{ "model": { "rank": 2138, "code": "thing-q11538", "model_id": "vr.tr.mathematical-proof", "name": "mathematical proof", "purpose": "Let an agent explain what a mathematical proof is and the main proof techniques, help construct, check and explain proofs, describe formal and computer-assisted proof, and support learning proof writing.", "family": "Thing Registry", "category": "Information and virtual systems", "status": "research-draft", "kind": "thing", "plane": "INF", "domain": "INF.KNW", "industry": "", "version": "", "url": "/models/thing/q11538/", "tier": 2, "score": 114, "payload": { "layer": "wikidata", "aliases": [ "computer-assisted proof", "strategy-stealing argument", "proof by contrapositive", "Proofs of Fermat's theorem on sums of two squares", "indirect proof", "proof of the pythagorean theorem", "exponent combination laws", "well-founded induction", "non-constructive proof", "proof by contradiction", "non-surveyable proof", "termination proof", "proofs of minimax theorem", "direct proof", "combinatorial proof", "proofs of Fermat's little theorem", "analytic proof", "inductive step", "Cantor's diagonal argument", "statistical proof", "umbral moonshine", "probabilistically checkable proof", "proof that e is irrational", "proof by exhaustion", "The Fermat method" ], "aliasCount": 25, "merged": 26, "knownIn": 114, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q11538", "url": "https://www.wikidata.org/wiki/Q11538", "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-Q11538", "registryId": "vr.tr.mathematical-proof", "name": "mathematical proof", "version": "0.2.0-wave.2", "entryKind": "thing", "family": "Thing Registry", "domain": [ "INF.KNW" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q11538/", "model": { "registry_id": "vr.tr.mathematical-proof", "name": "mathematical proof", "purpose": "Let an agent explain what a mathematical proof is and the main proof techniques, help construct, check and explain proofs, describe formal and computer-assisted proof, and support learning proof writing.", "definition": "A rigorous argument that establishes the truth of a mathematical statement from axioms, definitions and previously proved results by valid inference, in forms such as direct proof, proof by contradiction, proof by contrapositive, induction, construction, exhaustion, strategy-stealing arguments and computer-assisted proofs; proofs are the standard of certainty in mathematics, are written for human readers or checked by proof assistants, and famous examples include proofs of the Pythagorean theorem and of Fermat theorem on sums of two squares.", "what_it_is_for": "Rigorous arguments establishing mathematical truths.", "affordances": [ "explain proof and techniques", "construct and check proofs", "describe formal proof", "support learning" ], "distinguishing_features": [ "Deductive rigour", "Techniques", "Formalisable", "Peer review" ], "appearance": "Not physical; written arguments.", "visual_identification": [ "Deductive argument from axioms", "Establishes a theorem", "Evidence and heuristics are not proofs; a conjecture is unproved" ], "physical_properties": [], "families_and_kinds": [ "direct proofs", "proofs by contradiction and contrapositive", "proofs by induction", "constructive and existence proofs", "computer-assisted and formal proofs" ], "related_models": [ { "relation": "is a kind of", "target": "argument", "why": "category" }, { "relation": "is a kind of", "target": "mathematical reasoning", "why": "category" }, { "relation": "is related to", "target": "inequality", "why": "statements often proved" }, { "relation": "is related to", "target": "imaginary", "why": "objects proofs concern" } ], "identifiers": [], "standards_and_regulation": [ "Conventions of mathematical writing", "Peer review in journals", "Proof assistant standards" ], "failure_modes_and_hazards": [ "Circular reasoning and gaps", "Confusing examples with proofs", "Errors in long proofs" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What proofs are.", "rationale": "Mathematics.", "layers": [ { "id": "concept", "name": "Concept", "description": "Concept and standards.", "findings": [ { "id": "concept-finding", "name": "Concept", "description": "Concept.", "questions": [ { "text": "What makes an argument a mathematical proof, and how do axioms, definitions and inference rules function in it?", "kind": "definition" }, { "text": "Which entry fits logic?", "kind": "action" } ] } ] }, { "id": "techniques", "name": "Techniques", "description": "Techniques.", "findings": [ { "id": "techniques-finding", "name": "Techniques", "description": "Techniques.", "questions": [ { "text": "How do direct proof, contradiction, contrapositive, induction and construction work, and when is each used?", "kind": "definition" }, { "text": "Which entry fits a specific technique?", "kind": "action" } ] } ] } ] }, { "id": "write", "name": "Write", "description": "Constructing and checking.", "rationale": "Practice.", "layers": [ { "id": "construct", "name": "Construct", "description": "Constructing a proof.", "findings": [ { "id": "construct-finding", "name": "Construct", "description": "Constructing.", "questions": [ { "text": "How can this statement be proved, and which technique fits?", "kind": "action" }, { "text": "Is the statement precisely stated?", "kind": "boundary" } ] } ] }, { "id": "check", "name": "Check", "description": "Checking a proof.", "findings": [ { "id": "check-finding", "name": "Check", "description": "Checking.", "questions": [ { "text": "Is this proof valid, and where are its gaps or errors?", "kind": "action" }, { "text": "Which entry fits common proof errors?", "kind": "action" } ] } ] } ] }, { "id": "formal", "name": "Formal", "description": "Formal and computer proof.", "rationale": "Technology.", "layers": [ { "id": "assistants", "name": "Assistants", "description": "Proof assistants.", "findings": [ { "id": "assistants-finding", "name": "Assistants", "description": "Assistants.", "questions": [ { "text": "How do proof assistants and formal verification work, and which proofs have been formalised?", "kind": "provenance" }, { "text": "Which entry fits a specific proof assistant?", "kind": "action" } ] } ] }, { "id": "computer", "name": "Computer", "description": "Computer-assisted proofs.", "findings": [ { "id": "computer-finding", "name": "Computer", "description": "Computer.", "questions": [ { "text": "What are computer-assisted proofs, and what debates surround their acceptance, with positions attributed?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] } ] }, { "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 concept of proof develop from Euclid to modern foundations, and which proofs are famous?", "kind": "provenance" }, { "text": "Which entry fits a famous proof?", "kind": "action" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching proof.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How can proof writing be taught to beginners?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should each technique be a separate entry?", "How should proof assistants be linked?", "How should famous proofs be linked?" ], "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 mathematical proof, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain what a mathematical proof is and the main proof techniques, help construct, check and explain proofs, describe formal and computer-assisted proof, and support learning proof writing.", "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: computer-assisted proof, strategy-stealing argument, proof by contrapositive, Proofs of Fermat's theorem on sums of two squares, indirect proof, proof of the pythagorean theorem, exponent combination laws, well-founded induction, non-constructive proof, proof by contradiction, non-surveyable proof, termination proof.", "26 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 114 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 mathematical proof, 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": [ "computer-assisted proof", "strategy-stealing argument", "proof by contrapositive", "Proofs of Fermat's theorem on sums of two squares", "indirect proof", "proof of the pythagorean theorem", "exponent combination laws", "well-founded induction", "non-constructive proof", "proof by contradiction", "non-surveyable proof", "termination proof" ] } ], "standing": "Described in 114 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 26 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 mathematical proof from another, and a mathematical proof 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 mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof 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 mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof 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 mathematical proof between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a mathematical proof 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 mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof 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 mathematical proof contains and in what form it is held.", "questions": [ { "id": "content-and-format-q01", "text": "What identifies and describes the content of a mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof and the period it survives.", "questions": [ { "id": "access-rules-q01", "text": "What identifies and describes access to a mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof 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 mathematical proof 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 mathematical proof 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 mathematical proof 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 mathematical proof 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 mathematical proof, 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 mathematical proof 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 mathematical proof and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a mathematical proof, 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 mathematical proof, 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 mathematical proof 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 mathematical proof is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a mathematical proof, 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 mathematical proof, 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 mathematical proof 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 } } } }