{ "model": { "rank": 16267, "code": "thing-q782566", "model_id": "vr.tr.automorphism", "name": "automorphism", "purpose": "Let an agent explain automorphisms, relay definitions, automorphism groups and examples across algebra, graph theory, topology and dynamics from mathematics sources, describe the named forms, and distinguish automorphisms from endomorphisms, isomorphisms between different objects, homomorphisms and symmetries in physics.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q782566/", "tier": 2, "score": 32, "payload": { "layer": "wikidata", "aliases": [ "inner automorphism", "hyperbolic toral automorphism", "group automorphism", "hyperbolic motion", "deck transformation", "graph automorphism", "holonomy" ], "aliasCount": 7, "merged": 7, "knownIn": 32, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q782566", "url": "https://www.wikidata.org/wiki/Q782566", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-14T00:01:21Z", "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-Q782566", "registryId": "vr.tr.automorphism", "name": "automorphism", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q782566/", "model": { "registry_id": "vr.tr.automorphism", "name": "automorphism", "purpose": "Let an agent explain automorphisms, relay definitions, automorphism groups and examples across algebra, graph theory, topology and dynamics from mathematics sources, describe the named forms, and distinguish automorphisms from endomorphisms, isomorphisms between different objects, homomorphisms and symmetries in physics.", "definition": "An isomorphism from a mathematical object to itself, preserving its structure, forming a group under composition called the automorphism group, including group automorphisms, with inner automorphisms given by conjugation, graph automorphisms that permute vertices while preserving edges and measure a graph s symmetry, field automorphisms central to Galois theory, deck transformations of covering spaces in topology, and in dynamical systems hyperbolic toral automorphisms such as Arnold s cat map; the registry alias hyperbolic motion names isometries of hyperbolic space, related symmetries. Automorphisms formalise the idea of symmetry across mathematics.", "what_it_is_for": "Describing symmetries of mathematical structures.", "affordances": [ "explain the definition", "relay automorphism groups", "describe named forms", "distinguish related maps" ], "distinguishing_features": [ "Bijective", "Structure-preserving", "Forms a group", "Captures symmetry" ], "appearance": "Not a physical object; visualised as symmetries of graphs, shapes and structures.", "visual_identification": [ "Structure-preserving self-isomorphism", "Inner automorphism, hyperbolic toral automorphism, group automorphism, hyperbolic motion, deck transformation, graph automorphism", "Endomorphisms need not be invertible; isomorphisms may map between objects; homomorphisms preserve structure without bijection; physical symmetries act on systems" ], "physical_properties": [ { "quantity": "Galois theory", "typical_range": "1830s", "unit": "note", "note": "field automorphisms" }, { "quantity": "Arnold cat map", "typical_range": "1960s", "unit": "note", "note": "hyperbolic toral automorphism" }, { "quantity": "registry parents", "typical_range": "endomorphism, isomorphism", "unit": "note", "note": "" } ], "families_and_kinds": [ "group automorphisms", "field automorphisms", "graph automorphisms", "deck transformations", "toral automorphisms" ], "related_models": [ { "relation": "is a kind of", "target": "isomorphism", "why": "in registry terms" }, { "relation": "forms", "target": "automorphism group", "why": "" }, { "relation": "includes", "target": "inner automorphism", "why": "" }, { "relation": "is contrasted with", "target": "endomorphism", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "No regulation" ], "failure_modes_and_hazards": [ "Confusing endomorphisms and automorphisms", "Ignoring the structure being preserved", "Loosely filed geometry alias" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What an automorphism is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is an automorphism, and how does it differ from endomorphisms, isomorphisms, homomorphisms and physical symmetries?", "kind": "definition" }, { "text": "Is the question about algebra, graphs, topology or dynamics?", "kind": "boundary" } ] } ] }, { "id": "forms", "name": "Forms", "description": "Named forms.", "findings": [ { "id": "forms-finding", "name": "Forms", "description": "Forms.", "questions": [ { "text": "What are inner, group, graph and hyperbolic toral automorphisms, deck transformations and hyperbolic motions?", "kind": "definition" }, { "text": "Which entry fits the specific form?", "kind": "action" } ] } ] } ] }, { "id": "algebra", "name": "Algebra", "description": "Algebra.", "rationale": "Science.", "layers": [ { "id": "groups", "name": "Groups", "description": "Group automorphisms.", "findings": [ { "id": "groups-finding", "name": "Groups", "description": "Groups.", "questions": [ { "text": "How do inner and outer automorphisms of groups differ?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "galois", "name": "Galois", "description": "Galois theory.", "findings": [ { "id": "galois-finding", "name": "Galois", "description": "Galois.", "questions": [ { "text": "How do field automorphisms explain solvability of equations?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "other", "name": "Other fields", "description": "Other fields.", "rationale": "Science.", "layers": [ { "id": "graphs", "name": "Graphs", "description": "Graph symmetry.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "How are graph automorphism groups computed, and what is the graph isomorphism problem?", "kind": "provenance" }, { "text": "Which entry fits graph isomorphism problem?", "kind": "action" } ] } ] }, { "id": "dynamics", "name": "Dynamics", "description": "Toral automorphisms.", "findings": [ { "id": "dynamics-finding", "name": "Dynamics", "description": "Dynamics.", "questions": [ { "text": "What is Arnold s cat map, and why is it chaotic?", "kind": "provenance" }, { "text": "Which entry fits Arnold s cat map?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Topology.", "rationale": "Context.", "layers": [ { "id": "covering", "name": "Covering spaces", "description": "Deck transformations.", "findings": [ { "id": "covering-finding", "name": "Covering spaces", "description": "Covering spaces.", "questions": [ { "text": "What are deck transformations of covering spaces?", "kind": "provenance" }, { "text": "Which entry fits covering space?", "kind": "action" } ] } ] }, { "id": "symmetry", "name": "Symmetry", "description": "Symmetry concept.", "findings": [ { "id": "symmetry-finding", "name": "Symmetry", "description": "Symmetry.", "questions": [ { "text": "How do automorphisms formalise symmetry?", "kind": "provenance" }, { "text": "Is the information accurate?", "kind": "boundary" } ] } ] } ] } ] }, "openQuestions": [ "Should graph automorphism be a separate entry?", "Should hyperbolic motion be moved to hyperbolic geometry entries?", "How should algebra references 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 automorphism, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain automorphisms, relay definitions, automorphism groups and examples across algebra, graph theory, topology and dynamics from mathematics sources, describe the named forms, and distinguish automorphisms from endomorphisms, isomorphisms between different objects, homomorphisms and symmetries in physics.", "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: inner automorphism, hyperbolic toral automorphism, group automorphism, hyperbolic motion, deck transformation, graph automorphism, holonomy.", "7 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 32 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 automorphism, 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": [ "inner automorphism", "hyperbolic toral automorphism", "group automorphism", "hyperbolic motion", "deck transformation", "graph automorphism", "holonomy" ] } ], "standing": "Described in 32 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 7 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 automorphism from another, and a automorphism 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 automorphism, 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 automorphism, 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 automorphism 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 automorphism, 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 automorphism, 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 automorphism 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 automorphism 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 automorphism, 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 automorphism, 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 automorphism 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 automorphism from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a automorphism, 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 automorphism, 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 automorphism 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 automorphism 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 automorphism between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a automorphism, 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 automorphism, 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 automorphism 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 automorphism, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a automorphism 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 automorphism, 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 automorphism, 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 automorphism 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 automorphism 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 automorphism 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 automorphism 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 automorphism 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 automorphism?", "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 automorphism?", "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 automorphism 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 automorphism and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a automorphism, 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 automorphism, 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 automorphism 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 automorphism is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a automorphism, 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 automorphism, 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 automorphism 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 } } } }