Pi as Unified Expression of Observer Symmetry: Information Conservation Framework from Fourier Observation to Holographic Projection

Building on the Zeta--Bernoulli--k-Bonacci unified framework, this paper formalizes $\pi$'s role as the guardian of observer symmetry in the triadic information framework. Bernoulli sequences correspond to the curvature-like component $i_-$, the constant $e$ to the wave component $i_0$, the golden ratio $\phi$ to the ordered component $i_+$, while $\pi$ enforces invariance of the conservation law $i_+ + i_0 + i_- = 1$ across reference frames. Core contributions: (1) $\pi$-observer symmetry equivalence for $\mathrm{SO}(n)$ and $\mathrm{U}(1)$ rotation groups; (2) Fourier-$\pi$ necessity theorem proving that $2\pi$ period normalization is required for observational closure and the uncertainty relation $\Delta x\, \Delta p \geq \hbar/2$; (3) Statistical correspondence between $\pi$ and the wave information component through \textsc{gue} eigenvalue spacing distribution $p(s)=(32/\pi^2)s^2 e^{-4s^2/\pi}$; (4) Holographic encoding via black hole entropy $S_{\mathrm{BH}}=A/(4G)=\pi r_s^2/G$ and Ryu-Takayanagi formula; (5) Hawking temperature $T_H=\hbar c^3/(8\pi G M k_B)$ as manifestation of observer boundary thermodynamics. Numerical verification using mpmath (dps=50 precision) confirms $|e^{i\pi}+1|<10^{-50}$, Fourier normalization through $\delta$-function representation, \textsc{gue} distribution normalization $\int_0^{\infty} p(s)\,ds=1.0000$, and solar-mass black hole Hawking temperature $T_H \approx 6.169\times 10^{-8}$ K. Deviations with $\pi=3$ yield systematic errors exceeding $4\%$ in holographic observables, supporting $\pi$'s fundamental role as an observation-symmetry constant.