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HB∣b⟩=Eb∣b⟩H_B|b\rangle = E_b|b\rangle

Why this formula appears here

Two limits inside this one formula answer different questions about how confidently the ledger can be interpreted. First, an uncorrelated, non-interacting bath: set HSBH_{SB} = 0 and let ρ\rho = ρS\rho_S ⊗\otimes |b⟩\rangle⟨\langle b| for any system state ρS\rho_S and any bath eigenstate |b⟩\rangle with HBH_B|b⟩\rangle = EbE_b|b⟩\rangle . Then Δ\Delta EΛ(ρ)E_\Lambda(\rho) = EbE_b for every choice of ρS\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…

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HB∣b⟩=Eb∣b⟩H_B|b\rangle = E_b|b\rangle

Equation 68 · Evolutionary Physics

The Entry a Relabeling Cannot Write

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Two limits inside this one formula answer different questions about how confidently the ledger can be interpreted. First, an uncorrelated, non-interacting bath: set HSBH_{SB} = 0 and let ρ\rho = ρS\rho_S ⊗\otimes |b⟩\rangle⟨\langle b| for any system state ρS\rho_S and any bath eigenstate |b⟩\rangle with HBH_B|b⟩\rangle = EbE_b|b⟩\rangle . Then Δ\Delta EΛ(ρ)E_\Lambda(\rho) = EbE_b for every choice of ρS\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…

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