Polarity as a Structural Necessity: Lagrangian-Hamiltonian Duality, the Cartesian Blind Spot, the Antimatter Problem and the Mexican Hat Potential

We demonstrate that polarity, the existence of two complementary, discrete states, is not an empirical observation but a structural necessity, directly derivable from the algebraic relationship between the Lagrangian L=T +V and the Hamiltonian H=T-V of any physical system. Introducing a weight function F such that F.H = L immediately yields T(F-1)=V(F+1), a polarity equation, whose solutions form a Mobius transformation with two fixed points. For F=a+b x this generates the quartic Mexican Hat potential V(φ)=-μ 2 φ 2 +λφ 4 without any model assumption. The BEH field of the Standard Model is therefore a direct consequence of the L/H duality already present in classical mechanics since Kepler. Boltzmann's almost forgotten quartic correction of 1874 to Biot's law of optical rotation, derived by least squares, is the same mathematical step applied empirically to chirality. Descartes' actual error was not his choice of handedness (his quadrant sequence I, II, III, IV is itself left-handed) but the ontological exclusion of the negative square root as a second physical state. Spectral alnalysis connects these threads to the Kratzer potential, the Lyman series, and natural antihydrogen states.