%
\label{thm:three-pi}
The three constructions of~$\pi$
from Category~$\tau$ yield the same number:
\[
    \boxed{%
    \pi_{\mathrm{top}}
    \;=\;
    \pi_{\mathrm{geo}}
    \;=\;
    \pi_{\mathrm{spec}}
    \;=\;
    3.14159265\ldots}
\]
Specifically:
\begin{enumerate}
    \item[\textup{(T$=$G)}]
          The topological half-period
          equals the geometric circumference-to-diameter ratio:
          the figure-eight path on~$\mathbb{L}$
          traverses two lobes,
          each of which has Archimedean circumference~$\pi$
          (matching the limit of the Archimedes polygon sequence).
    \item[\textup{(G$=$S)}]
          The geometric ratio equals the spectral phase average:
          the Archimedes polygon perimeters converge
          to the same value
          as the normalized spectral phase accumulation
          (both are controlled by the equidistribution
          of primes on the solenoidal circle).
    \item[\textup{(T$=$S)}]
          The topological period equals the spectral phase:
          the lemniscate period is computed
          from character values on the boundary ring,
          and the normalization yields the same~$\pi$.
\end{enumerate}

[Proof outline]
\textbf{(T$=$G).}
The topological $\pi$ is the half-period
of the lemniscate path.
Each lobe of~$\mathbb{L}$ is a copy of~$S^1$
(Theorem~\ref{thm:torus-degeneration}, II.T13,
Chapter~\ref{ch:torus-degeneration}),
and the Archimedean circumference of~$S^1$
is $2\pi_{\mathrm{geo}}$
(by Definition~\ref{def:geometric-pi}).
Hence the half-period $\pi_{\mathrm{top}} = \pi_{\mathrm{geo}}$.

\smallskip
\noindent\textbf{(G$=$S).}
The Archimedes polygon perimeters
at stage~$k$ are determined by the chord lengths
between consecutive vertices of the level circle.
Each chord length depends on the angle subtended,
which is $2\pi / Q_k^X$.
The spectral phase accumulation
$\Theta_k^X$ tracks the same angular data
through character values.
The normalization ensures
$\pi_{\mathrm{geo}} = \pi_{\mathrm{spec}}$
in the limit $k \to \infty$.

\smallskip
\noindent\textbf{(T$=$S).}
Transitivity: $\pi_{\mathrm{top}} = \pi_{\mathrm{geo}} = \pi_{\mathrm{spec}}$.