{
  "vercy": "1.0-draft",
  "publication": {
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    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T14:42:13Z",
    "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-Q1530271",
    "registryId": "vr.tr.equilibrium-point",
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    "entryKind": "thing",
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    "domain": [
      "XCT.STA"
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  "model": {
    "registry_id": "vr.tr.equilibrium-point",
    "name": "equilibrium point",
    "purpose": "Let an agent explain equilibrium points and their classification, relay stability analysis and examples from dynamical systems and economics sources, describe hyperbolic and market equilibria, and distinguish equilibrium points from equilibrium in thermodynamics and chemistry and from steady states of open systems.",
    "definition": "A state of a dynamical system in which the system remains at rest unless disturbed, a point where the rate of change is zero, classified as stable, unstable or saddle points and, when eigenvalues have no zero real part, as hyperbolic equilibrium points, with applications from mechanics and ecology to economics, where market equilibrium is the price and quantity at which supply equals demand; equilibrium points are the starting point for stability analysis.",
    "what_it_is_for": "Analysing rest states and stability.",
    "affordances": [
      "explain the concept",
      "relay classification and analysis",
      "describe applications",
      "distinguish related concepts"
    ],
    "distinguishing_features": [
      "Zero rate of change",
      "Stability types",
      "Linearisation",
      "Cross-field use"
    ],
    "appearance": "Not a visible object; a fixed point on a phase portrait.",
    "visual_identification": [
      "State where the system s rate of change is zero",
      "Stable and unstable points, hyperbolic equilibrium point, market equilibrium",
      "Thermodynamic equilibrium is a macroscopic state; chemical equilibrium balances reactions; steady states may involve flows"
    ],
    "physical_properties": [
      {
        "quantity": "condition",
        "typical_range": "derivative equals zero",
        "unit": "note",
        "note": ""
      },
      {
        "quantity": "hyperbolic",
        "typical_range": "no eigenvalue with zero real part",
        "unit": "note",
        "note": ""
      },
      {
        "quantity": "Hartman-Grobman theorem",
        "typical_range": "linearisation valid near hyperbolic points",
        "unit": "note",
        "note": ""
      }
    ],
    "families_and_kinds": [
      "stable equilibrium points such as nodes and foci",
      "unstable and saddle points",
      "hyperbolic and non-hyperbolic equilibria",
      "market equilibrium in economics",
      "Nash equilibrium in game theory",
      "equilibria in ecological and epidemic models"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "equilibrium",
        "why": "in registry terms"
      },
      {
        "relation": "is a kind of",
        "target": "state",
        "why": "in registry terms"
      },
      {
        "relation": "is analysed by",
        "target": "stability theory",
        "why": "through linearisation"
      },
      {
        "relation": "is exemplified by",
        "target": "market equilibrium",
        "why": "in economics"
      }
    ],
    "identifiers": [],
    "standards_and_regulation": [
      "No regulation"
    ],
    "failure_modes_and_hazards": [
      "Confusing equilibrium points with thermodynamic equilibrium",
      "Linearising non-hyperbolic points",
      "Registry aliases mixing mathematics and economics"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": []
  },
  "sources": [],
  "structure": {
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      {
        "id": "understand",
        "name": "Understand",
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            "description": "Definition.",
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                "id": "definition-finding",
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                  {
                    "text": "What is an equilibrium point, and how does it differ from thermodynamic and chemical equilibrium and steady states?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about mathematics, physics, ecology or economics?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "types",
            "name": "Types",
            "description": "Types.",
            "findings": [
              {
                "id": "types-finding",
                "name": "Types",
                "description": "Types.",
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                    "text": "What are stable, unstable, saddle and hyperbolic equilibrium points?",
                    "kind": "definition"
                  },
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                    "text": "Which entry fits the specific type?",
                    "kind": "action"
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                ]
              }
            ]
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        ]
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      {
        "id": "analysis",
        "name": "Analysis",
        "description": "Stability analysis.",
        "rationale": "Science.",
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          {
            "id": "linearisation",
            "name": "Linearisation",
            "description": "Linearisation.",
            "findings": [
              {
                "id": "linearisation-finding",
                "name": "Linearisation",
                "description": "Linearisation.",
                "questions": [
                  {
                    "text": "How do eigenvalues and the Hartman-Grobman theorem classify equilibria?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "lyapunov",
            "name": "Lyapunov",
            "description": "Lyapunov methods.",
            "findings": [
              {
                "id": "lyapunov-finding",
                "name": "Lyapunov",
                "description": "Lyapunov.",
                "questions": [
                  {
                    "text": "How do Lyapunov functions establish stability without linearisation?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "applications",
        "name": "Applications",
        "description": "Applications.",
        "rationale": "Application.",
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          {
            "id": "economics",
            "name": "Economics",
            "description": "Market equilibrium.",
            "findings": [
              {
                "id": "economics-finding",
                "name": "Economics",
                "description": "Economics.",
                "questions": [
                  {
                    "text": "What is market equilibrium, and how is it analysed for stability?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits economic equilibrium?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "science",
            "name": "Science",
            "description": "Science and engineering.",
            "findings": [
              {
                "id": "science-finding",
                "name": "Science",
                "description": "Science.",
                "questions": [
                  {
                    "text": "How are equilibria used in mechanics, ecology, epidemiology and control?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits control theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "Beyond points.",
        "rationale": "Context.",
        "layers": [
          {
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            "name": "Bifurcation",
            "description": "Bifurcations.",
            "findings": [
              {
                "id": "bifurcation-finding",
                "name": "Bifurcation",
                "description": "Bifurcation.",
                "questions": [
                  {
                    "text": "How do equilibria appear, disappear and change stability in bifurcations?",
                    "kind": "provenance"
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                    "text": "Which entry fits bifurcation theory?",
                    "kind": "action"
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              }
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          {
            "id": "history",
            "name": "History",
            "description": "History.",
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              {
                "id": "history-finding",
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                  {
                    "text": "How did Poincare, Lyapunov and others develop equilibrium and stability theory?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of dynamical systems?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should market equilibrium and hyperbolic equilibrium be separate primary entries?",
    "How should dynamical systems sources be linked?",
    "The registry entry has merged aliases from mathematics and economics; should they be split off?"
  ],
  "statistics": {
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    "findings": 8,
    "questions": 16
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}