> If you want to make your own definitions of macrostate sure.

Note that I was the first one to propose a particular way to define the macrostate of the system - using energy - trying to understand what you meant. Your reply was "that's not the definition of macrostate. It's just one arbitrary choice you have chosen." And you gave your own arbitrary choice "I'll choose something arbitrary. Temperature as measured by a thermometer".

The point is that they are all arbitrary. (Energy is a conserved quantity for a closed system so it's arguably more natural - but let's say that any choice is equally arbitrary.)

> But no matter your definition of macrostate, this remains true:
> Entropy is independent of knowledge.

Maybe be can agree that [Gibbs] entropy [which is a property of the distribution of microstates corresponding to our description of the system based one some macroscopic properties] is independent of knowledge [other than about that particular macroscopic description].

The discussion started with me trying to understand what did you mean by "low entropy" when you said:

"Let's say those balls all have a random initial velocity at a random direction but all those balls are initially positioned near one corner in the cube. Thus the balls from a position stand point start with low entropy."

We agree that if the box is isolated and we define the state using N,V,E there is one single macrostate and one single value for the entropy - independent of the position of the particles. We agree that we can define thermodynamical systems using other variables and calculate other entropies. We agree that the entropy is not determined by the configuration of the particles alone.

(Even when a thermodynamical system is defined the entropy is not necesarily a function of the microstate when it refers to only a part of the system. The microstate may not fully determine the macroscopic description. If that box is in equilibrium in a heat bath the temperature is fixed and there is some corresponding entropy. But the same microstate can happen for equilibrium systems at different temperatures and therefore with different entropies.)

You said that you're using temperature as measured by a thermometer as the macroscopic property describing the system. And I'm still trying to understand how "higher concentration" means "lower entropy" even in the context of your own arbitrary choice.

Temperature works well as a state variable for a system in thermal equilibrium which has the same temperature everywhere. You mention the level of mercury and the way a thermometer works is by reaching thermal equilibrium between the mercury and the thing being measured. Note that if for some reason all the particles in the gas go away from the thermometer for a second the temperature of the mercury won't change (ignoring that it will radiate energy over time). The reading of the thermometer doesn't drop to zero just because there is nothing there.

Let's assume that you are somehow measuring the local temperature in a small region around some specific point. What you measure will not depend in any way on the particles that are elsewhere. The particles in the rest of the box could be well spread or all near one corner and you would have no reason to say that the latter configuration is lower entropy than the former based on your macrostate.

Another way to look at it: you reasoning seems to be that the state with the particles near one corner has low entropy because they are close to each other and there is some temperature reading in your far-away thermometer lower than the equilibrium temperature and then the entropy is low. If a similar cluster of particles was close to the thermometer rather than far from it I guess that you will tell me that the temperature reading would be higher than in equilibrium (and the entropy would also be lower).

But if you put that cluster of particles at some intermediate position the reading of the thermometer will be the same as in equilibrium! Your macrostate will be the same as in equilibrium. Your entropy will not be lower than in equilibrium. Even though you said -if I understood you correctly- that the entropy should be lower because the higher concentration has a lower probability of occurring.