Referring to the time in about 5 billion years when the Sun is supposed to expand into a red giant and engulf Mercury and Venus and possibly Earth (the expansion would probably make Earth uninhabitable in less than 1 billion years) - It's entirely possible that there may not even be a red giant phase for the Sun. This relies on entropy being looked at from another angle - with the apparent randomness in quantum and cosmic processes obeying Chaos theory, in which there's a hidden order behind apparent randomness. Expansion to a Red Giant (and so much else) could then be described with the Information Theory vital to the Internet, mathematics, deep space, etc. In information theory, entropy is defined as a logarithmic measure of the rate of transfer of information. This definition introduces a hidden exactness, removing superficial probability. It suggests it's possible for information to be transmitted to objects, processes, or systems and restore them to a previous state - like refreshing (reloading) a computer screen. Potentially, the Sun could be prevented from becoming a red giant and returned to a previous state in a billion years (or far less) - and repeatedly every few billion years - so Earth could remain habitable permanently.
Entropy is not a measure of the heat transfer between two objects and states. While dS≥dQ/T, this is a result, not the definition. The equation s=k*log(W) is chiseled into Boltzmann's tombstone. W is the number of microstates (or some other vague quantity). Einstein chided Boltzmann because Boltzmann could never prove this relationship nor give a concrete definition of W. The intersection of macro thermodynamics, statistical thermodynamics, and information theory is often misunderstood. Information theory uses many metaphors when describing processes and relationships. It's important to know what is a metaphor and what is not. Every metaphor (or analogy) breaks down at some point. Macroscopic thermodynamics also breaks down at some point (loss of continuum). There is only so much we can do with statistical thermodynamics. Neither Boltzmann nor Einstein were able to completely describe or prove either one of their theories and associated equations, but this doesn't mean we can't learn from and utilize the results productively.
Thanks for that reply - it was really interesting. You might notice that I've updated my Discussion (there's no mention of measuring heat transfer now). I guess I've joined Boltzmann and Einstein, since I can't prove my ideas either. But I've tried to give people who live millions and millions of years in the future something they can learn from as a way of keeping Earth habitable when the Sun starts becoming a red giant. And I've tried to support this idea by the description in my article "Mathematizing the Universe" https://doi.org/10.6084/m9.figshare.9830828.v1
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