{ "model": { "rank": 16924, "code": "thing-q438842", "model_id": "vr.tr.alternating-series", "name": "alternating series", "purpose": "Let an agent explain alternating series by definition, convergence tests, error bounds and examples for learners and practitioners.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.REL", "industry": "", "version": "", "url": "/models/thing/q438842/", "tier": 2, "score": 31, "payload": { "layer": "wikidata", "aliases": [], "aliasCount": 0, "merged": 0, "knownIn": 31, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q438842", "url": "https://www.wikidata.org/wiki/Q438842", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T13:03:37Z", "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": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q438842", "registryId": "vr.tr.alternating-series", "name": "alternating series", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.REL" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q438842/", "model": { "registry_id": "vr.tr.alternating-series", "name": "alternating series", "purpose": "Let an agent explain alternating series by definition, convergence tests, error bounds and examples for learners and practitioners.", "definition": "An infinite series whose terms alternate in sign, such as 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln 2; by the alternating series (Leibniz) test, such a series converges if the absolute values of its terms decrease monotonically to zero, and the error of a partial sum is at most the first omitted term.", "what_it_is_for": "Analysis, approximation and numerical computation.", "affordances": [ "test an alternating series for convergence", "bound the error of a partial sum", "distinguish absolute and conditional convergence", "use alternating series in approximations" ], "distinguishing_features": [ "Signs alternate", "Leibniz test applies", "Simple error bound", "Can converge conditionally" ], "appearance": "Not physical; series written with alternating plus and minus signs.", "visual_identification": [ "Factor (-1)^n in terms", "Alternating signs", "Series with all positive terms are not alternating" ], "physical_properties": [], "families_and_kinds": [ "alternating harmonic series", "alternating series from Taylor expansions", "absolutely convergent alternating series", "conditionally convergent alternating series" ], "related_models": [ { "relation": "is a kind of", "target": "numerical series", "why": "category" }, { "relation": "is tested by", "target": "alternating series test", "why": "test" }, { "relation": "includes", "target": "alternating harmonic series", "why": "example" }, { "relation": "is studied in", "target": "mathematical analysis", "why": "field" } ], "identifiers": [], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Applying the test when terms do not decrease", "Rearranging conditionally convergent series" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "test", "name": "Test", "description": "Convergence.", "rationale": "Conditions matter.", "layers": [ { "id": "leibniz", "name": "Leibniz", "description": "Alternating series test.", "findings": [ { "id": "leibniz-finding", "name": "Leibniz", "description": "Leibniz test.", "questions": [ { "text": "Do the absolute values of the terms decrease to zero?", "kind": "boundary" }, { "text": "Does the series therefore converge?", "kind": "definition" } ] } ] }, { "id": "absolute", "name": "Absolute", "description": "Absolute convergence.", "findings": [ { "id": "absolute-finding", "name": "Absolute", "description": "Absolute convergence.", "questions": [ { "text": "Does the series converge absolutely or only conditionally?", "kind": "boundary" }, { "text": "Which test shows it?", "kind": "definition" } ] } ] } ] }, { "id": "error", "name": "Error", "description": "Approximation.", "rationale": "Error bounds are simple.", "layers": [ { "id": "bound", "name": "Bound", "description": "Error bound.", "findings": [ { "id": "bound-finding", "name": "Bound", "description": "Error bound.", "questions": [ { "text": "How large can the error of this partial sum be?", "kind": "measurement" }, { "text": "How many terms are needed for a given accuracy?", "kind": "measurement" } ] } ] }, { "id": "sum", "name": "Sum", "description": "Known sums.", "findings": [ { "id": "sum-finding", "name": "Sum", "description": "Known sums.", "questions": [ { "text": "What does this alternating series sum to, such as ln 2 or pi/4?", "kind": "provenance" }, { "text": "How is it derived?", "kind": "definition" } ] } ] } ] }, { "id": "pitfalls", "name": "Pitfalls", "description": "Common errors.", "rationale": "Care is needed.", "layers": [ { "id": "rearrange", "name": "Rearrange", "description": "Rearrangements.", "findings": [ { "id": "rearrange-finding", "name": "Rearrange", "description": "Rearrangements.", "questions": [ { "text": "What happens if a conditionally convergent series is rearranged?", "kind": "definition" }, { "text": "What does the Riemann rearrangement theorem say?", "kind": "definition" } ] } ] }, { "id": "conditions", "name": "Conditions", "description": "Test failure.", "findings": [ { "id": "conditions-finding", "name": "Conditions", "description": "When the test fails.", "questions": [ { "text": "What if terms do not decrease monotonically?", "kind": "boundary" }, { "text": "Which other tests apply?", "kind": "definition" } ] } ] } ] }, { "id": "learning", "name": "Learning", "description": "Teaching.", "rationale": "Examples help.", "layers": [ { "id": "examples", "name": "Examples", "description": "Worked examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Worked examples.", "questions": [ { "text": "Can a worked example of the test be given step by step?", "kind": "action" }, { "text": "Which exercises help?", "kind": "action" } ] } ] }, { "id": "computation", "name": "Computation", "description": "Numerics.", "findings": [ { "id": "computation-finding", "name": "Computation", "description": "Numerical use.", "questions": [ { "text": "How are alternating series used to compute values numerically?", "kind": "definition" }, { "text": "How can convergence be accelerated?", "kind": "definition" } ] } ] } ] } ] }, "openQuestions": [ "Should convergence tests be separate entries?", "How should worked examples be linked?", "How should the Riemann rearrangement theorem be linked?" ], "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 alternating series, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain alternating series by definition, convergence tests, error bounds and examples for learners and practitioners.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Described in 31 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 alternating 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": [], "standing": "Described in 31 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one alternating series from another, and a alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating series from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a alternating 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 alternating 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 alternating 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 alternating 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 alternating series between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a alternating 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 alternating 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 alternating 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 alternating series, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating 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 alternating series and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a alternating 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 alternating 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 alternating 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 alternating series is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a alternating 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 alternating 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 alternating 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 } } } }