{
  "artifact_id": "KERNEL-ARITHMETICA",
  "version": "1.0.0-KERNEL-ARITHMETICA-EXECUTABLE",
  "status": "ACTIVE_RUNTIME_KERNEL",
  "branch": "ARITHMETICA",
  "class": "SIGMA_SATOR",
  "authority": "AGLA_CONTROL_PLANE",
  "mutation_policy": "VERSION_CONTROLLED_ONLY",
  "runtime_order": ["TENET", "AREPO", "OPERA", "SATOR"],
  "rotas": {
    "id": "ROTA-QUANTITAS-ET-QUALITAS",
    "center": {
      "index": "PRE",
      "number": 1,
      "quality": "identity",
      "role": "non_relational_anchor"
    },
    "chambers": [
      { "index": "I", "number": 2, "quality": "distinction", "slot": null },
      { "index": "II", "number": 3, "quality": "mediation", "slot": null },
      { "index": "III", "number": 4, "quality": "stabilization", "slot": null },
      { "index": "IV", "number": 5, "quality": "asymmetry", "slot": null },
      { "index": "V", "number": 6, "quality": "synthesis", "slot": null },
      { "index": "VI", "number": 7, "quality": "rupture", "slot": null },
      { "index": "VII", "number": 8, "quality": "amplification", "slot": null },
      { "index": "VIII", "number": 9, "quality": "closure", "slot": null },
      { "index": "IX", "number": 10, "quality": "transition", "slot": null }
    ],
    "relations": {
      "center_to_all": true,
      "complete_graph": true,
      "relation_types": ["GEN", "MAN", "UG", "PC", "PRESUPPOSES", "MEDIATES", "INTEGRATES"],
      "default_edges": [
        { "from": 2, "to": 3, "relation": "MEDIATES", "reading": "distinction opens mediation" },
        { "from": 3, "to": 4, "relation": "STABILIZES", "reading": "mediation stabilizes" },
        { "from": 4, "to": 5, "relation": "DISTURBS", "reading": "stabilization exposes asymmetry" },
        { "from": 2, "to": 3, "result": 6, "relation": "INTEGRATES", "operator": "multiplication", "reading": "distinction × mediation yields synthesis" },
        { "from": 3, "to": 3, "result": 9, "relation": "CLOSES", "operator": "multiplication", "reading": "repeated mediation yields closure" },
        { "from": 9, "to": 10, "relation": "TRANSITIONS", "reading": "closure opens transition" }
      ]
    }
  },
  "tenet": {
    "intentiones": {
      "prima": "being_as_such",
      "secunda": "mode_of_intelligibility",
      "laws": [
        "prima_intentio != secunda_intentio",
        "relation_does_not_create_being",
        "structure_and_notation_express_or_articulate_but_do_not_create_number"
      ]
    },
    "numeric_causality": {
      "unit": 1,
      "number_as_ens": "additive_plurality",
      "addition": {
        "role": "causa_constituens",
        "intentio": "prima"
      },
      "multiplication": {
        "role": "causa_ordinans",
        "intentio": "secunda",
        "constraint": "operates_over_additive_being"
      },
      "base": {
        "role": "grouping_visibility",
        "constraint": "does_not_create_number"
      },
      "notation": {
        "role": "expression",
        "constraint": "does_not_create_number"
      }
    },
    "ontologia_cardinalis": {
      "mapping": {
        "1": { "index": "PRE", "quality": "identity" },
        "2": { "index": "I", "quality": "distinction" },
        "3": { "index": "II", "quality": "mediation" },
        "4": { "index": "III", "quality": "stabilization" },
        "5": { "index": "IV", "quality": "asymmetry" },
        "6": { "index": "V", "quality": "synthesis" },
        "7": { "index": "VI", "quality": "rupture" },
        "8": { "index": "VII", "quality": "amplification" },
        "9": { "index": "VIII", "quality": "closure" },
        "10": { "index": "IX", "quality": "transition" }
      },
      "inverse_mapping": {
        "identity": 1,
        "distinction": 2,
        "mediation": 3,
        "stabilization": 4,
        "asymmetry": 5,
        "synthesis": 6,
        "rupture": 7,
        "amplification": 8,
        "closure": 9,
        "transition": 10
      },
      "laws": [
        "number_carries_quality",
        "number != quality",
        "quality_is_base_neutral",
        "1_is_PRE_non_relational",
        "2_to_10_are_I_to_IX_relational_field"
      ]
    }
  },
  "arepo": {
    "global": {
      "required": ["input", "operation"],
      "must_declare": ["layer"],
      "base_required_when": ["base_manifest", "positional"],
      "reject_if": [
        "layer_missing",
        "base_missing_when_required",
        "invalid_carrier",
        "number_quality_identity_collapse",
        "quality_as_ontological_cause",
        "positional_string_as_ontology"
      ]
    },
    "ex_quantitas": {
      "operation": "NUMERUS_TO_QUALITAS",
      "admissible_input": ["integer", "base_manifest_numeric", "positional_numeric_with_base"],
      "normalization_required": true,
      "output_requirement": "canonical_quality_exists"
    },
    "ex_qualitas": {
      "operation": "QUALITAS_TO_NUMERUS",
      "admissible_input": ["canonical_quality", "compound_quality"],
      "normalization_required": true,
      "output_requirement": "numeric_carrier_exists"
    },
    "quantitas_et_qualitas": {
      "operation": "NUMERUS_QUALITAS_TO_LOGOS",
      "admissible_input": ["numeric", "quality", "mixed", "operation"],
      "requires": ["valid_numeric_layer", "valid_quality_mapping", "no_logical_contradiction"]
    }
  },
  "opera": {
    "ex_quantitas": {
      "id": "OPERA-EX-QUANTITAS",
      "direction": "NUMERUS_TO_QUALITAS",
      "execute": {
        "steps": [
          "validate_with_arepo_ex_quantitas",
          "normalize_to_integer",
          "lookup_quality_in_ontologia_cardinalis",
          "return_trace"
        ]
      }
    },
    "ex_qualitas": {
      "id": "OPERA-EX-QUALITAS",
      "direction": "QUALITAS_TO_NUMERUS",
      "execute": {
        "steps": [
          "validate_with_arepo_ex_qualitas",
          "normalize_to_canonical_quality",
          "lookup_number_in_ontologia_cardinalis",
          "return_trace"
        ]
      }
    },
    "quantitas_et_qualitas": {
      "id": "OPERA-QUANTITAS-ET-QUALITAS",
      "direction": "NUMERUS_QUALITAS_TO_LOGOS",
      "execute": {
        "steps": [
          "validate_with_arepo_quantitas_et_qualitas",
          "normalize_inputs",
          "map_number_quality_pairs",
          "bind_relation_from_rota",
          "render_with_sator_quantitas_et_qualitas",
          "return_trace"
        ]
      }
    }
  },
  "sator": {
    "language_constraints": {
      "allowed_verbs": ["carries", "expresses", "yields", "resolves_to", "maps_to", "renders"],
      "forbidden_identity_forms": ["is", "equals"],
      "forbidden_causal_forms": ["creates", "causes_as_being"]
    },
    "ex_quantitas": {
      "forms": {
        "direct": "{n} -> {quality}",
        "declarative": "{n} carries {quality}",
        "explicit": "{n} carries {quality} under secunda intentio"
      }
    },
    "ex_qualitas": {
      "forms": {
        "direct": "{quality} -> {n}",
        "declarative": "{quality} is carried by numeric carrier {n}",
        "explicit": "{quality} resolves to numeric carrier {n}"
      }
    },
    "quantitas_et_qualitas": {
      "forms": {
        "proposition": "{numeric_statement} yields {result}, which carries {quality}",
        "relational_proposition": "{a} relates to {b} through {relation}, rendering {quality}",
        "syllogism": {
          "major": "{general_rule}",
          "minor": "{numeric_instance}",
          "conclusion": "{conclusion}"
        }
      },
      "must_expose": ["numeric_layer", "qualitative_layer", "logical_layer"]
    }
  },
  "execution_api": {
    "input_schema": {
      "operation": [
        "EX_QUANTITAS",
        "EX_QUALITAS",
        "QUANTITAS_ET_QUALITAS"
      ],
      "payload": "object",
      "layer": ["intrinsic", "base_manifest", "positional", "qualitative", "mixed"],
      "base": "optional_integer",
      "output_mode": ["minimal", "explicit", "full_trace"]
    },
    "output_schema": {
      "status": ["ACCEPTED", "REJECTED"],
      "operation": "string",
      "result": "object",
      "exposition": "string",
      "trace": "object",
      "errors": "array"
    }
  },
  "runtime_functions_pseudocode": {
    "normalize_number": "if layer=='intrinsic' return n; if layer=='base_manifest' sum declared carriers; if layer=='positional' decode with declared base; else reject",
    "normalize_quality": "map synonym to canonical quality; if not canonical reject",
    "ex_quantitas": "n=normalize_number(input); q=mapping[n]; return sator_ex_quantitas(n,q)",
    "ex_qualitas": "q=normalize_quality(input); n=inverse_mapping[q]; return sator_ex_qualitas(q,n)",
    "quantitas_et_qualitas": "validate mixed input; derive missing side via ex_quantitas or ex_qualitas; bind relation; render proposition_or_syllogism"
  },
  "sample_runs": [
    {
      "input": {
        "operation": "EX_QUANTITAS",
        "payload": { "n": 8 },
        "layer": "intrinsic",
        "output_mode": "explicit"
      },
      "output": {
        "status": "ACCEPTED",
        "result": { "number": 8, "quality": "amplification" },
        "exposition": "8 carries amplification under secunda intentio.",
        "trace": {
          "numeric_layer": "8",
          "mapping": "8 -> amplification",
          "source": "TENET_ONTOLOGIA_CARDINALIS"
        }
      }
    },
    {
      "input": {
        "operation": "EX_QUALITAS",
        "payload": { "quality": "synthesis" },
        "layer": "qualitative",
        "output_mode": "explicit"
      },
      "output": {
        "status": "ACCEPTED",
        "result": { "quality": "synthesis", "number": 6 },
        "exposition": "Synthesis resolves to numeric carrier 6.",
        "trace": {
          "qualitative_layer": "synthesis",
          "mapping": "synthesis -> 6",
          "source": "TENET_ONTOLOGIA_CARDINALIS"
        }
      }
    },
    {
      "input": {
        "operation": "QUANTITAS_ET_QUALITAS",
        "payload": {
          "numeric_statement": "3 * 3 = 9"
        },
        "layer": "mixed",
        "output_mode": "full_trace"
      },
      "output": {
        "status": "ACCEPTED",
        "result": {
          "numeric": "3 * 3 = 9",
          "qualities": {
            "3": "mediation",
            "9": "closure"
          },
          "logical_form": "proposition"
        },
        "exposition": "3 × 3 yields 9, which carries closure through repeated mediation.",
        "trace": {
          "numeric_layer": "3 * 3 = 9",
          "qualitative_layer": "3 -> mediation; 9 -> closure",
          "logical_layer": "Repeated mediation yields closure."
        }
      }
    }
  ]
}