= Solution
Nuclear burning explains both the large available energy and its release under stellar-core conditions. Converting hydrogen into helium releases approximately $0.007$ of the fuel rest-mass energy. The net <proton–proton chain> reaction can be written $4p+2e^-\to{}^4\mathrm{He}+2\nu_e$, with about $26.7$ MeV available per helium nucleus, including the neutrino energy loss. The <CNO cycle> achieves the same net hydrogen-to-helium conversion with catalysts. Measured nuclear reaction rates and stellar-core temperatures give rates capable of supplying the observed <luminosity>.
The energy budget distinguishes <stellar nuclear fusion> from alternatives. <Kelvin-Helmholtz contraction> has a characteristic <Kelvin-Helmholtz cooling time> $t_{\rm KH}\sim GM^2/(RL)$, only tens of millions of years for a solar-type star. Geological ages and stellar-population ages are much longer. Chemical binding energy per unit mass is smaller still. Nuclear fuel in a substantial fraction of the core gives a <stellar nuclear timescale> of billions of years, explaining long-lived main-sequence stars.
<Main sequence> luminosities, the correlation of lifetime with mass, and cluster turnoffs are reproduced by models in which central hydrogen is consumed and helium accumulates. Spectroscopic abundances and the appearance of helium-, carbon- and heavier-element-rich evolved material agree with sequential nuclear processing and <stellar dredge-up>. These observations support the reaction network and evolutionary consequences, rather than independently measuring every reaction in every star.
<Solar neutrinos> provide direct information from the burning core because they escape without the long radiative diffusion delay of photons. Their measured energies and fluxes test weak reactions in the hydrogen-burning network. The historical deficit of electron neutrinos did not require abandoning nuclear power; flavor conversion changes the measured electron component. Later flavor-sensitive measurements confirmed the total flux expected from solar nuclear burning, as described in https://sno.phy.queensu.ca/first_results/[the SNO collaboration's first-results report]. Thus the evidence combines an adequate energy source, quantitatively successful stellar models and a direct nuclear by-product. Cooling remnants and brief gravitationally powered evolutionary phases are exceptions to the statement that a star's instantaneous luminosity is nuclear.
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