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A proof-driven route from simple roots to the finite Dynkin diagrams, using positive-definite Cartan forms, spectral radius, Schur complements, and a determinant recurrence to explain both the infinite families and the exceptional cases.
A single tangency forces a change in what we mean by geometric space, leading from polynomial zero sets to spectra, nilpotents, infinitesimal probes, and families over a base.
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