\section*{Identifiability and Convexification of a Band-Limited Unassigned Distance-Geometry Problem}

Let \(X=(x_1,\dots,x_N)\in(\mathbb{R}^3)^N\) carry labels \(e_i\in E\) and weights
\(b_i>0\). Quotient by rigid motions (translations and rotations, but
\emph{not} reflections):
\[
  \mathcal{X}=(\mathbb{R}^3)^N/E(3).
\]
The weighted pair-distance measure and its measurement are
\[
  \rho_X=\sum_{i\lt j} w_{ij}\,\delta_{\|x_i-x_j\|},\quad w_{ij}=b_i b_j\ge 0,
  \qquad
  F(X)=R_{Q_{\max},\sigma}\,\rho_X\in Y,
\]
where \(R_{Q_{\max},\sigma}\) is a known compact smoothing operator: a
band-limit \(Q_{\max}\) (real-space resolution \(\Delta r\approx\pi/Q_{\max}\))
followed by convolution of width \(\sigma>0\). Its \emph{information capacity}
\[
  m_{\mathrm{eff}}(Q_{\max},\sigma)
  =\dim\operatorname{span}\{R_{Q_{\max},\sigma}\delta_t:t\in[0,R_{\max}]\}
  \approx\lceil R_{\max}/\Delta r\rceil
\]
is finite, with \(m_{\mathrm{eff}}\to\infty\) as \(Q_{\max}\to\infty\)
(exact-distance limit) and \(m_{\mathrm{eff}}\to 0\) as \(Q_{\max}\to 0\), and
\[
  \operatorname{rank}(DF_X)\le\min(d_C,m_{\mathrm{eff}}).
\]
A \emph{prior} is a closed \(C\subset\mathcal{X}\), a smooth manifold of
dimension \(d_C\) realizable in internal coordinates (a graph of fixed edge
lengths and angles with \(d_C\) free dihedrals). \(F\) is an instance of the
unassigned distance geometry problem \cite{Billinge2018}; in the motivating
application \(C\) encodes chemical constraints, which play no formal role below.

\paragraph{Known facts.}
\begin{enumerate}
\item For \(R=\mathrm{id}\), injectivity of \(\rho\) up to rigid motion is the
global-rigidity problem; in \(3\)D generic global rigidity requires redundant
rigidity and stressability and is subtler than edge-counting, and special
configurations are generically \emph{not} globally rigid (flip ambiguities
persist).
\item In \(1\)D (turnpike/beltway) with unassigned noisy distances, projected
gradient descent with a spectral initializer converges locally linearly to the
global optimizer under explicit conditions \cite{HuangDokmanic2021}; for
integer sets, sparsity yields polynomial-time recovery up to shift and reversal
with high probability \cite{JaganathanHassibi2012}.
\item A configuration and its mirror image have identical \(F(X)\), so
identifiability holds at best modulo \(\mathbb{Z}_2\) (fiber \(K\ge2\) for
chiral configurations).
\end{enumerate}

\paragraph{Open problem.}
For \(y=F(X_\star)\) and \(\mathcal{L}_y(X)=\|F(X)-y\|_Y^2|_C\):
\begin{itemize}
\item[\textbf{(P1)}] \emph{Identifiability:} \(F|_C\) injective modulo
reflection (fibers of size \(1\), or \(2\) = the mirror pair).
\item[\textbf{(P2)}] \emph{Local identifiability:} the Fisher operator
\(I_X=(DF_X)^{*}(DF_X)\) has full rank on \(T_XC\) for all \(X\in C\).
\item[\textbf{(P3)}] \emph{\(\varepsilon\)-convexity:} \(\mathcal{L}_y|_C\) is
geodesically \(\varepsilon\)-convex on the component of \(X_\star\) with a
unique stationary point (no spurious local minima).
\end{itemize}

\begin{problem}
Characterize the priors \(C\), as a function of \(m_{\mathrm{eff}}\), for which
\(\text{(P1)}\Leftrightarrow\text{(P2)}\Leftrightarrow\text{(P3)}\) hold for
generic \(X_\star\in C\), and find the \emph{minimal} such prior. In particular:
\begin{enumerate}
\item[\textbf{(a)}] Prove or disprove a sharp threshold at
\(m_{\mathrm{eff}}=d_C\) (modulo reflection), with counterexamples at
\(m_{\mathrm{eff}}=d_C-1\).
\item[\textbf{(b)}] Find the maximal admissible \(d_C\) (least external
information) and the trade-off curve \(d_C^{\,*}(Q_{\max})\).
\item[\textbf{(c)}] Show the threshold coincides with vanishing of
\(I_{\mathrm{deficit}}=h(X\mid C)-I(F(X);X\mid C)\) (up to \(\log 2\)).
\end{enumerate}
\end{problem}

This is generic-global-rigidity theory for band-limited, weighted, soft data
on a constraint manifold - the regime open at the intersection of
\cite{HuangDokmanic2021,JaganathanHassibi2012,Billinge2018}.

\paragraph{Remark.}
For more details on the chemical side of the problem, see
\url{https://github.com/grebenyyk/pdf-convexification}.

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Sets from Distance Distributions}, IEEE Trans.\ Signal Process.\ \textbf{69}
(2021), 1811--1827. arXiv:1804.02465.
\bibitem{JaganathanHassibi2012} K.~Jaganathan, B.~Hassibi, \emph{Reconstruction
of Integers from Pairwise Distances}, arXiv:1212.2386 (2012).
\bibitem{Billinge2018} S.~J.~L.~Billinge, P.~M.~Duxbury, D.~S.~Goncalves,
C.~Lavor, A.~Mucherino, \emph{Recent results on assigned and unassigned
distance geometry with applications to protein molecules and nanostructures},
Ann.\ Oper.\ Res.\ \textbf{271}(1) (2018), 161--203. DOI:10.1007/s10479-018-2989-6.
\bibitem{Billinge2004} S.~J.~L.~Billinge, M.~G.~Kanatzidis, \emph{Beyond
crystallography: the study of disorder, nanocrystallinity and
crystallographically challenged materials with pair distribution functions},
Chem.\ Commun.\ (2004), 749--760. DOI:10.1039/b311073n.
\end{thebibliography}