= Solution
Two major suppressions are missing from the one-femtometre rectangular estimate. First, the <Coulomb barrier> extends far beyond nuclear distances and is not rectangular. A semiclassical <integral> through it produces the <Gamow factor> $\exp(-2\pi\eta)$, with $\eta=\alpha_{\rm fs}\sqrt{\mu c^2/(2E)}$; at the illustrative thermal <energy> it is already of order $10^{-11}$. A thermal average additionally weights the high-energy tail and is not the same as evaluating a single thermal <energy>.
Second, two <protons> cannot simply form a stable bound diproton. The <proton–proton chain> starts with a weak-interaction conversion producing a <deuteron>, a <Positron> and a <neutrino>. The weak transition probability supplies a further very large suppression. Thus the quoted tiny fusion probability is not a pure one-dimensional barrier transmission probability with universal energy-independent meaning.
The Sun contains of order $10^{57}$ nucleons in a hot dense core and offers enormously many encounters over astronomical <times>. A very small reaction probability per encounter can therefore supply its <luminosity> while consuming only a small fraction of the fuel per unit <time>. The small weak reaction rate is also crucial for the long stellar lifetime; a larger core <temperature> increases burning and triggers expansion and cooling.
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