fix(browser): probe pid liveness through the platform-safe helper

_process_is_alive used os.kill(pid, 0). That probe is POSIX-only:
CPython's Windows os.kill calls TerminateProcess(handle, sig) for any
signal other than CTRL_C/CTRL_BREAK, so it terminates the process it is
asked about. This function is only reached when there is no procfs to
read a command line from, which is exactly the macOS and Windows case
the rest of this change exists to handle.

core/platform_compat.py already owns that probe and documents the
hazard; its module docstring asks callers to import from there rather
than spell a POSIX-only call out locally. Delegate to it.

pid_alive answers False where os.kill raises PermissionError — a live
process owned by another user. Both call sites want that reading: the
sweep only unlinks a pid file it wrote itself, and a pid it cannot
confirm is not the daemon it is looking for.

tests/test_private_browser_tool.py

@@ -1923,3 +1923,27 @@ def test_procfs_host_still_matches_on_the_command_line(monkeypatch, tmp_path) ->
 
     _pid_file_for(tmp_path, monkeypatch, "clawmm-test", "session-6", 6666)
     assert PrivateBrowserTool._owned_daemon_exists({}, "session-6") is False
+
+
+def test_liveness_probe_goes_through_the_platform_safe_helper(monkeypatch) -> None:
+    """The no-procfs path must not reach a bare ``os.kill(pid, 0)``.
+
+    CPython's Windows ``os.kill`` calls ``TerminateProcess(handle, sig)`` for
+    any signal other than CTRL_C / CTRL_BREAK, so probing liveness with signal
+    0 terminates the process it asks about — and the only hosts that reach this
+    probe are the ones with no procfs, Windows among them.
+    ``core.platform_compat.pid_alive`` is the tree's platform-safe answer.
+    """
+
+    asked: list[int] = []
+    monkeypatch.setattr(
+        web_tools, "pid_alive", lambda pid: asked.append(pid) or True
+    )
+    monkeypatch.setattr(
+        web_tools.os,
+        "kill",
+        lambda *a, **kw: pytest.fail("os.kill must not be used to probe liveness"),
+    )
+
+    assert web_tools._process_is_alive(4242) is True
+    assert asked == [4242]