[STCT-PROJECTION] Algebraic Holographic Projection and Information Conservation via Rough Operator Algebra: Resolving the Dimensional Reduction Anomaly

This paper resolves the long-standing problem of information loss and entropy divergence during the dimensional reduction (projection) of high-dimensional objects to lower dimensional manifolds, utilizing the framework of Rough Operator Algebra (ROA). Unlike traditional geometric projection theories that suffer from irreversible information deficits, this study introduces a topological phase transition within the Type III1 factor of von Neumann algebras. We rigorously prove that the geometric information ostensibly lost during the N →N−1projection is not destroyed; rather, it is perfectly encoded into the topologicaltime phase of the modular automorphism group, mediated by the Golden Ratio resonance frequency (ωφ). This mechanism not only establishes a new algebraic holographic principle that resolves the Hawking information paradox but also provides a concrete theoretical foundation for designing holographic outer shells in macroscopic spacetime engineering.