Nested p-adic golden bells on E8

ABSTRACT================================================================================== Write $K=\Q(\sqrt5)$, $\Phi=(1+\sqrt5)/2$, $\Psi=1-\Phi=-\Phi^{-1}$, $\cO_1=\Z[\Phi]$, and $F_n$, $L_n$ for the Fibonacci and Lucas numbers.$P_n$ are the Pell numbers ($P_0=0$, $P_1=1$, $P_{n+1}=2P_n+P_{n-1}$) and $Q_n$ the Pell--Lucas numbers ($Q_0=Q_1=2$, same recurrence), so$Q_n=\sigma^n+\bar\sigma^n$ is the trace of the $n$-th power of the silver unit $\sigma=1+\sqrt2$; $H_n=Q_n/2=1,1,3,7,17,41,\dots$ is the half trace.For a metallic index $k\ge1$, $U_n(k)$ and $V_n(k)$ are the base and trace sequences of $x^2=kx+1$ ($U_0=0$, $U_1=1$; $V_0=2$, $V_1=k$),so $(U,V)=(F,L)$ at $k=1$ and $(P,Q)$ at $k=2$. $\Tch(x)=x^2-2$ is Chebyshev doubling. For a prime $p$ we say \emph{split}, \emph{inert}or \emph{ramified} for the behaviour of $p$ in $K$: $p\equiv\pm1$, $p\equiv\pm2\pmod5$ or $p=2$, and $p=5$.