Supplementary Material
======================
Manuscript: Time and Gravity from a Layered Phase Substrate
Author:     Ryosuke Sano

This archive contains the simulation code used to generate Figures 1
and 2 of the manuscript.

Files
-----
fig1_simulation.py   Generates Figure 1: emergence of macroscopic
                     time on the K_N-coupled substrate, with
                     verification of the sqrt(N) stabilisation.

fig2_simulation.py   Generates Figure 2: gravitational time dilation
                     on a 2D substrate lattice with a localised mass.

Requirements
------------
Python 3.8 or later, with:
    numpy
    matplotlib

Both packages are available via pip:
    pip install numpy matplotlib

How to run
----------
From a terminal:
    python3 fig1_simulation.py
    python3 fig2_simulation.py

Each script runs in approximately 30 seconds to 5 minutes on a
standard laptop. Output figures are saved as .png and .pdf files
in the working directory.

Reproducibility
---------------
Both scripts use fixed random seeds. Re-running the scripts with
no modification reproduces the figures of the manuscript exactly,
including the fitted exponent (-0.487 for the sqrt(N) test) and
the chi-squared values (2.5 vs 47 for the gravitational test).