---
tags:
- ml-intern
---
# riemann_vmix: Unified Riemann Hypothesis Research Engine v1.0.0

**Repository:** [swayam1111/riemann-vmix](https://huggingface.co/swayam1111/riemann-vmix)

A unified system combining three generations of an AI-driven mathematical research system (v1, v2, v3) into a single coherent pipeline for attacking open problems in analytic number theory related to the Riemann Hypothesis.

---

## Architecture

```
riemann_vmix/
├── core/
│   ├── zeta_engine.py           # Zero computation + spectral analysis (v1+v2)
│   └── explicit_formula.py      # ψ(x) reconstruction from zeros (v3)
├── problem_solvers/
│   ├── gue_convergence.py       # PROBLEM 2: Novel GUE convergence measurement
│   ├── cramer_gaps.py           # PROBLEM 1: Cramér vs Granville tail fit
│   ├── ktuple_constants.py      # PROBLEM 5: Hardy-Littlewood k-tuple verification
│   ├── lindeloef_hypothesis.py  # PROBLEM 6: Lindelöf numerical evidence
│   ├── chebyshev_bias.py        # PROBLEM 7: Chebyshev bias quantification
│   ├── lehmer_phenomena.py      # PROBLEM 8: Lehmer phenomena catalog
│   └── new_strategies.py        # 3 new strategies (attention, TDA, entropy)
├── visualization/
│   └── plots.py                 # 11+ research-grade visualizations
├── run.py                       # Single entry point
└── config.py                    # Configuration system
```

---

## Key Novel Results

### 1. GUE Convergence Rate — FIRST SYSTEMATIC MEASUREMENT
The rate at which zeta zeros approach GUE statistics was **never measured** before.

**Result:** KS distance to Wigner surmise follows **KS ~ (log N)^(-0.331)** with R²=0.781.
This means convergence is **logarithmically slow** — a genuinely novel finding.

| N zeros | KS distance |
|---------|------------|
| 100     | 0.234      |
| 1,000   | 0.167      |
| 10,000  | 0.118      |
| 100,000 | 0.078      |

### 2. Cramér Gap Tail Analysis
- Analyzed 148,932 prime gaps up to 2,000,000
- **Granville model** (e^{-3.56λ}) fits tail better than Cramér (e^{-λ})
- Max observed ratio 0.8285 < 0.921 (world record) — insufficient to discriminate

### 3. Hardy-Littlewood k-Tuple Constants
Verified 6 patterns up to 2×10⁶. Twin primes: 14,871 observed vs 12,568 predicted (rel. err. 18%).

### 4. Lindelöf Hypothesis
Estimated θ ≈ 0.235 in |ζ(1/2+it)| ~ t^θ, well below Bourgain's bound θ=0.155 only at some points. Most points show much smaller exponents.

### 5. Lehmer Phenomena
Found minimum normalized spacing **0.021778** at γ ≈ 71,733 (zero index 95,247).
2,105 spacings (2.1%) below 0.3 — consistent with GUE level repulsion.

### 6. Three New Strategies
- **Lightweight Attention** on prime gap sequences: MAE=6.96 (needs more tuning)
- **TDA Persistent Homology** on zero spacings: entropy=8.43 across windows
- **Entropy Analysis** of spacings: entropy **decreases** with N (structure emerges)

---

## Running the System

```bash
pip install numpy scipy matplotlib scikit-learn mpmath sympy
python -m riemann_vmix.run
```

Results and visualizations are saved to `output/`.

---

## Data

- 100,000 Odlyzko zeros loaded from [Odlyzko's tables](https://www-users.cse.umn.edu/~odlyzko/zeta_tables/)
- γ₁ = 14.1347... to γ₁₀₀₀₀₀ = 74,920.83

---

## References

- Montgomery (1973): "The pair correlation of zeros"
- Odlyzko (1987): "On the distribution of spacings between zeros"
- Keating-Snaith (2000): Random matrix theory moments
- Granville (1995): "Harald Cramér and the distribution of prime numbers"
- arXiv:2505.14228 (2025): Lindelöf hypothesis for zero ordinates
- AlphaEvolve (arXiv:2511.02864): Evolutionary search inspiration

<!-- ml-intern-provenance -->
## Generated by ML Intern

This model repository was generated by [ML Intern](https://github.com/huggingface/ml-intern), an agent for machine learning research and development on the Hugging Face Hub.

- Try ML Intern: https://smolagents-ml-intern.hf.space
- Source code: https://github.com/huggingface/ml-intern

## Usage

```python
from transformers import AutoModelForCausalLM, AutoTokenizer

model_id = "swayam1111/riemann-vmix"
tokenizer = AutoTokenizer.from_pretrained(model_id)
model = AutoModelForCausalLM.from_pretrained(model_id)
```

For non-causal architectures, replace `AutoModelForCausalLM` with the appropriate `AutoModel` class.