%
\label{def:orbit-set-map}
The \textbf{orbit-set map}
$\mathrm{Set} \colon \Obj(\tau) \to \mathcal{P}(\Obj(\tau))$
is defined orbit-by-orbit as follows.
Let $p_n$ denote the $n$-th prime.

\medskip
\noindent
\textbf{(i) $\alpha$-orbit} ($\alpha_n \in O_\alpha$,\; $n \geq 1$):
\[
    \boxed{%
    \mathrm{Set}(\alpha_n)
    \;:=\;
    \{\alpha_k : k \mid n\}.}
\]
This is the \emph{divisor set} of~$n$:
every divisor of $n$ produces a set-member,
and nothing else.
This is exactly $\tau$-membership from
Chapter~\ref{ch:membership-divisibility}:
$\alpha_k \in \mathrm{Set}(\alpha_n)
\iff k \in_\tau n \iff k \mid n$.
For a prime $p$: $\mathrm{Set}(\alpha_p) = \{\alpha_1, \alpha_p\}$
(atoms --- the minimal non-trivial sets).
For composite $n$: the set includes all divisors.

\medskip
\noindent
\textbf{(ii) $\pi$-orbit} ($\pi_n \in O_\pi$,\; $n \geq 0$):
\[
    \boxed{%
    \mathrm{Set}(\pi_n)
    \;:=\;
    \{\alpha_1\}
    \;\cup\;
    \{\alpha_{p_k} : 1 \leq k \leq n\}.}
\]
This is the \emph{first $n$ primes}
(as $\alpha$-orbit elements) plus the unit.
The definition uses ``first $n$ primes''
rather than ``primes $\leq n$,''
ensuring injectivity and aligning
with the rank transfer map $RT_\pi$
(Definition~\ref{def:rank-transfer}).

\medskip
\noindent
\textbf{(iii) $\gamma$-orbit} ($\gamma_n \in O_\gamma$,\; $n \geq 1$):
\[
    \boxed{%
    \mathrm{Set}(\gamma_n)
    \;:=\;
    \{\alpha_{p_n^k} : k \geq 0\}
    \;=\;
    \{\alpha_1, \alpha_{p_n}, \alpha_{p_n^2},
    \alpha_{p_n^3}, \ldots\}.}
\]
The powers of the $n$-th prime: a \emph{vertical} chain
exploring DEPTH along a single prime tower.
These are the \textbf{first infinite sets}
in $\tau$'s internal set theory.

\medskip
\noindent
\textbf{(iv) $\eta$-orbit} ($\eta_n \in O_\eta$,\; $n \geq 1$):
\[
    \boxed{%
    \mathrm{Set}(\eta_n)
    \;:=\;
    \{\alpha_{p_k^n} : k \geq 1\}
    \;=\;
    \{\alpha_{p_1^n}, \alpha_{p_2^n},
    \alpha_{p_3^n}, \ldots\}.}
\]
All primes raised to power~$n$: a \emph{horizontal} sweep
exploring BREADTH across all prime bases.
These are also infinite.

\medskip
\noindent
\textbf{(v) Crossing point} ($\omega$):
\[
    \boxed{%
    \mathrm{Set}(\omega)
    \;:=\;
    O_\alpha \cup \{\omega\}.}
\]
The \emph{one-point compactification} of $\mathbb{N}^+$:
all $\alpha$-orbit elements together with $\omega$ itself.
This is the \textbf{universal set} ---
the largest set in $\tau$'s internal universe.