Version of June 2nd, 2026, following the technical development within the Absolute Frame Theory programme. The companion paper of this programme derived a black-hole horizon, in the Absolute Frame Theory (AFT), as the active set of a variational saturation problem rather than from a curvature criterion, and left its Prediction P4 open: the explicit construction of the dual of that problem and the mechanism by which it preserves information during evaporation. This paper supplies that construction along a single thread. Working in the native Euclidean sector of AFT, where the substratum metric is positive definite and the Lorentzian signature of spacetime arises by analytic continuation, we prove that the embedding action is bounded below without any restriction to static configurations (Proposition 1) ; the Lagrange dual of the saturation constraint is therefore well posed and weak duality holds (Proposition 2). The Euclidean Schwarzschild geometry is the cigar, periodic in imaginary time, and the reconstructed state is therefore thermal at the Hawking temperature; the Bisognano--Wichmann and Sewell theorems then give, rigorously, that the modular flow of the exterior algebra is the horizon boost flow and that the modular conjugation J is the geometric reflection across the bifurcation surface (Proposition 3). Our main result assembles, from these conditions, the promotion of the exterior algebra from Type~III₁ to a Type~II von Neumann algebra by the crossed product with the modular flow, where the adjoined degree of freedom is the internal boost charge of the horizon---the saturated action of the embedding itself, not an external observer. Because the saturation bound makes that charge spectrum bounded above as well as below, the trace is finite for any weight: the algebra is Type~II₁ (Proposition 4), and the finiteness of the black-hole entropy follows from saturation rather than from a subtraction. The fourth axiom of the programme---informational irreversibility---sharpens this. An absolute entropy presupposes a trace, which excludes Type~III₁ rigorously; a bounded entropy possessing a maximum is then consistent only with the finite trace of Type~II₁. The type is thereby fixed by the arrow of time, as a structural exclusion of Type~III₁ together with a consistency argument for Type~II₁ over Type~II_, and, specialised to the horizon, the axiom is the generalized second law. Two conjectures are isolated and stated as such: that the modular commutant is the G\"odelian sector of AFT, and that the boost-charge crossed product is the correct Type~II promotion. We then give a falsifiability analysis of the formulation---including a discrete, equispaced horizon-area spectrum of the Bekenstein--Mukhanov type as its one concrete structural prediction---and close with the evaporating regime, where the Type~II₁ finiteness yields a Page curve without an island prescription and the regular core removes the singularity. The coefficient 1/4 in the area law is not derived; it remains a G\"odelian datum, and we state this plainly.
Patricio E. Valenzuela (Tue,) studied this question.