Formal Foundations of the Recursive Mapping Theorem (RMT)

This paper constructs the spacetime manifold as the emergent stable state of a concrete recursive matrix operator ϕ, defined as a nonlinear endomorphism on a Hilbertspace with a symmetry-protected spiked random connectivity matrix and stochasticnoise. From this single object we derive: the metric (via Normalized InformationDistance); the unique spatial dimensionality n = 3 (via a holographic saturation argument, stable under the holographic prior encoded in M); an emergent Lorentziancausal structure; inertial mass (via topological sealing at the minimum knotting dimension, grounded in Landauer’s Principle); and the tensorial Einstein field equations (via the Jacobson thermodynamic equation of state). The gravitational constantG emerges as a genuine spectral output G = c^5/(ℏω^2max), where ωmax is the tanhsaturation cutoff derived from the Nyquist–Shannon information volume using only ℏand c (no Planck-scale input). We predict a quadratic spectral decoherence signature∆ν/ν ≈ (1/√n)(E/EP)^2, yielding ≈ 0.577(E/EP)^2 at n = 3 and a numerical amplitude ≈3.9 × 10−31 at 10TeV, consistent with Fermi-LAT null results and testable byCTA. Note: The Hilbert space H is acknowledged as a structural prior; metric, causal,and curvature properties are derived rather than assumed.