diff --git a/docs/reference/language/declarations.md b/docs/reference/language/declarations.md
index f3bb9f71..4414b405 100644
--- a/docs/reference/language/declarations.md
+++ b/docs/reference/language/declarations.md
@@ -64,19 +64,23 @@ variables:
       upper: capacity
 ```
 
-| Field                           |                                                                                                                   |                        |
-| ------------------------------- | ----------------------------------------------------------------------------------------------------------------- | ---------------------- |
-| `dims`                          | required. The dimensions it is indexed by                                                                         |                        |
-| `where`                         | which coordinates exist ([absence](absence.md))                                                                   | default `null`         |
-| `bounds.lower` / `bounds.upper` | a number, or the name of a `float` or `int` parameter                                                             | default `-inf` / `inf` |
-| `domain`                        | `continuous`, `integer` or `binary`. `binary` carries fixed 0/1 bounds                                            | default `continuous`   |
-| `absence`                       | `undefined` or `zero`: what a masked-out coordinate means ([absence](absence.md#what-a-missing-coordinate-means)) | default `undefined`    |
-| `description`                   | free text                                                                                                         | default `null`         |
+| Field                           |                                                                                                                   |                      |
+| ------------------------------- | ----------------------------------------------------------------------------------------------------------------- | -------------------- |
+| `dims`                          | required. The dimensions it is indexed by                                                                         |                      |
+| `where`                         | which coordinates exist ([absence](absence.md))                                                                   | default `null`       |
+| `bounds.lower` / `bounds.upper` | a finite number, or the name of a `float` or `int` parameter. `null` leaves that side open                        | default `null`       |
+| `domain`                        | `continuous`, `integer` or `binary`. `binary` carries fixed 0/1 bounds                                            | default `continuous` |
+| `absence`                       | `undefined` or `zero`: what a masked-out coordinate means ([absence](absence.md#what-a-missing-coordinate-means)) | default `undefined`  |
+| `description`                   | free text                                                                                                         | default `null`       |
 
 !!! warning "A bound you omit leaves the variable unbounded on that side"
 
     You write non-negativity. The language does not assume it.
 
+An open side is `null`, as every other field a file may leave open is. A bound
+is never infinite: `.inf` and `-.inf` are refused, with `null` named as the
+rewrite.
+
 A bound is a name or a number: `upper: capacity` is accepted,
 and `upper: -rating` is refused. Ship the negated column as data. The dimensions of
 a bound parameter are a subset of the variable's.
diff --git a/schema/math-spec.schema.json b/schema/math-spec.schema.json
index 99300c13..78f55535 100644
--- a/schema/math-spec.schema.json
+++ b/schema/math-spec.schema.json
@@ -48,7 +48,7 @@
     },
     "BoundsBlock": {
       "additionalProperties": false,
-      "description": "Variable bounds \u2014 each side is a number or parameter name.\n\nAn omitted bound leaves the variable unbounded on that side, not\nimplicitly non-negative.",
+      "description": "Variable bounds \u2014 each side is a finite number, a parameter name, or ``None`` where it is open.\n\nAn omitted bound leaves the variable unbounded on that side, not\nimplicitly non-negative. An infinity is refused: an open side is ``null``,\nand the other infinity leaves no value at all.",
       "properties": {
         "lower": {
           "anyOf": [
@@ -57,8 +57,12 @@
             },
             {
               "type": "string"
+            },
+            {
+              "type": "null"
             }
           ],
+          "default": null,
           "title": "Lower"
         },
         "upper": {
@@ -68,8 +72,12 @@
             },
             {
               "type": "string"
+            },
+            {
+              "type": "null"
             }
           ],
+          "default": null,
           "title": "Upper"
         }
       },
@@ -617,7 +625,10 @@
         },
         "bounds": {
           "$ref": "#/$defs/BoundsBlock",
-          "default": {}
+          "default": {
+            "lower": null,
+            "upper": null
+          }
         },
         "description": {
           "anyOf": [
diff --git a/src/math_spec/boundedness.py b/src/math_spec/boundedness.py
index 018ccf87..95ab4aaa 100644
--- a/src/math_spec/boundedness.py
+++ b/src/math_spec/boundedness.py
@@ -13,7 +13,6 @@
 
 from __future__ import annotations
 
-import math
 from typing import TYPE_CHECKING, Literal, assert_never
 
 from math_spec.errors import Advice
@@ -50,10 +49,6 @@
 #: Which of a variable's two bounds a term drives it toward.
 BoundSide = Literal['lower', 'upper']
 
-#: The bound value that leaves each side open. A ``lower`` of ``+inf`` is not
-#: this — that model is empty, not unbounded — so the match is by value.
-_OPEN: dict[BoundSide, float] = {'lower': -math.inf, 'upper': math.inf}
-
 
 def unbounded_notes(program: Program) -> list[Advice]:
     """Name every variable the objective can drive to infinity unopposed.
@@ -90,7 +85,7 @@ def unbounded_notes(program: Program) -> list[Advice]:
                     'unbounded',
                     vname,
                     f"Variable '{vname}' makes this model unbounded: no constraint names it, and "
-                    f'bounds.{side} is {_OPEN[side]}, which is the direction a {sign}{vname} term '
+                    f'bounds.{side} is open, which is the direction a {sign}{vname} term '
                     f'improves a {program.objective.sense} objective in. No data can change that, so '
                     f'the solve would answer `unbounded` and name nothing.\n'
                     f'Give it a finite bounds.{side}, or the constraint that was meant to define it.',
@@ -100,12 +95,11 @@ def unbounded_notes(program: Program) -> list[Advice]:
 
 
 def _is_open(vdef: VariableDeclaration, side: BoundSide) -> bool:
-    """Whether *vdef*'s bound on *side* is the open value itself.
+    """Whether *vdef* states no bound on *side*.
 
     A bound naming a parameter is finite or not by data, so it does not count.
     """
-    bound = vdef.lower if side == 'lower' else vdef.upper
-    return bound == Constant(_OPEN[side])
+    return (vdef.lower if side == 'lower' else vdef.upper) is None
 
 
 def _flip(sign: Sign) -> Sign:
diff --git a/src/math_spec/lowering.py b/src/math_spec/lowering.py
index 0e4418a0..d95a8281 100644
--- a/src/math_spec/lowering.py
+++ b/src/math_spec/lowering.py
@@ -238,7 +238,9 @@ def _frame_of(name: str, entry: Named, schema: Spec) -> tuple[str, ...]:
     return tuple(d for d in schema.dimensions if d in carried)
 
 
-def _bound(value: float | str) -> Constant | Parameter:
+def _bound(value: float | str | None) -> Constant | Parameter | None:
+    if value is None:
+        return None
     if isinstance(value, str):
         return Parameter(value)
     return Constant(value)
diff --git a/src/math_spec/model.py b/src/math_spec/model.py
index 4cf5be00..119eb8db 100644
--- a/src/math_spec/model.py
+++ b/src/math_spec/model.py
@@ -217,16 +217,17 @@ class ParameterBlock(_StrictBlock):
 
 
 class BoundsBlock(_StrictBlock):
-    """Variable bounds — each side is a number or parameter name.
+    """Variable bounds — each side is a finite number, a parameter name, or ``None`` where it is open.
 
     An omitted bound leaves the variable unbounded on that side, not
-    implicitly non-negative.
+    implicitly non-negative. An infinity is refused: an open side is ``null``,
+    and the other infinity leaves no value at all.
     """
 
     _label: ClassVar[str] = 'a bounds block'
 
-    lower: float | str = float('-inf')
-    upper: float | str = float('inf')
+    lower: float | str | None = None
+    upper: float | str | None = None
 
     @field_validator('lower', 'upper', mode='before')
     @classmethod
@@ -237,6 +238,12 @@ def _a_number_or_a_name(cls, v: object, info: ValidationInfo[object]) -> object:
         if isinstance(v, float) and math.isnan(v):
             msg = f'bounds.{info.field_name} is nan, which no value compares to. Write a number, or omit the bound.'
             raise ValueError(msg)
+        if isinstance(v, float | int) and math.isinf(v):
+            msg = (
+                f'bounds.{info.field_name} is {v}, and a bound is finite. An open side is null: '
+                f'write {info.field_name}: null, or leave it out.'
+            )
+            raise ValueError(msg)
         return v
 
     @model_validator(mode='after')
@@ -689,10 +696,8 @@ def _without_absence(value: object) -> object:
 
 
 def _is_absent(value: object) -> bool:
-    """Whether *value* is a null or an infinite bound."""
-    if value is None:
-        return True
-    return isinstance(value, float) and math.isinf(value)
+    """Whether *value* is a null."""
+    return value is None
 
 
 class Spec(_StrictBlock):
@@ -798,7 +803,7 @@ def _check_version(cls, v: int) -> int:
 
     @model_serializer(mode='wrap')
     def _drop_absence(self, handler: SerializerFunctionWrapHandler) -> dict[str, object]:
-        """Absence is not serialised: a null, an infinite bound, a mapping that stripping emptied, a section declaring nothing.
+        """Absence is not serialised: a null, a mapping that stripping emptied, a section declaring nothing.
 
         An empty list stays, being a value rather than an absence (``dims:
         []`` is a scalar). On the serializer so that ``model_dump``,
diff --git a/src/math_spec/program.py b/src/math_spec/program.py
index e875184f..2122947b 100644
--- a/src/math_spec/program.py
+++ b/src/math_spec/program.py
@@ -21,7 +21,7 @@
 from __future__ import annotations
 
 from collections.abc import Mapping
-from dataclasses import dataclass, field, fields, replace
+from dataclasses import dataclass, fields, replace
 from functools import cached_property
 from typing import TYPE_CHECKING, Literal, assert_never
 
@@ -599,8 +599,11 @@ class ParameterDeclaration:
 class VariableDeclaration:
     dims: tuple[str, ...]
     where: Mask | None = None
-    lower: Expression = field(default_factory=lambda: Constant(float('-inf')))
-    upper: Expression = field(default_factory=lambda: Constant(float('inf')))
+    #: A number or a parameter, or ``None`` where that side is open. What stands
+    #: for an open side in a solve is the consumer's to choose.
+    lower: Expression | None = None
+    #: As :attr:`lower`, for the other side.
+    upper: Expression | None = None
     domain: VariableDomain = 'continuous'
     absence: VariableAbsence = 'undefined'
     description: str | None = None
diff --git a/src/math_spec/separability.py b/src/math_spec/separability.py
index de7ba06b..d31cb0c4 100644
--- a/src/math_spec/separability.py
+++ b/src/math_spec/separability.py
@@ -54,7 +54,8 @@ def _built_blocks(program: Program) -> Iterator[_Block]:
     for name, block in program.constraints.items():
         yield _Block(f"constraint '{name}'", name, (block.lhs, block.rhs), block.where, True)
     for name, variable in program.variables.items():
-        yield _Block(f"variable '{name}'", None, (variable.lower, variable.upper), variable.where, True)
+        bounds = tuple(side for side in (variable.lower, variable.upper) if side is not None)
+        yield _Block(f"variable '{name}'", None, bounds, variable.where, True)
     if program.objective is not None:
         yield _Block('the objective', None, (program.objective.expression,), None, False)
 
diff --git a/src/math_spec/sos.py b/src/math_spec/sos.py
index b24da870..3c04d60a 100644
--- a/src/math_spec/sos.py
+++ b/src/math_spec/sos.py
@@ -28,7 +28,7 @@
 Coefficients = tuple[float | str | None, float | str | None]
 
 
-def coefficients(domain: str, lower: float | str, upper: float | str) -> Coefficients:
+def coefficients(domain: str, lower: float | str | None, upper: float | str | None) -> Coefficients:
     """What a member's two linking rows multiply its binary by, ``None`` on a side the model leaves open.
 
     The 0 and 1 a binary's domain fixes, which no bounds block carries;
@@ -39,9 +39,9 @@ def coefficients(domain: str, lower: float | str, upper: float | str) -> Coeffic
     does not cap, and one above it is a looser row than the bound already
     states.
     """
-    fixed = domain == 'binary'
-    below = 0.0 if fixed else (None if lower == float('-inf') else lower)
-    return below, (1.0 if fixed else (None if upper == float('inf') else upper))
+    if domain == 'binary':
+        return 0.0, 1.0
+    return lower, upper
 
 
 @dataclass(frozen=True)
@@ -141,13 +141,11 @@ def _scaled(factor: float | str, picked: str) -> str:
 
 def _coefficients(member: dict[str, object]) -> tuple[float | str, float | str]:
     """The two coefficients as an expression writes them, read off the member."""
-    bounds: dict[str, object] = {'lower': float('-inf'), 'upper': float('inf')}
     declared = member.get('bounds')
     assert declared is None or isinstance(declared, dict), 'a validated model carries a bounds block as a mapping'
-    bounds.update(declared or {})
-    lower, upper = bounds['lower'], bounds['upper']
-    assert isinstance(lower, float | str) and isinstance(upper, float | str), (
-        'a bound is a number or the name of a parameter'
+    lower, upper = (declared.get('lower'), declared.get('upper')) if declared else (None, None)
+    assert isinstance(lower, float | str | None) and isinstance(upper, float | str | None), (
+        'a bound is a number, the name of a parameter, or open'
     )
     below, above = coefficients(str(member.get('domain', 'continuous')), lower, upper)
     assert below is not None and above is not None, 'a set with a side left open is refused at load'
diff --git a/src/math_spec/typesetting/walk.py b/src/math_spec/typesetting/walk.py
index 1ccef519..82da079c 100644
--- a/src/math_spec/typesetting/walk.py
+++ b/src/math_spec/typesetting/walk.py
@@ -801,18 +801,17 @@ def _variable(self, name: str) -> Line:
         if block.domain == 'binary':
             left, right = symbol, f'{self._op("in")} {self._op("binary_set")}'
         else:
-            below, above = lower == Constant(float('-inf')), upper == Constant(float('inf'))
-            if below and above:
-                domain = self._op('integers' if block.domain == 'integer' else 'reals')
-                left, right = symbol, f'{self._op("in")} {domain}'
-            elif below:
+            if lower is not None and upper is not None:
+                left = f'{self._bound(ctx, lower)} {self._op("le")} {symbol}'
+                right = f'{self._op("le")} {self._bound(ctx, upper)}'
+            elif upper is not None:
                 left, right = symbol, f'{self._op("le")} {self._bound(ctx, upper)}'
-            elif above:
+            elif lower is not None:
                 left, right = symbol, f'{self._op("ge")} {self._bound(ctx, lower)}'
             else:
-                left = f'{self._bound(ctx, lower)} {self._op("le")} {symbol}'
-                right = f'{self._op("le")} {self._bound(ctx, upper)}'
-            if block.domain == 'integer' and not (below and above):
+                domain = self._op('integers' if block.domain == 'integer' else 'reals')
+                left, right = symbol, f'{self._op("in")} {domain}'
+            if block.domain == 'integer' and (lower is not None or upper is not None):
                 right = f'{right}, {symbol} {self._op("in")} {self._op("integers")}'
         return Line(label=name, left=left, right=right, condition=condition)
 
diff --git a/tests/test_advice.py b/tests/test_advice.py
index 192a1499..fd83dd7b 100644
--- a/tests/test_advice.py
+++ b/tests/test_advice.py
@@ -70,7 +70,7 @@ def test_a_dimension_something_reaches_is_in_use(patch):
 
 #: A model with one note of each kind: nothing reaches `h`, and `p` is driven
 #: down by the objective with an open lower bound and no constraint on it.
-BOTH_KINDS = override(UNREACHED, **{'objective.expression': 'sum(p)', 'variables.p.bounds': {'lower': -float('inf')}})
+BOTH_KINDS = override(UNREACHED, **{'objective.expression': 'sum(p)', 'variables.p.bounds': {'lower': None}})
 
 
 def test_both_kinds_of_note_come_through_the_one_door():
@@ -84,9 +84,9 @@ def test_both_kinds_of_note_come_through_the_one_door():
 
 
 def _written(model: dict, tmp_path: Path) -> Path:
-    """The model as a file on disk — JSON is YAML, once infinity is spelled its way."""
+    """The model as a file on disk — JSON is YAML."""
     path = tmp_path / 'model.yaml'
-    path.write_text(json.dumps(model).replace('-Infinity', '-.inf'))
+    path.write_text(json.dumps(model))
     return path
 
 
diff --git a/tests/test_validation.py b/tests/test_validation.py
index 13c8fefe..e4d61ce3 100644
--- a/tests/test_validation.py
+++ b/tests/test_validation.py
@@ -1363,11 +1363,6 @@ class TestRulesDecidedWithoutData:
                 ('bounds.lower 5.0 is above bounds.upper 1.0, so no value satisfies them',),
                 id='literal-bounds-that-cross',
             ),
-            pytest.param(
-                {'variables.p.bounds': {'lower': float('inf'), 'upper': float('-inf')}},
-                ('bounds.lower inf is above bounds.upper -inf',),
-                id='infinite-bounds-that-cross',
-            ),
             pytest.param(
                 {'variables.p.dims': ['g', 'g']},
                 ("Variable 'p' names dimension 'g' twice",),
@@ -2197,3 +2192,37 @@ def test_a_plain_entry_that_breaks_a_dim_rule_is_refused_at_load_under_its_own_n
     model = override(SMALL_MODEL, expressions={'bad': {'expression': 'sum(k, over=g)'}}, constraints=constraints)
     with pytest.raises(DimensionError, match=r"^Named expression 'bad': sum\(over=g\)"):
         to_spec(model)
+
+
+@pytest.mark.parametrize(
+    'upper',
+    [
+        pytest.param({}, id='omitted'),
+        pytest.param({'upper': None}, id='null'),
+    ],
+)
+def test_an_open_bound_is_null_in_the_file_and_in_the_program(upper):
+    """`upper: null` was refused, though every other field a file may leave open takes `null`."""
+    spec = to_spec(override(DISPATCH_MODEL, **{'variables.p.bounds': {'lower': 0, **upper}}))
+    assert spec.variables['p'].bounds.upper is None
+    assert spec.program.variables['p'].upper is None, 'the program says the side is open rather than infinite'
+    assert spec.to_dict()['variables']['p']['bounds'] == {'lower': 0}, 'an open bound is not written back out'
+
+
+@pytest.mark.parametrize(
+    ('side', 'value'),
+    [
+        pytest.param('upper', float('inf'), id='the-infinity-that-opens-the-upper-side'),
+        pytest.param('lower', float('-inf'), id='the-infinity-that-opens-the-lower-side'),
+        pytest.param('lower', float('inf'), id='a-lower-bound-no-value-meets'),
+        pytest.param('upper', float('-inf'), id='an-upper-bound-no-value-meets'),
+    ],
+)
+def test_an_infinite_bound_is_refused_with_the_null_that_opens_a_side(side, value):
+    """An infinity is either the open side, which is `null`, or a bound no value meets.
+
+    A lone `lower: .inf` loaded: only two literal bounds that cross were refused.
+    """
+    message = _refusal(DISPATCH_MODEL, **{f'variables.p.bounds.{side}': value})
+    assert f'bounds.{side} is {value}, and a bound is finite' in message
+    assert f'{side}: null' in message, 'the refusal names the spelling of an open side'