%
\label{prop:orbit-set-extensional}
The orbit-set map is \textbf{injective}:
\[
    \mathrm{Set}(X) = \mathrm{Set}(Y)
    \quad\Longrightarrow\quad
    X = Y.
\]
In particular, distinct $\tau$-elements
represent distinct sets.

We verify injectivity on each orbit
and then across orbits.

\medskip
\noindent
\textbf{$\alpha$-orbit.}
$\mathrm{Set}(\alpha_n) = \{\alpha_k : k \mid n\}$
is the divisor set.
For $n \neq 0$, $\alpha_n$ is the unique maximum
(every other $\alpha_k$ satisfies $k < n$).
If $\mathrm{Set}(\alpha_n) = \mathrm{Set}(\alpha_m)$,
then the two sets share the same maximum,
so $n = m$.

\medskip
\noindent
\textbf{$\pi$-orbit.}
$|\mathrm{Set}(\pi_n)| = n + 1$,
since the first $n$ primes are distinct
and $\alpha_1$ is not equal to any $\alpha_{p_k}$
for $k \geq 1$.
If $\mathrm{Set}(\pi_n) = \mathrm{Set}(\pi_m)$,
then $n + 1 = m + 1$, so $n = m$.

\medskip
\noindent
\textbf{$\gamma$-orbit.}
The smallest element strictly greater than $\alpha_1$
in $\mathrm{Set}(\gamma_n)$ is $\alpha_{p_n}$.
This determines $p_n$, and since
the prime-indexing function $n \mapsto p_n$
is injective,
$\mathrm{Set}(\gamma_n) = \mathrm{Set}(\gamma_m)$
implies $p_n = p_m$, hence $n = m$.

\medskip
\noindent
\textbf{$\eta$-orbit.}
Every element of $\mathrm{Set}(\eta_n)$
has the form $\alpha_{p_k^n}$.
From any single element $\alpha_{p_k^n}$,
the exponent $n$ is determined
(since $p_k$ is prime, the $n$-th power
is the unique factorization exponent).
If $\mathrm{Set}(\eta_n) = \mathrm{Set}(\eta_m)$,
then $n = m$.

\medskip
\noindent
\textbf{Crossing point.}
$\mathrm{Set}(\omega) = O_\alpha \cup \{\omega\}$
is the unique set containing $\omega$ as an element.
No other set family produces a set containing $\omega$.

\medskip
\noindent
\textbf{Cross-orbit injectivity.}
The five families are distinguishable by structural type:
\begin{itemize}
    \item $\alpha$-orbit sets are \emph{finite} divisor sets,
          each containing its representing element $\alpha_n$
          as the unique maximum.
    \item $\pi$-orbit sets are \emph{finite}
          and consist entirely of
          $\alpha_1$ plus elements $\alpha_{p_k}$
          indexed by \emph{consecutive} primes.
    \item $\gamma$-orbit sets are \emph{infinite},
          and every element is a power $\alpha_{p_n^k}$
          of a single prime.
    \item $\eta$-orbit sets are \emph{infinite},
          and the elements $\alpha_{p_k^n}$
          range over \emph{all} primes
          with a fixed exponent.
    \item $\mathrm{Set}(\omega)$ is the unique set
          containing $\omega$.
\end{itemize}
These structural signatures are mutually exclusive,
so no cross-orbit collision is possible.