1) In the meantime, I found explicit solutions to the Carnot engine for real gas equations which demonstrate that Evgenij Rudnyi's comments on the other discussion were right about the value of the efficiency:

https://ajouronline.com/index.php/AJAS/article/view/6572

However, this only requires the first law of thermodynamics as e.g. stated below equation 12 in the paper.

2) You are still wrong about what the second law says. It is not an equation. I will just copy-paste:

Let us go back to the original sources of what the second law says.

First one is Clausius, his original words are:

"Es gibt keine Zustandsänderung, deren einziges Ergebnis die Übertragung von Wärme von einem Körper niederer auf einen Körper höherer Temperatur ist."

Translation: There is no change of states whose sole result is the transfer of warmth from one entity of low to an entity of high temperature.

Kelvin:

"It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce only a net amount of work."

Planck suggested to refer to such a non-existing device as a perpetuum mobile of second order.

Concluding from this in a strict manner, the second law only has failed if a transfer of warmth from a low-temperature to a high-temperature entity has been observed experimentally or if a device producing only work has been built - not simulated, built. But let's expand this to a less narrow-minded scale:

Neither of these formulations provide an equation of any kind, both of them provide process ranges and ranges in mathematics are specified by inequations, therefore any mathematical reformulation of the second law must also be an inequation. An expression equating a quantity to another, e.g. an efficiency to an expression, may be a threshold, but not a reformulation of the law.

A straightforward related statement is Clausius' theorem of entropy increase:

"Bei jedem natürlichen Vorgang nimmt die Entropie zu."

Translation: in any natural process, entropy will increase.

Mathematically spoken, we have dS≥0.

Transferring this to the Carnot cycle, the ideal cycle has dS=0 and therefore the ideal Carnot cycle constitutes a theoretical optimum of efficiency. Therefore, from the second law we can conclude that no other cycle will beat the efficiency of Carnot.

The two major expressions of the second law of thermodynamics in textbooks do not contain quantitative predictions. The lack of quantitative prediction is a major flaw. The quantification of the second law of thermodynamics is as follows,

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Conclusion of the 2nd law of thermodynamics is that the Carnot efficiency of carbon dioxide, water vapor, liquid water, solid water... is 1-T2/T1. Do you believe it?

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and the two are linked together through a very dirty logic. These are included in thermodynamics textbooks.

You have been given the original formulations by the original authors of the second law and a logical deduction above. If your textbooks convert this into an odd quantitative statement, their authors or translators did a bad job.

The actual second law is, as shown in my previous post, a pretty basic statement, so what you are fighting against has actually never been the second law. i get that after all the time you wasted on this that is hard to grasp.

The quantified threshold (not prediction) of the second law with respect to the Carnot cycle is:

"Es gibt keine Wärmekraftmaschine, die bei gegebenen mittleren Temperaturen der Wärmezufuhr und Wärmeabfuhr einen höheren Wirkungsgrad hat als den aus diesen Temperaturen gebildeten Carnot-Wirkungsgrad."

There is no heat engine, which at given mean temperatures of of heat input and output provides a higher efficiency than the Carnot efficiency calculated from these temperatures.

You can infer from this, that an ideal reversible heat engine, thus a DeltaS=0 process, would have the Carnot efficiency, but that is a gedankenexperiment based on the second law, not a reformulation of it. The actual calculation of the efficiency comes from the first law.

You're just a bulldozer pushing a pile of text over, and I don't want to reply in detail. Those who understand, I understand as soon as I step out.

You have forgotten that I pointed out the misunderstanding of the textbook to you. In fact, there are many people like you in our country.