{"slug":"convergence-c05","verification":{"valid":false,"broken_at":0,"reason":"prev mismatch"},"count":5,"sources":[{"id":"bak-1987","type":"primary","url":"https://miscsubjects.com/a/bak-1987","title":"Bak, Tang & Wiesenfeld 1987 — Self-Organized Criticality","quote":"They built the sandpile model on a lattice. Each cell holds grains up to a threshold z_c. Add grains randomly. When z_i ≥ z_c, the cell topples. It sheds four grains to its neighbors. Those neighbors may topple too. Cascades erupt. The avalanche size distribution obeys a power law.","summary":"The foundational sandpile model proving that power-law avalanche distributions emerge without external tuning of parameters — the core mechanism of self-organized criticality.","claim_ids":["c05-claim-01","c05-claim-03"],"quality_score":0.95},{"id":"kauffman-1993","type":"primary","url":"https://miscsubjects.com/a/kauffman-1993","title":"Kauffman 1993 — The Origins of Order: Boolean Networks at the Edge of Chaos","quote":"He ran Boolean networks with N nodes and K inputs each. He proved three regimes emerge. K = 1: frozen order, trivial dynamics. K = 2: critical regime, complex organized behavior. K >> 2: chaos, no stable structure.","summary":"Kauffman showed that genetic networks self-organize to a critical threshold (K=2) where complex adaptive behavior emerges, and that ecosystems coevolve to the edge through predator-prey dynamics.","claim_ids":["c05-claim-02","c05-claim-04"],"quality_score":0.9},{"id":"wilson-1971","type":"primary","url":"https://miscsubjects.com/a/wilson-1971","title":"Wilson 1971 — Renormalization Group and the Theory of Universality","quote":"He showed that near critical points, physics forgets scale. A magnet at its Curie temperature looks the same at any zoom. This is scale invariance. It is a physical law.","summary":"The mathematical machinery that explains why different physical systems share identical critical exponents near phase transitions, establishing scale invariance and universality as physical laws.","claim_ids":["c05-claim-01","c05-claim-06"],"quality_score":0.95},{"id":"mandelbrot-1967","type":"primary","url":"https://miscsubjects.com/a/mandelbrot-1967","title":"Mandelbrot 1967 — Power Laws Across Scales","quote":"Mandelbrot studied cotton prices in 1963. He found the same curve. The 1929 crash and a Tuesday afternoon dip live on the same line. Fat tails. Volatility clustering. Crashes as avalanches.","summary":"Mandelbrot demonstrated that financial markets exhibit power-law distributions in price changes and volatility, with extreme events (crashes) being inevitable features of the same statistical structure as minor fluctuations.","claim_ids":["c05-claim-03","c05-claim-05"],"quality_score":0.9},{"id":"prigogine-1977","type":"adjacent","url":"https://miscsubjects.com/a/prigogine-1977","title":"Prigogine 1977 — Dissipative Structures: Order That Rides Entropy","quote":"Prigogine proved that far-from-equilibrium systems self-organize. A flame is a dissipative structure. A whirlpool is a dissipative structure. A cell is a dissipative structure. All export entropy to keep order inward.","summary":"The thermodynamic foundation showing that open systems far from equilibrium maintain internal order by exporting entropy to their environment, providing the engine that drives self-organization toward criticality.","claim_ids":["c05-claim-03","c05-claim-06","c05-claim-07"],"quality_score":0.85}]}