= Solution
In mixed-valence manganites, mobile $e_g$ electrons hop between sites whose localized $t_{2g}$ core spins are coupled to them by strong <Hund's coupling>. Projecting onto local spin-aligned states gives the <double-exchange mechanism>
$$
\boxed{|t_{ij}^{\rm eff}|=|t|\cos(\theta_{ij}/2).}
$$
Alignment maximizes the hopping bandwidth and lowers carrier kinetic energy, favouring <ferromagnetism>. This involves real charge transfer, unlike insulating <superexchange>. Heating disorders core spins and reduces coherent transport; a <magnetic field> aligns spins, increasing mobility and reducing resistivity. The large negative response near the magnetic transition is <colossal magnetoresistance>.
Double exchange supplies the conduction–magnetism link, but its simplest spin-only model is insufficient for the full resistivity change. Lattice distortions, <Jahn-Teller effect> and carrier localization can amplify the effect; materials need not obey a universal resistivity-versus-magnetization law.<Interaction between the d-Shells in the Transition Metals. II. Ferromagnetic Compounds of Manganese with Perovskite Structure by Clarence Zener>[Double exchange and magnetic conduction]{ref}<Double Exchange Alone Does Not Explain the Resistivity of La1-xSrxMnO3 by A. J. Millis and others>[1995 abstract: limitations of the spin-only model and electron–phonon coupling]{ref}
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