RTT Agentic Module: A Quadratic Form Generalization of Rational dinv

quadratic-rational-dinv_module.json — Agentic module schema role assignments

Module ID: quadratic_rational_dinv_rtt
Source paper: https://arxiv.org/pdf/2604.13238

This module wraps the paper “A quadratic form generalization of rational dinv” in RTT operator grammar.
It preserves the authors’ mathematics while exposing the structural regimes that govern the finiteness and stability of Q-dinv.

1. Purpose#

Make the paper agentic and machine-navigable.
Clarify the regime structure behind the quadratic-form generalization of dinv.
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 authors generalize the classical rational dinv statistic by:

replacing the linear slope comparison with a positive definite quadratic form Q,
proving that the resulting statistic is finite,
and showing that it is stable across rational Dyck-path families.

This involves three interacting structures:

Dyck-path combinatorics
Quadratic-form geometry
Finiteness and stability arguments

The proof moves between these regimes, often implicitly.
This module makes those transitions explicit.

3. RTT structures in this module#

Regimes#

dyck_path_regime
quadratic_form_regime
finiteness_regime
stability_regime
combinatorial_geometry_regime

Tensions#

linear_vs_quadratic
combinatorial_vs_geometric
finiteness_vs_growth
local_comparison_vs_global_stability

Transitions#

linear_to_quadratic_transition
quadratic_to_finiteness_transition
finiteness_to_stability_transition
combinatorial_to_geometric_transition

4. Operators#

q_dinv_operator — computes Q-weighted dinv.
quadratic_region_operator — identifies Q-defined geometric regions.
finiteness_operator — bounds contributing pairs using positivity.
stability_operator — determines stability under rational scaling.

5. How to use this module#

Students:
Use this README alongside the PDF to understand how quadratic forms interact with Dyck-path combinatorics.
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:2604.13238.
License: Open educational use permitted.

✅ diagram.txt#

(ASCII regime–tension–transition map)

      +-----------------------------------------------------------+
      | quadratic_rational_dinv_rtt                               |
      +-----------------------------------------------------------+
 
REGIMES
  [R1] dyck_path_regime
  [R2] quadratic_form_regime
  [R3] finiteness_regime
  [R4] stability_regime
  [R5] combinatorial_geometry_regime
 
TENSIONS
  [T1] linear_vs_quadratic              (R1 <--> R2)
  [T2] combinatorial_vs_geometric       (R1 <--> R5)
  [T3] finiteness_vs_growth             (R2 <--> R3)
  [T4] local_comparison_vs_global_stability (R2/R3 <--> R4)
 
TRANSITIONS
  [X1] linear_to_quadratic_transition
  [X2] quadratic_to_finiteness_transition
  [X3] finiteness_to_stability_transition
  [X4] combinatorial_to_geometric_transition
 
FLOW
  dyck_path_regime (R1)
        |
        v
  quadratic_form_regime (R2)
        |
        v
  finiteness_regime (R3)
        |
        v
  stability_regime (R4)
        |
        v
  combinatorial_geometry_regime (R5)

Updated Jun 20, 2026