{
  "ai.module": "paucity_lattice_triangles_rtt",
  "ai.version": "1.0",
  "ai.purpose": "RTT structural map of the paper 'On the Paucity of Lattice Triangles'.",
  "ai.source.pdf": "https://arxiv.org/pdf/2603.23928",
  "ai.keywords": [
    "lattice triangles",
    "translation surfaces",
    "Veech surfaces",
    "Mirzakhani-Wright rank obstruction",
    "Ramanujan sums",
    "Fourier analysis",
    "density results",
    "AxiomProver",
    "Lean formalization"
  ],

  "ai.module.summary": "This module exposes the regime structure behind the density-zero result for obtuse rational lattice triangles, connecting geometric rank obstructions, modular arithmetic criteria, and Fourier-analytic cancellation. It preserves the authors' results while adding RTT operators for agentic reasoning.",

  "ai.regimes": {
    "geometric_rank_regime": {
      "description": "Mirzakhani–Wright rank obstruction restricts which translation surfaces can be lattice surfaces.",
      "evidence": "Paper reformulates rank obstruction into modular arithmetic conditions. "
    },
    "modular_obstruction_regime": {
      "description": "Larsen–Norton–Zykoski criterion reduces lattice property to inequalities involving residues modulo n.",
      "evidence": "Proposition 2.1 gives modular inequalities ruling out lattice triangles. "
    },
    "fourier_ramanujan_regime": {
      "description": "Counting function S(p,q) decomposed into main term and error term using Fourier analysis and Ramanujan sums.",
      "evidence": "Section 3 develops Fourier expansion and Ramanujan-sum decomposition. "
    },
    "large_prime_factor_regime": {
      "description": "Presence of a large prime factor in n forces strong cancellation in Ramanujan sums.",
      "evidence": "Proposition 4.1 shows error suppression when P+(n) is large. "
    },
    "density_one_regime": {
      "description": "Smooth-number theory implies denominators with large prime factors form a density-1 set.",
      "evidence": "Corollary 1.2 uses Dickman–de Bruijn theory. "
    },
    "formalization_regime": {
      "description": "AxiomProver autoformalizes the analytic engine (Theorem 6.1) in Lean.",
      "evidence": "Section 6 describes autonomous Lean verification. "
    }
  },

  "ai.tensions": {
    "geometry_vs_arithmetic": {
      "description": "Geometric rank obstruction vs modular arithmetic reformulation.",
      "paper_alignment": "Section 2 transitions from geometry to arithmetic. "
    },
    "main_term_vs_error_term": {
      "description": "Dominant main term vs oscillatory error term in S(p,q).",
      "paper_alignment": "Section 3–4 analyze M(p,q) and E(p,q). "
    },
    "large_prime_vs_exceptional_set": {
      "description": "Large prime factor yields cancellation; small exceptional residue classes remain.",
      "paper_alignment": "Proposition 4.1 constructs exceptional sets Bq(R). "
    },
    "analytic_vs_combinatorial_density": {
      "description": "Analytic cancellation vs combinatorial counting in Hn.",
      "paper_alignment": "Theorem 5.1 compares main term and error term. "
    },
    "informal_vs_formal_proof": {
      "description": "Human narrative vs Lean autoformalization.",
      "paper_alignment": "Section 6 describes AxiomProver protocol. "
    }
  },

  "ai.transitions": {
    "geometry_to_modular_transition": {
      "description": "Rank obstruction becomes modular inequalities.",
      "paper_alignment": "Proposition 2.1. "
    },
    "modular_to_fourier_transition": {
      "description": "Counting usable units becomes Fourier/Ramanujan analysis.",
      "paper_alignment": "Section 3. "
    },
    "fourier_to_large_prime_transition": {
      "description": "Ramanujan-sum vanishing controlled by large prime factors.",
      "paper_alignment": "Section 4. "
    },
    "error_to_density_transition": {
      "description": "Error suppression implies density-zero lattice triangles.",
      "paper_alignment": "Theorem 1.1 and Corollary 1.2. "
    },
    "informal_to_lean_transition": {
      "description": "AxiomProver converts natural-language proof into Lean code.",
      "paper_alignment": "Section 6. "
    }
  },

  "ai.operators": {
    "rank_obstruction_operator": {
      "type": "operator",
      "role": "Encodes Mirzakhani–Wright rank obstruction in modular form."
    },
    "fourier_decomposition_operator": {
      "type": "operator",
      "role": "Splits S(p,q) into main term and error term."
    },
    "ramanujan_cancellation_operator": {
      "type": "operator",
      "role": "Detects vanishing of Ramanujan sums based on prime-power structure."
    },
    "density_operator": {
      "type": "operator",
      "role": "Evaluates density of admissible triangles in Hn."
    },
    "formalization_operator": {
      "type": "operator",
      "role": "Maps informal proofs to Lean-verifiable structures."
    }
  },

  "ai.audience": "Researchers in translation surfaces, Teichmüller dynamics, analytic number theory, and formal verification.",
  "ai.contact.github": "https://github.com/TriadicFrameworks",
  "ai.license": "Open educational use permitted"
}