| \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. |
|
|
