diff --git a/CHANGELOG.md b/CHANGELOG.md
index 076db77b..ad5c7285 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -12,6 +12,7 @@ it releases that version ([RELEASING.md](https://github.com/energy-models/mathsp
 
 ## Upcoming version
 
+- feat(program): the axis a join opens names the relation column it stands for ([#665](https://github.com/energy-models/mathspec/pull/665))
 - feat(language): a call names the columns of a relation as relation[column], and a sum through a relation is a sum over the axes its join opens ([#664](https://github.com/energy-models/mathspec/pull/664))
 - feat(language): a parameter value that is null or NaN is refused when the data is attached ([#788](https://github.com/energy-models/mathspec/pull/788))
 - fix(language): a where string names several columns in at's over= and into=, as an expression does ([#782](https://github.com/energy-models/mathspec/pull/782))
diff --git a/docs/about/relations-as-linear-maps.md b/docs/about/relations-as-linear-maps.md
index 1ac29638..e9530465 100644
--- a/docs/about/relations-as-linear-maps.md
+++ b/docs/about/relations-as-linear-maps.md
@@ -133,9 +133,10 @@ $`x`$ over generators and $`y`$ over buses,
 
 which is why the program lowers `at` to a `Join` node and `sum(by=)` to the
 same `Join` under a `Sum`. The join keeps the column it sums away as an axis
-named for the relation's column, `gen_zone.generator`, and the `Sum` stands
-over that axis. So a map into its own dimension, which drops and adds one
-dimension, still has two axes between the join and the sum. A bare relation has
+of its own, `Axis('generator', Column('gen_zone', 'generator'))`: it runs over
+`generator` and stands for the relation's column. The `Sum` stands over that
+axis. So a map into its own dimension, which drops and adds one dimension,
+still has two axes between the join and the sum. A bare relation has
 the same matrix without the functional claim. A column of $`M`$ may hold several
 ones, so the sum fans out and no group is one row. That is why `at` through a bare relation is refused.
 
diff --git a/docs/contributing.md b/docs/contributing.md
index cc36caeb..fc9dcbe7 100644
--- a/docs/contributing.md
+++ b/docs/contributing.md
@@ -111,7 +111,7 @@ to two nodes, so the file's spelling cannot decide the name.
 
 | File verb          | Node                | What the node names                                  |
 | ------------------ | ------------------- | ---------------------------------------------------- |
-| `sum(over=)`       | `Sum`               | dims removed from the result                         |
+| `sum(over=)`       | `Sum`               | axes removed from the result                         |
 | `sum(by=)`         | `Sum` over a `Join` | a join, and the sum over the axes it opens           |
 | `at(by=)`          | `Join`              | a join whose groups are one row, with no sum over it |
 | `shift(along=)`    | `Translate`         | a re-index along one dimension                       |
diff --git a/src/mathspec/_expression_resolver.py b/src/mathspec/_expression_resolver.py
index ddd8d25c..560776d8 100644
--- a/src/mathspec/_expression_resolver.py
+++ b/src/mathspec/_expression_resolver.py
@@ -43,6 +43,7 @@
 )
 from mathspec.program import (
     Add,
+    Axis,
     Constant,
     Divide,
     Dual,
@@ -295,14 +296,14 @@ def _sum(
             if over is None:
                 return self._bare_sum(operand)
             assert not isinstance(over, ColumnsNode), 'over=relation[column] is read only beside by='
-            return Sum(operand, over)
+            return Sum(operand, tuple(Axis(d) for d in over))
         assert over is not None, 'the call shape requires over= beside by='
         grouping = self._grouping(columns, over)
         if grouping is None:
             return None
         join, plain = grouping
         grouped = Sum(Join(operand, join), join.axes)
-        return Sum(grouped, plain) if plain else grouped
+        return Sum(grouped, tuple(Axis(d) for d in plain)) if plain else grouped
 
     def _at(self, operand: Expression, columns: ColumnsNode) -> Expression | None:
         """``at(x, by=relation[column])``: *operand* read at the value of *columns* each row of the relation holds."""
@@ -356,7 +357,7 @@ def _bare_sum(self, operand: Expression) -> Expression | None:
                 f'scalar. Drop the sum.'
             )
             return None
-        return Sum(operand, tuple(sorted(inner)))
+        return Sum(operand, tuple(Axis(d) for d in sorted(inner)))
 
     def _edge_fits(self, operand: Expression, offset: int | str, *, wrap: bool, fill: float | None) -> bool:
         """What a ``shift``'s ``edge=`` may say, and where saying nothing is an answer.
diff --git a/src/mathspec/dimensions.py b/src/mathspec/dimensions.py
index 825356f8..64b176f3 100644
--- a/src/mathspec/dimensions.py
+++ b/src/mathspec/dimensions.py
@@ -85,10 +85,11 @@ def dims_of(node: Expression, schema: Spec, context: str) -> frozenset[str]:
     if isinstance(node, Negate | Add | Multiply | Power | Divide):
         return frozenset().union(*(dims_of(child, schema, context) for child in children(node)))
 
-    inner = dims_of(node.operand, schema, context)
     if isinstance(node, Sum):
-        return _sum_dims(node, inner, context)
+        return _sum_dims(node, schema, context)
+    inner = dims_of(node.operand, schema, context)
     if isinstance(node, Join):
+        assert not node.columns.axes, 'a join that opens axes is read only by the sum that closes them'
         return join_dims(node.columns, inner, context, 'the expression')
     if isinstance(node, Translate | WindowSum):
         return _translation_dims(node, inner, schema, context)
@@ -112,21 +113,28 @@ def _not_carried(context: str, call: str, inner: frozenset[str], rewrite: str) -
     )
 
 
-def _sum_dims(node: Sum, inner: frozenset[str], context: str) -> frozenset[str]:
-    """``sum`` reduces each named dim away, so the operand carries every one."""
+def _sum_dims(node: Sum, schema: Spec, context: str) -> frozenset[str]:
+    """``sum`` reduces each dim in ``over`` away, so the operand carries every one.
+
+    The axes a join opens never reach a frame: the sum over the join closes
+    every one, so its frame is the join's frame less them.
+    """
+    if isinstance(node.operand, Join) and node.operand.columns.axes:
+        join = node.operand
+        assert node.over == join.columns.axes, 'a sum through a relation closes exactly the axes its join opens'
+        return join_dims(join.columns, dims_of(join.operand, schema, context), context, 'the expression')
+    inner = dims_of(node.operand, schema, context)
     for summed in node.over:
-        if summed not in inner:
+        if summed.dimension not in inner:
             raise DimensionError(_not_carried(context, f'sum(over={summed})', inner, 'drop the sum, or fix the dim'))
-    return inner - frozenset(node.over)
+    return inner - {axis.dimension for axis in node.over}
 
 
 def join_dims(columns: JoinColumns, inner: frozenset[str], context: str, operand: str) -> frozenset[str]:
     """The dims *inner* has once *columns* joins it, an expression's or a predicate's alike.
 
-    The dims joined on go, the dims grouped by arrive, and each column joined
-    on and not grouped by opens its own axis ([`JoinColumns.axes`][mathspec.program.JoinColumns.axes]), which
-    the [`Sum`][mathspec.program.Sum] over the join takes away. A column
-    both joined on and grouped by keeps its dim. The call a refusal quotes is
+    The dims joined on go and the dims grouped by arrive. A column both
+    joined on and grouped by keeps its dim. The call a refusal quotes is
     ``at`` where each group is one row, and ``sum`` otherwise.
 
     Raises:
@@ -155,7 +163,7 @@ def join_dims(columns: JoinColumns, inner: frozenset[str], context: str, operand
             f'or group by a column over another dimension.'
         )
     _check_joined(call, columns, inner, context)
-    return (inner - set(columns.joined_dims)) | set(columns.grouped_dims) | set(columns.axes)
+    return (inner - set(columns.joined_dims)) | set(columns.grouped_dims)
 
 
 #: The verb a file writes each translation with, which its refusals quote.
diff --git a/src/mathspec/program.py b/src/mathspec/program.py
index 441d56c8..d518a8d9 100644
--- a/src/mathspec/program.py
+++ b/src/mathspec/program.py
@@ -39,8 +39,10 @@
     'Add',
     'And',
     'Assumption',
+    'Axis',
     'BooleanLiteral',
     'Cases',
+    'Column',
     'Connective',
     'Constant',
     'ConstraintDeclaration',
@@ -245,19 +247,50 @@ class Divide:
     divisor: Expression
 
 
+@dataclass(frozen=True)
+class Column:
+    """One column of a relation: the relation's name, and the column's name in its declaration."""
+
+    relation: str
+    name: str
+
+    def __str__(self) -> str:
+        """The column as a file writes it, ``relation[column]``."""
+        return f'{self.relation}[{self.name}]'
+
+
+@dataclass(frozen=True)
+class Axis:
+    """A position in a frame: the dimension whose labels it runs over, and the relation column it stands for.
+
+    A declared dimension is the axis ``Axis(dimension)``. A [`Join`][]
+    opens ``Axis(dimension, column)`` for each column it drops, so a column
+    dropped and a column added over one dimension are two axes of one frame.
+    Two axes are equal where both fields are, so a frame holds a dimension's
+    own axis at most once however many join axes run over the same labels.
+    """
+
+    dimension: str
+    column: Column | None = None
+
+    def __str__(self) -> str:
+        """The dimension's name, or the column's ``relation[column]`` for a join's axis."""
+        return self.dimension if self.column is None else str(self.column)
+
+
 @dataclass(frozen=True)
 class Sum:
-    """Sum ``operand`` over the named dims, removing them from the result.
+    """Sum ``operand`` over the axes in ``over``, removing them from the result.
 
-    A name in ``over`` is a dimension, or one of the axes a [`Join`][]
-    under it opens ([`JoinColumns.axes`][]): ``sum(x, over=a,
-    by=relation[b])`` lowers to a ``Sum`` over a ``Join``, over the axis of
-    each column the join does not group by. The join is the join, and this node is the
-    group-by that follows it.
+    An axis in ``over`` is a dimension's own, or one a [`Join`][] under it
+    opens ([`JoinColumns.axes`][]): ``sum(x, over=a, by=relation[b])``
+    lowers to a ``Sum`` over a ``Join``, over the axis of each column the join
+    does not group by. The join is the join, and this node is the group-by
+    that follows it.
     """
 
     operand: Expression
-    over: tuple[str, ...]
+    over: tuple[Axis, ...]
 
 
 @dataclass(frozen=True)
@@ -266,9 +299,9 @@ class Join:
 
     The operand carries every dim joined on. The result has the operand's
     dims, less the dims joined on, plus the dims grouped by, plus one axis
-    per column joined on and not grouped by ([`JoinColumns.axes`][]), named
-    for the relation's column and not for its dimension. So a column dropped
-    and a column added over one dimension stay two axes. A [`Sum`][] over
+    per column joined on and not grouped by ([`JoinColumns.axes`][]), which
+    stands for the relation's column and runs over its dimension. So a column
+    dropped and a column added over one dimension stay two axes. A [`Sum`][] over
     those axes is the group-by that follows the join, which is how
     ``sum(x, over=a, by=relation[b])`` lowers. Where the grouped columns
     hold the relation's whole key, they determine every other column, the
@@ -540,16 +573,17 @@ def one_row_per_group(self) -> bool:
         return set(self.relation.key) <= set(self.grouped)
 
     @property
-    def axes(self) -> tuple[str, ...]:
-        """The axis the join opens for each column it drops, ``relation.column``, empty where each group is one row.
+    def axes(self) -> tuple[Axis, ...]:
+        """The axis the join opens for each column it drops, empty where each group is one row.
 
-        A dimension's name holds no dot, so an axis never meets one. A column
-        the grouped columns determine opens none: in a lookup they hold the
-        key, and the key determines every column.
+        Each runs over the column's dimension and stands for the column, so it
+        never equals that dimension's own axis. A column the grouped columns
+        determine opens none: in a lookup they hold the key, and the key
+        determines every column.
         """
         if self.one_row_per_group:
             return ()
-        return tuple(f'{self.name}.{role}' for role in self.dropped)
+        return tuple(Axis(self.dim(role), Column(self.name, role)) for role in self.dropped)
 
 
 @dataclass(frozen=True)
diff --git a/src/mathspec/separability.py b/src/mathspec/separability.py
index a5fda511..0619b65c 100644
--- a/src/mathspec/separability.py
+++ b/src/mathspec/separability.py
@@ -98,22 +98,21 @@ def waits_on(dimension: str, label: str, name: str, kind: Literal['offset', 'par
             if isinstance(node, Cases):
                 masks.extend(region.when for region in node.regions)
             elif isinstance(node, Sum):
-                join = node.operand.columns if isinstance(node.operand, Join) and node.operand.columns.axes else None
-                grouped = dict(zip(join.axes, join.dropped_dims, strict=True)) if join is not None else {}
-                for name in node.over:
-                    if join is not None and name in grouped:
+                for axis in node.over:
+                    if axis.column is not None:
+                        assert isinstance(node.operand, Join), 'a join axis is opened by the join the sum stands over'
                         report(
                             'coupled',
-                            grouped[name],
+                            axis.dimension,
                             label,
-                            f'groups {grouped[name]} into {", ".join(join.added_dims)} — window that dimension instead, or cut only at the group edges',
+                            f'groups {axis.dimension} into {", ".join(node.operand.columns.added_dims)} — window that dimension instead, or cut only at the group edges',
                         )
                     elif reductions_couple:
                         report(
                             'coupled',
-                            name,
+                            axis.dimension,
                             label,
-                            f'sums over {name} — a rolling sum_back(window=n) windows, a total over the horizon does not',
+                            f'sums over {axis.dimension} — a rolling sum_back(window=n) windows, a total over the horizon does not',
                         )
             elif isinstance(node, Join) and not node.columns.axes:
                 for dimension in node.columns.dropped_dims:
diff --git a/src/mathspec/typesetting/walk.py b/src/mathspec/typesetting/walk.py
index 4a2d4a8c..29408da9 100644
--- a/src/mathspec/typesetting/walk.py
+++ b/src/mathspec/typesetting/walk.py
@@ -422,7 +422,7 @@ def _sum(self, node: Sum, ctx: _Context) -> tuple[str, int]:
             return self._grouped_sum(node.operand, ctx)
         memberships = []
         inner = ctx
-        for d in self._sorted(frozenset(node.over)):
+        for d in self._sorted(frozenset(axis.dimension for axis in node.over)):
             dummy, inner = inner.reducing(d)
             memberships.append(self._membership(d, dummy))
         domain = self.format.joined(memberships, '')
diff --git a/tests/test_dimensions.py b/tests/test_dimensions.py
index 7607eef8..96c8ab21 100644
--- a/tests/test_dimensions.py
+++ b/tests/test_dimensions.py
@@ -12,7 +12,7 @@
 
 from mathspec.dimensions import DimensionError, _check_where_dims, dims_of
 from mathspec.errors import SchemaError
-from mathspec.program import Join, Mask, RelationPairComparison, Sum
+from mathspec.program import Axis, Column, Join, Mask, RelationPairComparison, Sum
 from mathspec.resolution import Namespace
 from mathspec.validation import to_spec
 from tests.fixtures import DISPATCH_MODEL, expression_of, schema_of, varied, where_of
@@ -235,19 +235,16 @@ def test_a_lookup_carries_the_whole_key_and_what_the_operand_brings_beside_it():
 
 
 def test_a_join_opens_an_axis_for_the_column_it_drops_and_the_sum_over_it_closes_it():
-    """A map into its own dimension drops and adds one dimension, so the join names the dropped column for the relation.
+    """A map into its own dimension drops and adds one dimension, so the axis the join opens stands for the column.
 
-    Named for its dimension, the column the join drops and the column it
-    groups by are one name, and the sum over the join takes away the dim the
-    row keeps.
+    The dropped column's axis runs over `snapshot` and stands for
+    `rep_of[snapshot]`, so it is not the dimension's own axis. The sum over
+    the join closes it, and the frame keeps the `snapshot` the row groups by.
     """
     s = _schema()
     node = expression_of('sum(p, over=snapshot, by=rep_of[rep])', Namespace(s), 't')
     assert isinstance(node, Sum) and isinstance(node.operand, Join)
-    assert node.over == ('rep_of.snapshot',), 'the sum stands over the axis the join opens'
-    assert dims_of(node.operand, s, 't') == {'generator', 'snapshot', 'rep_of.snapshot'}, (
-        'the join keeps the dropped column beside the dimension it groups by'
-    )
+    assert node.over == (Axis('snapshot', Column('rep_of', 'snapshot')),), 'the sum stands over the axis the join opens'
     assert dims_of(node, s, 't') == {'generator', 'snapshot'}, 'and the sum over it leaves the frame the row keeps'
 
 
diff --git a/tests/test_expansion.py b/tests/test_expansion.py
index c3f96a68..0f8178f4 100644
--- a/tests/test_expansion.py
+++ b/tests/test_expansion.py
@@ -14,7 +14,7 @@
 from mathspec import to_spec
 from mathspec.errors import LanguageError
 from mathspec.expansion import parse_and_expand
-from mathspec.program import Multiply, Named, Parameter, Sum, Translate, Variable
+from mathspec.program import Axis, Multiply, Named, Parameter, Sum, Translate, Variable
 from mathspec.resolution import Namespace
 from tests.fixtures import DISPATCH_MODEL, SMALL_MODEL, comparison_of, expression_of, schema_of
 
@@ -141,7 +141,7 @@ def test_a_call_expands_to_core_ast(expressions, macros, call, want):
 def test_a_named_expression_arrives_under_the_node_carrying_its_name():
     ns = Namespace(schema(expressions={'gen_cost': 'p * cost'}))
     resolved = expression_of('sum(gen_cost, over=generator)', ns, 'e')
-    assert resolved == Sum(Named('gen_cost', Multiply(Variable('p'), Parameter('cost'))), ('generator',)), (
+    assert resolved == Sum(Named('gen_cost', Multiply(Variable('p'), Parameter('cost'))), (Axis('generator'),)), (
         'the body is inlined resolved and the name kept, for the typesetter to define it once'
     )
     assert isinstance(resolved, Sum)
diff --git a/tests/test_lowering.py b/tests/test_lowering.py
index b44811bb..33fd7837 100644
--- a/tests/test_lowering.py
+++ b/tests/test_lowering.py
@@ -22,8 +22,10 @@
     Add,
     And,
     Assumption,
+    Axis,
     BooleanLiteral,
     Cases,
+    Column,
     Constant,
     CountComparison,
     DimensionComparison,
@@ -138,13 +140,13 @@ def test_program_structure(dispatch_program):
     ((cname, c),) = dispatch_program.constraints.items()
     assert cname == 'power_balance'
     assert c.dims == ('snapshot',), 'the frame is the dims, in the order the file wrote it'
-    assert c.lhs == Sum(Variable('dispatch'), ('generator',))
+    assert c.lhs == Sum(Variable('dispatch'), (Axis('generator'),))
     assert c.sense == '==', "the comparison crosses as the file's own operator, untranslated"
     assert c.rhs == Parameter('load')
 
     assert dispatch_program.objective.sense == 'minimize', "the program carries the language's spelling, untranslated"
     assert dispatch_program.objective.expression == Sum(
-        Multiply(Variable('dispatch'), Parameter('cost')), ('generator', 'snapshot')
+        Multiply(Variable('dispatch'), Parameter('cost')), (Axis('generator'), Axis('snapshot'))
     ), 'the objective carries the sum the file wrote, over the dims it named none of'
 
 
@@ -566,11 +568,15 @@ def test_a_power_resolves_to_a_node_of_its_own(dispatch_schema):
 @pytest.mark.parametrize(
     ('expression', 'expected'),
     [
-        pytest.param('sum(q)', Sum(Variable('q'), ('g', 'h')), id='a-bare-sum-sums-away-every-dim-the-operand-carries'),
-        pytest.param('sum(q, over=h)', Sum(Variable('q'), ('h',)), id='an-over-sums-away-the-dim-it-names'),
+        pytest.param(
+            'sum(q)',
+            Sum(Variable('q'), (Axis('g'), Axis('h'))),
+            id='a-bare-sum-sums-away-every-dim-the-operand-carries',
+        ),
+        pytest.param('sum(q, over=h)', Sum(Variable('q'), (Axis('h'),)), id='an-over-sums-away-the-dim-it-names'),
         pytest.param(
             'sum(p, over=g, by=lk[h])',
-            Sum(Join(Variable('p'), LK_JOIN), ('lk.g',)),
+            Sum(Join(Variable('p'), LK_JOIN), (Axis('g', Column('lk', 'g')),)),
             id='a-grouped-sum-is-a-sum-over-the-axis-its-join-opens',
         ),
         pytest.param(
@@ -709,7 +715,7 @@ def test_a_relation_lowers_with_the_join_each_call_names():
     assert program.relations == {'zone_of': declared}, 'the relation sits once in the program, under its name'
     zonal = program.constraints['zonal'].lhs
     columns = JoinColumns('zone_of', declared, ('generator', 'snapshot'), ('zone', 'snapshot'))
-    assert zonal == Sum(Join(Variable('p'), columns), ('zone_of.generator',)), (
+    assert zonal == Sum(Join(Variable('p'), columns), (Axis('generator', Column('zone_of', 'generator')),)), (
         'a grouped sum is a sum over a join: the join names the column over the over= dim and the unnamed key '
         'column as joined on, the by= column and that key column as grouped by, and the sum stands over the axis '
         'the join opens for the column it drops'
@@ -725,7 +731,7 @@ def test_a_relation_lowers_with_the_join_each_call_names():
     )
     assert program.constraints['history'].lhs == Sum(
         Join(Variable('p'), JoinColumns('zone_of', declared, ('snapshot', 'generator'), ('zone', 'generator'))),
-        ('zone_of.snapshot',),
+        (Axis('snapshot', Column('zone_of', 'snapshot')),),
     ), 'the same table joined on its other key column'
     priced = program.constraints['priced'].rhs
     assert priced == Join(
@@ -762,7 +768,7 @@ def test_a_divisor_under_a_join_is_still_named():
     looked_up = Join(quotient, JoinColumns('component_of', component_of, ('component',), ('flow',)))
 
     assert parameters_of(looked_up) == frozenset({'rate'}), 'the walk descends through `Join`'
-    assert parameters_of(Sum(looked_up, ('flow',))) == frozenset({'rate'}), 'and through a `Sum` over it'
+    assert parameters_of(Sum(looked_up, (Axis('flow'),))) == frozenset({'rate'}), 'and through a `Sum` over it'
 
 
 def test_a_divisor_under_a_power_is_still_named():
diff --git a/tests/typesetting/test_golden.py b/tests/typesetting/test_golden.py
index da588018..b716caa3 100644
--- a/tests/typesetting/test_golden.py
+++ b/tests/typesetting/test_golden.py
@@ -150,11 +150,11 @@ def _rendered_trees() -> Iterator[object]:
 
 #: A dataclass the walk steps *through* rather than renders: a region has no
 #: branch of its own — its ``when`` and ``value`` do — the columns a join
-#: names and the relation it reads are the facts a node carries rather than
-#: nodes, and a ``Mask`` is the wrapper a leaf carries a predicate in. None is a
+#: names, the axes a sum is over and the relation it reads are the facts a
+#: node carries rather than nodes, and a ``Mask`` is the wrapper a leaf carries a predicate in. None is a
 #: member of any node union, so they are subtracted from what the tree walk
 #: finds rather than added to what the vocabulary declares.
-CARRIERS = {'Region', 'JoinColumns', 'Mask', 'Partition', 'RelationDeclaration'}
+CARRIERS = {'Axis', 'Column', 'Region', 'JoinColumns', 'Mask', 'Partition', 'RelationDeclaration'}
 
 
 def test_the_golden_model_carries_every_node_kind_the_walk_renders():