diff --git a/docs/about/file-and-program.md b/docs/about/file-and-program.md
index dda0ffc1..86b6d377 100644
--- a/docs/about/file-and-program.md
+++ b/docs/about/file-and-program.md
@@ -39,7 +39,7 @@ kept as one declaration.
 - **The program keeps the model the author wrote.** A curve is one declaration
   to print and one to explain. Its rows are one formulation of it, so the rows
   are a second model, which a caller asks for with
-  [`spec.expand()`](../reference/reading.md#formulations-written-out).
+  [`spec.expand()`](../reference/api.md#math_spec.Spec.expand).
 - **The spec keeps the text.** A tool that rewrites a model needs the file as
   written: `to_yaml()` writes it back, and `expand()` rewrites it. A tree does
   not give the text back.
diff --git a/docs/about/what-counts-as-public-api.md b/docs/about/what-counts-as-public-api.md
index 140531b0..e2ba29c4 100644
--- a/docs/about/what-counts-as-public-api.md
+++ b/docs/about/what-counts-as-public-api.md
@@ -32,7 +32,7 @@ diff, the typesetter prints it, and an engine in another language reads it.
   talks about a file the language accepts, and changes nothing.
 - **Nothing is written out unasked.** A `piecewise:` or `sos:` block stays the
   block until a caller calls
-  [`spec.expand()`](../reference/reading.md#formulations-written-out).
+  [`spec.expand()`](../reference/api.md#math_spec.Spec.expand).
 
 What a solver or file format can take, how the numbers attach to the names, and
 which solver runs are each engine's to decide
diff --git a/docs/howto/see-an-expansion.md b/docs/howto/see-an-expansion.md
index 6bab0b94..8b72c2b7 100644
--- a/docs/howto/see-an-expansion.md
+++ b/docs/howto/see-an-expansion.md
@@ -428,7 +428,7 @@ the set out too.
 <!-- expansion:curve:end -->
 <!-- prettier-ignore-end -->
 
-[`Spec.expand()`](../reference/reading.md#formulations-written-out) lists what
+[`Spec.expand()`](../reference/api.md#math_spec.Spec.expand) lists what
 the call accepts, and
 [writing a formulation out](../reference/language/piecewise.md#writing-a-formulation-out)
 says what each block emits.
diff --git a/docs/reference/language/piecewise.md b/docs/reference/language/piecewise.md
index 43b08b31..034abd2c 100644
--- a/docs/reference/language/piecewise.md
+++ b/docs/reference/language/piecewise.md
@@ -165,7 +165,7 @@ expansion writes that the file already declares is refused at load too.
 ## Writing a formulation out
 
 Writing a formulation out replaces the block with the variables and constraints
-it states. [`Spec.expand()`](../reading.md#formulations-written-out) is the
+it states. [`Spec.expand()`](../api.md#math_spec.Spec.expand) is the
 call, and [see what a curve or a set expands to](../../howto/see-an-expansion.md)
 shows a model before and after.
 
diff --git a/docs/reference/reading.md b/docs/reference/reading.md
index 302cae1e..42cc7395 100644
--- a/docs/reference/reading.md
+++ b/docs/reference/reading.md
@@ -76,31 +76,11 @@ is one the file declared.
 
 ## Formulations written out
 
-`spec.expand(*kinds)` returns a new `Spec` with each `piecewise:` and `sos:`
-block replaced by the variables and constraints it states.
-[Writing a formulation out](language/piecewise.md#writing-a-formulation-out)
-says what those are.
-
-```python
-expanded = spec.expand()
-expanded == spec  # False
-expanded.expand() is expanded  # True
-spec.expand('sos') is spec  # True
-```
-
-- **The kinds are `'piecewise'` and `'sos'`, and no argument means both.** Any
-  other string raises `ValueError`, naming the two. Curves go first whatever
-  the order of the arguments, so the set a `method: sos2` curve states is
-  written out too.
-- **The expansion is a different model.** It declares more variables and
-  constraints, so it does not compare equal to the model it came from. It
-  declares the same dimensions and parameters, so the same data attaches to both.
-- **A model with nothing to write out comes back as itself.** So does an
-  expansion asked for the same kinds again.
-- **The spec keeps no expansion.** A second call builds it again.
-- **Nothing expands a model unasked.** A program holds its curves until
-  `expand()` writes them out. The expansion is a model like any other:
-  `to_yaml()` writes it, and its `program` holds the rows and no curve.
+A program holds each curve and each set as one declaration until
+[`Spec.expand()`](api.md#math_spec.Spec.expand) writes it out. An engine that
+builds rows reads the program of `spec.expand('piecewise')` if it takes a set,
+and the program of `spec.expand()` if it does not. The program of an expansion
+holds no curve:
 
 ```python
 sorted(rows.piecewise)  # []
diff --git a/docs/reference/typeset.md b/docs/reference/typeset.md
index 934bdcf2..09fbe747 100644
--- a/docs/reference/typeset.md
+++ b/docs/reference/typeset.md
@@ -46,7 +46,7 @@ a flag. The [Python API](api.md#typesetting) gives each signature.
 - The model's `description:` opens the document.
 - A `piecewise:` block prints as one line: the curve it states, over the frame
   it states one curve per coordinate of. To print its rows, print
-  [`spec.expand()`](reading.md#formulations-written-out) or pass `--expand`
+  [`spec.expand()`](api.md#math_spec.Spec.expand) or pass `--expand`
   ([see an expansion](../howto/see-an-expansion.md)).
 - An [`assumptions:`](language/assumptions.md) entry prints under an
   **Assumptions** heading, last, beside what each curve assumes of its
diff --git a/schema/math-spec.schema.json b/schema/math-spec.schema.json
index f461aa8a..399403c9 100644
--- a/schema/math-spec.schema.json
+++ b/schema/math-spec.schema.json
@@ -681,7 +681,7 @@
   },
   "$schema": "https://json-schema.org/draft/2020-12/schema",
   "additionalProperties": false,
-  "description": "The declared math \u2014 one YAML file, or one dict, validated. Nothing here has seen data.\n\nA ``Spec`` that exists has passed the whole language: constructing one by\nany route \u2014 ``to_spec``, :meth:`model_validate`, the constructor \u2014 runs\nevery load-time check, expression pass included, and raises\n:class:`~math_spec.errors.LanguageError` on a model the language refuses.\nHolding one is the proof, so nothing downstream checks it again.\n\nThe API is the eleven declaration sections plus ``version`` and\n``description``, three ways back out \u2014 :meth:`to_dict` for the model as\ndata, :meth:`to_yaml` for the file a reviewer reads, :meth:`expand` for the\nsame math with its formulations written out \u2014 and :attr:`program`, the\nmodel typed, which every reader after load walks. Everything else on this\nclass is pydantic's, not a contract this package keeps.",
+  "description": "The declared math \u2014 one YAML file, or one dict, validated. Nothing here has seen data.\n\nA ``Spec`` that exists has passed the whole language: constructing one by\nany route \u2014 ``to_spec``, :meth:`model_validate`, the constructor \u2014 runs\nevery load-time check, expression pass included, and raises\n:class:`~math_spec.errors.LanguageError` on a model the language refuses.\nHolding one is the proof, so nothing downstream checks it again.\n\nThe API is the eleven declaration sections plus ``version`` and\n``description``, three ways back out \u2014 :meth:`to_dict` for the model as\ndata, :meth:`to_yaml` for the file a reviewer reads, :meth:`expand` for the\nmodel with its formulations written out as plain rows \u2014 and :attr:`program`, the\nmodel typed, which every reader after load walks. Everything else on this\nclass is pydantic's, not a contract this package keeps.",
   "properties": {
     "assumptions": {
       "additionalProperties": {
diff --git a/src/math_spec/model.py b/src/math_spec/model.py
index 9910cf17..dccd6feb 100644
--- a/src/math_spec/model.py
+++ b/src/math_spec/model.py
@@ -712,7 +712,7 @@ class Spec(_StrictBlock):
     The API is the eleven declaration sections plus ``version`` and
     ``description``, three ways back out — :meth:`to_dict` for the model as
     data, :meth:`to_yaml` for the file a reviewer reads, :meth:`expand` for the
-    same math with its formulations written out — and :attr:`program`, the
+    model with its formulations written out as plain rows — and :attr:`program`, the
     model typed, which every reader after load walks. Everything else on this
     class is pydantic's, not a contract this package keeps.
     """
@@ -825,11 +825,13 @@ def expand(self, *kinds: Formulation) -> Spec:
         """This model with its formulations written out as plain variables and constraints.
 
         A formulation states rows rather than being one — ``piecewise:`` states
-        a curve, ``sos:`` states which members of a family may be nonzero — and
-        expanding one writes those rows under names prefixed with the block's
-        own, then drops the block. The math is the same afterwards, and so is
-        the data attached to it: neither a set nor a curve emits a parameter,
-        and a curve's rows sit on ``where`` predicates over the file's own.
+        a curve, ``sos:`` states which members of a family may be nonzero.
+        Expanding one writes those rows under names prefixed with the block's
+        own, and drops the block. The result is a different model: it declares
+        more variables and constraints, so it does not compare equal to this
+        one. It declares the same dimensions and parameters, so the same data
+        attaches to both. Nothing is cached, so a second call builds the
+        expansion again.
 
         Args:
             kinds: Which formulations to write out — ``'piecewise'``,
@@ -839,9 +841,10 @@ def expand(self, *kinds: Formulation) -> Spec:
                 curve.
 
         Returns:
-            The model those blocks wrote out, or this one where it declares
-            none of them. It is a model like any other: :meth:`to_yaml` writes
-            it, and the same data attaches to it as to the one it came from.
+            The model with those blocks written out, or this same object where
+            it declares none of them, so an expansion asked for the same kinds
+            again returns itself. It is a model like any other: :meth:`to_yaml`
+            writes it, and :attr:`program` holds its rows.
 
         Raises:
             ValueError: *kinds* names something that is not a formulation.
diff --git a/tests/test_expand.py b/tests/test_expand.py
index 6e22e588..a04d90d6 100644
--- a/tests/test_expand.py
+++ b/tests/test_expand.py
@@ -161,3 +161,18 @@ def test_the_same_sources_bind_a_model_and_its_expansion(model):
     written_out = set(spec.expand().program.parameters)
 
     assert written_out == supplied, 'writing a formulation out asks for data the model it came from did not'
+
+
+def test_an_expansion_is_a_different_model_and_has_nothing_left_to_write_out():
+    """What `expand()` returns: a new model, which a second expansion hands back unchanged."""
+    spec = schema_of(CURVE)
+    expanded = spec.expand()
+
+    assert expanded != spec, 'the expansion declares more rows, so it is a different model'
+    assert expanded.expand() is expanded, 'an expansion has no formulation left, so it comes back as itself'
+
+
+def test_a_model_with_no_formulation_expands_to_itself():
+    spec = schema_of(DISPATCH_MODEL)
+
+    assert spec.expand() is spec, 'nothing to write out returns the same object, not a copy'
diff --git a/tests/test_reading_page.py b/tests/test_reading_page.py
index 909569e0..7f932ecf 100644
--- a/tests/test_reading_page.py
+++ b/tests/test_reading_page.py
@@ -55,6 +55,6 @@ def test_the_page_shows_the_declarations_the_expansion_emits(tmp_path, monkeypat
         exec(compile(code, str(PAGE), 'exec'), namespace)
         claims.extend(_claims(code))
 
-    assert len(claims) == 22, 'every `expression  # value` line on the page is checked; one without one is not'
+    assert len(claims) == 19, 'every `expression  # value` line on the page is checked; one without one is not'
     for expression, claimed in claims:
         assert eval(expression, namespace) == claimed, f'reading.md says `{expression}` is {claimed}'