feat(session,llm): restore session runtime on resume; align Responses API details

pkg/llm/responses.go

@@ -124,6 +124,9 @@ func buildResponsesRequest(req ChatRequest) map[string]any {
 		"input":  buildResponsesInput(req.Messages),
 		"stream": true,
 	}
+	if instructions := responsesInstructions(req.Messages); instructions != "" {
+		payload["instructions"] = instructions
+	}
 	if len(req.Tools) > 0 {
 		payload["tools"] = buildResponsesTools(req.Tools)
 	}
@@ -153,7 +156,9 @@ func buildResponsesInput(messages []Message) []any {
 	for _, msg := range messages {
 		switch msg.Role {
 		case RoleSystem:
-			items = append(items, responsesMessageInput{Role: "system", Content: msg.Content})
+			// System instructions are sent as the top-level "instructions"
+			// field and must not be duplicated inside the input array.
+			continue
 		case RoleUser:
 			items = append(items, responsesMessageInput{Role: "user", Content: msg.Content})
 		case RoleAssistant:
@@ -179,6 +184,22 @@ func buildResponsesInput(messages []Message) []any {
 	return items
 }
 
+// responsesInstructions extracts system messages into a single instructions
+// string, which is the OpenAI Responses API way to pass developer/system
+// guidance (and the most prompt-cache friendly).
+func responsesInstructions(messages []Message) string {
+	var instructions []string
+	for _, msg := range messages {
+		if msg.Role != RoleSystem {
+			continue
+		}
+		if text := strings.TrimSpace(msg.Content); text != "" {
+			instructions = append(instructions, text)
+		}
+	}
+	return strings.Join(instructions, "\n\n")
+}
+
 func buildResponsesTools(tools []Tool) []responsesTool {
 	out := make([]responsesTool, 0, len(tools))
 	for _, tool := range tools {