fix(gateway): a quoted local show_reasoning "false" stays off

The local overlay copied display.platforms.cli.show_reasoning unchanged, so a
quoted "false" reached Python/Ink truthiness checks as a non-empty string and
showed reasoning, while Desktop hid it. Normalise the winning value with the
setting's existing normaliser; precedence and no-value defaults are unchanged.

gateway/display_config.py

@@ -226,5 +226,5 @@ def local_display_config(display: Any) -> dict[str, Any]:
             continue
         if isinstance(value, dict) and isinstance(display.get(key), dict):
             value = {**display[key], **value}
-        display[key] = value
+        display[key] = _normalise(key, value)
     return display

tests/gateway/test_local_display_key.py

@@ -93,6 +93,23 @@ def test_launch_default_reads_the_local_key():
     assert launch_shows_reasoning(_Policy({'show_reasoning': False}), None) is False
 
 
+def test_quoted_local_false_stays_off():
+    """A quoted ``"false"`` under the local key is off on every surface, not a truthy string."""
+    from gateway.display_config import local_display_config
+    from gateway.local_viewer_events import launch_shows_reasoning
+
+    display = {'show_reasoning': False, 'platforms': {'cli': {'show_reasoning': 'false'}}}
+
+    class _Policy:
+        source = 'tui'
+
+        def config(self, _authority, secrets=False):
+            return {'display': display}
+
+    assert local_display_config(display)['show_reasoning'] is False
+    assert launch_shows_reasoning(_Policy(), None) is False
+
+
 def test_classic_cli_display_reads_the_local_key():
     from gateway.display_config import local_display_config