RTT Agentic Module: Almost All Primes Are Partially Regular

• partial-regular-primes_module.json — Agentic module schema role assignments

Module ID: partial_regular_primes_rtt
Source paper: https://arxiv.org/pdf/2602.05090

This module wraps the paper “Almost all primes are partially regular” in RTT operator grammar.
It preserves the authors’ results while exposing the structural regimes that govern irregularity, partial regularity, and density-one phenomena.

1. Purpose#

• Make the paper agentic and machine-navigable.
• Clarify the regime structure behind partial regularity of primes.
• 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 primes are partially regular, meaning:

• A prime may be irregular at some Bernoulli indices,
• but for a positive proportion of admissible indices, it behaves regularly,
• and the set of such primes has density 1.

This involves three interacting structures:

• Bernoulli irregularity regime:
  Irregularity is defined by divisibility of Bernoulli numerators.

• Partial regularity regime:
  A relaxed notion of regularity that allows some irregular pairs.

• Analytic density regime:
  Density-one results require analytic number theory tools.

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

3. RTT structures in this module#

Regimes#

• bernoulli_irregularity_regime
• partial_regularity_regime
• analytic_density_regime

Tensions#

• local_vs_global_irregularity
• combinatorial_vs_analytic
• rare_irregularity_vs_density_one

Transitions#

• irregular_to_partial_transition
• local_condition_to_density_transition
• exception_control_transition

4. Operators#

• irregularity_operator — detects irregular pairs (p, k).
• partial_regularity_operator — measures proportion of regular indices.
• density_operator — evaluates analytic density of prime subsets.

5. How to use this module#

• Students:
  Use this README alongside the PDF to understand how irregularity and density 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:2602.05090.
• License: Open educational use permitted.

✅ diagram.txt#

(ASCII regime–tension–transition map)

        +---------------------------------------------+
        | partial_regular_primes_rtt                  |
        +---------------------------------------------+
 
REGIMES
  [R1] bernoulli_irregularity_regime
  [R2] partial_regularity_regime
  [R3] analytic_density_regime
 
TENSIONS
  [T1] local_vs_global_irregularity     (R1 <--> R2)
  [T2] combinatorial_vs_analytic        (R1 <--> R3)
  [T3] rare_irregularity_vs_density_one (R2 <--> R3)
 
TRANSITIONS
  [X1] irregular_to_partial_transition      (R1 -> R2)
  [X2] local_condition_to_density_transition (R1/R2 -> R3)
  [X3] exception_control_transition         (R3)
 
FLOW
  bernoulli_irregularity_regime (R1)
        |
        v
  partial_regularity_regime (R2)
        |
        v
  analytic_density_regime (R3)