The Cosmological Constant as a Sectoral Count in Horizon Relative Entropy: A Bridge to the Quantum-Information Derivation of the Einstein Equations

A recent result of Dorau and Much derives the semiclassical Einstein equations from the quantum relative entropy between the vacuum and coherent excitations on a local Rindler horizon, extending Jacobson's thermodynamic argument into rigorous algebraic quantum field theory. In their derivation the cosmological constant appears as an undetermined integration constant, free to take any value. We observe that the framework developed in our earlier work fixes precisely this constant, through the relation Λ = 6/S_dS, in which six is the number of sectors of the G2 symmetry breaking and S_dS is the de Sitter horizon entropy. The two results are complementary rather than competing: the quantum-information derivation supplies the Einstein equations and leaves the cosmological constant open, and our relation supplies the one quantity their method leaves free. We make the relationship precise, state honestly its current status — that their derivation permits, but does not imply, our value — and set out the mechanism that would tighten the connection: the identification of the integer six as a vacuum sectoral count, requiring the horizon-entropy form and the G2 sector structure as two ingredients that a multi-sector vacuum analysis would unify. We are explicit about the units in which the relation holds and about what remains to be proved.