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 estab
lishes the ultimate physical principle of “information first”, proving that all paths
derived from the axioms of information dynamics—the algebraic-spectral path,
the information-geometric path, the statistical-cosmological path, the Langlands
number theoretic path, the RG-dynamical path, the holographic-error correction
path, and the higher-categorical path—are mathematically strictly equivalent, and
inevitably give rise to the unified structure of Einstein gravity and quantum mechan
ics. The core achievement is the “Trinity Theorem”: any theory satisfying conserva
tion of information distinguishability and finite describability is, in the low-energy
long-wavelength limit, necessarily equivalent to a four-dimensional Lorentzian man
ifold, the Einstein-Hilbert action, and local quantum field theory. We construct the
complete “theory space” Mtheory, proving that all paths correspond to different
coordinate charts of this space, and that the critical value of 83 bits corresponds to
the curvature singularity of this space. This framework provides a mathematically
complete argument for “It from Bit/Qubit” and points the way toward the ultimate
unification of future physics.