{ "model": { "rank": 13638, "code": "thing-q217616", "model_id": "vr.tr.meromorphic-function", "name": "meromorphic function", "purpose": "Let an agent define meromorphic functions and their properties, give examples and distinguish them from holomorphic and entire functions, relay key theorems and their role in number theory, and separate the concept from the algebraic geometry sense of regular function listed as an alias.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q217616/", "tier": 2, "score": 37, "payload": { "layer": "wikidata", "aliases": [ "regular function", "Dirichlet L-function", "Artin L-function", "polygamma function", "balanced polygamma function", "L-function", "Virasoro conformal block", "automorphic L-function", "Rankin–Selberg method", "Shimizu L-function" ], "aliasCount": 10, "merged": 10, "knownIn": 37, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q217616", "url": "https://www.wikidata.org/wiki/Q217616", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T22:59:37Z", "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": 3, "engine": "claude" }, "metaModel": { "id": "THING-Q217616", "registryId": "vr.tr.meromorphic-function", "name": "meromorphic function", "version": "0.2.0-wave.3", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q217616/", "model": { "registry_id": "vr.tr.meromorphic-function", "name": "meromorphic function", "purpose": "Let an agent define meromorphic functions and their properties, give examples and distinguish them from holomorphic and entire functions, relay key theorems and their role in number theory, and separate the concept from the algebraic geometry sense of regular function listed as an alias.", "definition": "A function of a complex variable that is holomorphic on an open domain except at isolated points where it has poles, so that it can be written locally as a ratio of holomorphic functions; meromorphic functions include rational functions, the gamma and polygamma functions and L-functions such as the Riemann zeta, Dirichlet and Artin L-functions, and they are central to complex analysis, number theory and the study of Riemann surfaces.", "what_it_is_for": "A central class of functions in complex analysis.", "affordances": [ "define and characterise the class", "give examples and counterexamples", "relay key theorems", "distinguish related notions" ], "distinguishing_features": [ "Isolated poles only", "Field of functions on a domain", "Residues and Laurent expansions", "Examples from number theory" ], "appearance": "Not a visible object; functions with poles, visualised by domain colouring.", "visual_identification": [ "Holomorphic except at isolated poles", "Locally a ratio of holomorphic functions", "Holomorphic functions have no poles; essential singularities are excluded" ], "physical_properties": [], "families_and_kinds": [ "rational functions", "the gamma function and polygamma functions", "L-functions including Dirichlet and Artin L-functions and the Riemann zeta function", "elliptic functions", "meromorphic functions on Riemann surfaces", "regular functions in algebraic geometry, a related but distinct notion listed as an alias" ], "related_models": [ { "relation": "is a kind of", "target": "function of a complex variable", "why": "in registry terms" }, { "relation": "is a generalisation of", "target": "holomorphic function", "why": "allowing poles" }, { "relation": "includes", "target": "rational function", "why": "the simplest case" }, { "relation": "is related to", "target": "Riemann surface", "why": "where the theory extends" } ], "identifiers": [], "standards_and_regulation": [ "No regulation; standard definitions in complex analysis texts" ], "failure_modes_and_hazards": [ "Confusing poles with essential singularities", "Confusing meromorphic with holomorphic or entire", "Sense confusion with regular functions in algebraic geometry" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "define", "name": "Define", "description": "What a meromorphic function is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition and examples.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a meromorphic function, and what are examples and counterexamples?", "kind": "definition" }, { "text": "Is the function meromorphic, holomorphic, entire or none?", "kind": "boundary" } ] } ] }, { "id": "poles", "name": "Poles", "description": "Poles and residues.", "findings": [ { "id": "poles-finding", "name": "Poles", "description": "Poles.", "questions": [ { "text": "How are poles, orders and residues defined and computed?", "kind": "definition" }, { "text": "Which entry fits residue?", "kind": "action" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Theorems.", "rationale": "Theory.", "layers": [ { "id": "theorems", "name": "Theorems", "description": "Key theorems.", "findings": [ { "id": "theorems-finding", "name": "Theorems", "description": "Theorems.", "questions": [ { "text": "What do the residue theorem, the argument principle and Mittag-Leffler theorem say?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "surfaces", "name": "Surfaces", "description": "Riemann surfaces.", "findings": [ { "id": "surfaces-finding", "name": "Surfaces", "description": "Surfaces.", "questions": [ { "text": "How do meromorphic functions behave on compact Riemann surfaces, and what does Riemann-Roch say?", "kind": "provenance" }, { "text": "Which entry fits Riemann-Roch theorem?", "kind": "action" } ] } ] } ] }, { "id": "examples", "name": "Examples", "description": "Important examples.", "rationale": "Application.", "layers": [ { "id": "special", "name": "Special", "description": "Special functions.", "findings": [ { "id": "special-finding", "name": "Special", "description": "Special.", "questions": [ { "text": "Why are the gamma, polygamma and elliptic functions meromorphic, and where are their poles?", "kind": "provenance" }, { "text": "Which entry fits gamma function?", "kind": "action" } ] } ] }, { "id": "lfunctions", "name": "L-functions", "description": "L-functions.", "findings": [ { "id": "lfunctions-finding", "name": "Lfunctions", "description": "Lfunctions.", "questions": [ { "text": "How do the Riemann zeta, Dirichlet and Artin L-functions extend meromorphically, and why does it matter in number theory?", "kind": "provenance" }, { "text": "Which entry fits L-function?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "History and teaching.", "rationale": "Context.", "layers": [ { "id": "history", "name": "History", "description": "History.", "findings": [ { "id": "history-finding", "name": "History", "description": "History.", "questions": [ { "text": "How did the concept develop with Cauchy, Riemann and Weierstrass?", "kind": "provenance" }, { "text": "Which entry fits the history of complex analysis?", "kind": "action" } ] } ] }, { "id": "teach", "name": "Teach", "description": "Teaching.", "findings": [ { "id": "teach-finding", "name": "Teach", "description": "Teaching.", "questions": [ { "text": "How are meromorphic functions taught, and which misconceptions arise?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should L-functions be separate primary entries?", "How should reference texts be linked?", "The registry entry has merged aliases naming specific functions and a different notion; should they be split off?" ], "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 meromorphic function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent define meromorphic functions and their properties, give examples and distinguish them from holomorphic and entire functions, relay key theorems and their role in number theory, and separate the concept from the algebraic geometry sense of regular function listed as an alias.", "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: regular function, Dirichlet L-function, Artin L-function, polygamma function, balanced polygamma function, L-function, Virasoro conformal block, automorphic L-function, Rankin–Selberg method, Shimizu L-function.", "10 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 37 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 meromorphic function, 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": [ "regular function", "Dirichlet L-function", "Artin L-function", "polygamma function", "balanced polygamma function", "L-function", "Virasoro conformal block", "automorphic L-function", "Rankin–Selberg method", "Shimizu L-function" ] } ], "standing": "Described in 37 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 10 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 meromorphic function from another, and a meromorphic function 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 meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function 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 meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function 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 meromorphic function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a meromorphic function 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 meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function 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 meromorphic function 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 meromorphic function 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 meromorphic function 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 meromorphic function?", "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 meromorphic function?", "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 meromorphic function 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 meromorphic function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a meromorphic function, 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 meromorphic function, 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 meromorphic function 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 meromorphic function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a meromorphic function, 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 meromorphic function, 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 meromorphic function 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 } } } }