TriadicFrameworks
अवलोकन

RTT Agentic Module: Thakur’s Hypotheses on Power Sums over F_q[t]

Module ID: thakur_power_sums_rtt
Source paper: https://arxiv.org/pdf/2606.16239

This module wraps the paper “Thakur’s Hypotheses on Power Sums over F_q[t]” in RTT operator grammar.
It preserves the authors’ mathematics while exposing the structural regimes that govern the three hypotheses (H1–H3), their proofs, and their consequences.

1. Purpose#

  • Make the paper agentic and machine‑navigable.
  • Clarify the regime structure behind H1, H2, and H3.
  • Provide students with a clean conceptual map.
  • Support AI agents in reasoning over the paper without drift.

2. Core RTT view of the paper#

The paper studies the degrees
[ s_d(k) = -\deg_t S_d(k) ]
of power sums over ( \mathbb{F}_q[t] ), and proves:

  • H1 (prime fields):
    Carlitz expansion has a unique maximal-degree term.

  • H2 (prime fields):
    A recursion
    [ s_d(k) = s_{d-1}(s_1(k)) + s_1(k) ]
    obtained via reciprocal digit slots and block minimization.

  • H3 (all finite fields):
    Monotonicity in the exponent:
    [ s_d(k) < s_d(k+1) \quad (p \nmid k). ]

These yield:

  • strict Newton‑polygon convexity,
  • the Carlitz–Goss RH analogue,
  • nonvanishing of multizeta values.

The appendix documents Lean formalizations of H1–H3.

3. RTT structures in this module#

Regimes#

  • carlitz_expansion_regime
  • greedy_assignment_regime
  • reciprocal_slot_regime
  • sheats_uniqueness_regime
  • asymptotic_consequence_regime
  • formalization_regime

Tensions#

  • digit_local_vs_degree_global
  • prime_field_vs_extension_field
  • combinatorial_vs_analytic_degree
  • positive_power_vs_negative_power_expansions

Transitions#

  • carlitz_to_greedy_transition
  • digits_to_slots_transition
  • slots_to_recursion_transition
  • positive_power_to_uniqueness_transition
  • recursion_to_convexity_transition

4. Operators#

  • carlitz_operator — expands Sd(k) via Carlitz.
  • greedy_assignment_operator — solves H1.
  • reciprocal_slot_operator — constructs Slotsp(k−1).
  • block_minimization_operator — proves H2.
  • sheats_operator — ensures uniqueness for H3.
  • degree_recursion_operator — implements the H2 recursion.

5. How to use this module#

  • Students:
    Use this README alongside the PDF to see how digit combinatorics control analytic degree.

  • Researchers:
    Query the module’s regimes and operators to explore structural dependencies.

  • Agents:
    Treat module.json as the canonical structural map of the paper.

6. Provenance#

  • Module authoring: TriadicFrameworks (RTT / agentic mapping).
  • Original content: Authors of arXiv:2606.16239.
  • License: Open educational use permitted.

✅ diagram.txt#

     +--------------------------------------------------------------+
     | thakur_power_sums_rtt                                        |
     +--------------------------------------------------------------+
 
REGIMES
  [R1] carlitz_expansion_regime
  [R2] greedy_assignment_regime
  [R3] reciprocal_slot_regime
  [R4] sheats_uniqueness_regime
  [R5] asymptotic_consequence_regime
  [R6] formalization_regime
 
TENSIONS
  [T1] digit_local_vs_degree_global
  [T2] prime_field_vs_extension_field
  [T3] combinatorial_vs_analytic_degree
  [T4] positive_power_vs_negative_power_expansions
 
TRANSITIONS
  [X1] carlitz_to_greedy_transition
  [X2] digits_to_slots_transition
  [X3] slots_to_recursion_transition
  [X4] positive_power_to_uniqueness_transition
  [X5] recursion_to_convexity_transition
 
FLOW
  carlitz_expansion_regime (R1)
        |
        v
  greedy_assignment_regime (R2)
        |
        v
  reciprocal_slot_regime (R3)
        |
        v
  sheats_uniqueness_regime (R4)
        |
        v
  asymptotic_consequence_regime (R5)
        |
        v
  formalization_regime (R6)

Updated

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