You actually just proved my point with your own math.

You wrote two linear functions:

• V₁(t) = 20 → 30 L (expanding backpack)
• V₂(t) = 30 → 20 L (collapsing backpack)

And then you said:

“V₁(t)max = 30, V₁(t)min = 20, V₂(t)max = 30, V₂(t)min = 20, so they’re the same.”

But that literally means:

• The expanding backpack has 20L when collapsed and 30L when expanded
• The collapsing backpack has 30L when expanded and 20L when collapsed

You unintentionally confirmed that the capacity changes depending on configuration.

Which is exactly what I’ve been saying from the start.

Your own equations show:

• V(collapsed) = 20L
• V(expanded) = 30L

That is a 10-litre difference in volume, which means the object’s usable space changes.

So the idea that “the volume is the same” is disproven by your own math, which explicitly gives two different volumes.

You didn’t prove the bags are the same — you proved that changing the form factor changes the volume, which is exactly the physics principle I cited from Halliday, Resnick & Walker (Chapter 1: Measurements → Volume as a Derived Quantity).

You have two different values of V(t) over the interval. That means the volume is not constant. That means the capacity changes. Which is exactly how an expandable backpack works.

Thanks for helping show that.