Information-Theoretic Reconstruction of Quantum Mechanics - NGV Randomness, Species Primes, Dual-Observer Manifestation, and Three-Observer Chaos

We present a complete framework of \textbf{information-theoretic reconstruction of quantum mechanics}: \begin{enumerate} \item \textbf{Statistical Layer}: Computable construction of NGV-indistinguishable\\ randomness for finite observer families (cylinder sets) through\\ "primes$\to$blocks$\to$permutations", with explicit TV upper bound\\ (exponential convergence rate under RH) \item \textbf{Dynamical Layer}: Three-observer bit stream driving switching\\ tent map on circle, yielding positive Lyapunov exponent and chaos \item \textbf{Measurement/Entanglement Layer}: Propose Species Prime\\ Framework (SPF) and dual-observer manifestation function, using\\ deterministic threshold selection to reproduce Born frequencies, and\\ reproduce quantum entanglement correlations while maintaining\\ no-signaling (requiring abandonment of Bell's local factorization\\ or measurement independence in ontological layer) \end{enumerate} We further bridge number theory and quantum chaos through \textbf{$\zeta$-triadic information}, prove three-observer-driven Lyapunov positivity theorem, and report high-precision $\zeta$ fixed point and entropy inequality verification. Results are a set of NGV information systems equivalent to quantum mechanics experimental predictions \cite{neumann1932, bell1964} and computable \cite{church1936, turing1937}, equipped with executable experimental protocols and "species-level micro-fingerprint" sample complexity lower bounds.