Stellar-evolution models leave little room for doubt that the Sun of four billion years ago radiated roughly seventy percent of its present output; a star fuses hydrogen into denser helium, contracts, and brightens as it ages. Yet the geological record of that same era is emphatic that Earth's surface was not frozen. Sedimentary rocks that can form only in liquid water, and microfossils of organisms that required it, are found in strata older than three billion years. A planet receiving so much less warmth than today should, by the simplest energy balance, have been locked in ice. This is the faint young Sun paradox, and its resolution is not a matter of deciding whether Earth was warm but of explaining how it could have been. The orthodox answer invokes the greenhouse. If the early atmosphere held far more carbon dioxide, or a persistent haze of methane, than it does now, the surface could have retained enough heat to stay above freezing despite the dim Sun. The appeal of the hypothesis is that it requires no exotic physics: it asks only that the atmosphere's composition differed, which independent evidence makes likely. Yet the quantitative demands are severe. To offset a thirty-percent shortfall in sunlight, carbon dioxide concentrations would have had to exceed present levels by orders of magnitude - and here the record resists. Paleosols, ancient soils whose mineralogy is sensitive to the carbon dioxide they formed under, imply concentrations too low, by some estimates, to close the gap on their own. Confronted with this shortfall, some researchers have not abandoned the greenhouse but supplemented it. A weaker Sun warms a darker planet more efficiently, they observe, and the early Earth may have been darker: fewer continents to reflect light, and, crucially, fewer of the microscopic marine organisms whose byproducts seed the bright, reflective clouds that now cool the planet. On this account no single factor need do all the work; a modest greenhouse, a lower albedo, and perhaps other contributions together suffice. Critics counter that this pluralism, however reasonable, purchases explanatory adequacy at the price of testability, since a hypothesis assembled from several adjustable factors can accommodate almost any reconstruction of the early surface. The dispute is instructive precisely because the paradox admits no escape through the front door. One cannot lower the estimate of the young Sun's dimness; the stellar physics is among the best-constrained in the science. Nor can one dismiss the evidence for early liquid water without discarding rocks whose origin is not seriously contested. The pressure of the paradox therefore falls entirely on the atmosphere and surface - the one part of the system for which the record is fragmentary enough to permit competing reconstructions. That the debate has not been settled reflects less a failure of ingenuity than the stubborn incompleteness of the archive: where the evidence is thinnest, the hypotheses are most numerous, and the paradox endures not because no answer will serve but because too many might.