{"slug":"convergence-c20","verification":{"valid":false,"broken_at":0,"reason":"prev mismatch"},"count":5,"sources":[{"id":"turing-1936","type":"primary","url":"https://miscsubjects.com/a/turing-1936","title":"On Computable Numbers, with an Application to the Entscheidungsproblem (1936)","quote":"","summary":"Turing proved that a single abstract machine can simulate any other computable machine. This is the foundational theorem of computer science. The halting problem is undecidable. The Church-Turing thesis is introduced as a hypothesis.","claim_ids":["c1","c2","c7","c8"],"quality_score":1},{"id":"godel-1931","type":"primary","url":"https://miscsubjects.com/a/godel-1931","title":"On Formally Undecidable Propositions of Principia Mathematica and Related Systems (1931)","quote":"","summary":"Gödel proved that any formal system powerful enough to express arithmetic is either incomplete or inconsistent. Truth exceeds proof. This sets the logical limit within which computation operates.","claim_ids":[],"quality_score":1},{"id":"shannon-1948","type":"primary","url":"https://miscsubjects.com/a/shannon-1948","title":"A Mathematical Theory of Communication (1948)","quote":"","summary":"Shannon unified communication, cryptography, and computation in one framework. The bit is the unit of information. Information has a rate and a limit. The same math governs telegraphs, genes, and neurons.","claim_ids":["c5"],"quality_score":1},{"id":"landauer-1961","type":"primary","url":"https://miscsubjects.com/a/landauer-1961","title":"Irreversibility and Heat Generation in the Computing Process (1961)","quote":"","summary":"Landauer proved that erasing one bit of information requires at least kT ln 2 of energy. Computation is physical. Information is physical. There is no free computation.","claim_ids":["c3","c5"],"quality_score":0.95},{"id":"prigogine-1977","type":"primary","url":"https://miscsubjects.com/a/prigogine-1977","title":"Self-Organization in Nonequilibrium Systems (1977)","quote":"","summary":"Prigogine showed that open systems far from equilibrium spontaneously organize into dissipative structures. These structures maintain order by exporting entropy. Observed in chemistry, biology, and meteorology.","claim_ids":["c4","c5"],"quality_score":0.9}]}