This repository contains the finite-dimensional numerical supplement for the theorem paper "Leakage-Induced Emergence in Projected Linear and Nonlinear Cognitive Fields. " The provided Python package implements reproducible, projected linear and nonlinear dynamics to illustrate the mechanisms of emergence in finite-dimensional complex vector spaces. It is designed to generate paper-ready figures and provide transparent numerical diagnostics that support the infinite-dimensional operator-theoretic proofs presented in the main manuscript. Theoretical Context: The accompanying mathematical framework studies an emergence problem for projected dynamical systems on Hilbert space. Given a complex state space and a bounded observation map, the system decomposes into a hidden ("preconscious") sector and an orthogonal observable sector. The central structural object of this theory is the "leakage operator, " which specifically measures how hidden states are transported into the observable sector over time. What the Code Illustrates: The software package provides controlled, finite-dimensional analogues of the paper's core theorems. It includes automated scripts to reproduce the following phenomena: Exact Hiddenness: Demonstrates that if the leakage operator is zero, trajectories starting in the hidden sector remain completely hidden. First-Order Emergence: Validates the analytical prediction that observable amplitude emerges linearly over time when a hidden initial state is actively coupled to the observable sector. Duhamel Reconstruction: Verifies that direct observable dynamics precisely match the variation-of-constants integral formula. Thresholded Observation: Computes the exact first crossing time for a given observation norm threshold. Spectral & Resolvent Leakage: Tests hidden eigenmodes and evaluates the leakage resolvent norm to detect hidden-to-observable mixing in the frequency domain. Energy Onset Scaling: Confirms the distinct early-time scaling laws where observable amplitude scales linearly with time, while observable-sector energy scales quadratically with time. Nonlinear Dynamics: Integrates cubic nonlinearities using SciPy to contrast hiddenness-preserving nonlinearities against leak-inducing ones. Reproducibility and Usage: The package (leakage-emergence) requires Python 3. 11+ and relies purely on standard scientific libraries (numpy, scipy, matplotlib, and pytest). It contains straightforward execution scripts (runₗinearₑxamples. py, runₙonlinearₑxamples. py, and generatefigures. py) that instantly regenerate the exact numerical validations, metrics, and publication-ready PDF and PNG figures used in the manuscript.
Prithvidev Kamboj (Wed,) studied this question.