| \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. |
|
|
