{
  "ai.module": "thakur_power_sums_rtt",
  "ai.version": "1.0",
  "ai.purpose": "RTT structural map of the paper 'Thakur’s Hypotheses on Power Sums over F_q[t]'.",
  "ai.source.pdf": "https://arxiv.org/pdf/2606.16239",
  "ai.keywords": [
    "Carlitz–Goss zeta",
    "finite fields",
    "power sums",
    "Thakur hypotheses",
    "carry-free decompositions",
    "Lucas theorem",
    "Sheats theorem",
    "Newton polygon",
    "multizeta values",
    "Lean formalization"
  ],

  "ai.module.summary": "This module exposes the regime structure behind Thakur’s three hypotheses (H1–H3) on power sums over F_q[t], connecting carry-free combinatorics, Carlitz expansions, reciprocal slot geometry, and Sheats’ uniqueness theorem. It preserves the authors’ results while adding RTT operators for agentic reasoning.",

  "ai.regimes": {
    "carlitz_expansion_regime": {
      "description": "Power sums Sd(k) expanded via Carlitz’s multinomial formula with carry-free constraints.",
      "evidence": "Section 3 uses Carlitz expansion (3) and Lucas’ theorem to characterize admissible decompositions."
    },
    "greedy_assignment_regime": {
      "description": "Prime-field H1 is solved by a weighted assignment problem with a unique greedy minimizer.",
      "evidence": "Algorithm 3.3 and Lemma 3.4 show the greedy fill minimizes Φ."
    },
    "reciprocal_slot_regime": {
      "description": "H2 uses complementary digit-slot multisets and block minimization.",
      "evidence": "Theorem 5.1 expresses sd(k) via reciprocal blocks βr(k−1)."
    },
    "sheats_uniqueness_regime": {
      "description": "H3 and the positive-power expansion rely on Sheats’ uniqueness of maximal weight decompositions.",
      "evidence": "Lemma 4.4 and Theorem 4.2 ensure unique minimal exponent."
    },
    "asymptotic_consequence_regime": {
      "description": "Newton-polygon convexity, RH analogue, and multizeta nonvanishing follow from H2/H3.",
      "evidence": "Corollaries 1.5–1.7 derive convexity, RH, and nonvanishing."
    },
    "formalization_regime": {
      "description": "Lean formalizations of H1–H3 generated by AxiomProver.",
      "evidence": "Appendix describes Lean files and dependency graph."
    }
  },

  "ai.tensions": {
    "digit_local_vs_degree_global": {
      "description": "Local base-p digit constraints vs global degree of Sd(k).",
      "paper_alignment": "Carry-free decompositions determine global degree."
    },
    "prime_field_vs_extension_field": {
      "description": "H1/H2 hold only for q=p; extension fields exhibit pathologies.",
      "paper_alignment": "Examples in Appendix A.11 of [8] show failures for q=4."
    },
    "combinatorial_vs_analytic_degree": {
      "description": "Combinatorial digit assignments vs analytic consequences (RH, convexity).",
      "paper_alignment": "H2 recursion yields Newton-polygon convexity."
    },
    "positive_power_vs_negative_power_expansions": {
      "description": "Carlitz expansion (negative powers) vs Sheats expansion (positive powers).",
      "paper_alignment": "H1 uses Carlitz; H2/H3 use Lemma 4.1."
    }
  },

  "ai.transitions": {
    "carlitz_to_greedy_transition": {
      "description": "Translate Carlitz expansion into a weighted assignment problem.",
      "paper_alignment": "Section 3 converts (3) into Φ-minimization."
    },
    "digits_to_slots_transition": {
      "description": "Convert base-p digits of k−1 into reciprocal slot multisets.",
      "paper_alignment": "Definition (18) and Theorem 5.1."
    },
    "slots_to_recursion_transition": {
      "description": "Block minimization yields sd(k)=sd−1(s1(k))+s1(k).",
      "paper_alignment": "Corollary 1.4."
    },
    "positive_power_to_uniqueness_transition": {
      "description": "Use Sheats’ theorem to ensure unique minimal exponent.",
      "paper_alignment": "Lemma 4.4 + Theorem 4.2."
    },
    "recursion_to_convexity_transition": {
      "description": "H2 recursion implies strict Newton-polygon convexity.",
      "paper_alignment": "Corollary 1.5."
    }
  },

  "ai.operators": {
    "carlitz_operator": {
      "type": "operator",
      "role": "Expands Sd(k) via Carlitz multinomial formula and carry-free constraints."
    },
    "greedy_assignment_operator": {
      "type": "operator",
      "role": "Computes the unique Φ-minimizing decomposition for H1."
    },
    "reciprocal_slot_operator": {
      "type": "operator",
      "role": "Constructs Slotsp(k−1) and block sums βr(k−1)."
    },
    "block_minimization_operator": {
      "type": "operator",
      "role": "Applies block rearrangement inequality to derive H2 recursion."
    },
    "sheats_operator": {
      "type": "operator",
      "role": "Applies Sheats’ uniqueness to positive-power expansions."
    },
    "degree_recursion_operator": {
      "type": "operator",
      "role": "Implements sd(k)=sd−1(s1(k))+s1(k)."
    }
  },

  "ai.audience": "Researchers in finite-field arithmetic, zeta functions, combinatorics, and formal verification.",
  "ai.contact.github": "https://github.com/TriadicFrameworks",
  "ai.license": "Open educational use permitted"
}