{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.theorem","code":"thing-q65943","url":"https://ver.cy/models/thing/q65943/","name":"theorem","alternateNames":["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"],"kind":"thing","status":"published","version":"0.2.0-wave.1","language":"en","classifiers":{"family":"Thing Registry","category":"Information and virtual systems","entryKind":"thing","plane":"INF","domain":["INF.KNW"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"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.","purpose":"Let an agent explain theorems by statement, proof, history and use, distinguish related terms, and point to formal verification resources.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Proved statement","Depends on axioms","Builds on earlier results","Can be formally verified"],"structure":{"bundles":[{"id":"statement","name":"Statement","description":"What it says.","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.","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.","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.","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"}]}]}]}]},"agentConduct":{"may":["State a theorem with its hypotheses and cite where it is proved.","Check or help write proofs at the request of the person doing the work."],"mustNot":["Present a conjecture or unproved claim as a theorem.","Drop hypotheses of a theorem so that it appears to apply where it does not.","Attribute a theorem or proof to the wrong person.","Pass off machine-generated proofs as a student's own work in assessed settings."],"requiresHuman":["Accepting a machine-checked proof as the basis for a safety-critical system."]},"ethics":{"considerations":["Credit for results matters to mathematicians' careers and has often been denied to some groups.","Misapplied theorems can make flawed engineering or statistics look rigorous."],"affectedParties":["Mathematicians and students","Users of results in science and engineering"]},"owners":{"steward":"Nobody owns a theorem; it belongs to mathematics, and its discoverers are credited.","roles":[],"masterSystems":[]},"relations":[{"target":"Q108163","type":"parent","note":"registry parent class"},{"target":"Q20665666","type":"parent","note":"registry parent class"},{"target":"proposition","type":"related","note":"category"},{"target":"mathematical proof","type":"related","note":"method"},{"target":"conjecture","type":"related","note":"unproved statement"},{"target":"axiom","type":"related","note":"foundation"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.theorem","Wikidata: Q65943 (https://www.wikidata.org/wiki/Q65943)"]},"properties":{"applicability":"not-applicable","items":[]},"recognition":{"applicability":"optional","items":["Named results with proofs","Lemmas and corollaries","Conjectures are unproved","Not physical; statements and proofs in texts and formal libraries."]},"capabilities":{"applicability":"required","items":["state and explain a theorem","outline or check a proof","distinguish lemma, corollary and conjecture","find formalised proofs"]},"hazards":{"applicability":"required","items":["Misstating hypotheses","Misattributing theorems","Treating conjectures as proved"]},"interfaces":{"applicability":"required","items":["No specific regulation"]},"context":{"applicability":"required","items":["Building mathematical knowledge.","major named theorems","lemmas","corollaries","existence and classification theorems","formally verified theorems"]}},"sources":[{"title":"Wikidata item Q65943: theorem","url":"https://www.wikidata.org/wiki/Q65943","note":"identity and sense of the item"},{"title":"Wikipedia: Theorem","url":"https://en.wikipedia.org/wiki/Theorem","note":"general description of the item"}],"openQuestions":["Should each theorem be a separate entry?","How should formal libraries be linked?","How should naming disputes be recorded?"],"resources":{"spec":"/models/things/publications/thing-q65943/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q65943/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-11T12:56:33Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:52:01+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane INF: no invented physical properties."},"score":1.0}}