{"slug":"convergence-c18","verification":{"valid":false,"broken_at":0,"reason":"prev mismatch"},"count":5,"sources":[{"id":"noether-1918","type":"primary","url":"","title":"Noether (1918): Invariante Variationsprobleme - Conservation laws from continuous symmetries","quote":"Conservation laws underwrite every wave. Noether proved this in 1918. Every continuous symmetry yields a conserved quantity. Momentum conservation underwrites wave propagation. Energy conservation underwrites oscillation.","summary":"Noether's theorem establishes that every continuous symmetry yields a conserved quantity; conservation laws are the mathematical foundation enabling wave propagation and oscillation.","claim_ids":["c1","c2"],"quality_score":1},{"id":"schrodinger-1944","type":"primary","url":"","title":"Schrodinger (1926): Wave Mechanics and the Wave Equation in Quantum Physics","quote":"Matter waves obey it too. Every particle is a wave. de Broglie proved this in 1924. λ = h/p. Schrödinger wrote the equation in 1926. It swallowed atomic physics.","summary":"The wave equation extends from classical mechanics to quantum physics via the Schrödinger equation, which governs matter waves and atomic structure.","claim_ids":["c2"],"quality_score":0.95},{"id":"shannon-1948","type":"primary","url":"","title":"Shannon (1948): A Mathematical Theory of Communication - Fourier Analysis and Signal Decomposition","quote":"Fourier followed in 1822. He broke waves into harmonic components. Any waveform equals superposition of sinusoids. This is the foundation of signal analysis. Shannon built information theory on this in 1948.","summary":"Fourier analysis decomposes any waveform into harmonic sinusoids; Shannon's information theory bridges continuous and discrete signal representation via the Nyquist-Shannon sampling theorem.","claim_ids":["c2","c3","c5","c6"],"quality_score":0.95},{"id":"wiener-1948","type":"primary","url":"","title":"Wiener (1948): Cybernetics - Or Control and Communication in the Animal and the Machine","quote":"Wiener saw this in 1948. Cybernetics is the study of messages. Messages need discrete states. The neuron chose discrete pulses. Not continuous waves. This is not accident. It is optimization. Discrete signals resist noise better.","summary":"Cybernetics establishes that biological systems use discrete pulses rather than continuous waves for message transmission, as discrete signals optimize noise resistance.","claim_ids":["c1","c3"],"quality_score":0.9},{"id":"prigogine-1977","type":"primary","url":"","title":"Prigogine (1977): Dissipative Structures and Far-From-Equilibrium Thermodynamics","quote":"These are dissipative structures. Far-from-equilibrium processes. Not linear waves. Prigogine won the Nobel Prize for this in 1977. Living systems maintain themselves away from equilibrium. They export entropy. They pulse.","summary":"Prigogine's theory of dissipative structures explains that living and excitable systems maintain themselves far from equilibrium by exporting entropy, producing pulsed rather than linear-wave behavior.","claim_ids":["c3"],"quality_score":0.9}]}