\section{Optimal completion with nontrivial flat transport}

Let $M$ be a connected smooth finite-dimensional manifold and let
$(V,\nabla^V)$ be a finite-rank real flat vector bundle.  Fix a four-form
$B\in\Omega^4(M;V)$ and its defect
\[
 F=d_{\nabla^V}B\in\Omega^5(M;V).
\]
In particular $d_{\nabla^V}F=0$.  An admissible central completion consists
of another flat vector bundle $(U,\nabla^U)$, a parallel bundle map
$\partial:U\to V$, and a closed
$G\in\Omega^5(M;U)$ with $\partial G=F$.  The four-bracket data $B$ and
the flat actions are kept fixed, the new unary operation is $\partial$,
and the only additional operation of arity at least three is the
five-bracket represented by $-G$.  All operations of arity at least three
vanish when any input is in a coefficient bundle.  Binary module actions
remain present.  These are restrictions on the completion problem, not
claims about unrestricted higher algebroids.

Fix $p\in M$ and let $P_\gamma$ denote flat parallel transport.  Define
the transport-saturated defect space
\begin{equation}\label{eq:transport-defect-space}
 W_p=\operatorname{span}_{\mathbb R}
 \left\{P_\gamma F_x(X_1,\ldots,X_5):
 x\in M,\ X_i\in T_xM,\ \gamma:x\longrightarrow p\right\}\subseteq V_p.
\end{equation}
All paths are allowed.  Thus $W_p$ is invariant under the full holonomy
representation at $p$.  Parallel transport of $W_p$ produces a
well-defined flat subbundle $W\subseteq V$, independent of path, and
$W$ is the smallest parallel subbundle containing every value of $F$.

\begin{theorem}[Transport-optimal central completion]
An admissible completion exists, and its minimum auxiliary rank is
\[
 \boxed{\quad\min\operatorname{rank}U=\dim W_p=\operatorname{rank}W.\quad}
\]
It is attained by $U=W$, the inclusion $\partial:W\hookrightarrow V$,
and $G=F$ regarded as a $W$-valued form.  Every rank-minimal completion
has $\partial$ a flat isomorphism from $U$ onto $W$; hence its coefficient
pair and its five-form are uniquely identified with this construction.
\end{theorem}

\begin{proof}
Appending a loop at $p$ to any path in
\eqref{eq:transport-defect-space} proves holonomy invariance.  The
equivalence between holonomy-invariant fiber subspaces and parallel
subbundles therefore gives $W$.  Alternatively, choose local parallel
frames: the transported fiber subspaces are constant in each frame and
agree on overlaps.  They have constant rank because $M$ is connected.
Any parallel subbundle containing the values of $F$ contains every
transported value in \eqref{eq:transport-defect-space}, proving minimality
of $W$.

A parallel map has constant rank, since its matrix in parallel frames is
locally constant.  Its image is a parallel subbundle.  In an admissible
completion $F=\partial G$ takes values in that image, so
$W\subseteq\operatorname{im}\partial$.  Consequently
\[
 \operatorname{rank}U\ge\operatorname{rank}\partial
 \ge\operatorname{rank}W.
\]
Conversely, restriction of the connection to $W$ is flat.  Because the
inclusion is injective and parallel, the identity
$d_{\nabla^V}F=0$ implies that $F$, regarded as a $W$-valued form, is
$d_{\nabla^W}$-closed.  Thus $U=W$, $G=F$ attains the lower bound.
At equality the two rank inequalities are equalities, making
$\partial:U\to W$ a flat isomorphism and forcing
$G=\partial^{-1}F$.
\end{proof}

\subsection{The actual graded differential and the residual quotient}

In local parallel frames, adjoin generators $u$ of degree $3$ dual to $V$
and generators $v$ of degree $4$ dual to $U$.  On ordinary forms let
$Q=d$.  Define the degree-one derivation by
\[
 Qu=B+\partial v,\qquad Qv=-G.
\]
Here components are evaluated against the dual frame; the notation
$\partial v$ means the dual of the unary map on the generator spaces.
Parallel changes of frame are constant and the coefficients transform
tensorially, so these formulas glue to a global graded algebra.  In
arbitrary frames $Q$ includes the induced flat dual connections.
Its square on the new generators is
\[
 Q^2u=F-\partial G,\qquad Q^2v=-d_{\nabla^U}G.
\]
Thus the admissibility conditions are exactly $Q^2=0$.

Write $L=V/W$, with its induced flat connection, and let
$\overline B$ be the projection of $B$ to $L$.  It is closed because
$F$ has values in $W$.

\begin{theorem}[Quotient reduction without a parallel complement]
For the rank-minimal completion, the graded CE algebra is
quasi-isomorphic to the skeletal central extension with coefficient
bundle $L=V/W$ and four-cocycle $\overline B$.  The residual datum is
\[
 [\overline B]\in H^4_{\nabla^L}(M;L).
\]
No splitting of $V\to L$ by flat subbundles is required.  In particular
the result applies to nonsemisimple holonomy representations.
\end{theorem}

\begin{proof}
The dual quotient $L^*$ is canonically the annihilator of $W$ in $V^*$.
Its degree-three generators therefore form a canonical graded
subalgebra.  For such a generator the unary term is zero and its
$B$-term is evaluation on $\overline B$.  The subalgebra is preserved
by $Q$ and is exactly the stated skeletal CE algebra.

Filter both algebras by ordinary differential-form degree on $M$.
The filtration is finite, from zero to $\dim M+1$, and is preserved by
$Q$.  On the associated graded algebra the only differential with
form degree zero is the unary Koszul differential induced by
$W\hookrightarrow V$.  A smooth, not necessarily parallel, complement
of $W$ exists.  In that complement, the form-degree-zero complex is the
tensor product of the quotient-generator algebra with contractible pairs
\[
 \delta u_W=v,\qquad\delta v=0.
\]
For each odd degree-three $u_W$ and even degree-four $v$, the derivation
$h=u_W\partial_v$ satisfies $\delta h+h\delta=N$, where $N$ counts
generator multiplicity.  Division by the positive integer $N$ contracts
every nonconstant component.  Hence the inclusion of the quotient
algebra is an isomorphism on the first page of the form-degree spectral
sequence.  The finite-filtration comparison theorem makes it a
quasi-isomorphism on global complexes.  This argument only uses a
smooth complement to identify the fiberwise Koszul cohomology; no
parallel complement is invoked.

Finally, shifting a remaining degree-three generator by a three-form
changes its four-form differential by an exact $L$-valued form.  This
gives the stated residual class within the fixed coefficient and
identity-linear-part gauge class.
\end{proof}

\subsection{An arbitrarily large overhead on a fixed five-dimensional base}

Let $M=S^1_t\times\mathbb R^4$, with $t$ of period one.  The one-form
$dt$ is global although $t$ is not a global real-valued function.  Take
the smoothly trivial bundle $V=M\times\mathbb R^d$, with orthonormal
frame $e_0,\ldots,e_{d-1}$, and set
\[
 Ne_0=0,\qquad Ne_j=e_{j-1}\quad(1\le j<d),
 \qquad v=e_{d-1},\qquad
 \nabla_\varepsilon=d+\varepsilon N\,dt.
\]
The connection is flat and its holonomy around the circle is
$H_\varepsilon=\exp(-\varepsilon N)$.  Define the same global potential
for every real $\varepsilon$:
\[
 B=-x^1v\,dt\wedge dx^2\wedge dx^3\wedge dx^4.
\]
The connection term contains $dt\wedge dt$ and vanishes.  Therefore
\begin{equation}\label{eq:constant-rank-one-defect}
 F=d_{\nabla_\varepsilon}B
 =v\,dt\wedge dx^1\wedge dx^2\wedge dx^3\wedge dx^4
\end{equation}
is independent of $\varepsilon$.  Its pointwise image has dimension one.

\begin{theorem}[Sharp completion-rank jump]
For $d\ge2$, the minimum admissible auxiliary rank in this family is
\[
 \min\operatorname{rank}U=
 \begin{cases}
 1,&\varepsilon=0,\\
 d,&\varepsilon\ne0.
 \end{cases}
\]
In particular there is no bound on completion rank in terms of the
dimension of the base and the pointwise image rank of the defect alone.
The jump occurs with identical $B$ and $F$ and with connection forms
converging smoothly on every compact set as $\varepsilon\to0$.
\end{theorem}

\begin{proof}
At $\varepsilon=0$ the constant line $\mathbb Rv$ is parallel and
contains $F$, so the optimum is one.  For $\varepsilon\ne0$, transport
around integer numbers of circle loops produces
\[
 H_\varepsilon^jv=\exp(-j\varepsilon N)e_{d-1},
 \qquad 0\le j<d.
\]
In the ordered frame $e_0,\ldots,e_{d-1}$ their column matrix has entries
\[
 M_{ij}=\frac{(-j\varepsilon)^{d-1-i}}{(d-1-i)!}
 \quad(0\le i,j<d).
\]
The Vandermonde determinant gives
\[
 \det M=\varepsilon^{d(d-1)/2}.
\]
Indeed reversing the rows and extracting the signs from the powers
contribute the same sign and cancel; the integer-node Vandermonde
product $\prod_{j=1}^{d-1}j!$ cancels the factorial denominator.
Thus $W_p=V_p$ for every nonzero $\varepsilon$.  The optimality theorem
proves the claim.  The global connection difference is exactly
$\varepsilon Ndt$, whereas \eqref{eq:constant-rank-one-defect} and $B$
do not change.
\end{proof}

\begin{corollary}[The jump survives strict curvature error tolerance]
Fix $\varepsilon\ne0$ and use the displayed Euclidean fiber metric and
product metric on $M$.  Let $U$ have rank less than $d$ and
$\partial:U\to V$ be any parallel map.  For every $U$-valued five-form
$G$, closed or not, the pointwise operator norm of
$F-\partial G$ is at least one.  Thus permitting any uniform curvature
error strictly below one still requires auxiliary rank $d$.
More precisely, let $R_\varepsilon(\tau)$ be the minimum auxiliary rank
among flat $U$, parallel $\partial$, and closed $G$ with uniform error
$\lVert F-\partial G\rVert_\infty\le\tau$.  For every $\tau\ge0$,
\[
 R_\varepsilon(\tau)=
 \begin{cases}
 0,&\tau\ge1,\\
 1,&0\le\tau<1\text{ and }\varepsilon=0,\\
 d,&0\le\tau<1\text{ and }\varepsilon\ne0.
 \end{cases}
\]
\end{corollary}

\begin{proof}
The image of $\partial$ is a proper parallel subbundle, so its fiber at
any point is invariant under $H_\varepsilon$.  The finite logarithm
identity
\[
 -\varepsilon N=\log H_\varepsilon
 =\sum_{j=1}^{d-1}\frac{(-1)^{j+1}}j(H_\varepsilon-I)^j
\]
makes it $N$-invariant.  For a single Jordan block the only invariant
subspace of dimension $r$ is
$\operatorname{span}(e_0,\ldots,e_{r-1})$: regard the block as the cyclic
$\mathbb R[z]$-module $\mathbb R[z]/(z^d)$, whose submodules are the
ideals generated by powers of $z$.  Every proper such subspace is
orthogonal to $v=e_{d-1}$.  Evaluate the difference on the product
orthonormal five-frame.  Its $v$-component is one regardless of $G$,
so its norm is at least one.  The assumption that $\partial$ is exactly
parallel is essential; an approximately parallel map is a different
completion problem.
The zero auxiliary bundle attains error one.  At $\varepsilon=0$ the
parallel line $\mathbb Rv$ gives error zero, and at
$\varepsilon\ne0$ the rank-$d$ inclusion gives error zero.  These
constructions and the lower bound establish every branch of the formula.
\end{proof}

\subsection{A strict local-to-global version}

The preceding polynomial example can be localized so that every chart
of a suitable cover admits rank at most one, while the global optimum
remains $d$.  Choose a small contractible coordinate ball $D\subset M$
and a local parallel section $s$ whose value at one point is cyclic for
$H_\varepsilon$.  Choose an ordinary four-form
$\beta$ compactly supported in $D$ such that $d\beta$ is nonzero at
that point.  Extend $B=s\beta$ by zero outside $D$.  Smoothness follows
from compact support, and
\[
 F=d_{\nabla_\varepsilon}B=s\,d\beta\quad\hbox{on }D,
 \qquad F=0\quad\hbox{outside }D.
\]
On $D$, the parallel line generated by $s$ gives a rank-one completion;
away from the support of $F$, rank zero suffices.  These sets can be
refined to a good cover with local auxiliary ranks at most one.
Globally, the holonomy orbit of the single nonzero cyclic defect value
already spans $V$, so every admissible global completion has rank $d$.
Thus bounded local completion ranks do not control the global optimum
without transport compatibility.

\subsection{The same sharp frontier on a fixed closed five-torus}

\begin{theorem}[Compact-base resource discontinuity]\label{thm:compact-rank-frontier}
For every $d\ge2$ there are a fixed smooth rank-$d$ bundle on the closed
manifold $T^5$, a smooth one-parameter family of flat connections
$\nabla_\varepsilon$, and fixed smooth forms $B\in\Omega^4(T^5;V)$ and
$F\in\Omega^5(T^5;V)$ such that
\[
 d_{\nabla_\varepsilon}B=F,\qquad
 \lVert F\rVert_\infty=1
 \quad\hbox{for every }\varepsilon\in\mathbb R,
\]
while the exact parallel-boundary curvature-error frontier is
\[
 R_\varepsilon(\tau)=
 \begin{cases}
 0,&\tau\ge1,\\
 1,&0\le\tau<1\text{ and }\varepsilon=0,\\
 d,&0\le\tau<1\text{ and }\varepsilon\ne0.
 \end{cases}
\]
The forms may be supported in one fixed coordinate ball.  A good cover
admits local auxiliary ranks at most one, independently of $d$.
Moreover $\nabla_\varepsilon\to\nabla_0$ smoothly as
$\varepsilon\to0$, while $B$ and $F$ do not vary at all.
\end{theorem}

\begin{proof}
Let $dt$ denote the global closed one-form of the first unit-circle
factor of $T^5$.  Choose nested coordinate balls with a smaller ball
$D$ whose closure lies in a chart carrying a real local coordinate
$t_{\rm loc}$ for that circle.  Let $\chi$ be a smooth cutoff supported
in the chart and equal to one on a neighborhood of $\overline D$.
The product $\chi t_{\rm loc}$, extended by zero, is a global smooth
function.  Therefore
\[
 \eta=dt-d(\chi t_{\rm loc})
\]
is a global closed one-form, has the same periods as $dt$, and vanishes
on a neighborhood of $\overline D$.

Use the smoothly trivial bundle $V=T^5\times\mathbb R^d$, the same
single Jordan block $N$, and $v=e_{d-1}$.  Define
\[
 \nabla_\varepsilon=d+\varepsilon N\eta.
\]
Flatness follows from $d\eta=0$ and $\eta\wedge\eta=0$.  Parallel
transport along a curve $\gamma$ is
$\exp(-\varepsilon N\int_\gamma\eta)$, so a first-circle loop has
holonomy $\exp(-\varepsilon N)$.  This remains true when the loop is
based in $D$, because the connection matrices are all scalar multiples
of the same $N$ and $\eta$ has the periods of $dt$.

Choose an ordinary four-form $\beta$ compactly supported in $D$, with
$d\beta\ne0$, and rescale it so that
$\lVert d\beta\rVert_\infty=1$ in a fixed product metric.  Such a form
exists by taking a compactly supported scalar function with a nonzero
coordinate derivative and multiplying by the complementary coordinate
four-form.  Put
\[
 B=v\beta,\qquad F=v\,d\beta.
\]
The connection term in $d_{\nabla_\varepsilon}B$ vanishes because
$\eta$ is zero on the support of $\beta$.  Hence $B$ and $F$ are fixed
throughout the family, and $\lVert F\rVert_\infty=1$ for the fixed
Euclidean fiber metric.

For nonzero $\varepsilon$, the image of a parallel map of rank less
than $d$ is, at every base point, a proper invariant subspace of
$\exp(-\varepsilon N)$.  The finite logarithm and single-block
classification identify it with a proper Jordan prefix.  In the
displayed global frame this prefix is orthogonal to $v$.  At a point
where $\lVert d\beta\rVert=1$, evaluation on a unit oriented
five-frame therefore gives
$\lVert F-\partial G\rVert\ge1$, for every $G$.  At
$\varepsilon=0$ the constant line $\mathbb Rv$ is parallel and attains
zero error.  At nonzero $\varepsilon$ the full bundle $U=V$,
$\partial=\mathrm{id}$, $G=F$ attains zero error.  Top degree ensures
$G$ is closed; equivalently it follows from flatness and
$F=d_{\nabla_\varepsilon}B$.  Finally the zero auxiliary bundle
attains error one.  These bounds and constructions prove every branch
of the frontier.

On $D$ the connection is trivial and $F$ is line-valued, giving a
rank-one local completion.  Outside the support of $F$, rank zero
suffices.  Refining these sets to a good cover proves the stated local
bound.  The connection difference is the fixed smooth tensor
$\varepsilon N\eta$, so the convergence is smooth on the compact
manifold.  No noncompact coordinate growth is used.
\end{proof}

\subsection{A unitary compact model with an exact error--rank staircase}

The transport-optimality argument also holds for complex flat bundles,
with ranks taken over $\mathbb C$.  The next model has a parallel positive
Hermitian metric, so the large resource requirement does not depend on
nonunitary transport.

Keep the same closed five-torus, the closed one-form $\eta$ vanishing
near $\overline D$, and a real four-form $\beta$ compactly supported in
$D$, normalized by $\lVert d\beta\rVert_\infty=1$.  Take positive
weights $w_1,\ldots,w_d$ with $\sum_jw_j=1$, and put
\[
 V=T^5\times\mathbb C^d,\qquad
 v=(\sqrt{w_1},\ldots,\sqrt{w_d}),\qquad
 D_d=\operatorname{diag}(1,2,\ldots,d).
\]
Define the flat unitary connections
\[
 \nabla_\varepsilon=d+i\varepsilon D_d\eta,
 \qquad B=v\beta,\qquad F=v\,d\beta.
\]
Again $B$ and $F$ are fixed, and
$d_{\nabla_\varepsilon}B=F$ for every real $\varepsilon$.  The metric
is the fixed standard Hermitian metric.  Let
$w_{(1)}\ge\cdots\ge w_{(d)}$ be the weights in decreasing order.

\begin{theorem}[Exact unitary error--rank frontier]\label{thm:unitary-staircase}
Suppose $0<|\varepsilon|<2\pi/d$.  For $0\le r\le d$, the smallest
uniform curvature error achievable with auxiliary complex rank at most
$r$ is exactly
\[
 E_\varepsilon(r)=
 \left(1-\sum_{j=1}^r w_{(j)}\right)^{1/2}.
\]
The optimum is attained by the parallel coordinate subbundle associated
with the $r$ largest weights, with its inclusion as unary boundary and
the orthogonal projection of $F$ as five-form.  Thus for every $\tau\ge0$,
\[
 R_\varepsilon(\tau)
 =\min\left\{r\in\{0,\ldots,d\}:
          \sum_{j=1}^r w_{(j)}\ge1-\tau^2\right\}.
\]
For equal weights $w_j=1/d$, this becomes
\[
 R_\varepsilon(\tau)=
 \begin{cases}
 \lceil d(1-\tau^2)\rceil,&0\le\tau<1,\\
 0,&\tau\ge1.
 \end{cases}
\]
At $\varepsilon=0$, by contrast, $E_0(0)=1$ and $E_0(r)=0$ for every
$r\ge1$, so $R_0(\tau)=1$ when $0\le\tau<1$.
\end{theorem}

\begin{proof}
The unitary holonomy around the first circle is
\[
 H_\varepsilon
 =\operatorname{diag}(e^{-i\varepsilon},e^{-2i\varepsilon},
                      \ldots,e^{-di\varepsilon}).
\]
The displayed bound on $\varepsilon$ ensures that these eigenvalues
are distinct: any nonzero eigenphase difference has absolute value
less than $2\pi$.  Every invariant subspace of an operator with simple
complex spectrum is spanned by a subset of its eigenvectors.  To see
this directly, each eigenvector projection is a polynomial in
$H_\varepsilon$ by Lagrange interpolation.  An invariant subspace is
preserved by those projections and therefore decomposes into its
coordinate eigenspaces.

If $\partial:U\to V$ is parallel and
$\operatorname{rank}_{\mathbb C}U\le r$, its image at a base point of
$D$ is an invariant coordinate subspace of dimension at most $r$.
It is consequently orthogonal to the omitted components of $v$.
At a point where $\lVert d\beta\rVert=1$, every possible
$\partial G$ has error at least the distance of $v$ to that subspace,
which is
\[
 \left(\sum_{j\notin J}w_j\right)^{1/2}
 \ge\left(1-\sum_{j=1}^r w_{(j)}\right)^{1/2}.
\]
The parallel transports are diagonal in the global frame, so the
coordinate subspace selected by the $r$ largest weights extends to a
parallel subbundle.  Its orthogonal projection $P_J$ is parallel and
$G=P_JF$ is closed.  The inclusion gives error
\[
 \lVert F-P_JF\rVert_\infty
 =\lVert v-P_Jv\rVert\,\lVert d\beta\rVert_\infty,
\]
which equals the stated lower bound.  The rank frontier follows by
solving this exact error inequality.  At zero connection the constant
line $\mathbb Cv$ is parallel and carries all of $F$, proving the
remaining statements.
\end{proof}

\begin{corollary}[An arbitrarily large cost at a fixed nonzero tolerance]
For equal weights and uniform curvature tolerance $\tau=1/2$,
\[
 R_\varepsilon(1/2)=\lceil3d/4\rceil
 \quad(0<|\varepsilon|<2\pi/d),
 \qquad R_0(1/2)=1.
\]
This holds for the same fixed $B$ and $F$ on a closed five-dimensional
base, with unitary connections converging smoothly to the trivial
connection.  More generally, for every fixed $\tau<1$, the nonzero
connection cost grows linearly in $d$ while the zero-connection cost
is one.  The resource is complex coefficient rank; no conversion into
physical memory, energy, or sample complexity is implicit.
\end{corollary}

\subsection{A quantitative boundary: approximately parallel rank one}

Exact parallelity of the unary boundary is an essential assumption of the
resource frontier.  Its relaxation has a simple quantitative effect in
the unitary example.  Give the trivial complex line $U$ the flat unitary
connection
\[
 \nabla^U_{\varepsilon,\mu}=d+i\varepsilon\mu\eta,
 \qquad\mu\in\mathbb R,
\]
and take the constant isometric boundary $\partial(1)=v$ and the scalar
five-form $G=d\beta$.  Then $G$ is covariantly closed in dimension five
and
\[
 \partial G=F
\]
holds exactly.  However the compatibility tensor of the two connections
is
\[
 \nabla^{\operatorname{Hom}}\partial
 =i\varepsilon(D_d-\mu I)v\,\eta.
\]
Writing $\overline\lambda=\sum_jw_jj$, its uniform operator norm is
\begin{align*}
 \lVert\nabla^{\operatorname{Hom}}\partial\rVert_\infty
 &=|\varepsilon|\left(\sum_jw_j(j-\mu)^2\right)^{1/2}
                  \lVert\eta\rVert_\infty\\
 &=|\varepsilon|\left(
       \sum_jw_j(j-\overline\lambda)^2
          +(\mu-\overline\lambda)^2\right)^{1/2}
                  \lVert\eta\rVert_\infty.
\end{align*}
Consequently the weighted mean $\mu=\overline\lambda$ minimizes this
compatibility error within the displayed family of constant boundary
$\partial(1)=v$ and unitary line connections
$d+i\varepsilon\mu\eta$.  For equal weights, the minimum is
\[
 |\varepsilon|\sqrt{\frac{d^2-1}{12}}\,
                  \lVert\eta\rVert_\infty.
\]
This is not an optimization over all variable rank-one boundaries or
all possible line connections.

Thus a rank-one datum can cancel curvature exactly while its failure of
connection compatibility tends to zero with $\varepsilon$.  It is not
an admissible exact CE completion: in local graded coordinates the
square of the proposed differential contains the extra term
$(\nabla^{\operatorname{Hom}}\partial)v_{\rm aux}$.
The construction identifies the boundary between the proved
exact-parallel frontier and a different optimization problem that
allows compatibility error.  It does not extend the high-rank lower
bound to arbitrary approximate higher structures.

\paragraph{Contribution boundary.}
The construction uses standard flat transport, invariant subbundles,
Koszul contraction, and Vandermonde identities.  The precise optimal
completion problem, its canonical residual quotient without a flat
splitting, and the unbounded local-to-global and tolerance-preserving
rank jumps are the statements proved here.  They are not assertions
about unrestricted noncentral higher operations or a new general
theory of flat bundles.