= Solution
The first argument is the energy budget. The gravitational energy reservoir is of order $GM^2/R$, so the <Kelvin-Helmholtz cooling time> is $t_{\rm KH}\sim GM^2/(RL)$. For the <Sun>, $M\simeq2\times10^{30}\,\mathrm{kg}$, $R\simeq7\times10^8\,\mathrm m$ and $L\simeq3.8\times10^{26}\,\mathrm W$ give $t_{\rm KH}\sim3\times10^7$ years. This is much shorter than the approximately $4.6\times10^9$-year solar-system age. Chemical binding energies are only of order electronvolts per atom and provide an even smaller reservoir. Thus neither ordinary chemical energy nor continuing gravitational contraction can supply the long-term solar <luminosity>.
<Stellar nuclear fusion> provides an adequate reservoir. Conversion of four <hydrogen> atoms into one <helium-4> atom releases about $26.7\,\mathrm{MeV}$, corresponding to a mass defect of about $0.007$ of the initial mass; some energy leaves as <neutrinos>. If a fraction $f$ of stellar mass with hydrogen fraction $X$ is available for core burning, the energy scale is $0.007fXMc^2$. Taking $f\sim0.1$ and $X\sim0.7$ supplies roughly $9\times10^{43}\,\mathrm J$, enough for several billion years at the solar <luminosity>, consistent with a <stellar nuclear timescale> of order $10^{10}$ years. The core temperature and density inferred from stellar structure permit the <proton–proton chain>; hotter stellar cores favor the <CNO cycle>.
The most direct evidence is the detection of <solar neutrinos>. They are created in the weak reactions of the <proton–proton chain> and can escape from the core, unlike the <photons> whose transport through the interior is slow. Measurements accounting for <neutrino oscillation> recover a total neutrino flux consistent with nuclear-powered solar models. Their spectra and reaction-dependent fluxes test the core reactions rather than merely the surface <effective temperature>.
There is also independent evidence from <stellar evolution>: the <main sequence> is a long-lived core-hydrogen-burning phase, more massive luminous stars consume fuel more quickly, and the <main-sequence turnoff> of a star cluster records its age. Subsequent evolution and the production of heavier elements fit further <nuclear reactions>. Agreement among these observations, nuclear energy release and stellar-model lifetimes is much stronger than the energy-budget argument alone. \b[The combination of a sufficient nuclear reservoir, measured core neutrinos and the observed evolutionary sequence identifies nuclear reactions as the main long-term power source of ordinary stars.]
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