diff --git a/docs/changelog/175.feature.rst b/docs/changelog/175.feature.rst
new file mode 100644
index 0000000..c40387c
--- /dev/null
+++ b/docs/changelog/175.feature.rst
@@ -0,0 +1,3 @@
+Accept the free-threaded, debug, architecture and machine suffixes after a version specifier, as in ``>=3.13t`` or
+``cpython>=3.11,<3.14-64``, and raise ``ValueError`` for a spec that starts with a comparison operator but does not
+parse, such as ``>=3.13.``, instead of looking it up as an executable name.
diff --git a/docs/explanation.rst b/docs/explanation.rst
index 125532b..78009ed 100644
--- a/docs/explanation.rst
+++ b/docs/explanation.rst
@@ -364,10 +364,19 @@ and machine segments, leaves the string unrecognized, so it falls back to a lite
    * - ``cpython>=3.11``
      - `Version specifier <https://packaging.python.org/en/latest/specifications/version-specifiers/>`_
        restricted to CPython
+   * - ``>=3.13t``
+     - `Version specifier <https://packaging.python.org/en/latest/specifications/version-specifiers/>`_
+       restricted to free-threaded builds
 
 `Version specifiers <https://packaging.python.org/en/latest/specifications/version-specifiers/>`_
 (``>=``, ``<=``, ``~=``, ``!=``, ``==``, ``===``) are supported. Multiple specifiers can be comma-separated,
 for example ``>=3.11,<3.13``. Each version is a release of up to three numbers with an optional ``a``, ``b`` or
 ``rc`` pre-release suffix, such as ``3.12`` or ``3.13.0rc1``; ``==`` and ``!=`` also take a trailing ``.*``
-wildcard, as in ``==3.12.*``. A specifier with any item outside that form, such as ``>=3.x``, ``>=3.10.post1``,
-``>=3.12 <4`` (missing comma) or ``>=3.12,foo``, falls back to a literal executable-name match.
+wildcard, as in ``==3.12.*``.
+
+The ABI flags, ``-arch`` and ``-machine`` parts described above follow the last specifier and apply to the whole set:
+``>=3.13t`` asks for a free-threaded 3.13 or newer, ``cpython>=3.11,<3.14-64`` for a 64-bit CPython in that range.
+Without ``t`` a specifier accepts any build, while a plain ``3.13`` asks for the default (GIL) build. A string that
+starts with a comparison operator but does not parse this way, such as ``>=3.x``, ``>=3.10.post1``, ``>=3.12 <4``
+(missing comma) or ``>=3.13t,<3.15`` (a flag on an inner item), raises :exc:`ValueError` instead of being looked up as
+an executable name.
diff --git a/src/python_discovery/_py_spec.py b/src/python_discovery/_py_spec.py
index c263650..8e79c94 100644
--- a/src/python_discovery/_py_spec.py
+++ b/src/python_discovery/_py_spec.py
@@ -10,26 +10,29 @@
 from ._py_info import normalize_isa
 from ._specifier import SimpleSpecifier, SimpleSpecifierSet, SimpleVersion
 
-PATTERN = re.compile(
-    r"""
-    ^
-    (?P<impl>[a-zA-Z]+)?            # implementation (e.g. cpython, pypy)
-    (?P<version>[0-9.]+)?           # version (e.g. 3.12, 3.12.1)
+_MODIFIERS: Final[str] = r"""
     (?P<threaded>t)?                # free-threaded flag
     (?P<debug>d|(?:-dbg|-debug)(?=-|$))?  # debug build flag (d, -dbg or -debug)
     (?:-(?P<arch>32|64))?           # architecture bitness
     (?:-(?P<machine>[a-zA-Z0-9_.]+))?  # ISA (e.g. arm64, x86_64, i86pc.64bit)
     $
-    """,
+"""
+PATTERN = re.compile(
+    r"""
+    ^
+    (?P<impl>[a-zA-Z]+)?            # implementation (e.g. cpython, pypy)
+    (?P<version>[0-9.]+)?           # version (e.g. 3.12, 3.12.1)
+    """
+    + _MODIFIERS,
     re.VERBOSE,
 )
 SPECIFIER_PATTERN = re.compile(
     r"""
     ^
     (?:(?P<impl>[A-Za-z]+)\s*)?     # optional implementation prefix
-    (?P<spec>(?:===|==|~=|!=|<=|>=|<|>).+)  # PEP 440 version specifier
-    $
-    """,
+    (?P<spec>(?:===|==|~=|!=|<=|>=|<|>).*?)  # PEP 440 version specifier, lazy so the modifiers can follow it
+    """
+    + _MODIFIERS,
     re.VERBOSE,
 )
 
@@ -69,51 +72,60 @@ def _parse_version_parts(version: str) -> tuple[int | None, int | None, int | No
 def _parse_spec_pattern(string_spec: str) -> PythonSpec | None:
     if not (match := re.match(PATTERN, string_spec)):
         return None
-    groups = match.groupdict()
-    version = groups["version"]
-    major, minor, micro, threaded = None, None, None, None
-    debug = True if groups["debug"] else None  # unconstrained unless an explicit d/-dbg/-debug marker is present
-    if version is not None:
+    release, threaded = (None, None, None), None
+    if (version := match["version"]) is not None:
         try:
-            major, minor, micro = _parse_version_parts(version)
+            release = _parse_version_parts(version)
         except ValueError:
             return None
-        threaded = bool(groups["threaded"])
-    impl = groups["impl"]
+        threaded = bool(match["threaded"])
+    return _spec_from_match(string_spec, match, release, threaded=threaded, version_specifier=None)
+
+
+def _spec_from_match(
+    string_spec: str,
+    match: re.Match[str],
+    release: tuple[int | None, int | None, int | None],
+    *,
+    threaded: bool | None,
+    version_specifier: SpecifierSet | None,
+) -> PythonSpec:
+    impl = match["impl"]
     if impl in {"py", "python"}:
         impl = None
     if impl == "graalvm":
         impl = "graalpy"
-    arch = _int_or_none(groups["arch"])
-    machine = groups.get("machine")
-    if machine is not None:
-        machine = normalize_isa(machine)
     return PythonSpec(
-        string_spec, impl, major, minor, micro, arch, None, free_threaded=threaded, machine=machine, debug=debug
+        string_spec,
+        impl,
+        *release,
+        _int_or_none(match["arch"]),
+        None,
+        free_threaded=threaded,
+        machine=None if (machine := match["machine"]) is None else normalize_isa(machine),
+        debug=True if match["debug"] else None,  # unconstrained unless an explicit d/-dbg/-debug marker is present
+        version_specifier=version_specifier,
     )
 
 
 def _parse_specifier(string_spec: str) -> PythonSpec | None:
-    if not (specifier_match := SPECIFIER_PATTERN.match(string_spec.strip())):
-        return None
-    if SpecifierSet is None:  # pragma: no cover
+    if not (match := SPECIFIER_PATTERN.match(string_spec.strip())):
         return None
-    impl = specifier_match.group("impl")
-    spec_text = specifier_match.group("spec").strip()
-    try:
-        version_specifier = SpecifierSet.from_string(spec_text)
-    except InvalidSpecifier:  # pragma: no cover
-        return None
-    # the set never raises: it keeps an unparsable version as None and drops items without an operator, so fall back
-    # to a literal path, like any other unrecognized spec, unless every item parsed into a usable version
+    spec_text = match["spec"].strip()
+    version_specifier = SpecifierSet.from_string(spec_text)
+    # the set never raises: it keeps an unparsable version as None and drops items without an operator
     items = [item for item in spec_text.split(",") if item.strip()]
     if len(items) != len(version_specifier.specifiers) or any(spec.version is None for spec in version_specifier):
-        return None
-    if impl in {"py", "python"}:
-        impl = None
-    if impl == "graalvm":
-        impl = "graalpy"
-    return PythonSpec(string_spec, impl, None, None, None, None, None, version_specifier=version_specifier)
+        msg = f"invalid version specifier {string_spec!r}"
+        raise ValueError(msg)
+    return _spec_from_match(
+        string_spec,
+        match,
+        (None, None, None),
+        # the version bounds say nothing about the build, so without a trailing t any build qualifies
+        threaded=True if match["threaded"] else None,
+        version_specifier=version_specifier,
+    )
 
 
 class PythonSpec:
@@ -169,6 +181,7 @@ def from_string_spec(cls, string_spec: str) -> PythonSpec:
 
         :param string_spec: an interpreter spec — an absolute path, a version string, an implementation prefix,
             or a `version specifier <https://packaging.python.org/en/latest/specifications/version-specifiers/>`_.
+        :raises ValueError: if the spec starts with a version specifier operator but does not parse as one.
         """
         if pathlib.Path(string_spec).is_absolute():
             return cls(string_spec, None, None, None, None, None, string_spec)
diff --git a/tests/test_discovery.py b/tests/test_discovery.py
index 5d464ef..c552365 100644
--- a/tests/test_discovery.py
+++ b/tests/test_discovery.py
@@ -119,7 +119,7 @@ def test_discovery_fallback_fail(session_cache: DiskCache, caplog: pytest.LogCap
     assert "accepted" not in caplog.text
 
 
-@pytest.mark.parametrize("missing", ["magic-one", "python.", ">=3.12 <4"])
+@pytest.mark.parametrize("missing", ["magic-one", "python."])
 def test_discovery_fallback_ok(missing: str, session_cache: DiskCache, caplog: pytest.LogCaptureFixture) -> None:
     caplog.set_level(logging.DEBUG)
     result = get_interpreter([missing, sys.executable], cache=session_cache)
@@ -128,6 +128,11 @@ def test_discovery_fallback_ok(missing: str, session_cache: DiskCache, caplog: p
     assert "accepted" in caplog.text
 
 
+def test_discovery_invalid_specifier_raises(session_cache: DiskCache) -> None:
+    with pytest.raises(ValueError, match=r"^invalid version specifier '>=3\.12 <4'$"):
+        get_interpreter([">=3.12 <4", sys.executable], cache=session_cache)
+
+
 @pytest.fixture
 def mock_find_interpreter(mocker: MockerFixture) -> None:
     mocker.patch(
diff --git a/tests/test_py_info_extra.py b/tests/test_py_info_extra.py
index 17f0427..b421a05 100644
--- a/tests/test_py_info_extra.py
+++ b/tests/test_py_info_extra.py
@@ -195,6 +195,18 @@ def test_satisfies_version_specifier_fails() -> None:
     assert CURRENT.satisfies(spec, impl_must_match=False) is False
 
 
+@pytest.mark.parametrize(
+    ("suffix", "expected"),
+    [
+        pytest.param("t", CURRENT.free_threaded, id="free-threaded"),
+        pytest.param("", True, id="any-build"),
+    ],
+)
+def test_satisfies_version_specifier_free_threaded(suffix: str, expected: bool) -> None:
+    spec = PythonSpec.from_string_spec(f">={sys.version_info.major}.{sys.version_info.minor}{suffix}")
+    assert CURRENT.satisfies(spec, impl_must_match=False) is expected
+
+
 def test_satisfies_version_specifier_unparsable_version() -> None:
     specifier = SimpleSpecifierSet.from_string(">=3.x")
     spec = PythonSpec(">=3.x", None, None, None, None, None, None, version_specifier=specifier)
diff --git a/tests/test_py_spec.py b/tests/test_py_spec.py
index 36bacd2..d57c26c 100644
--- a/tests/test_py_spec.py
+++ b/tests/test_py_spec.py
@@ -1,5 +1,6 @@
 from __future__ import annotations
 
+import re
 import sys
 from typing import TYPE_CHECKING
 
@@ -23,15 +24,6 @@
         "python.",
         "cpython..",
         "pypy...",
-        ">=3.x",
-        ">=3.12 <4",
-        ">=3.10.post1",
-        "cpython>=v3.10",
-        ">=3.12,<4.x",
-        ">=3.x,<4",
-        "<,4",
-        ">=3.12,foo",
-        ">=3.12, 4",
     ],
 )
 def test_bad_py_spec(text: str) -> None:
@@ -45,6 +37,49 @@ def test_bad_py_spec(text: str) -> None:
     assert all(v is None for v in content.values())
 
 
+@pytest.mark.parametrize(
+    "text",
+    [
+        pytest.param(">=3.x", id="non-numeric-version"),
+        pytest.param(">=3.13.", id="trailing-dot"),
+        pytest.param(">=3.10.post1", id="post-release"),
+        pytest.param("cpython>=v3.10", id="v-prefix"),
+        pytest.param(">=3.12 <4", id="missing-comma"),
+        pytest.param(">=3.12,<4.x", id="bad-last-item"),
+        pytest.param(">=3.x,<4", id="bad-first-item"),
+        pytest.param("<,4", id="operator-without-version"),
+        pytest.param(">=3.12,foo", id="item-without-operator"),
+        pytest.param(">=", id="bare-operator"),
+        pytest.param(">=3.13t,<3.15", id="modifier-before-last-item"),
+    ],
+)
+def test_invalid_specifier_raises(text: str) -> None:
+    with pytest.raises(ValueError, match=f"^invalid version specifier {re.escape(repr(text))}$"):
+        PythonSpec.from_string_spec(text)
+
+
+@pytest.mark.parametrize(
+    ("text", "expected"),
+    [
+        pytest.param(">=3.13t", (">=3.13", True, None, None, None), id="free-threaded"),
+        pytest.param(">=3.13td", (">=3.13", True, True, None, None), id="free-threaded-debug"),
+        pytest.param(">=3.13-dbg", (">=3.13", None, True, None, None), id="debug"),
+        pytest.param(">=3.11,<3.14t", (">=3.11,<3.14", True, None, None, None), id="set"),
+        pytest.param("cpython>=3.12-64-arm64", (">=3.12", None, None, 64, "arm64"), id="arch-machine"),
+        pytest.param(">=3.13", (">=3.13", None, None, None, None), id="any-build"),
+    ],
+)
+def test_specifier_modifiers(text: str, expected: tuple[str, bool | None, bool | None, int | None, str | None]) -> None:
+    spec = PythonSpec.from_string_spec(text)
+    assert (
+        str(spec.version_specifier),
+        spec.free_threaded,
+        spec.debug,
+        spec.architecture,
+        spec.machine,
+    ) == expected
+
+
 def test_py_spec_first_digit_only_major() -> None:
     spec = PythonSpec.from_string_spec("278")
     assert spec.major == 2
diff --git a/tests/test_py_spec_extra.py b/tests/test_py_spec_extra.py
index 2250f07..571639a 100644
--- a/tests/test_py_spec_extra.py
+++ b/tests/test_py_spec_extra.py
@@ -101,11 +101,6 @@ def test_single_digit_version() -> None:
     assert spec.minor is None
 
 
-def test_specifier_with_invalid_inner() -> None:
-    spec = PythonSpec.from_string_spec(">=not_a_version")
-    assert spec.path is None or spec.version_specifier is not None or spec.path == ">=not_a_version"
-
-
 def test_two_digit_version() -> None:
     spec = PythonSpec.from_string_spec("python312")
     assert spec.major == 3