Theta–Loop Model – Appendix IV: Numerical Chronology Protection Diagram

This appendix presents a series of numerical experiments for the proposed Θ–loop model of self–consistent time–loops. Building on the analytical structure developed in Appendices I–III, we investigate three questions: (i) how the initial loop mismatch M0M_0M0 scales with paradox strength PPP and loop size kkk; (ii) whether loop stability is an artefact of a particular numerical scheme; and (iii) whether there exists a critical level of environmental noise above which self–consistent loops cease to exist. We find an empirical scaling law M0∝P2k2/3M_0 \propto P^2 k^{2/3}M0∝P2k2/3, strong algorithmic robustness across nine distinct update schemes, and a sharp noise–driven phase transition at an almost universal critical noise level σcrit∼2×10−3\sigma_{\text{crit}} \sim 2\times10^{-3}σcrit∼2×10−3, weakly dependent on loop size and practically independent of paradox strength. These results define a simple numerical „chronology–protection diagram” for the toy model and provide proof–of–concept evidence that the Θ–loop selection mechanism is internally consistent and structurally robust, while remaining agnostic about the existence of physical closed timelike curves in our Universe.