fix(gemini): Removes unsupported extension fields

Removes x-* extension fields from JSON schemas to ensure compatibility with the Gemini API.

These fields, while valid in OpenAPI/JSON Schema, are not recognized by the Gemini API and can cause issues.
The change recursively walks the schema, identifies these extension fields, and removes them, except when they define properties.

Amp-Thread-ID: https://ampcode.com/threads/T-019c0cd1-9e59-722b-83f0-e0582aba6914
Co-authored-by: Amp <amp@ampcode.com>

internal/util/gemini_schema_test.go

@@ -869,3 +869,129 @@ func TestCleanJSONSchemaForAntigravity_BooleanEnumToString(t *testing.T) {
 		t.Errorf("Boolean enum values should be converted to string format, got: %s", result)
 	}
 }
+
+func TestRemoveExtensionFields(t *testing.T) {
+	tests := []struct {
+		name     string
+		input    string
+		expected string
+	}{
+		{
+			name: "removes x- fields at root",
+			input: `{
+				"type": "object",
+				"x-custom-meta": "value",
+				"properties": {
+					"foo": { "type": "string" }
+				}
+			}`,
+			expected: `{
+				"type": "object",
+				"properties": {
+					"foo": { "type": "string" }
+				}
+			}`,
+		},
+		{
+			name: "removes x- fields in nested properties",
+			input: `{
+				"type": "object",
+				"properties": {
+					"foo": {
+						"type": "string",
+						"x-internal-id": 123
+					}
+				}
+			}`,
+			expected: `{
+				"type": "object",
+				"properties": {
+					"foo": {
+						"type": "string"
+					}
+				}
+			}`,
+		},
+		{
+			name: "does NOT remove properties named x-",
+			input: `{
+				"type": "object",
+				"properties": {
+					"x-data": { "type": "string" },
+					"normal": { "type": "number", "x-meta": "remove" }
+				},
+				"required": ["x-data"]
+			}`,
+			expected: `{
+				"type": "object",
+				"properties": {
+					"x-data": { "type": "string" },
+					"normal": { "type": "number" }
+				},
+				"required": ["x-data"]
+			}`,
+		},
+		{
+			name: "does NOT remove $schema and other meta fields (as requested)",
+			input: `{
+				"$schema": "http://json-schema.org/draft-07/schema#",
+				"$id": "test",
+				"type": "object",
+				"properties": {
+					"foo": { "type": "string" }
+				}
+			}`,
+			expected: `{
+				"$schema": "http://json-schema.org/draft-07/schema#",
+				"$id": "test",
+				"type": "object",
+				"properties": {
+					"foo": { "type": "string" }
+				}
+			}`,
+		},
+		{
+			name: "handles properties named $schema",
+			input: `{
+				"type": "object",
+				"properties": {
+					"$schema": { "type": "string" }
+				}
+			}`,
+			expected: `{
+				"type": "object",
+				"properties": {
+					"$schema": { "type": "string" }
+				}
+			}`,
+		},
+		{
+			name: "handles escaping in paths",
+			input: `{
+				"type": "object",
+				"properties": {
+					"foo.bar": {
+						"type": "string",
+						"x-meta": "remove"
+					}
+				},
+				"x-root.meta": "remove"
+			}`,
+			expected: `{
+				"type": "object",
+				"properties": {
+					"foo.bar": {
+						"type": "string"
+					}
+				}
+			}`,
+		},
+	}
+
+	for _, tt := range tests {
+		t.Run(tt.name, func(t *testing.T) {
+			actual := removeExtensionFields(tt.input)
+			compareJSON(t, tt.expected, actual)
+		})
+	}
+}