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\hypersetup{pdftitle={Proving Conjectures of Ashoke Sen: F-Theory Limits, Small Black Holes and Dyon Indices},pdfauthor={Deep Bhattacharjee}}

\title[Proving Conjectures of Ashoke Sen]{Proving Conjectures of Ashoke Sen:\\ F-Theory Limits, Small Black Holes and Dyon Indices}

\author{Deep Bhattacharjee\,\orcidlink{0000-0003-0466-750X}}
\address{Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India}
\address{Formerly, University of Calcutta (Heramba Chandra College), 23/49 Gariahat Road, Golpark, Kolkata 700029, West Bengal, India}
\email{d.bhattacharjee@erl-forschung.de}
\email{itsdeep@live.com}
\email{egspl@proton.me (GitHub-related contact)}
\urladdr{https://github.com/creelie}
\thanks{\sloppy Corresponding author: Deep Bhattacharjee. Email: \href{mailto:d.bhattacharjee@erl-forschung.de}{d.bhattacharjee@erl-forschung.de}, \href{mailto:itsdeep@live.com}{itsdeep@live.com}, \href{mailto:egspl@proton.me}{egspl@proton.me} (GitHub-related contact); GitHub \href{https://github.com/creelie}{creelie}; ORCID \href{https://orcid.org/0000-0003-0466-750X}{0000-0003-0466-750X}. Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India, and formerly, University of Calcutta (Heramba Chandra College), 23/49 Gariahat Road, Golpark, Kolkata 700029, West Bengal, India.}

\subjclass[2020]{81T30; 81T13; 83C57; 11F46; 35L67; 53C26}
\keywords{S-duality; tachyon condensation; black hole entropy; Siegel modular forms}
\date{28 October 2026}

\begin{document}

\begin{abstract}
\itshape
We examine thirty-six statements of Ashoke Sen on duality, dyon indices, tachyon condensation, F-theory and black hole entropy, and decide which are proved, which are supported but unproved, and where each stops. We prove that every degeneration of F-theory on an elliptic Calabi--Yau variety that is weakly coupled at the generic point of the base is, after base change and rescaling, Sen's family, and that one exists exactly when $-2K_B$ has a section. We prove that the entropy of a two-charge small black hole is fixed by one critical value for every curvature-corrected action, so that the Gauss--Bonnet truncation matches the microscopic count only in four and five dimensions and Prester's Lovelock completion is the unique one that matches it in every dimension. Exact product expansions reproduce the known negative dyon indices, and the tachyon-dust caustic concerns the pressureless equations, not the rolling solution.
\end{abstract}

\maketitle

\begin{quote}
\small\sloppy
\noindent\textbf{Note.} This paper was first made public on 25 October 2026, under the title \emph{Conjectures of Ashoke Sen: Scope, Exact Tests and Analytic Limits}, as release \texttt{v1.0.0} of the repository \url{https://github.com/creelie/paper_11} (concept DOI \href{https://doi.org/10.5281/zenodo.23652472}{10.5281/zenodo.23652472}, version DOI \href{https://doi.org/10.5281/zenodo.23652473}{10.5281/zenodo.23652473}); it was revised there on 26 October 2026 as \texttt{v1.0.1} (\href{https://doi.org/10.5281/zenodo.23652641}{10.5281/zenodo.23652641}) and \texttt{v1.1.0} (\href{https://doi.org/10.5281/zenodo.23656435}{10.5281/zenodo.23656435}). The first version counted two of the thirty-six statements as false and twenty-one negative charge triples; it used a charge cone with the discriminant in place of the linear inequality of Sen's equation (3.9), and its dust theorem replaced a supremum by a maximum without an attainment hypothesis. The later versions corrected the cone, extended the scan to thirty triples, restated the two verdicts as limits of scope, since the negative indices are full indices that the hair analysis of Chattopadhyaya and David already accounts for and the caustic concerns the pressureless effective equations rather than the homogeneous solution, and added the infrared appendices. The same mathematics was set in a single-column form for submission as release \texttt{v2.0.0} of the same repository (\href{https://doi.org/10.5281/zenodo.23669248}{10.5281/zenodo.23669248}); the two misprints of Chattopadhyaya and David recorded in Section~\ref{sec:positivity} and Appendix~\ref{app:computation} were added in release \texttt{v2.0.1} (\href{https://doi.org/10.5281/zenodo.23670581}{10.5281/zenodo.23670581}), with the exact program that computes every table, the Lean~4 certificates of the finite statements, and the scripts that check the explicit constants of the appendices. Release \texttt{v2.1.0} (\href{https://doi.org/10.5281/zenodo.23746042}{10.5281/zenodo.23746042}) proved claims~23 and~26, which the earlier versions recorded as limits of scope: Theorem~\ref{thm:normalform} with Corollaries~\ref{cor:existence} and~\ref{cor:k3cover}, and Theorems~\ref{thm:twocharge}, \ref{thm:prester} and~\ref{thm:rigidity}, with full proofs in Appendix~\ref{app:closures}. Release \texttt{v2.2.0} (\href{https://doi.org/10.5281/zenodo.23747112}{10.5281/zenodo.23747112}) defined the three highlights of the introduction, listed in Table~\ref{tab:unproved} what is left unproved, proved Theorem~\ref{thm:centralcharge} on the central charges of the heterotic string in $D\ge6$, and corrected the footnote on Sen's normalisation of the Gauss--Bonnet coupling, whose equation~(3.15) sets $C=1$. This version, release \texttt{v2.2.1} (\href{https://doi.org/10.5281/zenodo.23749469}{10.5281/zenodo.23749469}), corrects errors of that release, none of them in a statement of Sen. Parts~(iii) and~(iv) of Corollary~\ref{cor:k3cover} now assume a base-point-free anticanonical system, without which part~(iii) fails on $\FF_{12}$. The base change of Theorem~\ref{thm:normalform} has order at most two. The critical radii of Theorem~\ref{thm:descent}(i) are $1/\sqrt2$ for the brane pair and $\sqrt2$ for the non-BPS brane. The caption of Figure~\ref{fig:weakcoupling}, the endpoint argument for Corollary~\ref{cor:gb}, the indexing of $d_h$ in Section~\ref{sec:dyons}, the last sentence of the verdict on claim~18 and the scope of the condition $R_*=\alpha'$ are corrected as well.
\end{quote}

\input{sections/sec01_intro}
\input{sections/sec02_catalogue}
\input{sections/sec03_sduality}
\input{sections/sec04_dyons}
\input{sections/sec05_positivity}
\input{sections/sec06_tachyon}
\input{sections/sec07_rolling}
\input{sections/sec08_nonbps}
\input{sections/sec09_senlimit}
\input{sections/sec10_entropy}
\input{sections/sec11_quantum}
\input{sections/sec12_sft}
\input{sections/sec13_conclusion}

\appendix
\input{sections/appA_computation}
\input{sections/appB_dust}
\input{sections/appC_scopechecks}
\input{sections/appD_perturbative}
\input{sections/appE_infrared}
\input{sections/appF_softremainder}
\input{sections/appG_closures}

\input{sections/declarations}
\begingroup\sloppy\emergencystretch=5em\input{sections/bibliography}\endgroup

\end{document}