{ "model": { "rank": 6478, "code": "thing-q845060", "model_id": "vr.tr.maxima-and-minima", "name": "maxima and minima", "purpose": "Let an agent explain maxima and minima and how they are found, distinguish global, local and boundary extrema, support solving optimisation problems, and describe applications across fields.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q845060/", "tier": 2, "score": 60, "payload": { "layer": "wikidata", "aliases": [ "coordinate of northernmost point", "pricelessness", "comass", "maximum", "minimum", "maximal element", "maxima of a point set", "rated maximum", "RAM limit", "ruling gradient", "maximum profit at maturity", "maximum loss at maturity", "rated minimum", "minimal prime ideal", "wetting current", "global minimum", "local minimum", "maximal and minimal elements", "arg max and min", "local extremum", "global extremum" ], "aliasCount": 21, "merged": 22, "knownIn": 60, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q845060", "url": "https://www.wikidata.org/wiki/Q845060", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T00:49:34Z", "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-Q845060", "registryId": "vr.tr.maxima-and-minima", "name": "maxima and minima", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q845060/", "model": { "registry_id": "vr.tr.maxima-and-minima", "name": "maxima and minima", "purpose": "Let an agent explain maxima and minima and how they are found, distinguish global, local and boundary extrema, support solving optimisation problems, and describe applications across fields.", "definition": "The largest and smallest values of a function or set, either globally over the whole domain or locally in a neighbourhood, found in calculus at critical points where the derivative vanishes or fails to exist and at boundaries, and in ordered sets as maximal and minimal elements; maxima and minima are the subject of optimisation, appear in physics, economics, engineering and geography such as the northernmost point of a region, and are computed analytically and numerically.", "what_it_is_for": "Largest and smallest values.", "affordances": [ "explain extrema", "distinguish global and local", "support optimisation problems", "describe applications" ], "distinguishing_features": [ "Extreme values", "Critical points", "Global versus local", "Optimisation" ], "appearance": "Not physical; points on graphs or elements of sets.", "visual_identification": [ "Highest and lowest values", "Global or local", "Saddle points are critical but not extrema" ], "physical_properties": [], "families_and_kinds": [ "global maxima and minima", "local extrema", "boundary extrema", "maximal and minimal elements in orders", "constrained extrema" ], "related_models": [ { "relation": "is a kind of", "target": "critical point", "why": "category" }, { "relation": "is related to", "target": "curvature", "why": "second derivative tests" }, { "relation": "is related to", "target": "unit cost", "why": "cost minimisation" }, { "relation": "is related to", "target": "tripoint", "why": "extreme points of regions" } ], "identifiers": [], "standards_and_regulation": [ "Mathematical definitions and notation", "Numerical optimisation software conventions" ], "failure_modes_and_hazards": [ "Confusing local and global extrema", "Ignoring boundaries and non-differentiable points", "Numerical methods trapped in local optima" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What extrema are.", "rationale": "Mathematics.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definitions.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definitions.", "questions": [ { "text": "What are global and local maxima and minima, and how do they relate to critical points and boundaries?", "kind": "definition" }, { "text": "What are maximal and minimal elements in partially ordered sets?", "kind": "definition" } ] } ] }, { "id": "tests", "name": "Tests", "description": "Finding extrema.", "findings": [ { "id": "tests-finding", "name": "Tests", "description": "Tests.", "questions": [ { "text": "How are extrema found using first and second derivative tests, and in several variables using gradients and Hessians?", "kind": "definition" }, { "text": "Which entry fits calculus?", "kind": "action" } ] } ] } ] }, { "id": "solve", "name": "Solve", "description": "Solving problems.", "rationale": "Practice.", "layers": [ { "id": "calculate", "name": "Calculate", "description": "Computing extrema.", "findings": [ { "id": "calculate-finding", "name": "Calculate", "description": "Computing.", "questions": [ { "text": "What are the maxima and minima of this function on this domain?", "kind": "action" }, { "text": "Have boundaries and non-differentiable points been checked?", "kind": "boundary" } ] } ] }, { "id": "constrained", "name": "Constrained", "description": "Constrained optimisation.", "findings": [ { "id": "constrained-finding", "name": "Constrained", "description": "Constrained.", "questions": [ { "text": "How are extrema found under constraints using Lagrange multipliers or numerical methods?", "kind": "action" }, { "text": "Which entry fits optimisation?", "kind": "action" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Applications.", "rationale": "Context.", "layers": [ { "id": "fields", "name": "Fields", "description": "Extrema in fields.", "findings": [ { "id": "fields-finding", "name": "Fields", "description": "Fields.", "questions": [ { "text": "How do physics, economics and engineering use extrema, from least action to cost minimisation?", "kind": "provenance" }, { "text": "Which entry fits a specific application?", "kind": "action" } ] } ] }, { "id": "geography", "name": "Geography", "description": "Extreme points.", "findings": [ { "id": "geography-finding", "name": "Geography", "description": "Geography.", "questions": [ { "text": "How are extreme points such as the northernmost point of a country determined?", "kind": "provenance" }, { "text": "Which entry fits extreme points?", "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 theory of extrema develop from Fermat to modern optimisation?", "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 extrema be taught with graphs and applied problems?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should optimisation be a separate entry?", "How should textbooks be linked?", "How should software be linked?" ], "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 maxima and minima, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain maxima and minima and how they are found, distinguish global, local and boundary extrema, support solving optimisation problems, and describe applications across fields.", "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: coordinate of northernmost point, pricelessness, comass, maximum, minimum, maximal element, maxima of a point set, rated maximum, RAM limit, ruling gradient, maximum profit at maturity, maximum loss at maturity.", "22 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 60 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 maxima and minima, 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": [ "coordinate of northernmost point", "pricelessness", "comass", "maximum", "minimum", "maximal element", "maxima of a point set", "rated maximum", "RAM limit", "ruling gradient", "maximum profit at maturity", "maximum loss at maturity" ] } ], "standing": "Described in 60 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 22 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 maxima and minima from another, and a maxima and minima 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 maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima 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 maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima 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 maxima and minima between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a maxima and minima 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 maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima 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 maxima and minima 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 maxima and minima 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 maxima and minima 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 maxima and minima?", "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 maxima and minima?", "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 maxima and minima 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 maxima and minima and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a maxima and minima, 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 maxima and minima, 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 maxima and minima 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 maxima and minima is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a maxima and minima, 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 maxima and minima, 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 maxima and minima 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 } } } }