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This is clearly wrong. The bright side hasn't an uniform temperature T.
You’re right, the 2^(-3/4) (and the 2^1/4) is probably quantitatively wrong (unless each side is perfectly heat-conducing but both are isolated from each other. Or if the planet is a coin facing the sun. You know, spherical cows in a vacuum…). But I don’t think that changes the qualitative conclusion, which hold as long as the bright side is hotter but not twice as hotter than the perfectly-heat-conducing planet.