  For any $Q\in\{\frac{3}{2},2,\frac{5}{2},3,\dotsc\}$, we establish a
structure theory for the class $\mathcal{S}_Q$ of stable codimension 1
stationary integral varifolds admitting no classical singularities of density
$<Q$. This theory comprises three main theorems which describe the nature of a
varifold $V\in \mathcal{S}_Q$ when: (i) $V$ is close to a flat disk of
multiplicity $Q$ (for integer $Q$); (ii) $V$ is close to a flat disk of integer
multiplicity $<Q$; and (iii) $V$ is close to a stationary cone with vertex
density $Q$ and support the union of 3 or more half-hyperplanes meeting along a
common axis. The main new result concerns (i) and gives in particular a
description of $V\in \mathcal{S}_Q$ near branch points of density $Q$. Results
concerning (ii) and (iii) directly follow from parts of the work [Wic14] (and
are reproduced in Part 2).
  These three theorems, taken with $Q=p/2$, are readily applicable to
codimension 1 rectifiable area minimising currents mod $p$ for any integer
$p\geq 2$, establishing local structure properties of such a current $T$ as
consequences of little, readily checked, information. Specifically, applying
case (i) it follows that, for even $p$, if $T$ has one tangent cone at an
interior point $y$ equal to an (oriented) hyperplane $P$ of multiplicity $p/2$,
then $P$ is the unique tangent cone at $y$, and $T$ near $y$ is given by the
graph of a $\frac{p}{2}$-valued function with $C^{1,\alpha}$ regularity in a
certain generalised sense. This settles a basic remaining open question in the
study of the local structure of such currents near points with planar tangent
cones, extending the cases $p=2$ and $p=4$ of the result which have been known
since the 1970's from the De Giorgi--Allard regularity theory ([All72]) and the
structure theory of White ([Whi79]) respectively.
