Observer-Relative Finite-Screen Holography: A Conjectural Extension of Static-Patch Algebraic Holography

We propose that cosmological holography should be formulated not in terms of a single asymptotic boundary theory, but rather through a nested family of finite, observer-relative causal screens. In the first version of this proposal, the framework is explicitly algebra-first: each screen is associated with an effective observable algebra and state, and compatibility between screens is formulated in algebraic terms rather than by unspecified dynamical maps. We position the proposal as a conservative extension of existing observer-relative algebraic approaches to de Sitter holography, especially the static-patch framework of Chandrasekaran, Longo, Penington, and Witten. We formulate a precise conjectural framework, isolate its minimal assumptions, and identify baseline consistency checks in exact de Sitter, including recovery of the static-patch modular flow in the large-diamond limit. We also establish an interior realization of the algebraic nesting condition for conformal field theory in de Sitter space, and propose a modular crossed-product completion as the natural candidate for the horizon-limit extension. Finally, we present a geometric two-dimensional toy model illustrating how a nested family of finite diamonds develops a universal thermal modular structure in the large-diamond limit, while the associated entropy requires renormalized control.