Geometric Information Dynamics Construction N: The Trinity of Information, Number Theory, Geometry, and Physics

As the capstone of the Geometric Information Dynamics series, this paper establishes the ultimate physical principle of “information first”, proving that all pathsderived from the axioms of information dynamics—the algebraic-spectral path,the information-geometric path, the statistical-cosmological path, the Langlandsnumber theoretic path, the RG-dynamical path, the holographic-error correctionpath, and the higher-categorical path—are mathematically strictly equivalent, andinevitably give rise to the unified structure of Einstein gravity and quantum mechanics. The core achievement is the “Trinity Theorem”: any theory satisfying conservation of information distinguishability and finite describability is, in the low-energylong-wavelength limit, necessarily equivalent to a four-dimensional Lorentzian manifold, the Einstein-Hilbert action, and local quantum field theory. We construct thecomplete “theory space” Mtheory, proving that all paths correspond to differentcoordinate charts of this space, and that the critical value of 83 bits corresponds tothe curvature singularity of this space. This framework provides a mathematicallycomplete argument for “It from Bit/Qubit” and points the way toward the ultimateunification of future physics.