Subsurface lunar temp is constant -21 C at approx 1 meter depth, so our heat sink is... The Moon

It doesn't just have to be cold, it has to conduct several kilowatts of heat away continuously. The heat conductivity at that depth is less than 0.01 W/m/K (Xiao et al., 2022). Back of the envelope, that says 1 m^2 cooling surface with a temperature differential of 100K will dump 1 W of heat. 1/2

(That's assuming moving the heat 1 m away from the heat exchanger is sufficient, which is probably true, given the constant temperature at a depth of 1 m.). Stefan-Boltzmann radiation into space might be more efficient. But during the lunar day, scattered sunlight would probably kill you. 2/2

This is one huge advantage of solar panels; they're not heat engines.

I want you to spec out the weight on a battery that provides 40kW of power for 14 days of lunar night, and then spec out the mass required to build a few kilometers of moon-crete terracotta style piping to pump chill water through

And the equipment to bury kilometers of piping 1 m underground, in vacuo? Terracotta? You do know that’s porous? Gonna take a lot of water. Good think there are all those lunar seas (LOL). But if we’re talking 40 kW, that’s (optimistically) 100 kW of heat to be dissipated. Never mind.

I really mean, ‘never mind’. I’m not going to get into a discussion of the carbon footprint of the tooth fairy. I can’t see a practical way of dissipating the heat from a 1 kW nuclear reactor on the moon, and you’re talking about terracotta pipes. FFS.

Melted regolith wouldn't be terracotta, it would be melted regolith But it would kind of look like terracotta And to answer the question you avoided, a battery large enough for 14 days at 40kW would be about 1000 Powerwalls, weighing in at about 150 tons - assuming no self discharge

You could pack a lot of low pressure piping and shovels with a 290,000 lbs weight limit!

145 tons. And no source of water, beyond a few mL someone claims they found in a crater. Water that evaporates instantly in the lunar vacuum.

Oh, BTW, I got 40 tons, less if you assemble the batteries in situ. Not that any of this makes sense.