feat: add OpenAI Responses API support (POST /v1/responses)

Add full Responses API endpoint alongside existing Chat Completions,
with identical logging/tracking/cost pipeline.

New:
- internal/models/responses.go — request/response/stream types + ToUsage() bridge
- internal/providers/openai_responses.go — OpenAI Responses/ResponsesStream

Modified:
- provider.go — Responses()+ResponsesStream() added to Provider interface
- helpers.go — BuildOpenAIResponsesBody, parsers, SSE stream reader
- circuit_breaker.go — CB wraps Responses, passthrough for stream
- server.go — POST /v1/responses route + handleResponses handler
- all non-OpenAI providers — stub methods with clear error messages

Logging: ResponsesUsage.ToUsage() bridges to models.Usage, feeding same
logRequest() -> DB insert -> dashboard WS -> client stats -> cost calc
pipeline. No schema or logger changes needed.

internal/providers/openai_responses.go

@@ -0,0 +1,70 @@
+package providers
+
+import (
+	"context"
+	"encoding/json"
+	"fmt"
+
+	"gophergate/internal/models"
+)
+
+// Responses sends a non-streaming request to OpenAI's /v1/responses endpoint.
+func (p *OpenAIProvider) Responses(ctx context.Context, req *models.ResponsesRequest) (*models.ResponsesResponse, error) {
+	// Determine if streaming was requested
+	stream := req.Stream != nil && *req.Stream
+
+	body := BuildOpenAIResponsesBody(req, stream)
+
+	resp, err := p.client.R().
+		SetContext(ctx).
+		SetHeader("Authorization", "Bearer "+p.apiKey).
+		SetBody(body).
+		Post(fmt.Sprintf("%s/responses", p.config.BaseURL))
+
+	if err != nil {
+		return nil, fmt.Errorf("responses request failed: %w", err)
+	}
+
+	if !resp.IsSuccess() {
+		return nil, fmt.Errorf("OpenAI Responses API error (%d): %s", resp.StatusCode(), resp.String())
+	}
+
+	var respJSON map[string]interface{}
+	if err := json.Unmarshal(resp.Body(), &respJSON); err != nil {
+		return nil, fmt.Errorf("failed to parse responses response: %w", err)
+	}
+
+	return ParseOpenAIResponsesResponse(respJSON, req.Model)
+}
+
+// ResponsesStream sends a streaming request to OpenAI's /v1/responses endpoint.
+func (p *OpenAIProvider) ResponsesStream(ctx context.Context, req *models.ResponsesRequest) (<-chan *models.ResponsesStreamChunk, error) {
+	body := BuildOpenAIResponsesBody(req, true)
+
+	resp, err := p.client.R().
+		SetContext(ctx).
+		SetHeader("Authorization", "Bearer "+p.apiKey).
+		SetBody(body).
+		SetDoNotParseResponse(true).
+		Post(fmt.Sprintf("%s/responses", p.config.BaseURL))
+
+	if err != nil {
+		return nil, fmt.Errorf("responses stream request failed: %w", err)
+	}
+
+	if !resp.IsSuccess() {
+		return nil, fmt.Errorf("OpenAI Responses API error (%d): %s", resp.StatusCode(), resp.String())
+	}
+
+	ch := make(chan *models.ResponsesStreamChunk)
+
+	go func() {
+		defer close(ch)
+		err := StreamOpenAIResponses(resp.RawBody(), ch)
+		if err != nil {
+			fmt.Printf("Responses stream error: %v\n", err)
+		}
+	}()
+
+	return ch, nil
+}