-chernoff-strong-log-concavity","version":1},"title":"Pointwise strictness plus tail control","method":"Exponential peeling, Airy convolution and compactness","summary":"The one-sided curvature v is strictly positive and satisfies v(x) ≥ 4x. Factorization gives κ(x) = v(x)+v(−x) > 4|x|. Continuity controls the remaining compact interval, yielding a uniform positive lower bound.","limitation":"The proof does not report an optimal numerical constant. Authors’ checks and a manuscript argument are separate from proof-assistant execution.","dependencies":[{"id":"claim-chernoff-airy-factor","version":1},{"id":"claim-chernoff-curvature-tail","version":1}],"source_entry_ids":["chernoff-density:entry:theorem:v1"],"report_ids":["chernoff-density:report:strong:v1"]}],"bounds":[{"id":"chernoff-density-curvature-bound","claim":{"id":"claim-chernoff-curvature-tail","version":1},"quantity":"Negative log-curvature κ(x)","direction":"lower","strict":true,"value":"4|x|","conditions":"Standard Brownian variance, drift t² and every real x; the inequality is strict.","comparison_group":"chernoff-density-curvature","date":{"value":"2026-07-21"},"source_entry_ids":["chernoff-density:entry:theorem:v1"]}],"witnesses":[],"contributors":{"summary":"Fadoua Balabdaoui and Jon A. Wellner proved ordinary log-concavity and posed the stronger conjecture. Xianyang Zhang and Quan Zhou authored the 2026 note; they credit GPT-5.6 Sol with the proof and report checking it, revising the exposition and recognizing prior work on an Airy identity. Chernoff and Groeneboom supplied foundational distribution and factorization results; Menon and Srinivasan supplied the earlier convolution identity used in the final proof.","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1","chernoff-density:entry:factor:v1"]},"scopes":[{"id":"chernoff-density-ordinary-scope","claim":{"id":"claim-chernoff-ordinary-log-concavity","version":1},"label":"Ordinary log-concavity","source_entry_ids":["chernoff-density:entry:question:v1"],"facts":[{"label":"Distribution","value":"Z = argmax{W(t)−t²}, standard W"},{"label":"Curvature requirement","value":"κ(x) ≥ 0 for all x"},{"label":"Uniform positive constant","value":"Not required"}]},{"id":"chernoff-density-strong-scope","claim":{"id":"claim-chernoff-strong-log-concavity","version":1},"label":"Strong log-concavity","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1"],"facts":[{"label":"Distribution","value":"Z = argmax{W(t)−t²}, standard W"},{"label":"Curvature requirement","value":"κ(x) ≥ c for all x, for some c > 0"},{"label":"Uniform positive constant","value":"Required"}]},{"id":"chernoff-density-origin-minimum-scope","claim":{"id":"claim-chernoff-origin-minimum-curvature","version":1},"label":"Sharper original conjecture","source_entry_ids":["chernoff-density:entry:origin-minimum:v1"],"facts":[{"label":"Distribution","value":"Z = argmax{W(t)−t²}, standard W"},{"label":"Curvature requirement","value":"κ(x) ≥ κ(0) for every x"},{"label":"Uniform positive constant","value":"Specified by κ(0); numerical value ≈ 3.4052"}]}],"question":{"scope_summary":"Fix the Brownian scaling, define Z as a location, and distinguish ordinary log-concavity from a uniform positive curvature bound.","detail_title":"Exact conventions and hypotheses","aspects":[],"source_entry_ids":["chernoff-density:entry:question:v1"]},"remaining":{"mathematical":"The sharper 2014 conjecture κ(x) ≥ κ(0), with reported numerical κ(0) ≈ 3.4052, is not established by the selected 2026 theorem. Review that quantitative target and the analytic proof separately.","assessment":"Independent mathematical review is pending. External certificates and manuscript proofs are attributed to their sources.","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1","chernoff-density:entry:factor:v1","chernoff-density:entry:origin-minimum:v1"]},"connections":{"statements":[{"claim":{"id":"claim-chernoff-ordinary-log-concavity","version":1},"label":"Ordinary log-concavity"},{"claim":{"id":"claim-chernoff-strong-log-concavity","version":1},"label":"Uniform positive curvature"},{"claim":{"id":"claim-chernoff-origin-minimum-curvature","version":1},"label":"Minimum curvature at the origin"}],"relationships":[{"connection_id":"chernoff-old-and-new","title":"Ordinary and strong log-concavity","purpose":"comparison"},{"connection_id":"chernoff-uniform-versus-sharp","title":"Uniform positivity and the sharp constant","purpose":"variant"}]}},"representations":[{"hash_profile":"rfc8785","schema_version":"visual-representation-v1","id":"chernoff-density-setup","version":6,"subject":{"kind":"problem","id":"problem-chernoff-density","sha256":"ec8775fe6d3646f98d1047a6a57702c1e68390fa0dd8aebb5b14d0b5015e3d48"},"purpose":"setup","title":"A random path selects a location","caption":"An illustrative path with quadratic drift: Z is the location of the maximum, not the maximum height.","description":"The horizontal coordinate is t. A stylized continuous path represents W(t) − t², passes through zero at t = 0, and has its displayed maximum at Z. Repeating the random experiment produces the distribution whose density is f.","assumptions":["Layout is schematic; coordinates and line lengths carry no mathematical scale."],"limitation":"This authored sketch is not a Brownian sample, a numerical Chernoff density or a verification of log-curvature.","author":"OpenMath source curation","authoring_method":"ai_assisted_curation","authored_on":"2026-10-14","independent_review":"pending","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1"],"payload":{"kind":"diagram_schematic","marks":[{"x":40,"y":220,"id":"xaxis","kind":"segment","end":{"x":440,"y":220},"radius":null,"text":null,"anchor":"middle","dashed":false,"directed":true},{"x":240,"y":255,"id":"yaxis","kind":"segment","end":{"x":240,"y":40},"radius":null,"text":null,"anchor":"middle","dashed":false,"directed":true},{"x":440,"y":248,"id":"t","kind":"label","end":null,"radius":null,"text":"t","anchor":"middle","dashed":false},{"x":222,"y":242,"id":"zero","kind":"label","end":null,"radius":null,"text":"0","anchor":"middle","dashed":false},{"x":330,"y":35,"id":"target","kind":"label","end":null,"radius":null,"text":"argmax location Z","anchor":"middle","dashed":false},{"x":330,"y":58,"id":"location","kind":"segment","end":{"x":330,"y":220},"radius":null,"text":null,"anchor":"middle","dashed":true},{"x":50,"y":180,"id":"path0","kind":"segment","end":{"x":85,"y":160},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":85,"y":160,"id":"path1","kind":"segment","end":{"x":110,"y":195},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":110,"y":195,"id":"path2","kind":"segment","end":{"x":140,"y":120},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":140,"y":120,"id":"path3","kind":"segment","end":{"x":170,"y":155},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":170,"y":155,"id":"path4","kind":"segment","end":{"x":205,"y":80},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":205,"y":80,"id":"path5","kind":"segment","end":{"x":240,"y":220},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":240,"y":220,"id":"path6","kind":"segment","end":{"x":275,"y":95},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":275,"y":95,"id":"path7","kind":"segment","end":{"x":300,"y":125},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":300,"y":125,"id":"path8","kind":"segment","end":{"x":330,"y":65},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":330,"y":65,"id":"path9","kind":"segment","end":{"x":365,"y":130},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":365,"y":130,"id":"path10","kind":"segment","end":{"x":400,"y":145},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":400,"y":145,"id":"path11","kind":"segment","end":{"x":435,"y":190},"radius":null,"text":null,"anchor":"middle","dashed":false},{"x":140,"y":65,"id":"pathlabel","kind":"label","end":null,"radius":null,"text":"W(t) − t² (sketch)","anchor":"middle","dashed":false}],"features":[{"id":"argmax","label":"Z: where W(t) − t² is largest","claim":{"id":"claim-chernoff-strong-log-concavity","version":1},"binding":"scope","mark_ids":["xaxis","yaxis","t","zero","target","location","pathlabel","path0","path1","path2","path3","path4","path5","path6","path7","path8","path9","path10","path11"]}],"legend":"Z is the location of the maximum, not its height; the density of Z is the subject of the theorem."}}],"visuals":[{"representation":{"id":"chernoff-density-setup","version":6},"section":"question"}],"claim_relations":[{"claim":{"id":"claim-chernoff-strong-log-concavity","version":1},"relation":"primary_formulation","relationship_note":"The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.","result":"proved","report_ids":["chernoff-density:report:strong:v1"],"independent_review":"pending"},{"claim":{"id":"claim-chernoff-ordinary-log-concavity","version":1},"relation":"supporting_result","relationship_note":"Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.","result":"proved","report_ids":["chernoff-density:report:ordinary:v1"],"independent_review":"pending","reading":{"placement":"background","note":"Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound."}},{"claim":{"id":"claim-chernoff-curvature-tail","version":1},"relation":"bound","relationship_note":"Continuity, strict positivity on compact intervals and growth in the tails together yield the uniform lower bound. The numerical optimum of inf κ is not given.","result":"proved","report_ids":["chernoff-density:report:strong:v1"],"independent_review":"pending"},{"claim":{"id":"claim-chernoff-airy-factor","version":1},"relation":"supporting_result","relationship_note":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.","result":"proved","report_ids":["chernoff-density:report:factor:v1"],"independent_review":"pending","reading":{"placement":"background","note":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives."}},{"claim":{"id":"claim-chernoff-origin-minimum-curvature","version":1},"relation":"related_variant","relationship_note":"This is stronger than inf κ > 0. The selected 2026 theorem does not prove that the origin minimizes curvature, and this curation does not assert a comprehensive current-status assessment.","result":"unproved","report_ids":["chernoff-density:report:origin-minimum:v1"],"independent_review":"pending"}],"connections":[{"id":"chernoff-old-and-new","source":{"id":"claim-chernoff-ordinary-log-concavity","version":1},"target":{"id":"claim-chernoff-strong-log-concavity","version":1},"relation":"related_to","assertion":null,"explanation":"The newer theorem strengthens ordinary log-concavity for the same distribution to a uniform strictly positive curvature bound.","limitation":"A strictly positive pointwise second derivative alone does not give a uniform lower bound.","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1"],"independent_review":"pending"},{"id":"chernoff-uniform-versus-sharp","source":{"id":"claim-chernoff-strong-log-concavity","version":1},"target":{"id":"claim-chernoff-origin-minimum-curvature","version":1},"relation":"related_to","assertion":null,"explanation":"The qualitative question asks for some uniform positive constant; the sharper 2014 conjecture specifies the origin curvature as the best constant.","limitation":"The selected strong-log-concavity proof gives no origin-minimum conclusion. The old conjecture alone is not a current literature assessment.","source_entry_ids":["chernoff-density:entry:origin-minimum:v1","chernoff-density:entry:theorem:v1"],"independent_review":"pending"}],"milestones":[{"id":"chernoff-density:milestone:ordinary","date":{"value":"2014"},"date_basis":"publication","category":"related_result","title":"Log-concavity is proved and a stronger property proposed","summary":"Balabdaoui and Wellner publish ordinary log-concavity and conjecture uniform positive curvature.","limitation":"This is a selected source-supported milestone, not an exhaustive history.","claims":[{"id":"claim-chernoff-ordinary-log-concavity","version":1}],"source_entry_ids":["chernoff-density:entry:question:v1"],"independent_review":"pending"},{"id":"chernoff-density:milestone:strong","date":{"value":"2026-07-21"},"date_basis":"preprint","category":"related_result","title":"An analytic strong-curvature proof is submitted","summary":"Zhang and Zhou submit the selected preprint, crediting GPT-5.6 Sol and their own checking and exposition work.","limitation":"This is the arXiv submission date. The selected manuscript’s title page says 24 August 2026; neither is asserted to be the discovery date.","claims":[{"id":"claim-chernoff-strong-log-concavity","version":1},{"id":"claim-chernoff-curvature-tail","version":1}],"source_entry_ids":["chernoff-density:entry:theorem:v1"],"independent_review":"pending"}],"curation":{"checked_on":"2026-10-14","selection":"Canonical question, exact result scopes, proof evidence, contributors, history and explanatory examples.","search_scope":"Selected primary manuscript versions and their cited original literature, checked on 14 October 2026. This is not an exhaustive literature review.","formal_status":"No complete proof-assistant artifact was identified in the selected sources. The analytic manuscript and author checking are cited separately from independent mathematical assessment.","ai_status":"The final corrected proof, human checking and literature attribution are self-reported; an initial failed model attempt is not presented as mathematical evidence.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For Z = argmaxₜ∈ℝ{W(t)−t²}, with W two-sided standard Brownian motion and density f, infₓ∈ℝ −(log f)″(x) > 0.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For "},{"kind":"math","text":"Z = argmaxₜ∈ℝ{W(t)−t²}","latex":"Z=\\operatorname*{argmax}_{t\\in\\mathbb R}\\{W(t)-t^2\\}"},{"kind":"text","text":", with W two-sided standard Brownian motion and density f, "},{"kind":"math","text":"infₓ∈ℝ −(log f)″(x) > 0","latex":"\\inf_{x\\in\\mathbb R}-(\\log f)^{\\prime\\prime}(x)>0"},{"kind":"text","text":"."}]},"id":"claim-chernoff-strong-log-concavity","slug":"chernoff-strong-log-concavity","version":1,"kind":"conjecture","title":"Uniform positive curvature of the Chernoff density","assumptions":["The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound."],"scope":"The lower bound is uniform over the whole real line. Strict positivity at each point alone would not establish this infimum bound.","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1"],"formal_declarations":[]},{"statement":"The density of argmaxₜ∈ℝ{W(t)−t²}, for two-sided standard Brownian motion W, is log-concave.","id":"claim-chernoff-ordinary-log-concavity","slug":"chernoff-ordinary-log-concavity","version":1,"kind":"theorem","title":"The earlier ordinary log-concavity theorem","assumptions":["Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound."],"scope":"Ordinary log-concavity requires nonnegative curvature, without a uniform strictly positive lower bound.","source_entry_ids":["chernoff-density:entry:question:v1"],"formal_declarations":[]},{"statement":"For the standard Chernoff density f, κ(x) = −(log f)″(x) satisfies κ(x) > 4|x| for every real x, and infₓ∈ℝ κ(x) > 0.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For the standard Chernoff density f, κ(x) = −(log f)″(x) satisfies "},{"kind":"math","text":"κ(x) > 4|x|","latex":"\\kappa(x)>4|x|"},{"kind":"text","text":" for every real x, and "},{"kind":"math","text":"infₓ∈ℝ κ(x) > 0","latex":"\\inf_{x\\in\\mathbb R}\\kappa(x)>0"},{"kind":"text","text":"."}]},"id":"claim-chernoff-curvature-tail","slug":"chernoff-curvature-tail","version":1,"kind":"theorem","title":"Pointwise and tail curvature bounds","assumptions":["Continuity, strict positivity on compact intervals and growth in the tails together yield the uniform lower bound. The numerical optimum of inf κ is not given."],"scope":"Continuity, strict positivity on compact intervals and growth in the tails together yield the uniform lower bound. The numerical optimum of inf κ is not given.","source_entry_ids":["chernoff-density:entry:theorem:v1"],"formal_declarations":[]},{"statement":"For a positive normalized factor g with bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s), the standard Chernoff density has f(x) = C_f g(x)g(−x), where C_f = 2^(−1/3)/Ai(0)², and κ(x) = v(x)+v(−x) for v = −(log g)″.","id":"claim-chernoff-airy-factor","slug":"chernoff-airy-factor","version":1,"kind":"theorem","title":"Airy factorization of the density and curvature","assumptions":["The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives."],"scope":"The transform holds for Re(s) > −2^(1/3)a₁, where −a₁ is the largest Airy zero. Factor normalization is fixed; multiplication by a positive constant does not change logarithmic derivatives.","source_entry_ids":["chernoff-density:entry:factor:v1"],"formal_declarations":[]},{"statement":"For the standard Chernoff density f, κ(x) = −(log f)″(x) satisfies κ(x) ≥ κ(0) for every real x.","statement_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"For the standard Chernoff density f, "},{"kind":"math","text":"κ(x) = −(log f)″(x)","latex":"\\kappa(x)=-(\\log f)^{\\prime\\prime}(x)"},{"kind":"text","text":" satisfies "},{"kind":"math","text":"κ(x) ≥ κ(0)","latex":"\\kappa(x)\\ge\\kappa(0)"},{"kind":"text","text":" for every real x."}]},"id":"claim-chernoff-origin-minimum-curvature","slug":"chernoff-origin-minimum-curvature","version":1,"kind":"conjecture","title":"The sharper origin-minimum curvature conjecture","assumptions":["Two-sided standard Brownian motion and quadratic drift t².","The proposed optimal constant is the curvature at the origin; κ(0) ≈ 3.4052 is a numerical value reported in the 2014 source."],"scope":"This is stronger than inf κ > 0. The selected 2026 theorem does not prove that the origin minimizes curvature, and this curation does not assert a comprehensive current-status assessment.","source_entry_ids":["chernoff-density:entry:origin-minimum:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"proved","extent":"full_statement","summary":"The selected preprint proves existence of a uniformly positive curvature bound, beyond published ordinary log-concavity. It does not establish the sharper 2014 origin-minimum conjecture. Independent OpenMath mathematical review is pending.","report_ids":["chernoff-density:report:strong:v1"],"progress_claim_ids":[],"remaining":null},"statement_kind":"conjecture","problem_id":"problem-chernoff-density","slug":"chernoff-density","title":"Strong log-concavity of the Chernoff density","question":"Does the Chernoff density have a uniformly positive lower bound on its negative log-curvature?","scope":"Let W be two-sided standard Brownian motion with W(0) = 0, and Z = argmax over real t of W(t) − t². For its positive smooth density f, write κ(x) = −(log f)″(x). Strong log-concavity means there exists a constant c > 0 such that κ(x) ≥ c for every real x. This fixes the standard Brownian variance and unit quadratic drift; changing those conventions rescales the density and curvature.","scope_notation":{"schema_version":"mathematical-text-v1","segments":[{"kind":"text","text":"Let W be two-sided standard Brownian motion with W(0) = 0, and "},{"kind":"math","text":"Z = argmax over real t of W(t) − t²","latex":"Z=\\operatorname*{argmax}_{t\\in\\mathbb R}\\{W(t)-t^2\\}"},{"kind":"text","text":". For its positive smooth density f, write "},{"kind":"math","text":"κ(x) = −(log f)″(x)","latex":"\\kappa(x)=-(\\log f)^{\\prime\\prime}(x)"},{"kind":"text","text":". Strong log-concavity means there exists a constant c > 0 such that κ(x) ≥ c for every real x. This fixes the standard Brownian variance and unit quadratic drift; changing those conventions rescales the density and curvature."}]},"aliases":["Balabdaoui–Wellner conjecture","Chernoff distribution","Brownian parabolic drift"],"fields":["Probability","Statistics"],"msc":["60E05","62G20"],"topics":[],"background":"graduate","scale":"focused question","depth":"maintained","source_entry_ids":["chernoff-density:entry:question:v1","chernoff-density:entry:theorem:v1","chernoff-density:entry:factor:v1","chernoff-density:entry:origin-minimum:v1"],"status_report_ids":["chernoff-density:report:strong:v1"],"editorial":{"collection_id":"openmath-pilot","collection_title":"OpenMath pilot collection","reviewer":"OpenMath source curation","state":"known","scope":"Source comparison and presentation; independent mathematical review remains pending.","rationale":"The selected preprint proves existence of a uniformly positive curvature bound, beyond published ordinary log-concavity. It does not establish the sharper 2014 origin-minimum conjecture. Independent OpenMath mathematical review is pending.","decided_on":"2026-10-14","review_due_on":"2027-01-14","report_ids":["chernoff-density:report:strong:v1"],"independent_mathematical_review":"pending"},"literature":[{"title":"Chernoff’s density is log-concave","url":"https://arxiv.org/abs/1203.0828v2","note":"Published Bernoulli 20(1), 231–244; ordinary log-concavity and the stronger curvature conjecture."},{"title":"Chernoff’s Density Is Strongly Log-Concave","url":"https://arxiv.org/html/2607.18619v1","note":"Theorem 1; reciprocal-Airy representation, exponential peeling, convolution identity, compactness and AI Usage."}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"chernoff-density-impact","title":"Control over the whole distribution","summary":"The stronger theorem establishes a single positive curvature lower bound across the real line. The source explains why this places Chernoff’s law in the class supporting Gaussian-type concentration and associated functional inequalities.","review":"pending","kind":"impact","source_entry_ids":["chernoff-density:entry:theorem:v1"],"claims":[{"id":"claim-chernoff-strong-log-concavity","version":1},{"id":"claim-chernoff-curvature-tail","version":1}],"scenario":{"id":"claim-chernoff-strong-log-concavity","version":1},"limitation":"The theorem does not say the law is Gaussian or supply an optimal concentration constant."}],"coverage_gaps":["Independent OpenMath mathematical review and external proof execution remain pending.","The sharper origin-minimum curvature conjecture is not resolved by the selected theorem; later literature has not been exhaustively assessed."],"checked_on":"2026-10-14"},"sources":[{"work":{"authors":["Fadoua Balabdaoui","Jon A. Wellner"],"kind":"research_paper","publication":"published","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"not_reported","ai_note":"No AI contribution is reported in this selected source.","artifacts":[],"published":{"value":"2014-02-05"}},"id":"chernoff-density:source:original:v1","source_id":"chernoff-density-original","title":"Chernoff’s density is log-concave","owner":"Fadoua Balabdaoui, Jon A. Wellner","url":"https://arxiv.org/abs/1203.0828v2","subject_scope":"Published Bernoulli 20(1), 231–244; ordinary log-concavity and the stronger curvature conjecture.","revision":"v2","content_sha256":null,"update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/abs/1203.0828v2","attribution":"Fadoua Balabdaoui, Jon A. Wellner","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90},{"work":{"authors":["Xianyang Zhang","Quan Zhou"],"kind":"research_paper","publication":"preprint","publication_note":"Source-reported results and external checks are separate from independent OpenMath assessment.","ai_role":"self_reported","ai_note":"The authors attribute the proof to GPT-5.6 Sol, describe an initial algebraic failure and later corrected approaches, and report carefully checking the argument and revising the exposition.","artifacts":[],"published":{"value":"2026-07-21"}},"id":"chernoff-density:source:paper:v1","source_id":"chernoff-density-paper","title":"Chernoff’s Density Is Strongly Log-Concave","owner":"Xianyang Zhang, Quan Zhou","url":"https://arxiv.org/html/2607.18619v1","subject_scope":"Theorem 1; reciprocal-Airy representation, exponential peeling, convolution identity, compactness and AI Usage.","revision":"v1","content_sha256":"0929362634e98d635b7e7df132bbfc8b741d55b625f47df0162a54f395830a0c","update_method":"Compare the pinned source before publishing another immutable revision.","identifier_policy":"Stable source identity; exact source passages are versioned.","observed_on":"2026-10-14","source_updated_on":null,"rights":"metadata-link","license":"Source document rights apply; OpenMath publishes original paraphrases.","terms_url":"https://arxiv.org/html/2607.18619v1","attribution":"Xianyang Zhang, Quan Zhou","rights_note":"Linked proof prose is not republished. Any copied graph data is CC0, as specified by its archive.","check_interval_days":90}],"entries":[{"id":"chernoff-density:entry:question:v1","source_version_id":"chernoff-density:source:original:v1","item_id":"question","url":"https://arxiv.org/abs/1203.0828v2","wording":"Ordinary log-concavity is established, and uniform positive negative log-curvature is proposed as a stronger property of the same Chernoff density.","wording_kind":"paraphrase","locator":"Log-concavity theorem and Conjecture 3.1","supersedes":null},{"id":"chernoff-density:entry:theorem:v1","source_version_id":"chernoff-density:source:paper:v1","item_id":"theorem","url":"https://arxiv.org/html/2607.18619v1","wording":"The paper establishes v(x) > 0 and v(x) ≥ 4x for the normalized one-sided factor, hence κ(x) > 4|x| and inf κ > 0 for the Chernoff density.","wording_kind":"paraphrase","locator":"Theorem 1 and Section 5","supersedes":null},{"id":"chernoff-density:entry:factor:v1","source_version_id":"chernoff-density:source:paper:v1","item_id":"factor","url":"https://arxiv.org/html/2607.18619v1","wording":"The Chernoff density factors as f(x) = C_f g(x)g(−x), where the normalized one-sided factor has reciprocal-Airy bilateral Laplace transform. The curvature is v(x)+v(−x).","wording_kind":"paraphrase","locator":"Equations (2)–(6); Sections 2–4","supersedes":null},{"id":"chernoff-density:entry:origin-minimum:v1","source_version_id":"chernoff-density:source:original:v1","item_id":"origin-minimum","url":"https://arxiv.org/html/1203.0828v2#S3","wording":"The sharper original conjecture requires κ(x) ≥ κ(0) for every x. Its reported numerical value κ(0) ≈ 3.4052 is not established as a uniform lower bound by the selected 2026 theorem.","wording_kind":"paraphrase","locator":"Introduction; Section 3, Conjecture 3.1","supersedes":null}],"reports":[{"id":"chernoff-density:report:strong:v1","entry_id":"chernoff-density:entry:theorem:v1","state":"resolved","source_label":"Strong log-concavity proved in the selected preprint","scope":"Standard two-sided Brownian motion with unit quadratic drift; no explicit optimal uniform constant is asserted.","source_date":"2026-07-21","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.18619v1","supersedes":null},{"id":"chernoff-density:report:ordinary:v1","entry_id":"chernoff-density:entry:question:v1","state":"resolved","source_label":"Published ordinary log-concavity theorem","scope":"The standard Chernoff density.","source_date":"2014-02-05","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/abs/1203.0828v2","supersedes":null},{"id":"chernoff-density:report:factor:v1","entry_id":"chernoff-density:entry:factor:v1","state":"resolved","source_label":"Airy factorization and curvature identity","scope":"The normalized one-sided factor and fixed Brownian/drift convention.","source_date":"2026-07-21","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/2607.18619v1","supersedes":null},{"id":"chernoff-density:report:origin-minimum:v1","entry_id":"chernoff-density:entry:origin-minimum:v1","state":"unknown","source_label":"Sharper origin-minimum conjecture in the original paper","scope":"The 2014 conjecture is recorded as a distinct quantitative target. The selected 2026 result does not establish its sharp constant; no current resolution is inferred from the older source.","source_date":"2014-02-05","observed_on":"2026-10-14","evidence_url":"https://arxiv.org/html/1203.0828v2#S3","supersedes":null}]},"status":{"state":"proved","label":"Proved","extent":"full_statement","summary":"The selected preprint proves existence of a uniformly positive curvature bound, beyond published ordinary log-concavity. It does not establish the sharper 2014 origin-minimum conjecture. Independent OpenMath mathematical review is pending.","basis":"editorial","attributed_to":"OpenMath pilot collection · OpenMath source curation","as_of":"2026-10-14","progress_label":null,"progress_claim_ids":[],"remaining":null,"views":["resolved"]}},"selected_context":null}}