\section*{Foreword}\label{foreword:section}

With the exception of this foreword, the present paper is entirely AI-generated by
ChatGPT Astra 6, in response to prompts provided by the human authors. Some earlier
input came also from Claude Fable.

In \cite{KM-web}, an $\mathrm{SO}(3)$ instanton homology of webs was defined, and it was
conjectured there that the dimension of this instanton homology was equal to the
number of Tait colorings of the web if the web was planar. The correctness of this
conjecture would lead to a proof of the four-color theorem, independent of
the work of Appel and Haken. Since 2013, an important test case has been the
dodecahedron web. It has 60 Tait colorings. The dimension of its instanton
homology was eventually shown to be bounded below by 60 and bounded above by 64 (with
the latter bound being due to David Boozer in unpublished work). The paper here
presents a proposed proof that the instanton homology has dimension 64, contradicting
the conjecture.

The AI-generated text uses the term ``proposed theorem'' not because any gaps or
weaknesses in the argument are known, but because the paper has not been subject to
any human expert review. We use the same language, and for the same reason. In particular,
the authors make no claim to have examined the argument in any detail, though its
architecture is clear. The authors believe that a human-readable version of the
proof, written to high journal-ready standards, could be considerably shorter, and
the intention is to produce such a version in due course. The work in its present
form is being provided here because timeliness seems valuable.

The argument begins with algebraic steps which reduce the calculation of the
dimension to determining whether three mod-2 scalars vanish. Each scalar is a count of
instantons on a closed foam equipped with a 1-parameter family of metrics. All three
scalars are shown to vanish by demonstrating that the relevant instantons would live
in moduli spaces with negative action.

The paper is accompanied by the LaTeX source for this foreworded version, as well as
the original source from ChatGPT. The latter is in a folder which includes also
ChatGPT's extensive calculations which underlie the algebraic parts of the
argument.

\subparagraph{Acknowledgements.} The work of the first author was
supported by NSF grant
DMS-2304877. Both
authors were supported by Simons Foundation Awards \#994330 and \#14760-09,
\emph{Simons Collaboration on New Structures in Low-Dimensional
Topology}.
\clearpage