For any
Q
∈
{
3
2
,
2
,
5
2
,
3
,
…
}
Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \}
, we establish a structure theory for the class
S
Q
\mathcal {S}_Q
of stable codimension 1 stationary integral varifolds admitting no classical singularities of density
>
Q
>Q
. This theory comprises three main theorems which describe the nature of a varifold
V
∈
S
Q
V\in \mathcal {S}_Q
when: (i)
V
V
is close to a flat disk of multiplicity
Q
Q
(for integer
Q
Q
); (ii)
V
V
is close to a flat disk of integer multiplicity
>
Q
>Q
; and (iii)
V
V
is close to a stationary cone with vertex density
Q
Q
and supports 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
∈
S
Q
V\in \mathcal {S}_Q
near branch points of density
Q
Q
. Results concerning (ii) and (iii) directly follow from parts of previous work of the second author [Ann. of Math. (2) 179 (2014), pp. 843–1007].
These three theorems, taken with
Q
=
p
/
2
Q=p/2
, are readily applicable to codimension 1 rectifiable area minimising currents mod
p
p
for any integer
p
≥
2
p\geq 2
, establishing local structure properties of such a current
T
T
as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even
p
p
, if
T
T
has one tangent cone at an interior point
y
y
equal to an (oriented) hyperplane
P
P
of multiplicity
p
/
2
p/2
, then
P
P
is the unique tangent cone at
y
y
, and
T
T
near
y
y
is given by the graph of a
p
2
\frac {p}{2}
-valued function with
C
1
,
α
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
p=2
and
p
=
4
p=4
of the result which have been known since the 1970’s from the De Giorgi–Allard regularity theory [Ann. of Math. (2) 95 (1972), pp. 417–491] [Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, 1961] and the structure theory of White [Invent. Math. 53 (1979), pp. 45–58] respectively. If
P
P
has multiplicity
>
p
/
2
> p/2
(for
p
p
even or odd), it follows from case (ii) that
T
T
is smoothly embedded near
y
y
, recovering a second well-known theorem of White [Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1986, pp. 413–427]. Finally, the main structure results obtained recently by De Lellis–Hirsch–Marchese–Spolaor–Stuvard [arXiv:2105.08135, 2021] for such currents
T
T
all follow from case (iii).