\section{The exact rank--error frontier under unitary holonomy}
The exact completion rank can be discontinuous under perturbation.  A
different and quantitatively meaningful problem fixes an error tolerance
while retaining the parallel structure required by the binary module action.
This section gives the complete answer for unitary flat coefficient bundles.
All dimensions in this section are complex dimensions.

Let $M$ be a connected compact Riemannian manifold, and let
$(V,\nabla)$ be a rank-$n$ flat Hermitian vector bundle with unitary
holonomy.  Let $B\in\Omega^k(M;V)$, with $k\ge3$, and put
$F=d_\nabla B$.  An admissible subspace of coefficient directions is a
parallel subbundle $W\subseteq V$, with its parallel orthogonal projector
$P_W$.  Its projected model is $B_W=P_WB$.  Flatness and parallelism give
\[
 d_\nabla B_W=P_WF,\qquad d_\nabla(P_WF)=0.
\]
Thus the projected data admit the usual central completion with auxiliary
bundle $W$.  Equivalently, if one keeps $B$ fixed and uses $P_WF$ as the
cancelled defect, the uncancelled square of the graded differential is
$(I-P_W)F$.  These are two precise interpretations of the same error
criterion, not a claim that an incompletely cancelled model is homological.
Define
\[
 \mathcal E_r(F)=
 \min_{\substack{W\text{ parallel}\\\operatorname{rank}W\le r}}
 \|(I-P_W)F\|_{L^2}^2,
 \qquad 0\le r\le n.                                      \tag{U1}
\]
For a fixed $W$, the same minimum is obtained if one optimizes over all
closed $W$-valued forms $G$ in place of $P_WF$: the pointwise orthogonal
projection is already the best approximation, and it is closed.

Fix $o\in M$ and identify $V_o$ with $\mathbb C^n$.  Let $H\subseteq U(n)$
be the closure of the holonomy group.  For almost every $x$, choose a
measurable family of parallel transports $T_x:V_x\to V_o$.  If
$F_x=\sum_\alpha f_\alpha(x)\,e_\alpha(x)$ in an orthonormal frame of
exterior forms, define the seed Gramian and its compact-group average by
\[
 C=\int_M\sum_\alpha
  T_x f_\alpha(x)\,\bigl(T_xf_\alpha(x)\bigr)^*\,dx,
 \qquad
 \widehat C=\int_H h C h^*\,dh.                           \tag{U2}
\]
The Haar measure is normalized to have mass one.  One can construct the
measurable choices using a countable coordinate cover, one chosen path to
each chart, local flat frames, and a measurable partition of that cover.
The integral is finite because $\operatorname{tr}C=\|F\|_{L^2}^2$.
Changing the chosen paths conjugates each pointwise summand by an element
of the holonomy group, so its Haar average is unchanged.  Consequently
$\widehat C$ is independent of these path choices.

Choose the unitary isotypic decomposition
\[
 V_o=\bigoplus_{\lambda}
       \bigl(\mathbb C^{m_\lambda}\otimes E_\lambda\bigr),
 \qquad d_\lambda=\dim E_\lambda,
\]
where the $E_\lambda$ are pairwise inequivalent irreducible $H$-modules.
Let $C_\lambda$ be the \emph{unnormalized} partial trace over $E_\lambda$
of the corresponding diagonal block of $C$.  Write its eigenvalues as
\[
 \alpha_{\lambda,1}\ge\cdots\ge
 \alpha_{\lambda,m_\lambda}\ge0.
\]

\begin{theorem}[Unitary completion rank--error law]
The quantities above satisfy
\[
 \widehat C
  =\bigoplus_\lambda
    \left(\frac{C_\lambda}{d_\lambda}\otimes I_{E_\lambda}\right),
                                                               \tag{U3}
\]
and the exact error frontier is
\[
 \boxed{\displaystyle
 \mathcal E_r(F)=\operatorname{tr}C-
 \max_{\substack{0\le k_\lambda\le m_\lambda\\
                  \sum_\lambda d_\lambda k_\lambda\le r}}
      \sum_\lambda\sum_{j=1}^{k_\lambda}
              \alpha_{\lambda,j}.}                         \tag{U4}
\]
In particular, the exact minimum rank is
\[
 r_{\mathrm{exact}}
    =\sum_\lambda d_\lambda\operatorname{rank}C_\lambda
    =\operatorname{rank}\widehat C.                         \tag{U5}
\]
This is the dimension of the smallest parallel subbundle containing the
values of $F$.  Formula (U4) allows ranks with unused budget; not every
integer is necessarily a possible rank of a parallel subbundle.
\end{theorem}

\begin{proof}
An orthogonal projector determines a parallel subbundle precisely when
its value $P$ at $o$ commutes with the holonomy group, equivalently with
$H$.  The associated global projector is
$P_x=T_x^{-1}PT_x$, independent of the path choices.  Therefore
\[
 \|(I-P_W)F\|_{L^2}^2
  =\operatorname{tr}((I-P)C)
  =\operatorname{tr}((I-P)\widehat C).                     \tag{U6}
\]
The last equality uses $h^*Ph=P$ and cyclicity of trace.

Compact-group averaging annihilates off-diagonal blocks between
inequivalent irreducibles.  On an isotypic block it is scalar on the
irreducible factor by Schur's lemma.  Taking the partial trace identifies
that scalar factor and gives (U3).  Every orthogonal projector in the
commutant has the form
\[
 P=\bigoplus_\lambda(P_\lambda\otimes I_{E_\lambda}),
\]
where $P_\lambda$ is an arbitrary orthogonal projector on the multiplicity
space.  If its rank is $k_\lambda$, then the full rank contribution is
$d_\lambda k_\lambda$, while the captured energy is
$\operatorname{tr}(P_\lambda C_\lambda)$.

For a positive matrix with eigenvalues $\alpha_1\ge\cdots\ge\alpha_m$,
the maximum of $\operatorname{tr}(PC)$ over rank-$k$ orthogonal
projectors is $\sum_{j\le k}\alpha_j$.  Indeed in an eigenbasis the
diagonal entries $p_j$ of $P$ lie in $[0,1]$ and sum to $k$, so that
$\sum\alpha_jp_j\le\sum_{j\le k}\alpha_j$; the spectral projector
attains equality.  This is the classical Ky Fan variational principle.
Optimizing the integer ranks gives (U4).

Zero error means retaining every positive eigenvalue in every reduced
block.  Thus (U5) follows.  Alternatively, the range of $\widehat C$ is
invariant, and every vector orthogonal to it has zero integrated squared
pairing with the transported curvature and all its holonomy transforms.
Continuity of $F$ promotes almost-everywhere vanishing to pointwise
vanishing.  Its range is therefore exactly the required holonomy-saturated
coefficient subspace.
\end{proof}

\paragraph{Finite optimization and why a greedy rule is insufficient.}
Given the isotypic decomposition and the reduced spectra, put
$A_\lambda(k)=\sum_{j\le k}\alpha_{\lambda,j}$.  A budget dynamic
program, one isotypic block at a time, uses the recurrence
\[
 D_\lambda(r)=
  \max_{0\le k\le\min(m_\lambda,\lfloor r/d_\lambda\rfloor)}
  \{D_{\lambda-1}(r-d_\lambda k)+A_\lambda(k)\}.
                                                               \tag{U7}
\]
With $D_0(r)=0$, the maximum in (U4) is the final $D_\lambda(r)$.
At fixed spectra this requires at most $O(n^2)$ arithmetic comparisons
over all budgets $0,\ldots,n$.  Computing an isotypic decomposition or
exact spectra from arbitrary input is a separate problem.

An elementary example has two inequivalent irreducibles, each of
multiplicity one, with $(d_1,\alpha_{1,1})=(2,3)$ and
$(d_2,\alpha_{2,1})=(3,4)$.  For budget three the optimum captures four
units of energy by choosing the second block.  A rule that first chooses
the greater energy per dimension instead takes the first block and
captures only three.  Such irreducibles occur, for example, in the direct
sum of the defining two-dimensional representation of $SU(2)$ and the
defining three-dimensional representation of $SO(3)$, as a representation
of their product.  A flat holonomy group can have this product as its
compact closure.

\begin{corollary}[An arbitrarily large all-or-nothing barrier in dimension five]
For every $d\ge2$, there is a closed connected oriented five-manifold,
a rank-$d$ flat unitary coefficient bundle, and a smooth exact curvature
$F=d_\nabla B\ne0$ whose values lie in one local parallel line wherever
they are nonzero, such that
\[
 \mathcal E_r(F)=\|F\|_{L^2}^2\quad(0\le r<d),
 \qquad \mathcal E_d(F)=0.                              \tag{U8}
\]
For suitable measurable path choices, the untwirled Gramian $C$ has rank
one.  Its ordinary unconstrained rank-one approximation has zero error,
while every compatible parallel model of rank less than $d$ retains the
entire curvature error.
\end{corollary}

\begin{proof}
Take $M=(S^1\times S^4)\mathbin{\#}(S^1\times S^4)$, so
$\pi_1(M)$ is the free group on two generators.  On $\mathbb C^d$, let $X$ be the cyclic permutation of the standard
basis, and let $Y$ act on the first two coordinates by
\[
 \begin{pmatrix}0&-1\\1&0\end{pmatrix}
\]
and as the identity on all other coordinates.  These integer orthogonal
matrices define a flat Hermitian bundle.  Their common representation is
irreducible over $\mathbb C$.  For $d\ge3$, the conjugates
$X^jY^2X^{-j}$ negate the two coordinates of every cyclic adjacent pair.
Those sign patterns distinguish every pair of coordinates, so a matrix
commuting with all of them is diagonal.  Commuting also with $X$ forces
that diagonal matrix to be scalar.  For $d=2$, the direct equations for
commuting with $X$ and $Y$ have the same scalar solution.  Every invariant
subspace is reducing because the generators are unitary; its orthogonal
projector therefore belongs to this scalar commutant and is zero or the
identity.

Choose a coordinate ball with a flat frame, a nonzero constant coefficient
vector $v$, and a smooth compactly supported scalar function $\phi$ with
$\partial_5\phi\not\equiv0$.  In that ball set
\[
 B=\phi(x)v\,dx^1\wedge dx^2\wedge dx^3\wedge dx^4,
\]
and extend by zero.  This is a global smooth four-form.  Its curvature is
$F=(\partial_5\phi)v\,dx^1\wedge\cdots\wedge dx^5$, since
moving $dx^5$ past four factors has positive sign.  It is nonzero and
locally line-valued.  Choose the paths on this ball through the fixed
flat frame.  All nonzero transported coefficients are multiples of the
same vector, and hence $C$ has rank one.

Irreducibility implies that the only parallel subbundles are $0$ and the
whole coefficient bundle.  This proves (U8) directly.  Equivalently,
$\widehat C=(\operatorname{tr}C/d)I_d$ has full rank, but there is no
nontrivial projector in the commutant.
\end{proof}

\paragraph{Quantitative stability at fixed rank.}
For two seed Gramians $C,C'$, their optimal retained energies at budget
$r$ differ by at most $\|\widehat C-\widehat C'\|_*$, since every
admissible projector has operator norm at most one.  Combining this with
the trace difference gives
\[
 |\mathcal E_r(C)-\mathcal E_r(C')|
 \le 2\|\widehat C-\widehat C'\|_*.
\]
One may instead obtain the sharper direct bound
$|\mathcal E_r(C)-\mathcal E_r(C')|
\le\|\widehat C-\widehat C'\|_*$ by expressing the error as the
minimum of $\operatorname{tr}((I-P)\widehat C)$ and using
$\|I-P\|_{\mathrm{op}}\le1$.  Thus the full error curve is Lipschitz
in trace norm even when the exact zero-error rank is discontinuous.
The group average is trace-norm contractive on Hermitian matrices, so
the same bound holds with $\|C-C'\|_*$ for fixed holonomy representation.

\begin{proposition}[Stability under perturbing the holonomy generators]
Let $U_1,\ldots,U_m\in U(n)$ act irreducibly, and let
\[
 \mathcal L(S)=([S,U_1],\ldots,[S,U_m])
\]
on the traceless matrix space, with Hilbert--Schmidt norms.  Set
$s=\sigma_{\min}(\mathcal L)>0$.  If unitary matrices $U_i'$ satisfy
$\max_i\|U_i'-U_i\|_{\mathrm{op}}\le\epsilon$, then
\[
 \sigma_{\min}(\mathcal L')\ge s-2\sqrt m\,\epsilon.
\]
In particular irreducibility, and hence the all-or-nothing barrier,
persists for $\epsilon<s/(2\sqrt m)$.
If the $U_i$ are integer orthogonal matrices, the entirely explicit
sufficient condition
\[
 \epsilon<(4m)^{-(n^2-1)/2}
\]
also guarantees persistence.  This bound is deliberately conservative.
\end{proposition}
\begin{proof}
Schur's lemma makes the kernel of the commutator map consist precisely of
scalar matrices, so its restriction to the traceless subspace is
injective.  Moreover
$\|[S,U_i'-U_i]\|_{\mathrm{HS}}
\le2\epsilon\|S\|_{\mathrm{HS}}$.
Combining the $m$ inequalities and applying the reverse triangle
inequality gives the stated singular-value bound.  A nontrivial invariant
subspace of the perturbed unitary family would give a nonzero traceless
commuting projector, so injectivity implies irreducibility.

For the explicit estimate, the full positive operator
$T=\mathcal L^*\mathcal L$ has an integer matrix in the matrix-unit basis,
one zero eigenvalue, and $n^2-1$ positive eigenvalues.  Each eigenvalue is
at most $4m$.  The product of the nonzero eigenvalues is the absolute
value of the linear coefficient of its integer characteristic polynomial,
and hence is at least one.  Its least positive eigenvalue is therefore
at least $(4m)^{-(n^2-2)}$.  Taking a square root and dividing by
$2\sqrt m=\sqrt{4m}$ gives the displayed sufficient radius.
\end{proof}

\begin{corollary}[Arbitrarily weak unitary holonomy still forces full rank]
For every $d\ge2$ and every $\eta>0$, the all-or-nothing example in
(U8) can be chosen with both specified holonomy generators at operator-norm
distance less than $\eta$ from the identity.  More precisely, let
\[
 A=\operatorname{diag}(1,\ldots,d),\qquad J=(1)_{i,j=1}^d,
\]
and for $0<|\tau|<2\pi/d$ set
\[
 U_1(\tau)=e^{i\tau A},\qquad
 U_2(\tau)=e^{i\tau J}
 =I+\frac{e^{i d\tau}-1}{d}J.                         \tag{U9}
\]
The resulting representation of the free group on two generators is
irreducible, while
\[
 \|U_j(\tau)-I\|_{\mathrm{op}}\le d|\tau|\quad(j=1,2).
\]
On the fixed closed five-manifold
$(S^1\times S^4)\mathbin{\#}(S^1\times S^4)$, choose the same nonzero,
locally supported exact curvature in a common flat coordinate frame for
each parameter.  Every nonzero $\tau$ in (U9) then requires auxiliary
rank $d$ even to reduce its curvature error.  At $\tau=0$, the trivial
holonomy representation admits zero error with auxiliary rank one.
Thus the exact rank--error frontier is discontinuous at trivial holonomy,
even though the generators converge to the identity and the local
curvature data are fixed.
\end{corollary}
\begin{proof}
The diagonal entries of $U_1(\tau)$ are distinct under the displayed
parameter restriction.  A commuting orthogonal projector is therefore
diagonal.  Every off-diagonal entry of $U_2(\tau)$ is the same nonzero
number, so commuting with $U_2(\tau)$ forces all diagonal entries of
that projector to agree.  Hence it is zero or the identity, proving
irreducibility.  The exponential bounds follow from
$|e^{it}-1|\le|t|$ and $\|A\|_{\mathrm{op}}=\|J\|_{\mathrm{op}}=d$.
Taking $0<|\tau|<\min(2\pi/d,\eta/d)$ gives the first claim.

Use the compactly supported four-form from the previous corollary in a
fixed flat chart.  Its curvature values occupy the same coefficient line
for every parameter and have the same $L^2$ norm.  The irreducibility
argument gives (U8) at each nonzero parameter.  At zero, every fixed
coefficient subspace is parallel, and the one coefficient line suffices.
This is a statement about this explicit family of representations and
locally fixed data; no unmentioned global identification of different
flat connections is required.
\end{proof}

\paragraph{A perturbation radius proportional to holonomy strength.}
Define on traceless matrices
\[
 \mathcal L_0(S)=([S,A],[S,J]),\qquad
 s_0=\sigma_{\min}(\mathcal L_0)>0.
\]
The positivity follows by the same diagonal-then-all-entries commutant
argument.  The exponential remainder estimate gives
\[
 \|\mathcal L_\tau-i\tau\mathcal L_0\|
 \le\sqrt2\,d^2\tau^2.
\]
Consequently, when
$0<|\tau|\le s_0/(2\sqrt2\,d^2)$,
\[
 \sigma_{\min}(\mathcal L_\tau)\ge\tfrac12s_0|\tau|.
\]
By the preceding perturbation proposition, arbitrary unitary changes of
the two generators by less than
$s_0|\tau|/(4\sqrt2)$ preserve irreducibility and the all-or-nothing
barrier.  The robust radius therefore has a positive linear lower bound
in the nonzero holonomy parameter, although that radius necessarily
shrinks toward the singular trivial-holonomy limit.

\paragraph{Established ingredients and scope.}
Compact-group averaging, Schur's lemma, and the spectral variational
principle are established tools.  A primary reference for the latter is
K.~Fan, \emph{On a Theorem of Weyl Concerning Eigenvalues of Linear
Transformations I}, Proc.~Natl.~Acad.~Sci.~USA \textbf{35}(11) (1949),
652--655, \url{https://doi.org/10.1073/pnas.35.11.652}.
The result here is their explicit application to globally compatible
central defect cancellation, including its exact budget frontier and
the fixed-dimension all-or-nothing obstruction.  It does not solve a
general noncentral completion problem, and it does not treat a
nonunitary local system as though it had a parallel Hermitian metric.