TriadicFrameworks
Overview

RTT Agentic Module: Chebyshev Quotients, Demazure Multiplicities, and Dyck‑Path Models

This module wraps the paper “Chebyshev quotients, Demazure multiplicities, and Dyck‑path models” in RTT operator grammar.
It preserves the authors’ mathematics while exposing the structural regimes that govern:

  • the Chebyshev‑quotient formula for numerical Demazure multiplicities,
  • the eventual‑positivity dichotomy,
  • the signed matching/strip‑walk model, and
  • the Dyck‑path factorization families.

1. Purpose#

  • Make the paper agentic and machine‑navigable.
  • Clarify the regime structure behind Chebyshev quotients and positivity.
  • 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 shows that numerical Demazure multiplicities for sl₂[t] fusion products can be computed by extracting a single coefficient from a Chebyshev quotient.
This quotient exhibits a sharp dichotomy:

  • Either it becomes a polynomial (finite support),
  • Or its coefficients are eventually strictly positive.

The authors then:

  • give a signed combinatorial model using matchings and bounded strip walks,
  • identify infinite families where the quotient factors into Dyck‑path‑compatible pieces,
  • and translate these back into explicit formulas for Demazure multiplicities.

The appendix documents AxiomProver’s autonomous Lean formalization of the main theorems.

3. RTT structures in this module#

Regimes#

  • fusion_product_regime
  • chebyshev_quotient_regime
  • eventual_positivity_regime
  • matching_walk_regime
  • dyck_path_factorization_regime
  • formalization_regime

Tensions#

  • representation_vs_polynomial
  • signed_vs_unsigned
  • root_behavior_vs_combinatorics
  • formal_vs_informal

Transitions#

  • demazure_to_chebyshev_transition
  • quotient_to_positivity_transition
  • positivity_to_signed_model_transition
  • signed_to_unsigned_transition
  • informal_to_lean_transition

4. Operators#

  • chebyshev_coefficient_operator — extracts multiplicity coefficients.
  • root_analysis_operator — determines eventual positivity.
  • matching_operator — expands numerator via matchings.
  • strip_walk_operator — expands denominator via strip walks.
  • dyck_factor_operator — detects Dyck‑path factorizations.
  • formalization_operator — maps statements to Lean.

5. How to use this module#

  • Students:
    Use this README alongside the PDF to understand how Chebyshev polynomials, Dyck paths, and representation theory interact.

  • 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:2604.25246.
  • License: Open educational use permitted.

✅ diagram.txt#

(ASCII regime–tension–transition map)

     +--------------------------------------------------------------+
     | chebyshev_demazure_dyck_rtt                                  |
     +--------------------------------------------------------------+
 
REGIMES
  [R1] fusion_product_regime
  [R2] chebyshev_quotient_regime
  [R3] eventual_positivity_regime
  [R4] matching_walk_regime
  [R5] dyck_path_factorization_regime
  [R6] formalization_regime
 
TENSIONS
  [T1] representation_vs_polynomial        (R1 <--> R2)
  [T2] root_behavior_vs_combinatorics      (R2 <--> R3 <--> R4)
  [T3] signed_vs_unsigned                  (R4 <--> R5)
  [T4] formal_vs_informal                  (R5 <--> R6)
 
TRANSITIONS
  [X1] demazure_to_chebyshev_transition
  [X2] quotient_to_positivity_transition
  [X3] positivity_to_signed_model_transition
  [X4] signed_to_unsigned_transition
  [X5] informal_to_lean_transition
 
FLOW
  fusion_product_regime (R1)
        |
        v
  chebyshev_quotient_regime (R2)
        |
        v
  eventual_positivity_regime (R3)
        |
        v
  matching_walk_regime (R4)
        |
        v
  dyck_path_factorization_regime (R5)
        |
        v
  formalization_regime (R6)
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