You are the one that brought up the water cycle. if you want a cycle, then there has to be a cycle. You cannot go from saying there's a cycle, to saying there's no change of state thus no opportunity for anything remotely like the water cycle.
For water there is a water cycle, where the water evaporates and condenses.
For rock there is kind of a cycle like that, but the sun has nothing to do with it.
Further, you do not get to just boldly claim that the Earth wouldn't melt on RE timescales.
Good thing I didn't and instead I explained why.
On the other hand you just boldly claimed that it would melt, with no rational backing.
That is the whole discussion. The perpetual heating of both the Sun and core under RET ought to provide more than eneough energy on the vast expanse of time you rely on.
Sure, if that was all that was happening. But it isn't.
Instead Earth is also radiating heat out.
The sun and core need to provide more energy than Earth dissipates. But as the temperature rises the heat output of Earth also rises.
From a simple point of view, in order to double the temperature of Earth you would need to put in 16 times as much POWER. Not energy, POWER.
You can't just dump in the energy and expect Earth to keep it, you need to be providing 16 times the energy in a a given time (i.e. power) to match Earth outputting 16 times the energy in that same time.
Puddles indicate that this is possible. You can bring up the water cycle all you want, but where is this analogous cycles? You admit rocks don't fall from the sky.
No they don't.
Again, refer to my analogy with a pot on a stove.
I can put water in that pot and have the water boil away. I can put more water in and have it boil away. I can keep on doing this, for as long as I want, so I can just keep on providing energy. Yet no matter how long I leave it for, the iron (it is simpler to analyse than steel which has numerous compositions) pot wont melt.
Iron has a heat capacity of 460 J/kg K.
It has a latent heat of fusion of 272 kJ/kg.
It has a melting point of 1538 C.
So that means, going from 20 C up to its melting point would be 1518 K.
So if I take a steel pot that weighs 1 kg, it would take (1518 * 460 + 272000) J to melt.
That is 970 kJ.
What about water?
I'm not going to bother with the heating part, that will take more energy but I don't care, the vast majority of the energy is in boiling it.
Water has a latent heat of vaporisation of 2257 kJ/kg.
That means boiling 1 kg of water will take 2257 kJ of energy
So just boiling 1 kg of water, not even heating it up, instead starting with it at 100 C, takes over twice the energy that it takes to heat up and melt steel.
I can easily boil over 1 kg of water in an hour. Yet if I leave a 1kg steel pot on the stove for that long, it comes no where near to melting.
This alone indicates there is far more too it.
Do you know what it is? The fact that things will lose energy, they will radiate it away (or otherwise dissipate it).
In the case of the pot on the stove, the dominant factor is the heat lost to the air, which is proportional to the temperature difference.
So a pot at the same temperature as the room wont lose energy to the room.
Water at 100 C, or roughly 80 C hotter than the room will lose some significant energy (especially if it is in a steel pot)
The same steel pot at 820 C will be losing 10 times the energy and thus need 10 times as much power as simmering water to merely keep it at 820 C.
So no, puddles evaporating doesn't mean rocks should melt, just like boiling water on a stove doesn't mean the steel pots on the stove should melt.
It can radiate more than it takes in.
Ludicrous, plain and simple.
WHY?
It has energy, why can't it radiate it out?
If I take a glass of 100 C water, and put it in a cold room, is it incapable of cooling down because it isn't taking in energy and thus can't put out energy?
What if I mix sodium and water together so I don't even need to put in energy to heat the water up and instead it heats itself (akin to heat from the core), is the sodium water mix now incapable of cooling?
No.
It is your claim that it must radiate less than it takes in which is ludicrous.
Erosion does not change the state of anything, it turns a larger solid into smaller solids. Plate tectonics also have no connection to state, and vulcanism moves a liquid from one location to another, again not changing states. None of that is what you claim.
Actually, plate tectonics and volcanoes do result in state changes.
Subduction results in the rock going deeper into Earth and melting to a liquid. This liquid can come out of volcanoes and solidify.