\section{The optional ordinary rootwise scalar}
\label{common:sec-ordinary-scalar}

Only after the source cobordism has been identified do we ask what an
ordinary root-of-unity theory sees.  The source topology determines the scalar
calculation; the target theory is used here as a consequence, not as a
substitute for the geometric coherence proof.

All results through \cref{slide:sec:squares-order-closure} are statements in the
source category.  Only in this section do we apply a target theory.  By
\cref{fixed:thm-main-package}, the order-independent fixed-site double has
the marked source topology
\begin{equation}
 M^{\epsilon,\eta}
 \cong
 C(\operatorname{id}_{\Sigma_g^{\mathrm{out}}},N)
 \mathbin{\#}
 \left(
  \mathop{\#}_{t\in\mathcal M(\epsilon,\eta)}
  (S^1\times S^2,0)
 \right),
 \qquad
 b=\lvert\mathcal M(\epsilon,\eta)\rvert .
 \label{common:eq-source-double-topology}
\end{equation}
The separating spheres and the indexed match set in this formula are proof
witnesses.  Neither is retained as part of the resulting marked cobordism.

Fix an odd prime \(p\geq5\) and a primitive \(p\)-th root \(\zeta_p\).  Set
\begin{equation}
 A_p=-\zeta_p^{(p+1)/2},
 \qquad
 A_p^2=\zeta_p,
 \qquad
 K_p=\mathbb Q(\zeta_p).
 \label{common:eq-ordinary-root-data}
\end{equation}
Choose once and globally a correlated pair
\(D_p^{\mathrm{GM}},\kappa_p\neq0\) in a finite extension
\(\widetilde K_p/K_p\) such that
\begin{equation}
 (D_p^{\mathrm{GM}})^2
 =-\frac{p}{(A_p^2-A_p^{-2})^2},
 \qquad
 \kappa_p^2=A_p^{-6-p(p+1)/2}.
 \label{common:eq-ordinary-normalization}
\end{equation}
This is the ordinary BHMV--Gilmer--Masbaum normalization
\cite{BHMV1995,GilmerMasbaum2007,GilmerMasbaum2013}.  Let \(Z_p\) denote
the normalized ordinary BHMV theory and put
\[
 V_{p,g}^{\mathrm{out}}=Z_p(\Sigma_g^{\mathrm{out}}),
 \qquad
 \widetilde V_{p,g}^{\mathrm{out}}
 =\widetilde K_p\otimes_{K_p}V_{p,g}^{\mathrm{out}}.
\]
In this scalar-extended theory, increasing a Walker weight by one multiplies
the TQFT map by \(\kappa_p\)
\cite[arXiv v3, Section~11, p.~24]{GilmerMasbaum2013}.  The BHMV surgery
normalization gives a factor \(D_p^{\mathrm{GM}}\) for an internal connected
sum with one weight-zero \(S^1\times S^2\)
\cite{BHMV1995,GilmerMasbaum2007}.  The two signed choices belong to one
global normalization and are never selected independently at different
sites.

\Needspace{4\baselineskip}
\begin{corollary}[Optional ordinary scalar]
\label{common:cor-ordinary-scalar}
On \(\widetilde V_{p,g}^{\mathrm{out}}\),
\begin{equation}
 \boxed{
 Z_p(M^{\epsilon,\eta})
 =\kappa_p^N(D_p^{\mathrm{GM}})^b
  \operatorname{id}_{\widetilde V_{p,g}^{\mathrm{out}}}.}
 \label{common:eq-ordinary-scalar}
\end{equation}
This is an ordinary, unrefined, rootwise, scalar-extended conclusion.
\end{corollary}

\begin{proof}
 The geometric theorem first identifies the mixed double as an identity
 cylinder together with the actual matching \(S^1\times S^2\) summands.  The
ordinary functor then evaluates these two topological pieces separately:
Walker weight contributes \(\kappa_p\), while each genuine connected-sum
factor contributes the same globally chosen \(D_p^{\mathrm{GM}}\).  This is
why the exponent \(b\) counts the geometric match set and why the conclusion
remains an optional rootwise scalar rather than a refined state-level result.

Apply the normalized ordinary functor to
\eqref{common:eq-source-double-topology}.  The identity cylinder of weight
\(N\) acts by \(\kappa_p^N\) times the identity for every
\(N\in\mathbb Z\).  The connected-sum rule contributes one copy of the same
globally chosen \(D_p^{\mathrm{GM}}\) for each of the \(b\) actual
\(S^1\times S^2\) factors.  Successive application therefore gives
\((D_p^{\mathrm{GM}})^b\); no sign is selected independently at a site.
Multiplying the two contributions proves the formula.
\end{proof}

\begin{remark}[No refined exponentiation]
\label{common:rem-no-refined-power}
The refined genus-one-to-two scalar factor belongs to a different,
retained-label finite-coefficient theorem.  No input used here constructs
refined carriers in genus at least three, simultaneous refined handle maps,
cross-handle finite covariance, a refined connected-sum rule, or sewing.
Consequently neither \eqref{common:eq-source-double-topology} nor
\eqref{common:eq-ordinary-scalar} implies, defines, or even types a
\(b\)-fold power of such a refined factor.
\end{remark}