Geometric Information Dynamics Construction K: Information Weaving, the Holographic Principle, and Gravity

This paper establishes a rigorous correspondence between quantum error-correctingcodes and the emergence of gravity, demonstrating that the Einstein equations arethe geometric limit of holographic error-correction conditions that protect quantum information from local errors. The core innovation is the discovery that theradial direction of AdS space corresponds to the hierarchical redundancy of theerror-correcting code, and the Einstein equations are equivalent to the variationalcondition of “informational correctability”. We construct an explicit “informationweaving code” and prove that when the code rate exceeds the critical thresholdKc = 83ln2 bits, the steady-state logical space of the error-correcting code naturally emerges as a four-dimensional anti-de Sitter spacetime, whose geometric dynamics is described by the Einstein-Hilbert action. By generalizing the “tripodunit” on a hexagonal lattice to the “tetrahedral unit” in three-dimensional space,we achieve a dimensional extension from AdS/CFT to AdS/CFT. Furthermore,we reinterpret black hole evaporation as a gradual corruption process of the errorcorrecting code, with the Page curve corresponding to the decoding threshold. Nearthis threshold, an “error-correction phase transition” may leave observable signatures such as gravitational wave echoes and photon dispersion. Finally, we designa simulation scheme based on a 127-qubit superconducting processor to test thediscrete version of the Einstein equations. This framework provides a computablerealization of the “It from Qubit” program and opens new avenues for experimentaltests of quantum gravity.