\section{Introduction}\label{sec:intro}

Ashoke Sen has published conjectures for more than thirty years, and most of them were made in the same way: a computation in one regime, a symmetry that carries it to a regime where nothing can be computed, and a statement about that second regime sharp enough to be refuted. The statement that the heterotic string on a six-torus is invariant under $SL(2,\ZZ)$ required a bound state of two monopoles that nobody had seen \cite{Sen94duality,Sen94dyon}. The statement that the tachyon of an unstable D-brane rolls to a vacuum without open strings, at an energy equal to the tension of the brane \cite{Sen98bab,Sen99univ}, required an exact solution of open string field theory that did not exist for seven years. The statement that the Fourier coefficients of a Siegel modular form have a definite sign on a cone of charges \cite{Sen11positivity} required an argument about the angular momentum of a black hole. These are predictions in the strict sense, and they can be checked.

This paper examines a selected catalogue, not the complete publications of one author. For each entry we separate the original hypotheses, the result actually derived, and subsequent tests or proofs. An unproved conjecture is not a false theorem. A counterexample to a statement with an assumption removed is not a counterexample to the statement with that assumption retained.

The labels in the catalogue describe scope. \st{Proved} means established in the mathematical or perturbative framework identified in the corresponding section; it does not assert a nonperturbative construction of string theory. \st{Confirmed} means supported by the cited tests without a general proof supplied here. \st{Scope limited} means that an unrestricted restatement loses a necessary hypothesis or exceeds the approximation. The two \st{heuristic} entries are proposals rather than theorem statements. The catalogue is a reading guide, not a numerical ranking of scientific merit. Claims already settled or qualified in the cited literature are included as context, not presented as newly discovered errors. Three highlights mark what Sen's statements lack, and each is used for nothing else. A \wrong{rose highlight} marks an error: a formula or statement printed by Sen that is false as printed, while his conclusion survives its correction. Text in \fatal{burnt orange} marks a fatal error: a conclusion of Sen that fails under his own hypotheses. A \gap{yellow highlight} marks what is left unproved: a step, hypothesis or identification on which a claim rests and which neither Sen nor the later literature has proved, or a statement beyond his claims that the text leaves open. Every statement of the catalogue was compared with Sen's own text. None is false as printed and none fails under its hypotheses, so the paper has no rose and no burnt orange passage. The two statements counted as false in the first version are explained in the Note, and the two misprints corrected in Section~\ref{sec:positivity} and Appendix~\ref{app:computation} are in the tables of Chattopadhyaya and David, not in Sen's papers. Table~\ref{tab:unproved} collects the yellow passages. The highlights record the state of the mathematics and attribute no misconduct to an author.

Two exact tests receive detailed treatment. For the freely acting toroidal orbifolds, the negative full dyon indices were already found by Chattopadhyaya and David \cite{CD19}; their later hair analysis \cite{CD21} explains why the horizon index is a different quantity. Our contribution is a reproducible finite computation with the linear charge-cone inequality of \cite[Eq.~(3.9)]{Sen11positivity} retained. The corrected scan contains thirty negative triples in the stated range. The original twenty-one listed triples have a separate finite Lean certificate; the larger scan is checked by the supplied exact Python program. No inference from thirty charges to infinitely many charges is made without the published Fourier--Jacobi argument.

For tachyon matter, the characteristic calculation applies to the pressureless effective equations. Focusing is controlled by a positive eigenvalue of the Hessian of the initial tachyon profile on the branch $\partial_tT>0$. Concave profiles have no forward local focusing. Bounded nonconstant profiles have positive curvature somewhere and hence a finite focal time on some characteristic. The first time is a minimum only under an attainment hypothesis. This describes the limit of a classical effective description, not a counterexample to the homogeneous solution in \cite{Sen02rolling,Sen02matter}. Indeed \cite{Sen02field} also gives static dust with arbitrary nonconstant density, a useful example that must be retained.

The F-theory calculation similarly requires two spaces to be kept distinct: the base, where $\eta^2-h\chi=0$ can be smooth, and the orientifold double cover, where its pullback has a double locus. Non-Higgsable gauge symmetry alone is not an obstruction to weak coupling: $I_0^*$ is an exception, as the explicit family below illustrates. Two of Sen's statements are proved here in general form. Every degeneration of F-theory on an elliptic Calabi--Yau variety that is weakly coupled at the generic point of the base is, after a base change of order at most two and a rescaling, Sen's family, and one exists exactly when $-2K_B$ has a section (Theorem~\ref{thm:normalform}, Corollary~\ref{cor:existence}). For two-charge small black holes the entropy of every curvature-corrected action is fixed by one critical value; the Gauss--Bonnet truncation gives the ratio of Corollary~\ref{cor:gb}, equal to one only in $D=4,5$, and the Lovelock completion of Prester is the only one with the statistical entropy in every dimension (Theorems~\ref{thm:twocharge}, \ref{thm:prester}, \ref{thm:rigidity}). Full proofs are in Appendix~\ref{app:closures}.

\paragraphx{Plan.} Section~\ref{sec:catalogue} is the catalogue, Tables~\ref{tab:catalogue}, \ref{tab:catalogue_tachyons}, \ref{tab:catalogue2} and~\ref{tab:catalogue_sft}, with the verdict of every claim and the place where it is treated. Sections~\ref{sec:sduality} and~\ref{sec:dyons} treat duality: the monopole conjectures and the counting of dyons, with the exact verification of the wall-crossing formula. Section~\ref{sec:positivity} treats the positivity conjecture and its domain of applicability. Sections~\ref{sec:tachyon} to~\ref{sec:nonbps} treat the tachyon conjectures, the rolling tachyon with the caustic theorem, and the non-BPS branes. Section~\ref{sec:senlimit} treats the Sen limit, Sections~\ref{sec:entropy} and~\ref{sec:quantum} the entropy function, the small black holes and the quantum entropy function, Section~\ref{sec:sft} string field theory, D-instantons, the soft theorems and matrix theory. Section~\ref{sec:conclusion} records the remaining questions. Appendix~\ref{app:computation} describes the computation of the Fourier coefficients and Appendix~\ref{app:dust} completes the proof of the caustic theorem.


The primary-source register in Appendix~\ref{app:scopechecks} identifies the passages used for the principal scope corrections. Appendix~\ref{app:infraredlimit} constructs an explicit finite-energy soft model. Appendix~\ref{app:softremainder} derives a quantitative subtracted tree-emission bound and identifies the uniform loop estimates still needed for transfer to the complete theory. Literature-status statements concern the sources examined, rather than a claim to have read every citing paper.