Information-bearing Structures: Emergent Geometry from Compositional Information

We present Law IV of the modular research series Emergent Spacetime Dynamics, the capstone law in a four-paper hierarchy that derives geometry and spacetime from layered categorical and informational structures. Building on Law I (categorical primitives), Law II (phase-bound matter as functorial classes), and Law III (frequency-modulated processes as Floquet functors), we argue that the compositional structure produced by the prior three laws is sufficient to generate emergent information geometry, holographic codes, and ultimately spacetime itself. We formalise quantum error correction as a categorical functor and show that holographic codes realise AdS/CFT-style emergent geometry as an isometric embedding whose discrete Ryu–Takayanagi formula is exact.