RTT Agentic Module: Dominant Zeros of Nekrasov–Okounkov Polynomials

• nekr-okounkov-dominant-zeros_module.json — Agentic module schema role assignments

Module ID: nekr_okounkov_dominant_zeros_rtt
Source paper: https://arxiv.org/pdf/2606.15394

This module wraps the paper “Dominant Zeros of Nekrasov–Okounkov Polynomials” in RTT operator grammar.
It preserves the authors’ mathematics while exposing the structural regimes that govern the dominant-zero phenomenon, the bulk-zero cloud, and the asymptotic machinery behind them.

1. Purpose#

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

2. Core RTT view of the paper#

Nekrasov–Okounkov polynomials ( P_n(x) ) arise from weighted sums over partitions.
The authors show:

• For large ( n ), there is a unique dominant zero on the negative real axis.
• All other zeros form a bulk cloud with predictable scaling.
• The asymptotics are governed by a saddle point of the analytic continuation of the generating function.

The proof moves between partition combinatorics, analytic continuation, and steepest‑descent asymptotics.
This module makes those transitions explicit.

3. RTT structures in this module#

Regimes#

• partition_expansion_regime
• analytic_continuation_regime
• saddle_point_regime
• dominant_zero_regime
• bulk_zero_regime

Tensions#

• combinatorial_vs_analytic
• dominant_vs_bulk_zeros
• local_saddle_vs_global_distribution

Transitions#

• partition_to_analytic_transition
• analytic_to_saddle_transition
• saddle_to_zero_localization_transition

4. Operators#

• partition_weight_operator — builds the polynomials from partitions.
• analytic_continuation_operator — extends to complex x.
• saddle_point_operator — performs steepest‑descent.
• zero_localization_operator — identifies dominant and bulk zeros.

5. How to use this module#

• Students:
  Use this README alongside the PDF to understand how partition combinatorics produce analytic objects with rich zero structure.

• 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:2606.15394.
• License: Open educational use permitted.

✅ diagram.txt#

     +--------------------------------------------------------------+
     | nekr_okounkov_dominant_zeros_rtt                             |
     +--------------------------------------------------------------+
 
REGIMES
  [R1] partition_expansion_regime
  [R2] analytic_continuation_regime
  [R3] saddle_point_regime
  [R4] dominant_zero_regime
  [R5] bulk_zero_regime
 
TENSIONS
  [T1] combinatorial_vs_analytic
  [T2] dominant_vs_bulk_zeros
  [T3] local_saddle_vs_global_distribution
 
TRANSITIONS
  [X1] partition_to_analytic_transition
  [X2] analytic_to_saddle_transition
  [X3] saddle_to_zero_localization_transition
 
FLOW
  partition_expansion_regime (R1)
        |
        v
  analytic_continuation_regime (R2)
        |
        v
  saddle_point_regime (R3)
        |
        v
  dominant_zero_regime (R4)
        |
        v
  bulk_zero_regime (R5)