{ "model": { "rank": 11687, "code": "thing-q268493", "model_id": "vr.tr.elliptic-curve", "name": "elliptic curve", "purpose": "Let an agent explain the elliptic curve, relay its definition, group structure and uses from mathematics sources, describe the forms its aliases name, and distinguish it from an ellipse and from general cubic curves.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q268493/", "tier": 2, "score": 41, "payload": { "layer": "wikidata", "aliases": [ "elliptic curve over a finite field", "Mordell curve", "modular elliptic curve", "Edwards curve", "Jacobian curve", "Montgomery curve", "supersingular elliptic curve", "Tate curve", "twisted Edwards curve", "Koblitz curve" ], "aliasCount": 10, "merged": 10, "knownIn": 41, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q268493", "url": "https://www.wikidata.org/wiki/Q268493", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-14T03:25:52Z", "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": 4, "engine": "claude" }, "metaModel": { "id": "THING-Q268493", "registryId": "vr.tr.elliptic-curve", "name": "elliptic curve", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q268493/", "model": { "registry_id": "vr.tr.elliptic-curve", "name": "elliptic curve", "purpose": "Let an agent explain the elliptic curve, relay its definition, group structure and uses from mathematics sources, describe the forms its aliases name, and distinguish it from an ellipse and from general cubic curves.", "definition": "A smooth plane curve defined by a certain kind of cubic equation, which in mathematics carries a natural group structure that lets points be added together; elliptic curves are central to number theory and to modern cryptography, where they underpin elliptic-curve public-key systems. The registry aliases name forms and settings such as elliptic curves over finite fields, Mordell curves, modular elliptic curves and Edwards, Jacobian and Montgomery forms. Despite the name, an elliptic curve is not an ellipse.", "what_it_is_for": "Describing a cubic curve with a group structure.", "affordances": [ "explain its definition and group law", "relay its role in number theory", "relay its role in cryptography", "distinguish it from an ellipse" ], "distinguishing_features": [ "Cubic curve", "Group law on points", "Central to cryptography", "Not an ellipse" ], "appearance": "A smooth curve, often drawn as a looping shape with a separate arc, given by a cubic equation.", "visual_identification": [ "Smooth cubic curve with a group law", "Curves over finite fields, Mordell, modular, Edwards, Jacobian and Montgomery forms", "An ellipse is a conic section, not a cubic; a parabola and hyperbola are conics too" ], "physical_properties": [ { "quantity": "defined by", "typical_range": "a cubic equation", "unit": "note", "note": "" }, { "quantity": "structure", "typical_range": "abelian group on its points", "unit": "note", "note": "" }, { "quantity": "registry parents", "typical_range": "cubic plane curve, abelian variety", "unit": "note", "note": "" } ], "families_and_kinds": [ "curves over finite fields", "Mordell curves", "Edwards curves", "Montgomery curves" ], "related_models": [ { "relation": "is a kind of", "target": "cubic plane curve", "why": "in registry terms" }, { "relation": "is contrasted with", "target": "ellipse", "why": "" }, { "relation": "is used in", "target": "public-key cryptography", "why": "" }, { "relation": "is related to", "target": "number theory", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "Cryptographic standards for elliptic-curve systems" ], "failure_modes_and_hazards": [ "Confusing with an ellipse", "Overstating cryptographic detail", "Conflating the forms" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What an elliptic curve is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is an elliptic curve, and why is it not an ellipse?", "kind": "definition" }, { "text": "Is the question about a specific form such as an Edwards curve?", "kind": "boundary" } ] } ] }, { "id": "group", "name": "Group", "description": "Group law.", "findings": [ { "id": "group-finding", "name": "Group", "description": "Group.", "questions": [ { "text": "How does the group law add points on an elliptic curve?", "kind": "definition" }, { "text": "How are curves over finite fields counted?", "kind": "measurement" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Number theory.", "rationale": "Sources.", "layers": [ { "id": "numbertheory", "name": "Number theory", "description": "Number theory.", "findings": [ { "id": "numbertheory-finding", "name": "Number theory", "description": "Number theory.", "questions": [ { "text": "What role do elliptic curves play in number theory?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "modular", "name": "Modular", "description": "Modular curves.", "findings": [ { "id": "modular-finding", "name": "Modular", "description": "Modular.", "questions": [ { "text": "What are modular elliptic curves?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "cryptography", "name": "Cryptography", "description": "Cryptography.", "rationale": "Standards.", "layers": [ { "id": "ecc", "name": "ECC", "description": "Elliptic-curve cryptography.", "findings": [ { "id": "ecc-finding", "name": "ECC", "description": "ECC.", "questions": [ { "text": "How do elliptic curves underpin public-key cryptography?", "kind": "provenance" }, { "text": "Which standards apply?", "kind": "provenance" } ] } ] }, { "id": "forms", "name": "Forms", "description": "Curve forms.", "findings": [ { "id": "forms-finding", "name": "Forms", "description": "Forms.", "questions": [ { "text": "What are Edwards, Jacobian and Montgomery forms?", "kind": "provenance" }, { "text": "Which entry fits the specific form?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Context.", "rationale": "Context.", "layers": [ { "id": "curves", "name": "Curves", "description": "Curve family.", "findings": [ { "id": "curves-finding", "name": "Curves", "description": "Curves.", "questions": [ { "text": "How does an elliptic curve relate to other algebraic curves?", "kind": "provenance" }, { "text": "Which entry fits cubic curve?", "kind": "action" } ] } ] }, { "id": "finite", "name": "Finite", "description": "Finite fields.", "findings": [ { "id": "finite-finding", "name": "Finite", "description": "Finite.", "questions": [ { "text": "What are elliptic curves over finite fields used for?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should the named forms be separate entries?", "How should the group law be explained?", "How should cryptographic uses be scoped?" ], "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 elliptic curve, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain the elliptic curve, relay its definition, group structure and uses from mathematics sources, describe the forms its aliases name, and distinguish it from an ellipse and from general cubic curves.", "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: elliptic curve over a finite field, Mordell curve, modular elliptic curve, Edwards curve, Jacobian curve, Montgomery curve, supersingular elliptic curve, Tate curve, twisted Edwards curve, Koblitz curve.", "10 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 41 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 elliptic curve, 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": [ "elliptic curve over a finite field", "Mordell curve", "modular elliptic curve", "Edwards curve", "Jacobian curve", "Montgomery curve", "supersingular elliptic curve", "Tate curve", "twisted Edwards curve", "Koblitz curve" ] } ], "standing": "Described in 41 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 elliptic curve from another, and a elliptic curve 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 elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve 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 elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve 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 elliptic curve between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a elliptic curve 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 elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve 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 elliptic curve 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 elliptic curve 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 elliptic curve 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 elliptic curve?", "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 elliptic curve?", "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 elliptic curve 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 elliptic curve and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a elliptic curve, 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 elliptic curve, 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 elliptic curve 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 elliptic curve is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a elliptic curve, 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 elliptic curve, 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 elliptic curve 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 } } } }