# Reproducibility Guide

Author: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki**

## Requirements

Python 3 with:

- numpy
- scipy

No internet access is required for the verification scripts.

## Run all release checks

```bash
python code/run_release_checks.py
```

This performs:

1. exact arithmetic checks of composition counts and saturation-index logic;
2. finite-element spectral checks on several commensurate trees;
3. perturbed-metric checks confirming that the same tested eigenvalue moves above the Polya value;
4. mesh-refinement checks;
5. writing `data/numerical_checks.csv` and `data/release_check_summary.json`.

## Numerical method

`code/fem_metric_tree.py` implements standard piecewise-linear finite elements on each metric edge. It assembles

```text
K_e = (1/h) [[1,-1],[-1,1]]
M_e = (h/6) [[2,1],[1,2]]
```

on every mesh element, identifies common graph vertices, imposes Dirichlet conditions at every degree-one graph vertex, and solves the generalized symmetric eigenproblem

```text
K u = lambda M u.
```

The implementation is intentionally simple and transparent. It is a numerical regression test only.

## Why numerical checks are not proof dependencies

The principal equality theorem is analytic and topological. No finite-element estimate is used in any implication. Numerical checks are included only to detect indexing, normalization, and example-construction mistakes.

## PDF rebuild and visual verification

```bash
pdflatex -interaction=nonstopmode MANUSCRIPT.tex
pdflatex -interaction=nonstopmode MANUSCRIPT.tex
```

The release PDF was rendered after compilation and visually checked for clipping and malformed mathematics.