  This paper, and its companion [BCV24], are devoted to a negative resolution
of the Aldous--Lyons Conjecture [AL07, Ald07]. This conjecture, originated in
probability theory, is well known (cf. [Gel18]) to be equivalent to the
statement that every invariant random subgroup of the free group is co-sofic.
We disprove this last statement.
  In this part we introduce subgroup tests. These tests are finite
distributions over continuous functions from the space of subgroups of the free
group to $\{0,1\}$. Subgroup tests provide a general framework in which one can
study invariant random subgroups of the free group. Classical notions such as
group soficity and group stability arise naturally in this framework. By the
correspondence between subgroups of the free group and Schreier graphs, one can
view subgroup tests as a property testing model for certain edge-labeled
graphs. This correspondence also provides the connection to random networks.
  Subgroup tests have values, which are their asymptotic optimal expectations
when integrated against co-sofic invariant random subgroups. Our first main
result is that, if every invariant random subgroup of the free group is
co-sofic, then one can approximate the value of a subgroup test up to any
positive additive constant. Our second main result is an essentially value
preserving correspondence between certain non-local games and subgroup tests.
By composing this correspondence with a stronger variant of the reduction in
MIP*=RE [JNV+21], proved in the companion paper [BCV24], we deduce that
approximating the sofic value of a subgroup test is as hard as the Halting
Problem, and in particular, undecidable. The combination of our two main
results proves the existence of non co-sofic invariant random subgroups of the
free group.
