diff --git a/src/math_spec/program.py b/src/math_spec/program.py
index 7567aa09..52426fff 100644
--- a/src/math_spec/program.py
+++ b/src/math_spec/program.py
@@ -267,10 +267,11 @@ class GroupSum(Expression):
 
     ``walks`` says, per relation, which columns are consumed, which produced
     and which joined on, and is the one fact the node holds: ``coordinate``
-    names the relations, ``over`` is the dims every walk consumes and ``into``
-    the dims they produce, in walk order, so that several coordinates are
-    one grouping into a product of targets, consumed in a single join. The
-    result replaces every dim in ``over`` with every dim in ``into``. The
+    names the relations, ``over`` is the dims every walk consumes, ``into``
+    the dims they produce, in walk order, and ``joined`` the dims they join
+    on, so that several coordinates are one grouping into a product of
+    targets, consumed in a single join. The result replaces every dim in
+    ``over`` with every dim in ``into`` and keeps every dim in ``joined``. The
     join keys on the consumed columns and every joined column, and on a
     produced column too where the operand already carries its dimension.
     """
@@ -290,6 +291,11 @@ def over(self) -> tuple[str, ...]:
     def into(self) -> tuple[str, ...]:
         return tuple(dim for walk in self.walks for dim in walk.produced_dims)
 
+    @property
+    def joined(self) -> tuple[str, ...]:
+        """The dims the walks join on, each once — the key columns neither consumed nor produced, which the operand carries."""
+        return _joined_dims(self.walks)
+
 
 @dataclass(frozen=True)
 class At(Expression):
@@ -301,7 +307,7 @@ class At(Expression):
     (``Walk.is_function_read``). The join fans out, many ``over`` tuples
     sharing one ``into`` tuple — at each coordinate of the joined columns,
     which the operand carries and the result keeps. As on
-    :class:`GroupSum`, ``walks`` is the fact and the three are read off it.
+    :class:`GroupSum`, ``walks`` is the fact and the four are read off it.
     """
 
     operand: ExpressionNode
@@ -319,6 +325,16 @@ def over(self) -> tuple[str, ...]:
     def into(self) -> tuple[str, ...]:
         return tuple(dim for walk in self.walks for dim in walk.consumed_dims)
 
+    @property
+    def joined(self) -> tuple[str, ...]:
+        """The dims the walks join on, each once — the key columns neither consumed nor produced, which the operand carries."""
+        return _joined_dims(self.walks)
+
+
+def _joined_dims(walks: tuple[Walk, ...]) -> tuple[str, ...]:
+    """The dims *walks* join on, each once, in walk order — the rule :attr:`GroupSum.joined` and :attr:`At.joined` share."""
+    return tuple(dict.fromkeys(dim for walk in walks for dim in walk.joined_dims))
+
 
 @dataclass(frozen=True)
 class Translate(Expression):
diff --git a/tests/test_lowering.py b/tests/test_lowering.py
index 84ddb1f8..300833a4 100644
--- a/tests/test_lowering.py
+++ b/tests/test_lowering.py
@@ -521,9 +521,12 @@ def test_a_relation_lowers_with_the_walk_each_call_takes():
         'a grouped sum names the column it consumes, the one it produces and the one it joins on'
     )
     assert isinstance(zonal, GroupSum)
-    assert (zonal.over, zonal.into, zonal.coordinate) == (('generator',), ('zone',), ('zone_of',)), (
-        'the dims a consumer reads are read off the walk'
-    )
+    assert (zonal.over, zonal.into, zonal.joined, zonal.coordinate) == (
+        ('generator',),
+        ('zone',),
+        ('snapshot',),
+        ('zone_of',),
+    ), 'the dims a consumer reads are read off the walk'
     assert program.constraints['history'].lhs == GroupSum(
         Variable('p'), walks=(Walk(declared, ('snapshot',), ('zone',), ('generator',)),)
     ), 'the same table walked from its other key column'
@@ -532,8 +535,8 @@ def test_a_relation_lowers_with_the_walk_each_call_takes():
         'and its adjoint consumes the value column and produces the key column'
     )
     assert isinstance(priced, At)
-    assert (priced.over, priced.into) == (('generator',), ('zone',)), (
-        'an at produces the fine dims and consumes the coarse'
+    assert (priced.over, priced.into, priced.joined) == (('generator',), ('zone',), ('snapshot',)), (
+        'an at produces the fine dims, consumes the coarse, and joins on the rest of the key'
     )
     p_where = program.variable('p').where
     assert p_where is not None