Saelarien/saelarien-constraint-experiment-01-entropy-capacity-collapse

README — Saelariën Constraint Experiment 01

Entropy–Capacity Collapse Threshold Test

Author: Saelariën X

Date: February 19, 2026

DOI: https://doi.org/10.5281/zenodo.19212561

Theoretical basis

This dataset empiracally tests the Saelariën Constraint Theorem:

https://thesaelafield.com/preprints/the-saelarien-constraint

Overview

This dataset contains the full materials for Saelariën Constraint Experiment 01, a test exploring how increasing entropy (noise) affects the stability, coherence, and collapse threshold of a simple neural system.

The experiment trains a small neural network on a nonlinear function and injects different noise levels to measure when learning remains stable versus when representational collapse occurs.

The results show a consistent threshold:

systems only collapse once injected entropy exceeds internal interpretive capacity.

Files Included

1\. saelarien\_constraint.ipynb

A full, runnable Colab notebook containing:

• —model definition

• —training loop

• —noise injection

• —entropy–capacity tests

• —plotting code

• —export of raw results

Running the notebook reproduces the figure and the JSON results file.

2\. saelarien\_constraint\_results.json

A structured dictionary containing loss curves for each noise level.

Format example:

{ "0.0": \[...\], "0.1": \[...\], "0.2": \[...\], "0.4": \[...\], "0.6": \[...\], "0.8": \[...\], "1.0": \[...\] }

This file allows independent verification, re-plotting, and secondary analysis.

3\. Figure\_1\_Saelariën\_constraint.png

A plot titled:

“Saelariën Constraint Test: Entropy vs Collapse”

The figure shows:

• —Smooth convergence at low noise

• —Degradation at medium noise

• —Collapse at high noise

This visual is the primary supporting evidence of the collapse threshold.

Summary of Findings

A simple neural network (1–8–1 architecture) is trained on y \= x².

Noise injection reveals three learning regimes:

• —Stable coherence (0.0–0.2 noise): normal convergence

• —Critical instability (0.4 noise): oscillations but not collapse

• —Collapse (0.6+ noise): divergence, stagnation, or chaotic loss

These results align with the theoretical Saelariën Constraint:

collapse emerges only when entropy exceeds the system’s interpretive capacity.

How to Replicate

1. 1.Open the notebook in Google Colab.

1. 1.Run all cells.

1. 1.Inspect:

• —the plot

• —the all\_losses dictionary

• —the behavior of the system at each noise level

Dependencies: PyTorch, Matplotlib, JSON (standard library).