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"The bore is going to be just as toasty either way. "
Not necessarily. The same amount of heat is generated, true. But the rate of dissipation relative to heat created will determine net bore temp. Depends on many factors.
"It is possible, however, that the greater amount of metal in the fat barrel will draw heat from the surface of the bore faster than the smaller amount of metal in the thin barrel will."
Yes, exactly!
"But, on the other hand, the thin barrel may shed the heat to the air faster than the fat barrel will."
No. The fat barrel has more surface area (contact area with the ambient air), and thus can dissipate at a faster rate.
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