Non-Equilibrium Holographic Brane Cosmology in AdS5. I-A: Operator Algebras in the Static Patch, Type II_1 Reduction, and Holographic Reconstruction

The local description of cosmological horizons in quantum gravity faces a fundamental algebraic obstruction: in the static patch of four-dimensional de Sitter space (dS4), the algebra of local quantum observables forms a hyperfinite Type III1 von Neumann factor, precluding an invariant vacuum state, a well-defined density matrix, and a normalizable trace. In this work, the first of a dedicated series developing non-equilibrium holographic cosmology on Karch-Randall 3-branes embedded in asymptotically AdS5 spacetimes, we provide an analytical framework addressing cosmological horizon unitarity. By coupling the local quantum fields to the dynamical Hamiltonian of a timelike observer via a crossed product construction, we lift the algebra to a Type II∞ factor. Incorporating the non-perturbative Nariai black hole collapse limit provides a natural energy regularization that projects the extended algebra onto a finite, hyperfinite Type II1 von Neumann factor AdS equipped with a unique, faithful, normalized trace τ. We construct the physical Hilbert space HdS via the Gelfand-Naimark-Segal (GNS) representation and show that the maximum algebraic entropy rigorously saturates the Gibbons-Hawking area bound. Finally, we formulate the quantum isometric embedding V: HdS → Hphys into the asymptotic conformal boundary I+, establish the holographic dictionary via the canonical Petz recovery channel, and prove that the proper time of the comoving observer emerges directly as the canonical conjugate variable to the asymptotic entanglement modular flow through Jafferis-Lewkowycz-Maldacena-Suh (JLMS) duality.