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\begin{document}

\title{Universal Support Theorems for Resilient Coverage}

\author{Deep Bhattacharjee$^{*}$\,\orcidlink{0000-0003-0466-750X}}
\email{d.bhattacharjee@erl-forschung.de}
\thanks{$^{*}$\,Corresponding author. Further addresses:
\nolinkurl{itsdeep@live.com}, \nolinkurl{egspl@proton.me}.}
\affiliation{Formerly, Electro-Gravitational Space Propulsion Laboratory
(EGSPL), Bhubaneswar, Odisha 751030, India}
\affiliation{Formerly, University of Calcutta (Heramba Chandra College),
23/49 Gariahat Road, Golpark, Kolkata 700029, West Bengal, India}

\author{Ushashi Bhattacharya\,\orcidlink{0009-0002-2254-3914}}
\affiliation{Dortmund, Germany}
\affiliation{National Taiwan University, Taipei, Taiwan}

\author{Sanjeevan Singha Roy\,\orcidlink{0000-0002-6148-1421}}
\email{sanjeevan9905@gmail.com}
\affiliation{Formerly, Birla Institute of Technology (BIT) Mesra, India}

\date{28 October 2026}

\begin{abstract}
\itshape
We prove that one finite set of points decides whether sensors cover a
polygonal target under every failure rule in which an extra working sensor
never hurts. The set is the same for every such rule, and on a face of
dimension $f$ the weakest point is fixed by $f+1$ sensors. The theorem holds in
any dimension for functions that share a convex part and differ by affine
terms. For sensing disks it gives the exact radius and power that keep the
target covered under each rule, as rational numbers, the full trade-off between radius and
budget, the smallest list of sensor groups whose loss breaks coverage, exact
tests under independent placement errors, and protection plans proved optimal
by a matching dual certificate.
\end{abstract}

\maketitle

\begingroup
\renewcommand\thefootnote{}\footnotesize
\footnotetext{\footnotesize\emph{2020 Mathematics Subject Classification.}
Primary 68U05; Secondary 52A41, 90B25, 05C65, 55U10.
\emph{Keywords.} support theorem; sensor coverage; monotone failure rules;
resilience certificates.}
\footnotetext{\footnotesize The \LaTeX{} source and the finite checks of the
worked examples are archived on Zenodo, release \releaseTag,
\href{https://doi.org/\zenodoCodeID}{doi:\zenodoCodeID}.}
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\end{document}