Author: Luciano Benjamín Nieto
Affiliation: Independent Researcher, General Alvear, Mendoza, Argentina
Series: DDSD Part 2

The Convergence Frontier in Discrete Dynamical Systems: A Computational Study of Dissipative Families Beyond Collatz.

This repository is a companion to DDSD-Framework. While the parent repository establishes the DDSD (Discrete Dynamical Systems Dissipation) framework and applies it to Collatz, 5x+1, perturbed families, critical maps, 2-adic variable fields, toy cryptographic hashes, and evolved maps, this repository extends the framework by:

1. Formalizing an observable embedding $\Phi: \mathcal{D} \rightarrow \mathbb{R}^k$ that maps dynamical systems into a real vector space via computable statistics.
2. Defining the convergence frontier as an emergent geometric structure in embedding space, induced by density clustering of systems with distinct statistical behaviors.
3. Verifying Chang's (2026) one-bit mixing structure on pure Collatz and demonstrating its failure on strategically mixed maps.
4. Discovering the Ultra-Champion map via genetic algorithm — 3.6$\times$ more dissipative than Collatz, with 100% termination and 4$\times$ faster convergence.
5. Exploring the proportion space frontier through systematic sweep and simulated annealing.
6. Analyzing a toy cryptographic hash model as a computational proxy for hyper-dissipative behavior.

All claims are framed as empirical observations under defined sampling regimes. No proof of boundedness for general families is claimed.

# Install dependencies
pip install -r requirements.txt

# Run complete reproduction suite (takes ~10-15 minutes)
python src/master_simulation.py

# Verify data integrity against paper claims
python src/verify_submission.py

.
├── README.md                          # This file
├── LICENSE                            # MIT License
├── requirements.txt                   # Python dependencies
├── .gitignore                         # Git ignore rules
├── src/
│   ├── master_simulation.py           # Complete reproduction suite (all experiments)
│   └── verify_submission.py           # Automated verification script
├── data/
│   ├── chang_verification_results.txt   # Chang verification results
│   └── frontier_executive_summary.txt   # Frontier exploration summary
├── figures/
│   ├── fig15_chang_bit4_destruction.png
│   ├── fig16_chang_unified_verification.png
│   ├── fig17_even_coefficients_sa.png
│   ├── fig18_final_frontier_convergence.png
│   ├── fig19_frontier_proportion_space.png
│   ├── fig20_markov_memory_analysis.png
│   ├── fig21_original_map_deep_dive.png
│   ├── fig22_p3_p9_phase_diagram.png
│   ├── fig23_sa_deep_convergence.png
│   ├── fig24_sha256_dissipation.png
│   ├── fig25_ultra_champion_pattern.png
│   └── fig26_universe_comparison.png
└── paper/
    └── ddsd_convergence_frontier.md   # Markdown version (GitHub-ready)

All simulations use fixed random seed 42 for full reproducibility. The complete suite regenerates all data, figures, and verification reports.

Expected outputs:

• data/*.txt — Verification summaries
• figures/*.png — Figures 15-26
• Console report with all metrics

We define $\Phi: \mathcal{D} \rightarrow \mathbb{R}^4$ with components:

• $\Phi_1$: resolution-dependent decorrelation ($R^2$ at $K=6$)
• $\Phi_2$: intrafiber output dispersion (normalized entropy)
• $\Phi_3$: scale-dependent drift (Bonferroni-corrected)
• $\Phi_4$: pathwise recurrence frequency

The convergence frontier $\mathcal{F} \subset \mathbb{R}^4$ is the boundary induced by density clustering in this embedding space.

• Collatz pure: Satisfies Chang's one-bit mixing exactly — bit 4 determines gap outcome with probability ~0.505
• Ultra-Champion: Breaks Chang's structure — gap outcomes independent of bit 4 (~0.122 each)
• Implication: Collatz's rigidity is its difficulty; mixed maps with multi-bit mixing may be easier to prove

Discovered via genetic algorithm + intensive search:

Chromosome: [3, 7, 3, 5, 7, 3, 3, 9, 9, 7, 3, 5, 3, 3, 9, 9,
             7, 3, 3, 5, 5, 3, 7, 3, 7, 7, 3, 5, 5, 3, 5, 9]
Proportion: 13×3, 7×5, 7×7, 5×9

Exhaustive verification: 100% convergence for all 2,097,152 odd integers in [1, 2²²).

• Collatz is NOT optimal — sits in a local valley of the fitness landscape
• Disposition is everything — Same proportion, different arrangement: drift from +0.44 to -1.70
• Mod 32 is the sweet spot — Mod 64 and mod 128 do not improve dissipation
• No known theoretical lower bound — Empirical drift of -1.68 is 56× beyond the 2-adic theoretical limit

• Drift: -1.29 per round (15× stronger than Collatz)
• Perfect decorrelation (A1 R² ≈ 0)
• Maximum entropy mixing (A2 ≈ 0.998)
• Instant convergence, no recurrence

Disclaimer: No theoretical equivalence with dynamical invariants is claimed. The hash model is a computational proxy with structurally analogous features.

Before submission to any venue, run:

python src/verify_submission.py

This checks that all data files match the paper's claims. If it prints "VERIFICATION PASSED", the submission is internally consistent.

This work builds directly on:

• github.com/Rylow999/ddsd-framework — DDSD Framework v2.0

@article{Nieto2026Frontier,
  title={The Convergence Frontier in Discrete Dynamical Systems: A Computational Study of Dissipative Families Beyond Collatz},
  author={Nieto, Luciano Benjamín},
  year={2026},
  note={Companion repository to DDSD Framework v2.0}
}

For questions or issues, please open a GitHub issue or contact the authors.

Generated: 2026-06-19
Framework: DDSD Companion — Convergence Frontier
Seed: 42