The Variational Principle of Persistence: Mathematical Foundations and Computational Validation in Biological, Physical, and Topological Systems

This repository accompanies the manuscript “Variational Principle of Persistence (VPP)”, which proposes a structural framework for persistence and viability in systems operating under uncertainty. The core idea is that increasing structural closure/control can provide protective benefits but also introduces accelerating maintenance and coordination costs, yielding an interior optimum and a predictable “over-closure” failure regime called the Systemic Reduction Paradox (SRP).

The manuscript is validated computationally across three disjoint domains:

1. Bacterial thermal performance, using the Sharpe–Schoolfield model with cross-species prediction and survival simulations under stochastic thermal stress;

2. Phase transitions via the 2D Ising model, where the predicted optimum tracks the critical region under monotone reparameterizations;

3. Complex networks via the Watts–Strogatz model, showing an interior optimum and the SRP pattern when closure is excessive.

This release includes:

• The full manuscript (PDF);

• A reproducible scripts bundle used to generate key figures and statistics (including survival comparisons and confidence intervals);

• Supplementary assets required to rerun the analyses end-to-end.

Reproducibility: unzip the scripts bundle and run the Python scripts as indicated in their headers (requirements and commands are documented in-file). Outputs are saved as figures and summary statistics consistent with the manuscript.

Keywords: variational principle, persistence, viability, systemic reduction paradox, complex systems, Ising model, networks, thermal performance, autopoiesis, robustness.

**Note (Dec 2025):** A corrected PDF (V1) is included with improved figure placement and a minor cross-reference fix. Scientific content unchanged.