{ "model": { "rank": 2583, "code": "thing-q65943", "model_id": "vr.tr.theorem", "name": "theorem", "purpose": "Let an agent explain theorems by statement, proof, history and use, distinguish related terms, and point to formal verification resources.", "family": "Thing Registry", "category": "Information and virtual systems", "status": "research-draft", "kind": "thing", "plane": "INF", "domain": "INF.KNW", "industry": "", "version": "", "url": "/models/thing/q65943/", "tier": 2, "score": 102, "payload": { "layer": "wikidata", "aliases": [ "Johnson bound", "differentiation rule", "lemma", "pumping lemma", "isomorphism theorem", "Jacobi–Anger expansion", "Serre duality", "Newton's inequalities", "unexpected hanging paradox", "Mertens' theorems", "Abelian theorem", "Tauberian theorem", "Bierlein extension theorem", "network theorem", "fixed-point theorem", "Green's identities", "Weyl's inequality", "hypergeometric identity", "Siegel identity", "valuative criterion", "proof of impossibility", "polarization identity", "Abelian and tauberian theorems", "metatheorem", "universal coefficient theorem", "Fontené theorems", "tangent half-angle formula", "no-go theorem", "relationship among probability distributions", "spherical law of cosines", "Sahlqvist formula", "Minkowski–Steiner formula", "economic theorem", "fundamental theorem", "Cassini and Catalan identities", "base change theorem", "Clarkson's inequalities", "classification theorem", "Mac Lane's coherence theorem", "Generic flatness" ], "aliasCount": 57, "merged": 57, "knownIn": 102, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.theorem", "source": { "dataset": "wikidata", "item": "Q65943", "url": "https://www.wikidata.org/wiki/Q65943", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T12:56:33Z", "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": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q65943", "registryId": "vr.tr.theorem", "name": "theorem", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "INF.KNW" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q65943/", "model": { "registry_id": "vr.tr.theorem", "name": "theorem", "purpose": "Let an agent explain theorems by statement, proof, history and use, distinguish related terms, and point to formal verification resources.", "definition": "A mathematical statement proved true by logical deduction from axioms and previously established results, such as the Pythagorean theorem or the fundamental theorem of arithmetic; related terms include lemma, corollary and proposition.", "what_it_is_for": "Building mathematical knowledge.", "affordances": [ "state and explain a theorem", "outline or check a proof", "distinguish lemma, corollary and conjecture", "find formalised proofs" ], "distinguishing_features": [ "Proved statement", "Depends on axioms", "Builds on earlier results", "Can be formally verified" ], "appearance": "Not physical; statements and proofs in texts and formal libraries.", "visual_identification": [ "Named results with proofs", "Lemmas and corollaries", "Conjectures are unproved" ], "physical_properties": [], "families_and_kinds": [ "major named theorems", "lemmas", "corollaries", "existence and classification theorems", "formally verified theorems" ], "related_models": [ { "relation": "is a kind of", "target": "proposition", "why": "category" }, { "relation": "is established by", "target": "mathematical proof", "why": "method" }, { "relation": "is contrasted with", "target": "conjecture", "why": "unproved statement" }, { "relation": "is related to", "target": "axiom", "why": "foundation" } ], "identifiers": [], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Misstating hypotheses", "Misattributing theorems", "Treating conjectures as proved" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "statement", "name": "Statement", "description": "What it says.", "rationale": "Hypotheses matter.", "layers": [ { "id": "state", "name": "State", "description": "Precise statement.", "findings": [ { "id": "state-finding", "name": "State", "description": "Statement.", "questions": [ { "text": "What exactly does the theorem state, including all hypotheses?", "kind": "definition" }, { "text": "What happens if a hypothesis is dropped?", "kind": "boundary" } ] } ] }, { "id": "meaning", "name": "Meaning", "description": "Intuition.", "findings": [ { "id": "meaning-finding", "name": "Meaning", "description": "Intuition.", "questions": [ { "text": "How can the theorem be explained intuitively?", "kind": "action" }, { "text": "Which example illustrates it?", "kind": "action" } ] } ] } ] }, { "id": "proof", "name": "Proof", "description": "Why it is true.", "rationale": "Proofs justify.", "layers": [ { "id": "outline", "name": "Outline", "description": "Proof sketch.", "findings": [ { "id": "outline-finding", "name": "Outline", "description": "Proof outline.", "questions": [ { "text": "What are the main steps of a standard proof?", "kind": "definition" }, { "text": "Which earlier results does it use?", "kind": "provenance" } ] } ] }, { "id": "check", "name": "Check", "description": "Verifying.", "findings": [ { "id": "check-finding", "name": "Check", "description": "Checking proofs.", "questions": [ { "text": "Is this proposed proof correct, and where might it fail?", "kind": "boundary" }, { "text": "Is there a formalised proof in a proof assistant?", "kind": "provenance" } ] } ] } ] }, { "id": "terms", "name": "Terms", "description": "Related terms.", "rationale": "Terminology differs.", "layers": [ { "id": "lemma", "name": "Lemma", "description": "Lemmas and corollaries.", "findings": [ { "id": "lemma-finding", "name": "Lemma", "description": "Lemmas and corollaries.", "questions": [ { "text": "Is this result a lemma, corollary or main theorem?", "kind": "definition" }, { "text": "Why is it labelled that way?", "kind": "definition" } ] } ] }, { "id": "conjecture", "name": "Conjecture", "description": "Unproved statements.", "findings": [ { "id": "conjecture-finding", "name": "Conjecture", "description": "Conjectures.", "questions": [ { "text": "Has this statement actually been proved, or is it still a conjecture?", "kind": "provenance" }, { "text": "Which source confirms its status?", "kind": "provenance" } ] } ] } ] }, { "id": "history", "name": "History", "description": "Origins.", "rationale": "Attribution matters.", "layers": [ { "id": "who", "name": "Who", "description": "Discoverers.", "findings": [ { "id": "who-finding", "name": "Who", "description": "Attribution.", "questions": [ { "text": "Who first proved the theorem, according to historians of mathematics?", "kind": "provenance" }, { "text": "Is the naming historically accurate?", "kind": "boundary" } ] } ] }, { "id": "uses", "name": "Uses", "description": "Applications.", "findings": [ { "id": "uses-finding", "name": "Uses", "description": "Applications.", "questions": [ { "text": "Where is the theorem used in mathematics or applications?", "kind": "provenance" }, { "text": "Which fields depend on it?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should each theorem be a separate entry?", "How should formal libraries be linked?", "How should naming disputes be recorded?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "discipline or field of knowledge", "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 theorem, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain theorems by statement, proof, history and use, distinguish related terms, and point to formal verification resources.", "characteristics": { "substance": "information", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "read and interpreted" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: Johnson bound, differentiation rule, lemma, pumping lemma, isomorphism theorem, Jacobi–Anger expansion, Serre duality, Newton's inequalities, unexpected hanging paradox, Mertens' theorems, Abelian theorem, Tauberian theorem.", "57 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 102 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 theorem, and on what evidence are they told apart?" ], "whatItIsMadeOf": "content that has to be carried by something else", "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": [ "read it and act on what it says" ], "howItIsRecognised": [], "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": [ "Johnson bound", "differentiation rule", "lemma", "pumping lemma", "isomorphism theorem", "Jacobi–Anger expansion", "Serre duality", "Newton's inequalities", "unexpected hanging paradox", "Mertens' theorems", "Abelian theorem", "Tauberian theorem" ] } ], "standing": "Described in 102 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 57 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 theorem from another, and a theorem 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 theorem, 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 theorem, 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 theorem 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 theorem, 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 theorem, 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 theorem 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 theorem 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 theorem, 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 theorem, 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 theorem 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 theorem from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a theorem, 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 theorem, 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 theorem 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 theorem 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 theorem between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a theorem, 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 theorem, 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 theorem 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 theorem, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a theorem 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 theorem, 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 theorem, 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 theorem is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "content-and-access", "name": "Content, encoding and access", "description": "What a theorem says, how it is encoded and who may read it.", "rationale": "An informational thing carries content that can be copied, versioned and withheld, none of which its physical carrier explains.", "layers": [ { "id": "content-and-encoding", "name": "Content and encoding", "description": "The content itself, its format and its language.", "findings": [ { "id": "content-and-format", "name": "Content, format and language", "description": "What a theorem contains and in what form it is held.", "questions": [ { "id": "content-and-format-q01", "text": "What identifies and describes the content of a theorem, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "content-and-format-q02", "text": "Who or what asserted this about the content of a theorem, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "content-and-format-q03", "text": "What may an agent decide or do once the content of a theorem is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "access-and-rights", "name": "Access, rights and retention", "description": "Who may read, copy or change it, and for how long it is kept.", "findings": [ { "id": "access-rules", "name": "Access rules and retention", "description": "The permissions attached to a theorem and the period it survives.", "questions": [ { "id": "access-rules-q01", "text": "What identifies and describes access to a theorem, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "access-rules-q02", "text": "Who or what asserted this about access to a theorem, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "access-rules-q03", "text": "What may an agent decide or do once access to a theorem is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "subject-and-method", "name": "Subject matter and method", "description": "What theorem studies and how it establishes anything.", "rationale": "A field is identified by its questions and its methods, and those are what distinguish it from the fields next to it.", "layers": [ { "id": "subject", "name": "Subject matter", "description": "The questions it takes as its own.", "findings": [ { "id": "subject-record", "name": "Questions and objects of study", "description": "What it is about, stated so a neighbouring field can be told apart.", "questions": [ { "id": "subject-record-q01", "text": "What questions and objects does theorem take as its own?", "kind": "definition" }, { "id": "subject-record-q02", "text": "Which questions does it share with a neighbouring field, and who claims them?", "kind": "boundary" } ] } ] }, { "id": "method", "name": "Method and evidence", "description": "How claims are established and what counts as evidence.", "findings": [ { "id": "method-record", "name": "Methods and standards of evidence", "description": "The methods used and what they are taken to establish.", "questions": [ { "id": "method-record-q01", "text": "By what methods does theorem establish claims, and what counts as sufficient evidence?", "kind": "provenance" }, { "id": "method-record-q02", "text": "What kind of claim can this field not settle, and where should an agent look instead?", "kind": "action" } ] } ] } ] }, { "id": "institutions-and-classification", "name": "Institutions and classification", "description": "Where theorem is practised and how catalogues place it.", "rationale": "The institutional footprint is the evidence that a field exists as a field rather than as a topic.", "layers": [ { "id": "institutions", "name": "Institutional markers", "description": "Societies, journals, degrees and departments.", "findings": [ { "id": "institution-record", "name": "Bodies and venues", "description": "Where the field organises itself.", "questions": [ { "id": "institution-record-q01", "text": "Which societies, journals or degrees mark theorem as an established field?", "kind": "provenance" }, { "id": "institution-record-q02", "text": "What would show that it is emerging or dissolving rather than established?", "kind": "boundary" } ] } ] }, { "id": "classification", "name": "Classification schemes", "description": "Where library and research classifications put it.", "findings": [ { "id": "classification-record", "name": "Codes and their disagreements", "description": "The classification codes that place it, and where they differ.", "questions": [ { "id": "classification-record-q01", "text": "Which classification schemes place theorem, under what codes?", "kind": "definition" }, { "id": "classification-record-q02", "text": "Where do those schemes disagree about its scope?", "kind": "boundary" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a theorem 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 theorem and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a theorem, 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 theorem, 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 theorem 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 theorem is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a theorem, 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 theorem, 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 theorem is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 6, "layers": 12, "findings": 14, "questions": 38 } } } }