Thanks for your answer @Chemomecanics & Nullius In Verba. Second, sorry for my english, my phrasing can be imperfect. I think it is my mistake I forgot to mention one very important, but officially accepted fact: only memory erasure requires energy investment, memory write doesn't!

Correct me if I understand wrongly the Second Law of Thermodynamics, this is how I understand it:

"It is impossible to create an engine which working in cycle, results in no other than the extraction of heat energy without at least the same amount of external energy invested first."

The Demon does exactly this: without any energy investment is able to extract heat energy, because memory write has no cost. This is the paradox in short. The solution is, that yes yes, the Demon can extract without any energy investment, but when he needs to erase his memory, he needs to use at least the same amount of energy as he extracted, so he needs to use the energy he extracted to do the erasure. In short, in this way the problem is solved, no net energy is gained for no investment. However let's notice a cyclic energy flow is created either way, without any energy investment, and what prevents me to insert a heat engine to extract some energy, and do dome external useful work with it? Because when the Demon does the memory erasure, uses the energy extracted, so the energy flows through his memory back to the box-system. The Demon's memory becomes a heat source (the hot body) and the box-system becomes the cold body, actually becoming a possible energy-extraction point. Do you observe that the energy-neccessity-to-erase of the Demon is actually redundant?

Here are some presentations about the cycle:

      Box
      / ^
     /   \
    /     \
   /       \ <--- Possible energy extraction point
  /         \
 /           \
V             \

Ext.energy-------->Demon storage

Since here I suppose to write an answer, then my answer would be that Landauer's principle is wrong, memory write and erasure has the same cost, a Demon would be able to do this action. The classical thermodynamic Second Law was formulated, at a time when the atomic composition of the matter was still under heavy debate, many even rejected, and gases (and heat) was considered as "indivisible bodies", to which the original formulation of the Second Law what I wrote above was true. Science moved on, atoms were proven, thermodynamics haven't: physicists still today cling convulsively to the impossibility of extraction of the internal energy of the gases at no cost, because that would be something "too good to be true perpetuum mobile". To cut short, -here I can't explain in short-, to this adds the deep confusion between physical and informational entropy. The reason I opened this question is, if anyone could find a reason, what I haven't thought of, why the Demon's memory on erasure cannot be used as a heat source for energy extraction. I think it was never intended to ... by definition the heat generated on erasure is "waste heat" which supposed to be considered as lost in the environment (to increase back the global entropy), but one can argue why not to use it, since it can be considered coming from a source, the Demon's memory.

Thanks for your answer @Chemomecanics & Nullius In Verba. Second, sorry for my english, my phrasing can be imperfect. I think it is my mistake I forgot to mention one very important, but officially accepted fact: only memory erasure requires energy investment, memory write doesn't!

Correct me if I understand wrongly the Second Law of Thermodynamics, this is how I understand it:

"It is impossible to create an engine which working in cycle, results in no other than the extraction of heat energy without at least the same amount of external energy invested first."

The Demon does exactly this: without any energy investment is able to extract heat energy, because memory write has no cost. This is the paradox in short. The solution is, that yes yes, the Demon can extract without any energy investment, but when he needs to erase his memory, he needs to use at least the same amount of energy as he extracted, so he needs to use the energy he extracted to do the erasure. In short, in this way the problem is solved, no net energy is gained for no investment. However let's notice a cyclic energy flow is created either way, without any energy investment, and what prevents me to insert a heat engine to extract some energy, and do dome external useful work with it? Because when the Demon does the memory erasure, uses the energy extracted, so the energy flows through his memory back to the box-system. The Demon's memory becomes a heat source (the hot body) and the box-system becomes the cold body, actually becoming a possible energy-extraction point. Do you observe that the energy-neccessity-to-erase of the Demon is actually redundant?

Here are some presentations about the cycle:

      Box
      / ^
     /   \
    /     \
   /       \ <--- Possible energy extraction point
  /         \
 /           \
V             \

Ext.energy-------->Demon storage

Since here I suppose to write an answer, then my answer would be that Landauer's principle is wrong, memory write and erasure has the same cost, a Demon would be able to do this action. The classical thermodynamic Second Law was formulated, at a time when the atomic composition of the matter was still under heavy debate, many even rejected, and gases (and heat) was considered as "indivisible bodies", to which the original formulation of the Second Law what I wrote above was true. Science moved on, atoms were proven, thermodynamics haven't: physicists still today cling convulsively to the impossibility of extraction of the internal energy of the gases at no cost, because that would be something "too good to be true perpetuum mobile". To cut short, -here I can't explain in short-, to this adds the deep confusion between physical and informational entropy. The reason I opened this question is, if anyone could find a reason, what I haven't thought of, why the Demon's memory on erasure cannot be used as a heat source for energy extraction. I think it was never intended to ... by definition the heat generated on erasure is "waste heat" which supposed to be considered as lost in the environment (to increase back the global entropy), but one can argue why not to use it, since it can be considered coming from a source, the Demon's memory.