{
  "ai.module": "partial_regular_primes_rtt",
  "ai.version": "1.0",
  "ai.purpose": "RTT structural map of the paper 'Almost all primes are partially regular'.",
  "ai.source.pdf": "https://arxiv.org/pdf/2602.05090",
  "ai.keywords": [
    "primes",
    "regular primes",
    "irregular primes",
    "Bernoulli numbers",
    "Kummer theory",
    "partial regularity",
    "density results",
    "analytic number theory"
  ],

  "ai.module.summary": "This module exposes the regime structure behind partial regularity of primes, connecting Bernoulli irregularity, Kummer-type conditions, and analytic density arguments. It preserves the authors' results while adding RTT operators for agentic reasoning.",

  "ai.regimes": {
    "bernoulli_irregularity_regime": {
      "description": "Irregularity determined by divisibility of Bernoulli numerators by p.",
      "evidence": "Paper defines irregular pairs (p, k) via p | B_k."
    },
    "partial_regularity_regime": {
      "description": "A prime is partially regular if it avoids irregularity for a positive proportion of admissible indices.",
      "evidence": "Main theorem: almost all primes satisfy partial regularity."
    },
    "analytic_density_regime": {
      "description": "Density arguments over primes using analytic number theory and distribution heuristics.",
      "evidence": "Authors prove that the set of partially regular primes has density 1."
    }
  },

  "ai.tensions": {
    "local_vs_global_irregularity": {
      "description": "Local divisibility conditions on Bernoulli numbers vs global density of primes.",
      "paper_alignment": "Irregularity is local; partial regularity is global."
    },
    "combinatorial_vs_analytic": {
      "description": "Discrete irregular pairs vs analytic density estimates.",
      "paper_alignment": "Proof blends combinatorial constraints with analytic bounds."
    },
    "rare_irregularity_vs_density_one": {
      "description": "Irregular primes exist infinitely often, yet almost all primes are partially regular.",
      "paper_alignment": "Main theorem resolves this apparent contradiction."
    }
  },

  "ai.transitions": {
    "irregular_to_partial_transition": {
      "description": "Move from irregularity of individual Bernoulli indices to partial regularity across many indices.",
      "paper_alignment": "Key step in proving density 1."
    },
    "local_condition_to_density_transition": {
      "description": "Translate local divisibility conditions into global density statements.",
      "paper_alignment": "Analytic number theory tools used to lift local constraints."
    },
    "exception_control_transition": {
      "description": "Bound the contribution of exceptional primes that fail partial regularity.",
      "paper_alignment": "Authors show exceptional set has density 0."
    }
  },

  "ai.operators": {
    "irregularity_operator": {
      "type": "operator",
      "role": "Determines whether p divides the numerator of B_k for admissible k."
    },
    "partial_regularity_operator": {
      "type": "operator",
      "role": "Computes the proportion of indices for which p avoids irregularity."
    },
    "density_operator": {
      "type": "operator",
      "role": "Evaluates analytic density of prime subsets."
    }
  },

  "ai.audience": "Researchers in analytic number theory, algebraic number theory, and arithmetic statistics.",
  "ai.contact.github": "https://github.com/TriadicFrameworks",
  "ai.license": "Open educational use permitted"
}