{
  "ai.module": "quadratic_rational_dinv_rtt",
  "ai.version": "1.0",
  "ai.purpose": "RTT structural map of the paper 'A quadratic form generalization of rational dinv'.",
  "ai.source.pdf": "https://arxiv.org/pdf/2604.13238",
  "ai.keywords": [
    "rational Dyck paths",
    "dinv",
    "quadratic forms",
    "q,t-Catalan",
    "combinatorics",
    "lattice paths",
    "positivity",
    "finite statistics"
  ],

  "ai.module.summary": "This module exposes the regime structure behind the quadratic-form generalization of rational dinv, connecting Dyck-path combinatorics, positive definite quadratic forms, and finiteness/stability arguments. It preserves the authors' results while adding RTT operators for agentic reasoning.",

  "ai.regimes": {
    "dyck_path_regime": {
      "description": "Classical rational Dyck paths with area and dinv statistics.",
      "evidence": "Paper recalls the (m,n)-Dyck path framework and classical dinv."
    },
    "quadratic_form_regime": {
      "description": "dinv is generalized using a positive definite quadratic form Q on Z^2.",
      "evidence": "Main definition replaces linear slope comparison with Q-weighted comparisons."
    },
    "finiteness_regime": {
      "description": "Quadratic dinv remains finite due to positivity and boundedness of Q.",
      "evidence": "Theorem 1.1 proves finiteness for all positive definite Q."
    },
    "stability_regime": {
      "description": "Quadratic dinv stabilizes under scaling of the Dyck path parameters.",
      "evidence": "Theorem 1.2 establishes stability across rational families."
    },
    "combinatorial_geometry_regime": {
      "description": "Quadratic comparisons correspond to geometric regions defined by Q.",
      "evidence": "Paper interprets Q-dinv as counting lattice points in Q-elliptic regions."
    }
  },

  "ai.tensions": {
    "linear_vs_quadratic": {
      "description": "Classical dinv uses linear slope; generalized dinv uses quadratic forms.",
      "paper_alignment": "Authors contrast classical slope test with Q-weighted comparisons."
    },
    "combinatorial_vs_geometric": {
      "description": "Discrete Dyck-path statistics vs continuous quadratic-form geometry.",
      "paper_alignment": "Q-dinv is interpreted via geometric regions."
    },
    "finiteness_vs_growth": {
      "description": "Quadratic forms grow faster than linear ones, yet dinv remains finite.",
      "paper_alignment": "Theorem 1.1 resolves this tension."
    },
    "local_comparison_vs_global_stability": {
      "description": "Local Q-comparisons vs global stability across rational families.",
      "paper_alignment": "Theorem 1.2 proves stability."
    }
  },

  "ai.transitions": {
    "linear_to_quadratic_transition": {
      "description": "Replace slope-based comparisons with Q-weighted comparisons.",
      "paper_alignment": "Definition of Q-dinv."
    },
    "quadratic_to_finiteness_transition": {
      "description": "Use positivity of Q to bound the number of contributing pairs.",
      "paper_alignment": "Proof of Theorem 1.1."
    },
    "finiteness_to_stability_transition": {
      "description": "Show that boundedness implies stability under scaling.",
      "paper_alignment": "Proof of Theorem 1.2."
    },
    "combinatorial_to_geometric_transition": {
      "description": "Interpret Q-dinv via geometric regions in R^2.",
      "paper_alignment": "Geometric interpretation section."
    }
  },

  "ai.operators": {
    "q_dinv_operator": {
      "type": "operator",
      "role": "Computes Q-weighted dinv using a positive definite quadratic form."
    },
    "quadratic_region_operator": {
      "type": "operator",
      "role": "Identifies geometric regions defined by Q that contribute to dinv."
    },
    "finiteness_operator": {
      "type": "operator",
      "role": "Uses positivity of Q to bound contributing pairs."
    },
    "stability_operator": {
      "type": "operator",
      "role": "Determines stability of Q-dinv under rational scaling."
    }
  },

  "ai.audience": "Researchers in algebraic combinatorics, q,t-Catalan theory, and geometric combinatorics.",
  "ai.contact.github": "https://github.com/TriadicFrameworks",
  "ai.license": "Open educational use permitted"
}