{ "model": { "rank": 3164, "code": "thing-q170198", "model_id": "vr.tr.series", "name": "series", "purpose": "Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.REL", "industry": "", "version": "", "url": "/models/thing/q170198/", "tier": 2, "score": 90, "payload": { "layer": "wikidata", "aliases": [ "formal power series", "ultrapolynomial", "quasisymmetric function", "asymptotic expansion", "Cornish–Fisher expansion", "operator product expansion", "convergent series", "multipole expansion", "basic hypergeometric series", "Charlier series", "geometric series", "function series", "Lüroth series", "composition series", "perturbative series", "Edgeworth series", "general Dirichlet series", "Hahn series", "infinite arithmetic series", "Poincaré series", "binomial series", "divergent series", "numerical series", "arithmetic series", "absolutely convergent series", "conditionally convergent series", "finite geometric series", "infinite geometric series", "Puiseux series", "series diverging to infinity" ], "aliasCount": 30, "merged": 30, "knownIn": 90, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": "vr.tr.series-cognition", "source": { "dataset": "wikidata", "item": "Q170198", "url": "https://www.wikidata.org/wiki/Q170198", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T05:00:01Z", "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-Q170198", "registryId": "vr.tr.series", "name": "series", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.REL" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q170198/", "model": { "registry_id": "vr.tr.series", "name": "series", "purpose": "Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.", "definition": "In mathematics, the sum of the terms of a sequence, written a1 + a2 + a3 + ..., which may be finite or infinite and, if infinite, convergent or divergent.", "what_it_is_for": "Representing functions, approximating values and solving problems in analysis, physics and engineering.", "affordances": [ "sum it, when it converges", "test convergence, with standard tests", "expand functions as series, such as power or Fourier series", "truncate it, for approximation, with an error bound" ], "distinguishing_features": [ "A sum of terms, not the sequence itself", "Infinite series may converge or diverge", "Formal power series need not converge", "Asymptotic expansions can be useful even when divergent" ], "appearance": "Summation notation with a sigma, or terms with plus signs and ellipsis.", "visual_identification": [ "Sigma notation", "Named series such as geometric, harmonic, Taylor, Fourier", "TV and book series are different senses" ], "physical_properties": [], "families_and_kinds": [ "geometric and arithmetic series", "power, Taylor and Laurent series", "Fourier series", "formal power series", "asymptotic expansions" ], "related_models": [ { "relation": "is built from", "target": "sequence", "why": "terms of a sequence" }, { "relation": "is confused with", "target": "sequence", "why": "ordered terms, not their sum" }, { "relation": "is used in", "target": "analysis and physics", "why": "approximation and expansion" }, { "relation": "is confused with", "target": "series as a set of works", "why": "a homonym" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Summing a divergent series as if convergent", "Rearranging conditionally convergent series", "Truncating without an error bound" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "terms", "name": "Terms", "description": "What is summed.", "rationale": "The terms define the series.", "layers": [ { "id": "rule", "name": "Term rule", "description": "General term.", "findings": [ { "id": "rule-finding", "name": "Term rule", "description": "The general term.", "questions": [ { "text": "What is the general term of the series?", "kind": "definition" }, { "text": "Where does the summation start?", "kind": "definition" } ] } ] }, { "id": "kind", "name": "Kind", "description": "Geometric, power, Fourier.", "findings": [ { "id": "kind-finding", "name": "Kind", "description": "The kind of series.", "questions": [ { "text": "What kind of series is this?", "kind": "definition" }, { "text": "Is it formal or analytic?", "kind": "boundary" } ] } ] } ] }, { "id": "convergence", "name": "Convergence", "description": "Does it sum.", "rationale": "Convergence decides whether a sum exists.", "layers": [ { "id": "test", "name": "Tests", "description": "Ratio, root, comparison.", "findings": [ { "id": "test-finding", "name": "Convergence test", "description": "Test used.", "questions": [ { "text": "Does the series converge, by which test?", "kind": "measurement" }, { "text": "For which values of the variable?", "kind": "boundary" } ] } ] }, { "id": "type", "name": "Type of convergence", "description": "Absolute, conditional, uniform.", "findings": [ { "id": "type-finding", "name": "Convergence type", "description": "The type of convergence.", "questions": [ { "text": "Is the convergence absolute, conditional or uniform?", "kind": "definition" }, { "text": "What operations does that permit?", "kind": "boundary" } ] } ] } ] }, { "id": "sum", "name": "Sum", "description": "The value.", "rationale": "The sum is the goal when it exists.", "layers": [ { "id": "value", "name": "Value", "description": "Closed form.", "findings": [ { "id": "value-finding", "name": "Sum", "description": "The sum.", "questions": [ { "text": "What is the sum, in closed form if possible?", "kind": "measurement" }, { "text": "How was it derived?", "kind": "provenance" } ] } ] }, { "id": "approximation", "name": "Approximation", "description": "Truncation error.", "findings": [ { "id": "approximation-finding", "name": "Error bound", "description": "Error of partial sums.", "questions": [ { "text": "How many terms give the needed accuracy?", "kind": "measurement" }, { "text": "What bounds the truncation error?", "kind": "measurement" } ] } ] } ] }, { "id": "application", "name": "Application", "description": "Where it is used.", "rationale": "Series are tools for representing functions.", "layers": [ { "id": "expansion", "name": "Expansion", "description": "Functions as series.", "findings": [ { "id": "expansion-finding", "name": "Expansion", "description": "Function expansions.", "questions": [ { "text": "Which function does this series represent, and where?", "kind": "definition" }, { "text": "What is the radius of convergence?", "kind": "measurement" } ] } ] }, { "id": "computation", "name": "Computation", "description": "Numerical use.", "findings": [ { "id": "computation-finding", "name": "Numerics", "description": "Numerical behaviour.", "questions": [ { "text": "Is the series numerically stable to sum?", "kind": "boundary" }, { "text": "Should acceleration methods be used?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should named series be separate entries?", "How should convergence conditions be recorded?", "How should formal and analytic series be distinguished?" ], "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 series, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle series with their terms, convergence and sums, and separate a series from the sequence of its terms.", "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: formal power series, ultrapolynomial, quasisymmetric function, asymptotic expansion, Cornish–Fisher expansion, operator product expansion, convergent series, multipole expansion, basic hypergeometric series, Charlier series, geometric series, function series.", "30 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 90 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 series, 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": [ "formal power series", "ultrapolynomial", "quasisymmetric function", "asymptotic expansion", "Cornish–Fisher expansion", "operator product expansion", "convergent series", "multipole expansion", "basic hypergeometric series", "Charlier series", "geometric series", "function series" ] } ], "standing": "Described in 90 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 30 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 series from another, and a series 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 series, 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 series, 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 series 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 series, 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 series, 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 series 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 series 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 series, 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 series, 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 series 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 series from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a series, 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 series, 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 series 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 series 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 series between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a series, 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 series, 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 series 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 series, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a series 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 series, 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 series, 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 series 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 series 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 series 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 series 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 series 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 series?", "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 series?", "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 series 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 series and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a series, 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 series, 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 series 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 series is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a series, 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 series, 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 series 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 } } } }