\section{Fixed-site match-count topology and source dagger}
\label{fixed:sec-match-dagger}

Once order independence is known, the composite can be understood by asking
which local forward and reverse phases match.  This section turns that count
into the topology of the simultaneous double and separately records the
contravariant source-reversal operation.

First use the two
order-independence results to form the forward--reverse double with its two
independent phase records.  Then isolate each labelled neighborhood by a
separating sphere and classify its local double: matching phases retain a
punctured $S^1\times S^2$ piece, while mismatching phases retain the product
piece.  The disjoint connected-sum decomposition turns that local table into
the global match count and Betti-number formula.  The source dagger is a
separate contravariant operation on the marked cobordism, so it is recorded
after the topology of the double has been identified.

\subsection{The global match count}

\begin{definition}[Independent double and match data]
\label{fixed:def-global-double}
For independent order tags \(\alpha,\beta\in S_k\), define
\begin{equation}
 M_{\alpha,\beta}^{\epsilon,\eta}
 (\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-)
 :=R_\beta^\eta(\mathbf r,\mathbf u^-)
   J_\alpha^\epsilon(\mathbf c,\mathbf u^+)
 \in\operatorname{End}(\Sigma_g^{\mathrm{out}}).
 \label{fixed:eq-G0}
\end{equation}
Set
\begin{align}
 \mathcal M:=\mathcal M(\epsilon,\eta)&:=\{i\in[k]:\epsilon_i=\eta_i\},
 &b:=b(\epsilon,\eta)&:=|\mathcal M|,
 \label{fixed:eq-G1}\\
 N&:=\sum_{i=1}^k
 \bigl(c_i+r_i+\epsilon_i u_i^++\eta_i u_i^-\bigr)
 =\sum_{i=1}^k(a_i+b_i').
 \label{fixed:eq-G2}
\end{align}
Also write
\begin{equation}
 d_H(\epsilon,\eta):=
 |\{i\in[k]:\epsilon_i\ne\eta_i\}|
 \label{fixed:eq-hamming-definition}
\end{equation}
for the Hamming distance.
Inactive slope coordinates do not occur in \(N\).
\end{definition}

\begin{lemma}[Order independence of the double]
\label{fixed:lem-double-order-independence}
The morphism in \eqref{fixed:eq-G0} is independent of \(\alpha\) and
\(\beta\) separately.  Denote its common value by
\[
 M^{\epsilon,\eta}
 (\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-).
\]
We suppress the fixed label-attached weight arguments below.
\end{lemma}

\begin{proof}
Equations \eqref{fixed:eq-O3} and \eqref{fixed:eq-O4} are literal
equalities in their fixed Hom sets.  Substitute both into
\eqref{fixed:eq-G0} to prove the claim.  No local double is moved between
different Hom sets.
\end{proof}

\begin{theorem}[Simultaneous local pieces and match-count topology]
\label{fixed:thm-match-count}
For every allowed \(g,k\), phase pair, and active constituent weights,
\begin{equation}
 \boxed{
 M^{\epsilon,\eta}
 \cong
 C(\operatorname{id}_{\Sigma_g^{\mathrm{out}}},N)
 \mathbin{\#}
 \left(
  \mathop{\#}_{i\in \mathcal M}(S^1\times S^2,0)
 \right).}
 \label{fixed:eq-G3}
\end{equation}
Writing
\(\beta_1(W):=\dim_{\mathbb Q}H_1(W;\mathbb Q)\), the number of displayed
closed summands is
\begin{equation}
 \boxed{b=k-d_H(\epsilon,\eta),}
 \label{fixed:eq-G4}
\end{equation}
and
\begin{equation}
 \boxed{\beta_1(M^{\epsilon,\eta})=2g+b.}
 \label{fixed:eq-G5}
\end{equation}
Consequently,
\begin{equation}
 b=0\quad\Longrightarrow\quad
 M^{\epsilon,\eta}
 =C(\operatorname{id}_{\Sigma_g^{\mathrm{out}}},N),
 \label{fixed:eq-G6}
\end{equation}
whereas \(b>0\) implies that \(M^{\epsilon,\eta}\) is not an identity
cylinder of any Walker weight.
\end{theorem}

\paragraph{Local doubles and the global match count.}
Use the independently constructed simultaneous traces and surround each
labelled neighborhood by a disjoint separating sphere.  A matching phase
pair gives the local meridian-to-meridian double, hence one punctured
\(S^1\times S^2\); a mismatching pair gives the product piece.  Disjoint
connected-sum decomposition then counts the matching sites, while the
history calculation supplies the global Walker weight.

\begin{proof}
Compose the independently constructed traces
\(R_{\mathrm{sim}}^\eta J_{\mathrm{sim}}^\epsilon\).  In the geometric
construction, choose one auxiliary separating sphere around each \(N_i\).
These spheres are pairwise disjoint and lie in one actual global cobordism.
We inspect the pieces they cut off; we do not form, multiply, or commute the
typed symbols \(\mathcal D_{i\mid I}\) from different endomorphism sets.

Inside the \(i\)-th sphere the local word is the already proved same-site
word \(R_i^{\eta_i}J_i^{\epsilon_i}\), with its actual markings.  If
\(\epsilon_i=\eta_i\), the local table and its topology identify the piece
as punctured \(S^1\times S^2\).  This includes both the \(00\) and the
\(11\) match; in particular an all-one phase pair contributes at every
site.  If \(\epsilon_i\ne\eta_i\), the exact marked slope-word calculation
identifies the local piece as the punctured \(S^3\) product piece.  It
therefore disappears from connected sum with the outside product.

The Walker calculation is equally local but is made in this single global
trace.  Proposition \ref{fixed:prop-ordered-histories} gives the forward
and reverse internal and inter-factor corrections and their weights.
At the middle forward/reverse interface, the chosen direct-sum splitting
and the endpoint cases of
\eqref{fixed:eq-forward-partial-transport} and
\eqref{fixed:eq-reverse-partial-transport} give the exact correction
\begin{equation}
 \mu\bigl(\lambda_{[k]}^\epsilon,\lambda_{[k]},
          \lambda_{[k]}^\eta\bigr)
 =
 \mu(\lambda_g,\lambda_g,\lambda_g)
 +\sum_{i=1}^k
   \mu\bigl(L_i^{\epsilon_i},\langle m_i\rangle,
            L_i^{\eta_i}\bigr)=0.
 \label{fixed:eq-global-middle-seam}
\end{equation}
The four terms in the sum are exactly the four already proved local-table
seams and the four rows of
\eqref{fixed:eq-I2-spectator-table}; every one is zero.  In particular,
the matching \(11\) term is
\[
 \mu(\langle\ell_i\rangle,
     \langle m_i\rangle,
     \langle\ell_i\rangle)=0.
\]
A separating sphere has zero first homology and hence contributes no Maslov
term.  The total Walker weight is therefore exactly
\[
 \sum_i a_i+\sum_i b_i'=N.
\]
The central shift is recorded on the outside cylinder in
\eqref{fixed:eq-G3}; every displayed closed summand has weight zero.
The complement of the local spheres is the product on the old surface.
There is one \(S^1\times S^2\) summand for precisely each \(i\in \mathcal M\), which
proves \eqref{fixed:eq-G3}.

A coordinate is outside \(\mathcal M\) exactly when it contributes one to the
Hamming distance, proving \eqref{fixed:eq-G4}.  The product cylinder has
first Betti number \(2g\), and connected sum with each
\(S^1\times S^2\) adds one independent first-homology generator.  Hence
\eqref{fixed:eq-G5}.  If \(b=0\), every local piece is the exact punctured
\(S^3\) product with the prescribed markings, so removing all auxiliary spheres
leaves the literal identity cylinder of weight \(N\), proving
\eqref{fixed:eq-G6}.  If \(b>0\), then
\(\beta_1(M)=2g+b>2g\), whereas every weighted identity cylinder has first
Betti number \(2g\); no choice of Walker weight can make them equal.

The spheres and the subset \(\mathcal M\) index this proof only.  The construction
neither chooses nor retains a sphere system or an embedded copy of the match
subset as part of its output; the globally detected data here are the count
\(b\) and the independently retained total weight \(N\).
In particular, equal values of \(b\) do not erase different values of \(N\).
For \(k=1\), this is exactly the four-entry local table.  For \(k=2\), the
choice \(\epsilon=(0,1)\), \(\eta=(0,0)\) has \(b=1\).  Equal phase vectors,
including the all-one pair, have \(b=k\); complementary phase vectors have
\(b=0\).  When \(g=1\), formula \eqref{fixed:eq-G5} reads
\(\beta_1=2+b\).
\end{proof}

\begin{remark}[Limitations]
\label{fixed:rem-match-scope}
Every argument above uses one fixed chart of pairwise-disjoint,
distinctly labeled sites.  It does not cover repeated, overlapping, nested,
intersecting, slid, born, canceled, unlabelled, or relabeled sites.  The
elementary dagger used only to derive \eqref{fixed:eq-I1} is not a global
dagger theorem.  No scalar target value, refined finite carrier, physical
handle map, Cerf relation, Heegaard invariant, or \(\widehat Z\) comparison
is asserted here.
\end{remark}

\subsection{Typed global words and orientation reversal}
\label{fixed:subsec-global-dagger}

Fix integers \(g,k\geq1\), the tagged outgoing objects
\(\Sigma_I^{\mathrm{out}}\) for \(I\subseteq[k]\), the two phase vectors
\(\epsilon,\eta\in\{0,1\}^k\), and the four label-attached weight vectors
\[
 \mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-\in\mathbb Z^k.
\]
All products are right-factor first.  An inactive coordinate \(u_i^+\) when
\(\epsilon_i=0\), or \(u_i^-\) when \(\eta_i=0\), is ignored in every
arrow and every formula below.

For an order word \(\alpha=(\alpha_1,\ldots,\alpha_k)\in S_k\), set
\begin{equation}
 I_a^\alpha=\{\alpha_1,\ldots,\alpha_a\},
 \qquad I_0^\alpha=\varnothing.
 \label{fixed:eq-order-subsets-global}
\end{equation}
The two already constructed blocks are
\begin{align}
 J_\alpha^\epsilon(\mathbf c,\mathbf u^+)
 &:=J_{\alpha_k\mid I_{k-1}^\alpha}^{\epsilon_{\alpha_k}}
       (c_{\alpha_k},u_{\alpha_k}^+)\cdots
     J_{\alpha_1\mid\varnothing}^{\epsilon_{\alpha_1}}
       (c_{\alpha_1},u_{\alpha_1}^+)
 \notag\\[-2pt]
 &:\Sigma_\varnothing^{\mathrm{out}}
   \longrightarrow\Sigma_{[k]}^{\mathrm{out}},
 \label{fixed:eq-global-forward-word}\\
 R_\alpha^\eta(\mathbf r,\mathbf u^-)
 &:=R_{\alpha_1\mid\varnothing}^{\eta_{\alpha_1}}
       (r_{\alpha_1},u_{\alpha_1}^- )\cdots
     R_{\alpha_k\mid I_{k-1}^\alpha}^{\eta_{\alpha_k}}
       (r_{\alpha_k},u_{\alpha_k}^- )
 \notag\\[-2pt]
 &:\Sigma_{[k]}^{\mathrm{out}}
   \longrightarrow\Sigma_\varnothing^{\mathrm{out}}.
 \label{fixed:eq-global-reverse-word}
\end{align}
The arguments displayed in these formulas are understood by the inactive
coordinate convention.  Because products act from the right, the reverse
word chronologically removes \(\alpha_k,\ldots,\alpha_1\).  Thus \(\alpha\) is the
matching attachment tag, not an inverse-permutation tag.

The elementary source-dagger identities proved before any global argument
are, for \(\xi\in\{0,1\}\),
\begin{align}
 \bigl(J_{i\mid I}^{\xi}(c_i,u_i)\bigr)^\dagger
 &=R_{i\mid I}^{\xi}(-c_i,-u_i),
 \label{fixed:eq-elementary-J-dagger-recalled}\\
 \bigl(R_{i\mid I}^{\xi}(r_i,u_i)\bigr)^\dagger
 &=J_{i\mid I}^{\xi}(-r_i,-u_i).
 \label{fixed:eq-elementary-R-dagger-recalled}
\end{align}
They are handle-arrow identities.  They were obtained from
\(\Theta_{i\mid I}(u)^\dagger=C(\sigma_{i\mid I}^{-1},-u)\) only after the
proved half-turn absorption relations; no equality
\(C(\sigma^{-1},-u)=C(\sigma,-u)\) is being asserted.

\begin{proposition}[Exact dagger of the two global blocks]
\label{fixed:prop-global-dagger}
For every order tag \(\alpha\in S_k\),
\begin{equation}
 \boxed{
 \bigl(J_\alpha^\epsilon(\mathbf c,\mathbf u^+)\bigr)^\dagger
 =R_\alpha^\epsilon(-\mathbf c,-\mathbf u^+).}
 \tag{DG0}\label{fixed:eq-DG0}
\end{equation}
For every order tag \(\beta\in S_k\),
\begin{equation}
 \boxed{
 \bigl(R_\beta^\eta(\mathbf r,\mathbf u^-)\bigr)^\dagger
 =J_\beta^\eta(-\mathbf r,-\mathbf u^-).}
 \tag{DG1}\label{fixed:eq-DG1}
\end{equation}
Both are literal equalities of marked Walker extended cobordisms, including
the negated total integer weights.
\end{proposition}

\begin{proof}
Contravariance reverses the written factors of
\eqref{fixed:eq-global-forward-word}.  Applying
\eqref{fixed:eq-elementary-J-dagger-recalled} factor by factor gives
\begin{align*}
 \bigl(J_\alpha^\epsilon(\mathbf c,\mathbf u^+)\bigr)^\dagger
 &=
 R_{\alpha_1\mid\varnothing}^{\epsilon_{\alpha_1}}
       (-c_{\alpha_1},-u_{\alpha_1}^+)\cdots
 R_{\alpha_k\mid I_{k-1}^\alpha}^{\epsilon_{\alpha_k}}
       (-c_{\alpha_k},-u_{\alpha_k}^+)\\
 &=R_\alpha^\epsilon(-\mathbf c,-\mathbf u^+).
\end{align*}
This is exactly the reverse word with attachment tag \(\alpha\); no inverse
permutation is introduced.  The forward block has weight
\(\sum_i(c_i+\epsilon_i u_i^+)\).  Orientation reversal negates it, and
the block on the right has weight
\(\sum_i(-c_i-\epsilon_i u_i^+)\), so the Walker weights agree exactly.

The same calculation applied to
\eqref{fixed:eq-global-reverse-word}, now using
\eqref{fixed:eq-elementary-R-dagger-recalled}, yields
\begin{align*}
 \bigl(R_\beta^\eta(\mathbf r,\mathbf u^-)\bigr)^\dagger
 &=
 J_{\beta_k\mid I_{k-1}^\beta}^{\eta_{\beta_k}}
       (-r_{\beta_k},-u_{\beta_k}^-)\cdots
 J_{\beta_1\mid\varnothing}^{\eta_{\beta_1}}
       (-r_{\beta_1},-u_{\beta_1}^-)\\
 &=J_\beta^\eta(-\mathbf r,-\mathbf u^-).
\end{align*}
Its weight is the negative of
\(\sum_i(r_i+\eta_i u_i^-)\).  This proves both formulas with the endpoint
markings and integer weights retained.
\end{proof}

For independent tags \(\alpha,\beta\in S_k\), retain the typed double
\begin{equation}
 M_{\alpha,\beta}^{\epsilon,\eta}
 (\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-)
 :=R_\beta^\eta(\mathbf r,\mathbf u^-)
   J_\alpha^\epsilon(\mathbf c,\mathbf u^+)
 \in\operatorname{End}(\Sigma_g^{\mathrm{out}}).
 \label{fixed:eq-global-double-recalled}
\end{equation}
Its already proved total weight and match data are
\begin{equation}
 \begin{aligned}
 \mathcal M:=\mathcal M(\epsilon,\eta)&=\{i\in[k]:\epsilon_i=\eta_i\},
 & b:=b(\epsilon,\eta)&=|\mathcal M|,\\
 N&=\sum_{i=1}^k
   (c_i+r_i+\epsilon_i u_i^++\eta_i u_i^-).
 \end{aligned}
 \label{fixed:eq-N-b-recalled}
\end{equation}

\begin{corollary}[Exact dagger of the mixed double]
\label{fixed:cor-double-dagger}
One has
\begin{equation}
 \boxed{
 \begin{aligned}
 &\bigl(M_{\alpha,\beta}^{\epsilon,\eta}
 (\mathbf c,\mathbf r,\mathbf u^+,\mathbf u^-)\bigr)^\dagger\\
 &\qquad=
 M_{\beta,\alpha}^{\eta,\epsilon}
 (-\mathbf r,-\mathbf c,-\mathbf u^-,-\mathbf u^+).
 \end{aligned}}
 \tag{DG2}\label{fixed:eq-DG2}
\end{equation}
The transformation sends \(N\) to \(-N\) and leaves \(b\) unchanged.
\end{corollary}

\begin{proof}
Using contravariance and then
\eqref{fixed:eq-DG0}--\eqref{fixed:eq-DG1},
\begin{align*}
 \bigl(R_\beta^\eta J_\alpha^\epsilon\bigr)^\dagger
 &=(J_\alpha^\epsilon)^\dagger(R_\beta^\eta)^\dagger\\
 &=R_\alpha^\epsilon(-\mathbf c,-\mathbf u^+)
   J_\beta^\eta(-\mathbf r,-\mathbf u^-).
\end{align*}
By the definition \eqref{fixed:eq-global-double-recalled}, this last word
is the right side of \eqref{fixed:eq-DG2}: its forward parameter block is
\(( -\mathbf r,-\mathbf u^- )\), and its reverse parameter block is
\(( -\mathbf c,-\mathbf u^+ )\).  The phase pair and the order tags are
therefore exchanged exactly as displayed.  Substitution in
\eqref{fixed:eq-N-b-recalled} gives \(-N\), while equality of the two
phase bits is symmetric, so the match count remains \(b\).
\end{proof}

The symbol \(\dagger\) in this subsection denotes source orientation
reversal only.  None of the three displayed dagger identities is a
Hermitian-adjoint identity in a target theory.

\begin{proof}[Proof of \cref{fixed:thm-main-package}]
 The local phase table and typed interchanges reduce the ordered histories
to the simultaneous trace.  Its geometric decomposition gives the
match-count topology and total weight, after which block reversal gives the
dagger identities.  These steps retain the distinct conclusions of literal
equality, marked diffeomorphism, topological connected sum, and source
orientation reversal.

The common pulled-back chart and its Walker extension are
\cref{fixed:prop-common-walker-extension}.  The localized half-turn,
capping, absorption, and elementary source-reversal statements are
\cref{fixed:prop-halfturn,fixed:prop-capping,fixed:prop-reverse-absorption,fixed:prop-elementary-dagger}.
The four-entry weighted table and the topology distinguishing its matching
entries are \cref{fixed:thm-local-table,fixed:prop-local-topology}.

The separately typed forward and reverse interchanges are
\cref{fixed:thm-interchange}; their spectator corrections are checked in
\cref{fixed:prop-spectator-maslov}.  Literal order independence and
equality-only confluence then follow from
\cref{fixed:cor-order-independence,fixed:prop-coxeter-closure}.  The ordered
history calculation and its comparison with the independently constructed
simultaneous trace are
\cref{fixed:prop-ordered-histories,fixed:prop-simultaneous-traces}.  Finally,
\cref{fixed:lem-double-order-independence,fixed:thm-match-count} give the
order-independent mixed double, its match-count topology, and its first Betti
number, while
\cref{fixed:prop-global-dagger,fixed:cor-double-dagger} give the two block
reversals and the mixed-double reversal.  These results prove the seven
clauses in their stated order.
\end{proof}