{"slug":"convergence-c02","verification":{"valid":false,"broken_at":0,"reason":"prev mismatch"},"count":5,"sources":[{"id":"s1","type":"primary","url":"https://plato.stanford.edu/entries/least-action/","title":"Principle of Least Action","quote":"","summary":"Stanford Encyclopedia of Philosophy entry on the principle of least action, covering its history, formulation, and philosophical significance.","claim_ids":["c1","c6"],"quality_score":0.95},{"id":"s2","type":"adjacent","url":"https://en.wikipedia.org/wiki/Noether%27s_theorem","title":"Noether's theorem","quote":"","summary":"Noether's theorem links symmetries of the action to conserved quantities, providing the mathematical bridge between variational principles and conservation laws.","claim_ids":["c1"],"quality_score":0.9},{"id":"s3","type":"adjacent","url":"https://en.wikipedia.org/wiki/Path_integral_formulation","title":"Path integral formulation","quote":"","summary":"Feynman's path integral formulation sums all possible paths in quantum mechanics, with the classical least-action path emerging as the dominant contribution in the limit of large action.","claim_ids":["c3"],"quality_score":0.92},{"id":"s4","type":"rival","url":"https://arxiv.org/abs/quant-ph/0101082","title":"The Inverse Problem of the Calculus of Variations","quote":"","summary":"Mathematical analysis showing that any smooth differential equation can be derived from a Lagrangian, challenging the depth of the variational principle by suggesting it may be a formal dressing rather than a fundamental truth.","claim_ids":["c5"],"quality_score":0.85},{"id":"s5","type":"adjacent","url":"https://en.wikipedia.org/wiki/Natural_selection","title":"Natural selection","quote":"","summary":"Overview of natural selection as the mechanism of adaptation; the formal mapping to a variational optimization process is debated and not universally accepted.","claim_ids":["c4"],"quality_score":0.75}]}