diff --git a/src/lpspec/linopy/coverage.py b/src/lpspec/linopy/coverage.py
index 8e5c615e6..4b077db3d 100644
--- a/src/lpspec/linopy/coverage.py
+++ b/src/lpspec/linopy/coverage.py
@@ -44,20 +44,56 @@ def check_constant_side_covers(
 ) -> None:
     """A comparison's constant side must have values wherever the row is built.
 
-    A missing row is read as 0, and on a side with no variable that zero *is*
-    the bound — `x <= cap` becomes `x <= 0`, which binds rather than vanishing.
-    Keyed to the rows the declaration builds, not to the coordinate product: a
-    `where` that removed the coordinate has already answered the question.
+    A missing row is read as 0, and where that zero is a constant piece it *is*
+    the bound — `x <= cap` becomes `x <= 0`, and `x + hi <= 100` becomes
+    `x <= 100`, both binding rather than vanishing. The question is asked of
+    every constant piece, a parameter in a variable-free additive position,
+    whichever side it was written on and beside whatever term: a coefficient,
+    which a variable stands with in a product, is a zero when it is missing (the
+    absence rules) and is not one. Keyed to the rows the declaration builds, not
+    to the coordinate product: a `where` that removed the coordinate has already
+    answered the question.
     """
-    for side in (row.lhs, row.rhs):
-        if program.carries_variable(side):
-            continue
-        walk = _under_regions(side, ctx, mask)
-        found = [(node.name, where) for node, where in walk if isinstance(node, program.Parameter)]
-        for param, narrowed in sorted(found, key=itemgetter(0)):
-            missing = gaps_under(ctx.dataset[param], narrowed)
-            if missing:
-                raise DataError(uncovered_constant_message(param, missing, name))
+    found = [pair for side in (row.lhs, row.rhs) for pair in _constant_leaves(side, ctx, mask, coefficient=False)]
+    for param, narrowed in sorted(found, key=itemgetter(0)):
+        missing = gaps_under(ctx.dataset[param], narrowed)
+        if missing:
+            raise DataError(uncovered_constant_message(param, missing, name))
+
+
+def _constant_leaves(
+    node: program.ExpressionNode, ctx: EvaluationContext, mask: Any, coefficient: bool
+) -> Iterator[tuple[str, Any]]:
+    """Every parameter standing as a constant piece under *node*, with the rows it must cover.
+
+    A constant piece is a parameter in a variable-free additive position — `hi`
+    in `x + hi`, or in `sum(x) + sum(hi)`. A parameter a variable stands with in
+    a product is a coefficient instead, and a sparse coefficient is a zero the
+    absence rules allow, so ``coefficient`` records having passed through such a
+    product on the way down and suppresses the parameters below it. Additive
+    structure, a reduction and a division by it leave the flag as it was; a
+    ``cases:`` region narrows the mask as it does for the whole walk.
+    """
+    if isinstance(node, program.Variable):
+        return
+    if isinstance(node, program.Parameter):
+        if not coefficient:
+            yield node.name, mask
+        return
+    if isinstance(node, program.Cases):
+        for region in node.regions:
+            inside = evaluate_where(region.when, ctx)
+            yield from _constant_leaves(region.value, ctx, inside if mask is None else mask & inside, coefficient)
+        return
+    if isinstance(node, program.Multiply):
+        yield from _constant_leaves(node.left, ctx, mask, coefficient or program.carries_variable(node.right))
+        yield from _constant_leaves(node.right, ctx, mask, coefficient or program.carries_variable(node.left))
+        return
+    if isinstance(node, program.Divide):
+        yield from _constant_leaves(node.numerator, ctx, mask, coefficient)
+        return
+    for child in program.children(node):
+        yield from _constant_leaves(child, ctx, mask, coefficient)
 
 
 def _under_regions(
diff --git a/src/lpspec/relational/engines/polars/assembly.py b/src/lpspec/relational/engines/polars/assembly.py
index 2bba28855..34e71df3a 100644
--- a/src/lpspec/relational/engines/polars/assembly.py
+++ b/src/lpspec/relational/engines/polars/assembly.py
@@ -544,11 +544,7 @@ def _refuse_short_constant_parameters(
         short anywhere.
         """
         found = [
-            pair
-            for side in (c.lhs, c.rhs)
-            if not program.carries_variable(side)
-            for pair in _constant_parameters(side)
-            if pair[0] in self.measured.sparse
+            pair for side in (c.lhs, c.rhs) for pair in _constant_parameters(side) if pair[0] in self.measured.sparse
         ]
         for param, region in sorted(found, key=itemgetter(0)):
             missing = self._uncovered_coordinates(frame, param, c, region)
@@ -728,9 +724,16 @@ def short_parameters(program: program.Program, attached: AttachedSources) -> dic
 
 
 def _constant_parameters(
-    node: program.ExpressionNode, region: program.Mask | None = None
+    node: program.ExpressionNode, region: program.Mask | None = None, coefficient: bool = False
 ) -> Iterator[tuple[str, program.Mask | None]]:
-    """Every parameter under *node*, each with the region it stands in.
+    """Every parameter standing as a constant piece under *node*, each with the region it stands in.
+
+    A constant piece is a parameter in a variable-free additive position — `hi`
+    in `x + hi`, or in `sum(x) + sum(hi)`. A parameter a variable stands with in
+    a product is a coefficient instead, and a sparse coefficient is a zero the
+    absence rules allow, so ``coefficient`` records having passed through such a
+    product on the way down and suppresses the parameters below it. Additive
+    structure, a reduction and a division by it leave the flag as it was.
 
     A region narrows what the pieces under it owe, and regions compose by
     conjunction as the walk descends, which is what
@@ -741,16 +744,28 @@ def _constant_parameters(
         node: The expression to walk.
         region: The region *node* already stands in — the recursion's own
             accumulator, ``None`` at the call a caller writes.
+        coefficient: Whether *node* stands in a product with a variable — the
+            recursion's own accumulator, ``False`` at the call a caller writes.
     """
+    if isinstance(node, program.Variable):
+        return
     if isinstance(node, program.Parameter):
-        yield node.name, region
+        if not coefficient:
+            yield node.name, region
         return
     if isinstance(node, program.Cases):
         for r in node.regions:
-            yield from _constant_parameters(r.value, both_regions(region, r.when))
+            yield from _constant_parameters(r.value, both_regions(region, r.when), coefficient)
+        return
+    if isinstance(node, program.Multiply):
+        yield from _constant_parameters(node.left, region, coefficient or program.carries_variable(node.right))
+        yield from _constant_parameters(node.right, region, coefficient or program.carries_variable(node.left))
+        return
+    if isinstance(node, program.Divide):
+        yield from _constant_parameters(node.numerator, region, coefficient)
         return
     for child in program.children(node):
-        yield from _constant_parameters(child, region)
+        yield from _constant_parameters(child, region, coefficient)
 
 
 def declares_quadratic(c: program.ConstraintDeclaration) -> bool:
diff --git a/tests/test_absence.py b/tests/test_absence.py
index f510b0586..6ffd97c2a 100644
--- a/tests/test_absence.py
+++ b/tests/test_absence.py
@@ -140,6 +140,50 @@ def test_a_where_is_the_escape_from_the_constant_side_check():
         )
 
 
+def test_a_constant_piece_beside_a_term_is_refused_on_both_lanes():
+    """A parameter added beside a variable term is a constant piece, not skipped (#1521).
+
+    `w * x + c <= 100` reads `w * x <= 100` where the file says `w * x <= 100 - c`,
+    the missing `c` filled with the zero that is a bound. The eager check asked
+    its question of a side carrying no variable, so a piece beside a term went
+    unasked and the lane built the wrong row in silence; the relational lane,
+    asking of every constant fragment, refused alone.
+    """
+    spec = override(SPARSE_COEFFICIENT_SPEC, **{'constraints.cap.expression': 'w * x + c <= 100'})
+    both_lanes_refuse(spec, SPARSE_CONSTANT_DATA, match="parameter 'c' covers 1 fewer")
+
+
+def test_a_constant_piece_beside_a_term_is_refused_through_a_reduction():
+    """The same piece under a sum, where both lanes were blind (#1521).
+
+    `sum(w * x, over=t) + sum(c, over=t) <= 100` sums the gap away before either
+    lane's per-coordinate check can see it — the eager lane skipped the side for
+    its variable, the relational lane could not see the hole through the sum
+    (#1465's mechanism) — so the parameter is asked directly, of both lanes now.
+    """
+    spec = override(
+        SPARSE_COEFFICIENT_SPEC,
+        **{'constraints.cap': {'foreach': [], 'expression': 'sum(w * x, over=t) + sum(c, over=t) <= 100'}},
+    )
+    both_lanes_refuse(spec, SPARSE_CONSTANT_DATA, match="parameter 'c' covers 1 fewer")
+
+
+def test_a_sparse_coefficient_beside_a_constant_piece_is_still_a_zero():
+    """The widened check still reads a sparse coefficient as a zero (#1521).
+
+    `w * x + c <= 100` with `w` short and `c` whole builds and both lanes agree:
+    `w` is a coefficient wherever a variable stands with it, so its missing row
+    is the zero the absence rules allow rather than an uncovered bound — only the
+    piece `c`, which no variable stands with, is owed its coordinates.
+    """
+    data = {'t': [0, 1, 2], 'w': pd.Series({1: 1.0, 2: 1.0}), 'c': pd.Series({0: 5.0, 1: 4.0, 2: 5.0})}
+    spec = override(SPARSE_COEFFICIENT_SPEC, **{'constraints.cap.expression': 'w * x + c <= 100'})
+    with differential(spec, data, lp=True) as run:
+        assert run.result.objective == pytest.approx(10.0 + 10.0 + 10.0, rel=RTOL), (
+            't=0: w absent, so its term is zero and x runs to its bound; elsewhere the bound is slack'
+        )
+
+
 #: `south` is a load-only bus: both generators sit on `north`, so the group
 #: behind `south`'s constant side has no members at all.
 GROUPED_CONSTANT_SPEC = {