{ "model": { "rank": 5366, "code": "thing-q173091", "model_id": "vr.tr.transcendental-number", "name": "transcendental number", "purpose": "Let an agent handle transcendental numbers by definition, examples, proofs and open problems, stating proof status accurately.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q173091/", "tier": 2, "score": 68, "payload": { "layer": "wikidata", "aliases": [ "Liouville number", "uncomputable number" ], "aliasCount": 2, "merged": 2, "knownIn": 68, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.transcendental-number", "source": { "dataset": "wikidata", "item": "Q173091", "url": "https://www.wikidata.org/wiki/Q173091", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T10:36:56Z", "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-Q173091", "registryId": "vr.tr.transcendental-number", "name": "transcendental number", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q173091/", "model": { "registry_id": "vr.tr.transcendental-number", "name": "transcendental number", "purpose": "Let an agent handle transcendental numbers by definition, examples, proofs and open problems, stating proof status accurately.", "definition": "A real or complex number that is not algebraic, meaning it is not a root of any non-zero polynomial with rational coefficients, such as pi, e and Liouville numbers; almost all real numbers are transcendental, yet proving a specific number transcendental is often hard.", "what_it_is_for": "Number theory and understanding the structure of the real numbers.", "affordances": [ "explain transcendence versus irrationality", "list proven transcendental numbers", "state open problems accurately", "relate to classical construction problems" ], "distinguishing_features": [ "Non-algebraic", "Uncountably many", "Hard to prove for specific numbers", "Linked to impossibility of squaring the circle" ], "appearance": "Numbers such as pi and e, with non-repeating, non-algebraic expansions.", "visual_identification": [ "Not a root of any integer polynomial", "Examples pi, e, Liouville constant", "Irrational algebraic numbers like the square root of 2 are not transcendental" ], "physical_properties": [], "families_and_kinds": [ "Liouville numbers", "classical constants such as pi and e", "numbers from the Gelfond-Schneider theorem", "uncomputable numbers (related)" ], "related_models": [ { "relation": "is a kind of", "target": "irrational number", "why": "category" }, { "relation": "is contrasted with", "target": "algebraic number", "why": "complement" }, { "relation": "includes", "target": "pi and e", "why": "examples" }, { "relation": "is studied in", "target": "transcendental number theory", "why": "field" } ], "identifiers": [], "standards_and_regulation": [], "failure_modes_and_hazards": [ "Claiming unproven numbers are transcendental", "Confusing irrational and transcendental" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "What it means.", "rationale": "Precise definitions matter.", "layers": [ { "id": "meaning", "name": "Meaning", "description": "Non-algebraic.", "findings": [ { "id": "meaning-finding", "name": "Meaning", "description": "Definition.", "questions": [ { "text": "What does it mean for a number to be transcendental?", "kind": "definition" }, { "text": "How does it differ from irrational?", "kind": "definition" } ] } ] }, { "id": "status", "name": "Status", "description": "Proven or open.", "findings": [ { "id": "status-finding", "name": "Status", "description": "Proof status.", "questions": [ { "text": "Is this number proven transcendental, and by whom?", "kind": "provenance" }, { "text": "Or is its status open?", "kind": "boundary" } ] } ] } ] }, { "id": "examples", "name": "Examples", "description": "Known numbers.", "rationale": "Examples need proofs.", "layers": [ { "id": "classic", "name": "Classic", "description": "Pi and e.", "findings": [ { "id": "classic-finding", "name": "Classic", "description": "Classic examples.", "questions": [ { "text": "When were e and pi proven transcendental?", "kind": "provenance" }, { "text": "Which methods were used?", "kind": "provenance" } ] } ] }, { "id": "constructed", "name": "Constructed", "description": "Liouville.", "findings": [ { "id": "constructed-finding", "name": "Constructed", "description": "Constructed examples.", "questions": [ { "text": "How did Liouville construct the first transcendental numbers?", "kind": "definition" }, { "text": "Why do they work?", "kind": "definition" } ] } ] } ] }, { "id": "theorems", "name": "Theorems", "description": "Results.", "rationale": "Theorems extend knowledge.", "layers": [ { "id": "gelfond", "name": "Gelfond-Schneider", "description": "a^b.", "findings": [ { "id": "gelfond-finding", "name": "Gelfond-Schneider", "description": "Gelfond-Schneider theorem.", "questions": [ { "text": "What does the Gelfond-Schneider theorem say?", "kind": "provenance" }, { "text": "Which numbers does it cover?", "kind": "definition" } ] } ] }, { "id": "open", "name": "Open", "description": "Unknown cases.", "findings": [ { "id": "open-finding", "name": "Open", "description": "Open problems.", "questions": [ { "text": "Which well-known constants have unknown transcendence status?", "kind": "provenance" }, { "text": "Is that stated clearly?", "kind": "boundary" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Consequences.", "rationale": "Transcendence settles old problems.", "layers": [ { "id": "circle", "name": "Squaring the circle", "description": "Impossibility.", "findings": [ { "id": "circle-finding", "name": "Squaring the circle", "description": "Squaring the circle.", "questions": [ { "text": "Why does the transcendence of pi make squaring the circle impossible?", "kind": "definition" }, { "text": "Which proof established it?", "kind": "provenance" } ] } ] }, { "id": "measure", "name": "Measure", "description": "Almost all.", "findings": [ { "id": "measure-finding", "name": "Measure", "description": "Almost all reals.", "questions": [ { "text": "In what sense are almost all real numbers transcendental?", "kind": "definition" }, { "text": "Why is it hard to name examples?", "kind": "definition" } ] } ] } ] } ] }, "openQuestions": [ "Should key theorems be separate entries?", "How should proof status be tracked?", "How should open constants be listed?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "unit, quantity or measure", "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 transcendental number, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle transcendental numbers by definition, examples, proofs and open problems, stating proof status accurately.", "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: Liouville number, uncomputable number.", "2 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 68 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 transcendental number, 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: a unit is a convention. What is seen is an instrument reading in it." ], "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": [ "Liouville number", "uncomputable number" ] } ], "standing": "Described in 68 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 2 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 transcendental number from another, and a transcendental number 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 transcendental number, 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 transcendental number, 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 transcendental number 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 transcendental number, 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 transcendental number, 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 transcendental number 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 transcendental number 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 transcendental number, 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 transcendental number, 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 transcendental number 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 transcendental number from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a transcendental number, 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 transcendental number, 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 transcendental number 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 transcendental number 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 transcendental number between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a transcendental number, 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 transcendental number, 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 transcendental number 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 transcendental number, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a transcendental number 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 transcendental number, 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 transcendental number, 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 transcendental number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definition-and-realisation", "name": "Definition, dimension and realisation", "description": "What transcendental number is defined as and how it is realised in practice.", "rationale": "A unit is not the quantity it measures, and its definition has a history that older data still depends on.", "layers": [ { "id": "definition", "name": "Definition and dimension", "description": "The defining relation and the quantity it measures.", "findings": [ { "id": "definition-record", "name": "Definition, body and date", "description": "What defines it now, and what defined it before.", "questions": [ { "id": "definition-record-q01", "text": "How is transcendental number defined, by which body, and since when?", "kind": "definition" }, { "id": "definition-record-q02", "text": "What was the previous definition, and what data still depends on it?", "kind": "provenance" } ] } ] }, { "id": "conversion", "name": "Conversion and compatibility", "description": "Relations to other units, exact or approximate.", "findings": [ { "id": "conversion-record", "name": "Conversions and their exactness", "description": "Factors, and whether each is exact by definition.", "questions": [ { "id": "conversion-record-q01", "text": "What conversions relate transcendental number to other units, and which are exact by definition?", "kind": "measurement" }, { "id": "conversion-record-q02", "text": "Where does rounding a conversion change a decision, and how should that be recorded?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a transcendental number 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 transcendental number and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a transcendental number, 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 transcendental number, 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 transcendental number 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 transcendental number is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a transcendental number, 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 transcendental number, 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 transcendental number is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 4, "layers": 8, "findings": 10, "questions": 28 } } } }