What does this equation mean?

Two limits inside this one formula answer different questions about how confidently the ledger can be interpreted. First, an uncorrelated, non-interacting bath: set $H_{SB}$ = 0 and let $\rho$ = $\rho_S$ $\otimes$ |b$\rangle$$\langle$ b| for any system state $\rho_S$ and any bath eigenstate |b$\rangle$ with $H_B$|b$\rangle$ = $E_b$|b$\rangle$ . Then $\Delta$ $E_\Lambda(\rho)$ = $E_b$ for every choice of $\rho_S$ whatsoever — a pure constant, entirely independent of what the system is doing, unambiguously assignable to a named, located pile of energy sitting in a part of the world the coarse description agreed in advance not to look at. Nothing is missing in any troubling sense here; the coarse observer is simply declining to keep a receipt for energy that was never claimed to be theirs.