\section{Global dimension bounds with imperfect parallel compatibility}
\label{sec:global-imperfect-compatibility}

We now allow the unary boundary to fail parallel compatibility and quantify
the amount of failure required to reduce coefficient dimension.  The
boundary is normalized to be a fiberwise isometry.  Without a normalization,
rescaling a boundary by a small constant and its cancelling form by the
reciprocal constant would make an absolute compatibility norm arbitrarily
small without changing the cancelled curvature.

Let $M$ be a connected Riemannian manifold, let $(V,\nabla^V)$ be a
rank-$d$ flat Hermitian bundle with $d\ge2$, and let $(U,\nabla^U)$ be a rank-$r$
Hermitian bundle with a unitary connection.  Flatness of $\nabla^U$ is not
needed for the lower bound below, so that bound also applies to the more
restricted flat auxiliary systems.  Let $Q:U\to V$ be a smooth fiberwise
isometry and define
\[
 P=QQ^*,\qquad
 K=\nabla^{\operatorname{Hom}}Q
   =\nabla^VQ-Q\nabla^U,\qquad K_\perp=(I-P)K.
\]
For a unit tangent vector $X$, use the Hilbert--Schmidt norm of
$K_\perp(X):U_x\to V_x$, and put
\[
 \kappa_\perp=\sup_{x,\,\lVert X\rVert=1}
                         \lVert K_\perp(X)\rVert_{\rm HS}.
\]
The same lower bounds hold for the possibly larger norm defined using
$K$ instead of $K_\perp$.

Fix $p\in M$ and finitely many piecewise smooth loops
$\gamma_1,\ldots,\gamma_m$ based at $p$, of lengths $L_1,\ldots,L_m$,
with $\sum_iL_i^2>0$.
Write $H_i\in U(V_p)$ for their holonomies.  On the complex traceless
matrix space, with Hilbert--Schmidt norm, define
\[
 \mathcal L(S)=([S,H_1],\ldots,[S,H_m]),\qquad
 s=\inf_{\substack{\operatorname{tr}S=0\\\lVert S\rVert_{\rm HS}=1}}
       \left(\sum_i\lVert[S,H_i]\rVert_{\rm HS}^2\right)^{1/2}.
\]
If the $H_i$ act irreducibly, Schur's lemma and finite dimensionality imply
$s>0$.

\begin{theorem}[A global compatibility cost for every variable boundary]
\label{thm:global-compatibility-cost}
For every smooth isometric boundary $Q$ as above,
\[
 \boxed{\quad
 \kappa_\perp\ge
 \frac{s}{\sqrt{2\sum_iL_i^2}}
                 \sqrt{\frac{r(d-r)}d}.\quad}                 \tag{G1}
\]
In particular the bound is not restricted to constant boundaries or
constant coefficient subspaces.  A more detailed loopwise inequality is
\[
 s^2\frac{r(d-r)}d
 \le 2\sum_i
      \left(\int_{\gamma_i}\lVert K_\perp(\dot\gamma_i)\rVert_{\rm HS}
                                     \,dt\right)^2.          \tag{G2}
\]
The right side uses any piecewise smooth parametrizations, with the
velocity included as displayed.
\end{theorem}

\begin{proof}
Metric compatibility and $Q^*Q=I$ imply that $Q^*K(X)$ is skew-Hermitian.
Differentiating $P=QQ^*$ therefore cancels this tangential skew-Hermitian
part and gives
\[
 \nabla^{\operatorname{End}}_X P
       =K_\perp(X)Q^*+QK_\perp(X)^*,\qquad
 \lVert\nabla_X P\rVert_{\rm HS}^2
       =2\lVert K_\perp(X)\rVert_{\rm HS}^2.                 \tag{G3}
\]
The equality follows in an orthogonal splitting
$V_x=\operatorname{im}Q_x\oplus(\operatorname{im}Q_x)^\perp$:
the two summands occupy opposite off-diagonal blocks of equal norm.

Parallel-transport $P$ along a loop back into the fixed fiber $V_p$.
The resulting curve of projectors starts at $P_p$ and ends at
$H_i^{-1}P_pH_i$.  Unitary transport preserves Hilbert--Schmidt norm,
so integrating (G3) gives
\[
 \lVert[H_i,P_p]\rVert_{\rm HS}
 =\lVert H_i^{-1}P_pH_i-P_p\rVert_{\rm HS}
 \le\sqrt2\int_{\gamma_i}
          \lVert K_\perp(\dot\gamma_i)\rVert_{\rm HS}\,dt.
\]
Since $P_p$ is a rank-$r$ orthogonal projector,
\[
 \left\lVert P_p-\frac rd I\right\rVert_{\rm HS}^2
       =r-\frac{r^2}d=\frac{r(d-r)}d.
\]
The scalar matrix commutes with each holonomy.  Apply the definition of
$s$ to this traceless matrix, sum the squared loop bounds, and obtain
(G2).  Each integral is at most $L_i\kappa_\perp$, giving (G1).
\end{proof}

\begin{corollary}[A positive compatibility tolerance still forces full rank]
Suppose the holonomies above act irreducibly and a prescribed curvature
has a value of norm one at $p$.  If a cancellation $QG$ has curvature
error less than one at that value and
\[
 \kappa_\perp<
 \frac{s}{\sqrt{2\sum_iL_i^2}}\sqrt{\frac{d-1}d},           \tag{G4}
\]
then $r=d$.  Thus the all-or-nothing obstruction persists for a
nonzero compatibility budget.
\end{corollary}
\begin{proof}
Curvature error less than one excludes $r=0$.  For every integer
$1\le r\le d-1$ one has $r(d-r)\ge d-1$, so (G1) excludes every
remaining proper rank.  The full bundle with its identity boundary
attains zero compatibility and curvature error.
\end{proof}

\subsection{Matching-order transition on a fixed closed five-manifold}

The compatibility scale in (G4) is attained in order by an explicit
family with globally fixed potential and curvature.  This strengthening
uses two disjoint transport handles, so it does not require identifying
different flat bundles only in one local chart.

\begin{theorem}[The rank-reduction threshold has order $|\epsilon|$]
\label{thm:global-linear-compatibility-scale}
For every fixed $d\ge2$, there are a fixed closed connected oriented
five-manifold $M$, a fixed trivial Hermitian bundle $V=M\times\mathbb C^d$,
fixed smooth forms $B,F$ with $\lVert F\rVert_\infty=1$, and flat
unitary connections $\nabla^V_\epsilon$ converging smoothly to the trivial
connection, with
\[
 d_{\nabla^V_\epsilon}B=F.
\]
There are constants $c_d,C_d,\epsilon_d>0$ such that, for
$0<|\epsilon|<\epsilon_d$ and every fixed curvature tolerance
$0\le\tau<1$, the following statements hold.
\begin{enumerate}
\item Every smooth isometric boundary from any auxiliary Hermitian
      bundle of rank strictly between zero and $d$ that attains curvature
      error $\lVert F-QG\rVert_{L^\infty}\le\tau$ has
      $\lVert\nabla^{\operatorname{Hom}}Q\rVert_\infty
       \ge c_d|\epsilon|$.
\item A flat unitary auxiliary line, an isometric boundary, and a
      covariantly closed scalar five-form cancel $F$ exactly, with
      compatibility error at most $C_d|\epsilon|$.
\end{enumerate}
Consequently the compatibility budget necessary and sufficient, up to
fixed positive factors, for any rank reduction is $|\epsilon|$.
The constants depend on $d$ and the fixed geometry; no dimension-uniform
claim is made.
\end{theorem}

\begin{proof}
Let
$M=(S^1\times S^4)\mathbin{\#}(S^1\times S^4)$.
There are smooth closed one-forms $\eta_1,\eta_2$ with disjoint supports,
one in each punctured summand, with period one around the respective
circle generators and vanishing on a common coordinate ball $D$.
For example, on each circle choose a smooth function supported in a
proper interval with integral one and take $f(t)dt$ on the product with
$S^4$.  Remove the gluing ball away from that support.  The form vanishes
near the gluing sphere and extends by zero across the connected sum.
The two supports are disjoint, so $\eta_1\wedge\eta_2=0$.

Put
\[
 A=\operatorname{diag}(1,\ldots,d),\qquad
 J=(1)_{i,j=1}^d,\qquad
 v=d^{-1/2}(1,\ldots,1)^T,
\]
and define
\[
 \nabla^V_\epsilon=d+i\epsilon(A\eta_1+J\eta_2).
\]
The matrices $A,J$ are Hermitian.  The connection is flat because
$d\eta_i=0$ and every wedge product $\eta_i\wedge\eta_j$ vanishes.
Its two generating holonomies, based in $D$, are
\[
 H_1=e^{-i\epsilon A},\qquad
 H_2=e^{-i\epsilon J}
       =I+\frac{e^{-id\epsilon}-1}{d}J.
\]
For $0<|\epsilon|<2\pi/d$, the first has simple spectrum and the
second has every off-diagonal entry nonzero.  A common commuting
orthogonal projector must first be diagonal and then scalar, hence
zero or the identity.  Thus these holonomies act irreducibly.

Let $\beta$ be an ordinary four-form compactly supported in $D$, with
$\lVert d\beta\rVert_\infty=1$.  Set $B=v\beta$ and $F=v\,d\beta$.
Both are fixed, and the connection term vanishes on their support.
Hence $d_{\nabla^V_\epsilon}B=F$ for every parameter.  Choose the
base point $p\in D$ where the curvature norm is one and fix generating
loops of finite lengths $L_1,L_2$, independently of $\epsilon$.

On traceless matrices define
\[
 \mathcal L_0(S)=([S,A],[S,J]),\qquad
 s_0=\sigma_{\min}(\mathcal L_0)>0.
\]
Its kernel is zero by the same diagonal-then-scalar argument.
The unitary exponential remainder estimate yields
\[
 \lVert\mathcal L_\epsilon+i\epsilon\mathcal L_0\rVert
       \le\sqrt2\,d^2\epsilon^2.
\]
It follows that $s_\epsilon\ge s_0|\epsilon|/2$ for sufficiently
small $|\epsilon|$.  Apply (G1), using
$r(d-r)\ge d-1$, to obtain the first assertion with
\[
 c_d=\frac{s_0}{2\sqrt{2(L_1^2+L_2^2)}}
                              \sqrt{\frac{d-1}d}.
\]
This lower bound allows arbitrary smooth isometric boundaries and
even arbitrary unitary auxiliary connections.

For attainment in order, take the trivial complex line with flat
connection
\[
 \nabla^U_\epsilon
    =d+i\epsilon\left(\frac{d+1}{2}\eta_1+d\eta_2\right),
 \qquad Q(1)=v,\qquad G=d\beta.
\]
Then $QG=F$ exactly, and $G$ is covariantly closed by top degree.
Since $Jv=dv$,
\[
 \nabla^{\operatorname{Hom}}Q
 =i\epsilon\left(A-\frac{d+1}{2}I\right)v\eta_1,
\]
whose uniform norm is
\[
 |\epsilon|\sqrt{\frac{d^2-1}{12}}\,
                                  \lVert\eta_1\rVert_\infty.
\]
This proves the second assertion and the matching order.  The boundary
is generally not parallel, so the resulting small-error datum is not
an exact homological completion; its compatibility defect is the
specific error being optimized.
\end{proof}

The result gives a sharp order transition, not an exact formula for every
intermediate budget.  It also distinguishes two limits: for each fixed
nonzero $\epsilon$ there is a positive compatibility threshold below
which dimension $d$ is forced, while that threshold itself tends to zero
as the holonomy tends to the identity.