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

thakur-power-sums_module.json — Agentic module schema role assignments

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 Jun 20, 2026