We construct counterexamples to classical calculus facts such as the Inverse
and Implicit Function Theorems in Scale Calculus -- a generalization of
Multivariable Calculus to infinite dimensional vector spaces in which the
reparameterization maps relevant to Symplectic Geometry are smooth. Scale
Calculus is a cornerstone of Polyfold Theory, which was introduced by
Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli
spaces of pseudoholomorphic curves. We show how the novel nonlinear
scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit
Function Theorems, by formally establishing an often implicitly used fact: The
differentials of basic germs -- the local models for scale-Fredholm maps --
vary continuously in the space of bounded operators when the base point
changes. We moreover demonstrate that this continuity holds only in specific
coordinates, by constructing an example of a scale-diffeomorphism and
scale-Fredholm map with discontinuous differentials. This justifies the high
technical complexity in the foundations of Polyfold Theory.Comment: published in PNAS, final versio
