RTT Agentic Module: Parity of k-Differentials in Genus Zero and One

parity-k-differentials_module.json — Agentic module schema role assignments

Module ID: parity_k_differentials_rtt
Source paper: https://arxiv.org/pdf/2602.03722

This module wraps the paper “Parity of k-differentials in genus zero and one” in RTT operator grammar.
It preserves the authors’ mathematical results while exposing the structural regimes that govern parity behavior.

1. Purpose#

Make the paper agentic and machine-navigable.
Clarify the regime structure behind parity phenomena.
Provide students with a clean conceptual map.
Support AI agents in reasoning over the paper without drift.

2. Core RTT view of the paper#

Parity of k-differentials is governed by three interacting structures:

Combinatorial regime:
Parity depends on discrete data: orders of zeros/poles, k-residues.
Geometric regime:
The geometry of strata in moduli spaces shapes which configurations are possible.
Genus regime:
Genus 0 and genus 1 exhibit fundamentally different parity behaviors.

The paper’s proofs move between these regimes, often implicitly.
This module makes those transitions explicit.

3. RTT structures in this module#

Regimes#

combinatorial_parity_regime
geometric_strata_regime
genus_transition_regime

Tensions#

local_vs_global_parity
combinatorial_vs_geometric
genus_zero_vs_genus_one

Transitions#

strata_refinement_transition
genus_lift_transition
combinatorial_to_geometric_transition

Each item is backed by evidence from the paper.

4. Operators#

parity_operator — computes parity from local data.
strata_operator — identifies geometric strata.
genus_operator — encodes topological constraints.

These operators allow agents to reason structurally about the paper’s results.

5. How to use this module#

Students:
Use this README alongside the PDF to understand how parity depends on both combinatorics and geometry.
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:2602.03722.
License: Open educational use permitted.

✅ diagram.txt#

(ASCII regime–tension–transition map)

             +-------------------------------------------+
             | parity_k_differentials_rtt                |
             +-------------------------------------------+
 
REGIMES
  [R1] combinatorial_parity_regime
  [R2] geometric_strata_regime
  [R3] genus_transition_regime
 
TENSIONS
  [T1] local_vs_global_parity       (R1 <--> R3)
  [T2] combinatorial_vs_geometric   (R1 <--> R2)
  [T3] genus_zero_vs_genus_one      (R3)
 
TRANSITIONS
  [X1] strata_refinement_transition     (R2 -> R1)
  [X2] genus_lift_transition            (R1/R2 -> R3)
  [X3] combinatorial_to_geometric       (R1 -> R2)
 
FLOW
  combinatorial_parity_regime (R1)
        <--> geometric_strata_regime (R2)
                |
                v
        genus_transition_regime (R3)

Updated Jun 20, 2026