{
  "ai.module": "abc_tau_missing_primes_rtt",
  "ai.version": "1.0",
  "ai.purpose": "RTT structural map of the paper 'ABC implies that Ramanujan’s tau function misses almost all primes'.",
  "ai.source.pdf": "https://arxiv.org/pdf/2603.29970",
  "ai.keywords": [
    "Ramanujan tau function",
    "modular forms",
    "ABC conjecture",
    "prime divisors",
    "density zero",
    "Galois representations",
    "ℓ-adic methods",
    "Diophantine inequalities"
  ],

  "ai.module.summary": "This module exposes the regime structure behind the ABC-based proof that τ(n) ≡ 0 mod p occurs for density-zero many primes p. It preserves the authors' results while adding RTT operators for agentic reasoning.",

  "ai.regimes": {
    "modular_form_regime": {
      "description": "Ramanujan’s tau function arises as coefficients of the discriminant modular form Δ(z).",
      "evidence": "Paper recalls τ(n) as Fourier coefficients of Δ and its multiplicativity."
    },
    "galois_representation_regime": {
      "description": "τ(p) mod p corresponds to traces of Frobenius in a 2-dimensional ℓ-adic Galois representation.",
      "evidence": "Authors use Deligne’s ℓ-adic representation attached to Δ."
    },
    "abc_diophantine_regime": {
      "description": "ABC conjecture bounds integer solutions to a+b=c, enabling strong control over τ(p).",
      "evidence": "Main theorem uses ABC to bound prime solutions to τ(p) ≡ 0 mod p."
    },
    "sparsity_regime": {
      "description": "The set of primes p for which τ(p) ≡ 0 mod p has density zero.",
      "evidence": "Main result: ABC ⇒ τ misses almost all primes."
    },
    "formalization_regime": {
      "description": "AxiomProver formalizes the Diophantine inequalities and modular-form identities.",
      "evidence": "Appendix describes Lean verification pipeline."
    }
  },

  "ai.tensions": {
    "analytic_vs_diophantine": {
      "description": "Analytic modular-form bounds vs Diophantine inequalities from ABC.",
      "paper_alignment": "Proof blends Deligne bounds with ABC-based height estimates."
    },
    "local_trace_vs_global_density": {
      "description": "Local Frobenius trace conditions vs global density of primes.",
      "paper_alignment": "τ(p) mod p is local; density-zero is global."
    },
    "galois_vs_modular": {
      "description": "Galois representation viewpoint vs modular-form coefficient viewpoint.",
      "paper_alignment": "Authors move between τ(p) and Frobenius traces."
    },
    "informal_vs_formal_proof": {
      "description": "Narrative argument vs Lean formalization.",
      "paper_alignment": "Appendix describes automated formal proof."
    }
  },

  "ai.transitions": {
    "modular_to_galois_transition": {
      "description": "Translate τ(p) into Frobenius trace in ℓ-adic representation.",
      "paper_alignment": "Section 2 uses Deligne’s construction."
    },
    "galois_to_abc_transition": {
      "description": "Use ABC to bound integer solutions arising from trace conditions.",
      "paper_alignment": "Main theorem applies ABC to τ(p) ≡ 0 mod p."
    },
    "abc_to_density_transition": {
      "description": "ABC-based bounds imply density-zero set of primes.",
      "paper_alignment": "Final section proves density-zero result."
    },
    "informal_to_lean_transition": {
      "description": "AxiomProver converts the argument into Lean code.",
      "paper_alignment": "Appendix describes formalization."
    }
  },

  "ai.operators": {
    "tau_operator": {
      "type": "operator",
      "role": "Evaluates τ(p) and detects τ(p) ≡ 0 mod p."
    },
    "frobenius_trace_operator": {
      "type": "operator",
      "role": "Maps primes p to Frobenius traces in the ℓ-adic representation."
    },
    "abc_height_operator": {
      "type": "operator",
      "role": "Applies ABC to bound integer solutions arising from τ(p)."
    },
    "density_operator": {
      "type": "operator",
      "role": "Evaluates density of primes satisfying τ(p) ≡ 0 mod p."
    },
    "formalization_operator": {
      "type": "operator",
      "role": "Maps informal Diophantine arguments to Lean-verifiable structures."
    }
  },

  "ai.audience": "Researchers in modular forms, Galois representations, Diophantine geometry, and arithmetic statistics.",
  "ai.contact.github": "https://github.com/TriadicFrameworks",
  "ai.license": "Open educational use permitted"
}