{ "model": { "rank": 4447, "code": "thing-q28113351", "model_id": "vr.tr.inequality", "name": "inequality", "purpose": "Let an agent explain inequalities as mathematical relations, state and explain named inequalities, help solve and prove inequalities, and support learning.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q28113351/", "tier": 2, "score": 75, "payload": { "layer": "wikidata", "aliases": [ "Pedoe's inequality", "Weyl's inequality", "van den Berg–Kesten inequality", "Hölder's inequality for sums", "Hölder's inequality for integrals", "Minkowski inequality for integrals", "almost equal to", "geometric inequality", "interpolation inequality", "Whitney inequality", "triangle inequality theorem", "Cauchy's estimate", "strict inequality", "linear inequality", "Frobenius inequality", "variational inequality", "greater than", "apartness relation", "correlation inequality", "less than", "Erdős–Turán inequality", "less than or equal to", "greater than or equal to", "Shapiro inequality", "Hölder's inequality", "trace inequality", "Turán's inequalities", "Cauchy–Schwarz inequality for sums", "logarithmic Sobolev inequalities", "no later than", "no earlier than" ], "aliasCount": 31, "merged": 32, "knownIn": 75, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q28113351", "url": "https://www.wikidata.org/wiki/Q28113351", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T02:23:20Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 2, "engine": "claude" }, "metaModel": { "id": "THING-Q28113351", "registryId": "vr.tr.inequality", "name": "inequality", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q28113351/", "model": { "registry_id": "vr.tr.inequality", "name": "inequality", "purpose": "Let an agent explain inequalities as mathematical relations, state and explain named inequalities, help solve and prove inequalities, and support learning.", "definition": "A mathematical statement asserting that one quantity is less than, greater than, or at most or at least another, as a binary relation between numbers, functions or other objects, including elementary inequalities, named inequalities such as those of Cauchy-Schwarz, Holder, Minkowski, Weyl and Pedoe, and inequalities in probability and geometry; inequalities are central to analysis, optimisation and estimation, and are proved by techniques such as convexity, rearrangement and induction.", "what_it_is_for": "Order relations between mathematical quantities.", "affordances": [ "explain inequalities", "state named inequalities", "solve and prove inequalities", "support learning" ], "distinguishing_features": [ "Order relation", "Strict and non-strict forms", "Named results", "Proof techniques" ], "appearance": "Not physical; written mathematical statements.", "visual_identification": [ "Less than or greater than relation", "Named inequalities", "Equations assert equality; social inequality is another sense" ], "physical_properties": [], "families_and_kinds": [ "elementary and algebraic inequalities", "inequalities in analysis such as Holder and Minkowski", "inequalities in linear algebra such as Weyl", "geometric inequalities such as Pedoe", "probabilistic inequalities" ], "related_models": [ { "relation": "is a kind of", "target": "binary relation", "why": "category" }, { "relation": "is related to", "target": "mathematical proof", "why": "proving inequalities" }, { "relation": "is related to", "target": "imaginary", "why": "numbers not ordered by inequalities" }, { "relation": "is related to", "target": "value", "why": "quantities compared" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical notation conventions", "Textbook naming of inequalities" ], "failure_modes_and_hazards": [ "Sign errors when multiplying by negatives", "Confusing the mathematical and social senses", "Misapplying conditions of named inequalities" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What inequalities are.", "rationale": "Mathematics.", "layers": [ { "id": "concept", "name": "Concept", "description": "Concept and rules.", "findings": [ { "id": "concept-finding", "name": "Concept", "description": "Concept.", "questions": [ { "text": "What are inequalities, and what rules govern manipulating them?", "kind": "definition" }, { "text": "Is the mathematical relation or social inequality meant?", "kind": "boundary" } ] } ] }, { "id": "named", "name": "Named", "description": "Named inequalities.", "findings": [ { "id": "named-finding", "name": "Named", "description": "Named.", "questions": [ { "text": "What do Cauchy-Schwarz, Holder, Minkowski, Weyl, Pedoe and other named inequalities state, and when do they apply?", "kind": "definition" }, { "text": "Which entry fits a specific inequality?", "kind": "action" } ] } ] } ] }, { "id": "solve", "name": "Solve", "description": "Solving and proving.", "rationale": "Practice.", "layers": [ { "id": "solve", "name": "Solve", "description": "Solving inequalities.", "findings": [ { "id": "solve-finding", "name": "Solve", "description": "Solving.", "questions": [ { "text": "How is this inequality solved, and what is its solution set?", "kind": "action" }, { "text": "Which entry fits algebra?", "kind": "action" } ] } ] }, { "id": "prove", "name": "Prove", "description": "Proving inequalities.", "findings": [ { "id": "prove-finding", "name": "Prove", "description": "Proving.", "questions": [ { "text": "How can this inequality be proved, using convexity, rearrangement, induction or known results?", "kind": "action" }, { "text": "Which entry fits mathematical proof?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Context.", "layers": [ { "id": "analysis", "name": "Analysis", "description": "In analysis and optimisation.", "findings": [ { "id": "analysis-finding", "name": "Analysis", "description": "Analysis.", "questions": [ { "text": "How are inequalities used in analysis, optimisation and estimation?", "kind": "provenance" }, { "text": "Which entry fits optimisation?", "kind": "action" } ] } ] }, { "id": "probability", "name": "Probability", "description": "In probability.", "findings": [ { "id": "probability-finding", "name": "Probability", "description": "Probability.", "questions": [ { "text": "How are inequalities such as Markov, Chebyshev and concentration bounds used in probability?", "kind": "provenance" }, { "text": "Which entry fits probability theory?", "kind": "action" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "History and teaching.", "rationale": "Education.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did the study of inequalities develop, and who established the named results?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can inequalities be taught, including competition-style problems?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should each named inequality be a separate entry?", "How should proofs be linked?", "How should the social sense be separated?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a inequality, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain inequalities as mathematical relations, state and explain named inequalities, help solve and prove inequalities, and support learning.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: Pedoe's inequality, Weyl's inequality, van den Berg–Kesten inequality, Hölder's inequality for sums, Hölder's inequality for integrals, Minkowski inequality for integrals, almost equal to, geometric inequality, interpolation inequality, Whitney inequality, triangle inequality theorem, Cauchy's estimate.", "32 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 75 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a inequality, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "Pedoe's inequality", "Weyl's inequality", "van den Berg–Kesten inequality", "Hölder's inequality for sums", "Hölder's inequality for integrals", "Minkowski inequality for integrals", "almost equal to", "geometric inequality", "interpolation inequality", "Whitney inequality", "triangle inequality theorem", "Cauchy's estimate" ] } ], "standing": "Described in 75 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 32 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one inequality from another, and a inequality from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a inequality sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a inequality from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a inequality passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a inequality between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a inequality, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a inequality is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What inequality is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of inequality are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is inequality operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of inequality and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of inequality?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as inequality?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a inequality came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a inequality and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a inequality is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a inequality, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a inequality, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a inequality is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }