```
# Volume 2: 1D Tick‑Chain Double‑Flip Monte Carlo

This repository implements the Monte Carlo check of context‑flip invariance described in Volume 2 §22 of the Absolute Relativity theory.

---

## Repository Structure

```

vol2-tickchain-doubleflip/
├── README.md
├── LICENSE               # MIT License
├── .gitignore
├── environment.yml       # or requirements.txt
├── docs/
│   └── Section22\_spec.md
├── src/
│   └── tick\_chain.py
├── data/
│   └── init\_r.csv        # (optional empirical seed)
└── results/
├── tick\_chain\_results.csv   # generated after running
└── tick\_chain\_plot.png      # generated after running

````

- **docs/**  
  Contains the plain‑text extract of Volume 2 §22 that defines the state space and flip operators.

- **src/**  
  `tick_chain.py` is the uniform‑sampling Monte Carlo script.

- **data/**  
  `init_r.csv` (optional) for empirical seeding. If absent, the script defaults to uniform sampling.

- **results/**  
  Populated with the CSV and plot output after you run the simulation.

---

## Quick Start

1. **Clone the repo**  
   ```bash
   git clone https://github.com/<your‑org>/vol2-tickchain-doubleflip.git
   cd vol2-tickchain-doubleflip
````

2. **Install dependencies**
   Using Conda:

   ```bash
   conda env create -f environment.yml
   conda activate vol2-tickchain
   ```

   Or with pip:

   ```bash
   pip install -r requirements.txt
   ```

3. **(Optional) Provide empirical seed**
   If you have a real‑world distribution of initial states, place a CSV named `init_r.csv` in the `data/` folder with a single column header `r`.

4. **Run the simulation**

   ```bash
   python src/tick_chain.py
   ```

5. **Inspect results**

   * **CSV** → `results/tick_chain_results.csv`
   * **Plot** → `results/tick_chain_plot.png`

---

## What This Does

* Samples 10 000 initial states $r_0$ uniformly from $\{-N,\dots,+N\}$ (with $N=5$ by default).
* Applies two successive outward flips $F\circ F$ and inward flips $S\circ S$.
* Collapses the result back to $r=0$ via projector $B$.
* Records each intermediate and final state.
* Generates histograms to verify that $\,B(F(F(r_0)))$ and $B(S(S(r_0)))$ are always delta‑peaked at 0.

---

## Future Extensions

* **Empirical Sampling**: un-comment or extend the seed logic to draw from `data/init_r.csv`.
* **Parameter Tweaks**: change `N` (max radius) or `n_trials` in `src/tick_chain.py`.
* **Comparison Plots**: overlay results from uniform vs. empirical sampling.

---

## License

This project is released under the MIT License. See the LICENSE file for details.

```
```
