{
  "rules": [
    {
      "id": "modal_transitive_axiom4",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:transitive"],
      "entails": ["□A->□□A", "□a->□□a", "公理4"],
      "evidence": 2
    },
    {
      "id": "modal_euclidean_axiom5",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:euclidean"],
      "entails": ["◊A->□◊A", "◊a->□◊a", "公理5"],
      "evidence": 2
    },
    {
      "id": "modal_reflexive_axiomT",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:reflexive"],
      "entails": ["□A->A", "□a->a", "公理T"],
      "evidence": 2
    },
    {
      "id": "topo_sphere_connectivity",
      "domain": "topology",
      "type": "theorem",
      "requires": ["condition:n<k"],
      "entails": ["π_n(S^k)=0", "pi_n(S^k)=0", "自明", "trivial"],
      "evidence": 3
    },
    {
      "id": "topo_refute_n_gt_k_always_finite",
      "domain": "topology",
      "type": "counter_example",
      "requires": ["condition:n>k"],
      "match_all": ["常に", "有限"],
      "note": "Counterexample: pi_3(S^2)=Z is not finite.",
      "evidence": -10
    },
    {
      "id": "topo_s1_higher_vanish",
      "domain": "topology",
      "type": "theorem",
      "requires": ["condition:n>1"],
      "entails": ["π_n(S^1)=0", "pi_n(S^1)=0", "trivial"],
      "evidence": 2
    },
    {
      "id": "topo_refute_s1_all_Z",
      "domain": "topology",
      "type": "counter_example",
      "requires": ["condition:all_n"],
      "match_all": ["すべて", "Z", "同型"],
      "note": "Counterexample: for n>1, pi_n(S^1)=0.",
      "evidence": -10
    },
    {
      "id": "hybrid_characterize_serial_trans_notrefl_noteucl",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["logic:hybrid", "frame:serial", "frame:transitive", "frame:not_reflexive", "frame:not_euclidean"],
      "match_all": [
        "@i",
        "◊",
        "□",
        "¬@i□i"
      ],
      "entails": ["@i□p->@i□□p"],
      "evidence": 5
    },
    {
      "id": "nec_serial_for_box_implies_diamond",
      "domain": "modal_logic",
      "type": "necessity_counterexample",
      "target_then": "□A->◊A",
      "missing_assumption": "frame:serial",
      "evidence": -10,
      "note": "Without seriality, □A can hold vacuously at a dead-end while ◊A fails."
    },
    {
      "id": "modal_reflexive_axiomT",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:reflexive"],
      "entails": ["□A->A", "boxA->A"],
      "evidence": 2
    },
    {
      "id": "modal_transitive_axiom4",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:transitive"],
      "entails": ["□A->□□A", "boxA->boxboxA"],
      "evidence": 2
    },
    {
      "id": "modal_euclidean_axiom5",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:euclidean"],
      "entails": ["◊A->□◊A", "diamondA->boxdiamondA"],
      "evidence": 2
    },
    {
      "id": "modal_serial_axiomD",
      "domain": "modal_logic",
      "type": "theorem",
      "requires": ["frame:serial"],
      "entails": ["□A->◊A", "boxA->diamondA"],
      "evidence": 2
    },
    {
      "id": "refute_T_by_not_reflexive",
      "domain": "modal_logic",
      "type": "counter_example",
      "requires": ["frame:not_reflexive"],
      "match_all": ["□A->A"],
      "evidence": -10,
      "note": "T fails in non-reflexive frames."
    },
    {
      "id": "refute_4_by_not_transitive",
      "domain": "modal_logic",
      "type": "counter_example",
      "requires": ["frame:not_transitive"],
      "match_all": ["□A->□□A"],
      "evidence": -10,
      "note": "4 fails in non-transitive frames."
    },
    {
      "id": "refute_5_by_not_euclidean",
      "domain": "modal_logic",
      "type": "counter_example",
      "requires": ["frame:not_euclidean"],
      "match_all": ["◊A->□◊A"],
      "evidence": -10,
      "note": "5 fails in non-euclidean frames."
    },
    {
      "id": "refute_D_by_not_serial",
      "domain": "modal_logic",
      "type": "counter_example",
      "requires": ["frame:not_serial"],
      "match_all": ["□A->◊A"],
      "evidence": -10,
      "note": "D fails in non-serial frames."
    },
    {
      "id": "nec_reflexive_for_axiomT",
      "domain": "modal_logic",
      "type": "necessity_counterexample",
      "target_then": "□A->A",
      "missing_assumption": "frame:reflexive",
      "evidence": -10,
      "note": "Without reflexivity, axiom T fails."
    },
    {
      "id": "nec_transitive_for_axiom4",
      "domain": "modal_logic",
      "type": "necessity_counterexample",
      "target_then": "□A->□□A",
      "missing_assumption": "frame:transitive",
      "evidence": -10,
      "note": "Without transitivity, axiom 4 fails."
    },
    {
      "id": "nec_euclidean_for_axiom5",
      "domain": "modal_logic",
      "type": "necessity_counterexample",
      "target_then": "◊A->□◊A",
      "missing_assumption": "frame:euclidean",
      "evidence": -10,
      "note": "Without euclidean property, axiom 5 fails."
    },
    {
      "id": "schema_describe_features_food",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:describe_features", "entity_type:food_item"],
      "output_schema": {
        "overview": "string",
        "category": "string",
        "taste": "string",
        "texture": "string",
        "nutrition": "string",
        "origin": "string",
        "uses": "string",
        "notes": "string"
      },
      "evidence": 5
    },
    {
      "id": "schema_compare_general",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:compare", "comparison:true"],
      "output_schema": {
        "items_compared": "string",
        "key_differences": "string",
        "similarities": "string",
        "conclusion": "string"
      },
      "evidence": 4
    },
    {
      "id": "schema_why_general",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:why"],
      "output_schema": {
        "core_reason": "string",
        "background_context": "string",
        "implications": "string"
      },
      "evidence": 4
    },
    {
      "id": "schema_how_to_general",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:how_to"],
      "output_schema": {
        "goal": "string",
        "prerequisites": "string",
        "steps": "string",
        "tips": "string"
      },
      "evidence": 4
    },
    {
      "id": "schema_explain_general",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:explain"],
      "output_schema": {
        "definition": "string",
        "details": "string",
        "key_points": "string"
      },
      "evidence": 3
    },
    {
      "id": "schema_assumption_analysis_v1",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:assumption_analysis"],
      "output_schema": {
        "claim": "string",
        "assumptions": "string",
        "critical_assumptions": "string",
        "effect_if_dropped": "string",
        "counterexamples": "string",
        "final_verdict": "string"
      },
      "evidence": 5
    },
    {
      "id": "schema_scope_and_limits_v1",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:scope_and_limits"],
      "output_schema": {
        "topic": "string",
        "scope": "string",
        "non_scope": "string",
        "assumptions": "string",
        "limits": "string",
        "unknowns": "string"
      },
      "evidence": 5
    },
    {
      "id": "schema_failure_modes_v1",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:failure_modes"],
      "output_schema": {
        "system_or_topic": "string",
        "failure_modes": "string",
        "triggers": "string",
        "impact": "string",
        "mitigation": "string"
      },
      "evidence": 5
    },
    {
      "id": "schema_why_not_v1",
      "domain": "schema",
      "type": "schema_theorem",
      "requires": ["intent:why_not"],
      "output_schema": {
        "question": "string",
        "reason": "string",
        "missing_conditions": "string",
        "conflicts": "string",
        "what_would_enable": "string",
        "summary": "string"
      },
      "evidence": 5
    }
  ]
}