feat(openai): add Live gateway support

backend/internal/server/routes/gateway.go

@@ -180,6 +180,8 @@ func RegisterGatewayRoutes(
 		// Codex manifest format; other clients keep the OpenAI-style list.
 		gateway.GET("/models", modelsHandler)
 		gateway.GET("/usage", h.Gateway.Usage)
+		gateway.POST("/live", h.OpenAIGateway.Live)
+		gateway.GET("/live/:call_id", h.OpenAIGateway.LiveSideband)
 		// OpenAI Responses API: auto-route based on group platform
 		gateway.POST("/responses", func(c *gin.Context) {
 			if isOpenAIResponsesCompatibleGatewayPlatform(c) {
@@ -277,6 +279,8 @@ func RegisterGatewayRoutes(
 	codexDirect := r.Group("/backend-api/codex")
 	codexDirect.Use(bodyLimit, clientRequestID, opsErrorLogger, endpointNorm, gin.HandlerFunc(apiKeyAuth), compositeTarget, requireGroupAnthropic)
 	{
+		codexDirect.POST("/realtime/calls", h.OpenAIGateway.Live)
+		codexDirect.GET("/:call_id", h.OpenAIGateway.LiveSideband)
 		codexDirect.POST("/responses", responsesHandler)
 		codexDirect.POST("/responses/*subpath", responsesHandler)
 		codexDirect.POST("/alpha/search", textBodyLimit, h.OpenAIGateway.AlphaSearch)