{
  "$schema": "https://json-schema.org/draft/2020-12/schema",
  "$id": "state_moment.schema.json",
  "title": "State Civilizational Second Moment M(s)",
  "description": "M(s) = Sum_i w_s[i] * (mu_i - x_s)(mu_i - x_s)^T in R^{6x6}, symmetric positive semi-definite. Equal to Cov_w(mu;w) + weight_sum * (mu_bar - x_s)(mu_bar - x_s)^T where mu_bar = Sum_i w_s[i] * mu_i. Not a covariance: x_s is not the weighted barycentre. The historical 'tension tensor T(s)' name is deprecated; see docs/11_critiques_and_responses.md sections C9-C13.",
  "type": "object",
  "required": ["iso3", "M", "eigenvalues", "eigenvectors", "invariants", "anisotropy"],
  "properties": {
    "iso3": { "type": "string", "minLength": 3, "maxLength": 3 },
    "M": {
      "type": "array",
      "minItems": 6,
      "maxItems": 6,
      "description": "6x6 symmetric PSD matrix; weighted second moment of (mu_i - x_s) under w_s.",
      "items": {
        "type": "array",
        "minItems": 6,
        "maxItems": 6,
        "items": { "type": "number" }
      }
    },
    "eigenvalues": {
      "type": "array",
      "description": "Sorted descending: lambda_1 >= ... >= lambda_6 >= 0. Directional dispersion magnitudes.",
      "minItems": 6,
      "maxItems": 6,
      "items": { "type": "number" }
    },
    "eigenvectors": {
      "type": "array",
      "description": "Orthonormal basis: column k is e_k in B_score associated with lambda_k.",
      "minItems": 6,
      "maxItems": 6,
      "items": {
        "type": "array",
        "minItems": 6,
        "maxItems": 6,
        "items": { "type": "number" }
      }
    },
    "invariants": {
      "type": "object",
      "required": ["I1_trace", "I2_von_mises", "det"],
      "properties": {
        "I1_trace": { "type": "number", "description": "tr(M) - total second-moment magnitude" },
        "I2_von_mises": { "type": "number", "description": "sqrt(3/2 * s:s) with s = M - tr(M)/6 * I; standard von Mises invariant of the deviatoric part" },
        "det": { "type": "number", "description": "det(M)" }
      }
    },
    "anisotropy": {
      "type": "number",
      "minimum": 0,
      "maximum": 1,
      "description": "A = (lambda_1 - lambda_6) / lambda_1, or 0 when lambda_1 ~ 0."
    },
    "quality_flags": {
      "type": "object",
      "properties": {
        "monocivilizational": { "type": "boolean" },
        "low_evidence": { "type": "boolean" },
        "computed_from_imputed": { "type": "boolean" },
        "x_score_provenance": { "type": "string" },
        "diagonal_inflated_by_prior": { "type": "boolean" }
      }
    },
    "decomposition": {
      "type": "object",
      "description": "Decomposition M = Cov_w + bias_term + prior_variance_inflation. The third term is non-zero when x_score is imputed (centroid_prior or imputed_governance) and propagates the imputation uncertainty diagonally.",
      "required": ["weighted_barycentre_mu_bar", "intra_civilizational_covariance", "bias_term"],
      "properties": {
        "weighted_barycentre_mu_bar": {
          "type": "array",
          "minItems": 6,
          "maxItems": 6,
          "items": { "type": "number" }
        },
        "intra_civilizational_covariance": {
          "type": "array",
          "minItems": 6,
          "maxItems": 6,
          "items": {
            "type": "array",
            "minItems": 6,
            "maxItems": 6,
            "items": { "type": "number" }
          }
        },
        "bias_term": {
          "type": "array",
          "minItems": 6,
          "maxItems": 6,
          "items": {
            "type": "array",
            "minItems": 6,
            "maxItems": 6,
            "items": { "type": "number" }
          }
        },
        "prior_variance_inflation": {
          "type": "array",
          "minItems": 6,
          "maxItems": 6,
          "description": "diag(sigma_prior^2) added to M when x_score is imputed; zero matrix for observed states.",
          "items": {
            "type": "array",
            "minItems": 6,
            "maxItems": 6,
            "items": { "type": "number" }
          }
        },
        "trace_intra": { "type": "number" },
        "trace_bias": { "type": "number" },
        "trace_prior_inflation": { "type": "number" }
      }
    }
  },
  "additionalProperties": false
}