TriadicFrameworks
Panoramica

RTT Agentic Module: On the Paucity of Lattice Triangles

Module ID: paucity_lattice_triangles_rtt
Source paper: https://arxiv.org/pdf/2603.23928

This module wraps the paper “On the Paucity of Lattice Triangles” in RTT operator grammar.
It preserves the authors’ mathematics while exposing the structural regimes that govern the density-zero result for obtuse rational lattice triangles.

1. Purpose#

  • Make the paper agentic and machine-navigable.
  • Clarify the regime structure behind the Mirzakhani–Wright rank obstruction.
  • 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 proves that almost all obtuse rational triangles in the hard window are not lattice triangles, by showing:

  • The geometric rank obstruction can be reformulated arithmetically.
  • The modular inequalities can be analyzed using Fourier and Ramanujan sums.
  • Large prime factors in the denominator force strong cancellation.
  • The remaining exceptional sets have density zero.
  • Therefore, lattice triangles are extremely rare in this regime.

The proof moves between geometry, modular arithmetic, Fourier analysis, and density arguments.
This module makes those transitions explicit.

3. RTT structures in this module#

Regimes#

  • geometric_rank_regime
  • modular_obstruction_regime
  • fourier_ramanujan_regime
  • large_prime_factor_regime
  • density_one_regime
  • formalization_regime

Tensions#

  • geometry_vs_arithmetic
  • main_term_vs_error_term
  • large_prime_vs_exceptional_set
  • analytic_vs_combinatorial_density
  • informal_vs_formal_proof

Transitions#

  • geometry_to_modular_transition
  • modular_to_fourier_transition
  • fourier_to_large_prime_transition
  • error_to_density_transition
  • informal_to_lean_transition

4. Operators#

  • rank_obstruction_operator — geometric → modular obstruction.
  • fourier_decomposition_operator — splits S(p,q) into main/error terms.
  • ramanujan_cancellation_operator — detects cancellation from large prime factors.
  • density_operator — evaluates density of lattice triangles.
  • formalization_operator — maps informal proofs to Lean.

5. How to use this module#

  • Students:
    Use this README alongside the PDF to understand how geometry, arithmetic, and Fourier analysis 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:2603.23928.
  • License: Open educational use permitted.

✅ diagram.txt#

(ASCII regime–tension–transition map)

        +------------------------------------------------+
        | paucity_lattice_triangles_rtt                  |
        +------------------------------------------------+
 
REGIMES
  [R1] geometric_rank_regime
  [R2] modular_obstruction_regime
  [R3] fourier_ramanujan_regime
  [R4] large_prime_factor_regime
  [R5] density_one_regime
  [R6] formalization_regime
 
TENSIONS
  [T1] geometry_vs_arithmetic          (R1 <--> R2)
  [T2] main_term_vs_error_term         (R2 <--> R3)
  [T3] large_prime_vs_exceptional_set  (R3 <--> R4)
  [T4] analytic_vs_combinatorial       (R4 <--> R5)
  [T5] informal_vs_formal_proof        (R5 <--> R6)
 
TRANSITIONS
  [X1] geometry_to_modular_transition
  [X2] modular_to_fourier_transition
  [X3] fourier_to_large_prime_transition
  [X4] error_to_density_transition
  [X5] informal_to_lean_transition
 
FLOW
  geometric_rank_regime (R1)
        |
        v
  modular_obstruction_regime (R2)
        |
        v
  fourier_ramanujan_regime (R3)
        |
        v
  large_prime_factor_regime (R4)
        |
        v
  density_one_regime (R5)
        |
        v
  formalization_regime (R6)

Updated

Powered by Docsbook